Coupling between two mesoscopic systems towards the measurement of noise Thi Kim Thanh Nguyen To cite this version: Thi Kim Thanh Nguyen. Coupling between two mesoscopic systems towards the measurement of noise. Condensed Matter [cond-mat]. Université de la Méditerranée - Aix-Marseille II, 2007. English. �tel-00175563� HAL Id: tel-00175563 https://tel.archives-ouvertes.fr/tel-00175563 Submitted on 28 Sep 2007 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. 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DGLGLKMPNOLGRFVUÇJ!K ÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HÊ Ã ¨ K&J!KM}¶JVNCTL5C°GNRT$´ hFKMiDGKML}& iD4UTLqJ!DGÁ LGCENOp!KeNL2ÁKMp!CTp!}MCTBGN}e}MCTLGWGDG}¶JVCEFpIq WGCEDGIPKSiD4UTLqJVDÁ WGCTJ/ÈDGLG}¶JVNCTL ÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HÊ Ã[email protected]Ê G KxNOLGKMPUTpJ!N}"}&DGF!FKMLqJ ÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HÊ Ã$Ê Ã ="D4UÇLiJ!DGÁ BCENLqJH}&CELqJ UT}¶J°UÇp½UÅpCEDGF!}&KxC LGCTNp!K ÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HÊ H ²½CENOp!KSWGK¶JVK&}&JVNOCELlNLeK $BKMF!NOÁK&LqJVprÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HÊ [email protected]Ê G ²©CENpKF!KMWDG}&J!NCELeÁKjUTpDGF!K&ÁK&LiJ!p`ÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ Æ$NON G à à H 4 ; ¿ I ¿ D GjÄ GG G G G> [email protected]? [email protected]? GBA GBA D G!I !G I !G D ÃÇÄ ÃFG ÃTà ÃH ÃH ÃF ÃG? ÃG? HÊ HÊ Ã 4Ë DGPOP+}MCTDGLqJVNLRp¬J UÇJ!Np¬JVN}&pC}MDGFF!KMLqJ>4DG}¶JVD4UÇJ!NCELp ÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ Ã A HÊ H Ê H "Ì L$´ } GNBeWGK¶JVK&}&JVNOCELlCiD4UÇLiJ!DGÁ LGCENOp!KÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ Tà ¿ d X ^ A Aqg ^ AA ZT\ f Y d C GÊ@G ±µLqJVF!CWGDG}¶JVNOCELÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HTÄ GÊ Ã F!UTLGpNOJVNOCEL hFCEÁÁK&JVUTPPON}ÅJVC:JVDLGLGKMPONLGRÕF!KMRTNÁKMpÂNOL p!DGBKMF}MCELGWDG}&J!NLGRÁN}MFCÇ´ }MCELpJVFN}¶JVNCTLGpÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ H7G GÊ Ã[email protected]Ê G K>Î;CTRECEPDI C1Æq´µWGKS¸K&LGLGKMp©KMiD4U3JVNCTL¦ÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HEà GÊ Ã$Ê Ã FVUÇLGp!ÁNpp!NCTLzUÇLGWeF!%K GKM}&J!NCELp©C /B4UTF¬JVNO}MPK&pUÇJJ!GKr²"ÀÉNLqJVK&/F UT}&K ÊxÊrÊxÊ HH GÊ Ã$Ê H Kx}&CELGWGDG}¶J UTL}MKrC /UÅLGCTF!ÁUTP+ÁK&J UÇP ´µpDGBKMF!}&CELGWGD}&JVCTF ÈDGLG}¶JVNCTL ÊxÊrÊxÊ H ? d X ^ AiAig5^ A A ZT\ f Y [email protected]m 6 ^ 6Y A Y ] m4d \ ]{ d Y f @MA = Y4\1Y [email protected]@jf|@ \ A X C; [email protected]Ê G ±µLqJVF!CWGDG}¶JVNOCELÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HE¿ ?$Ê [email protected]Ê G ¨ K&JVK&}&J!CEF½}&CELGpNpJ!NLGRC UÅp!NOLGREPK"²"ÀxÈDGLG}&J!NCEL ÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HI ?$Ê GEÊ Ã ¨ K&JVK&}&J!CEF½}&CELGpNpJ!NLGRCU²"ÀxÈDGLG}¶JVNCTLlpKMB4UÇFVUÇJ!KMWlIilUÅiD4UTLqJ!DGÁ WGCTJ Ê HD ?$Ê Ã DLGLGKMPONLGR}&DGF!FKMLqJ½J!GF!CTDGREÉJVGKr²°À>ÈDGL}&JVNOCEL ÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ EÄ ?$Ê Ã[email protected]Ê G lC$WGK&P Ô UTÁNP JVCELNUTL¡ÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ EÄ ?$Ê Ã$Ê Ã DGLGLKMPNOLGR}MDGFF!K&LiJvÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ G ?$Ê Ã$Ê H ÀNLGREPOK"KMPOKM}¶JVF!CTLlJ!DGLGLGK&PNLRÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ qà ?$Ê Ã$<Ê & 7C KMPOKM}¶JVF!CTLGpQJVDGLLGKMPONLGRUTp*J 7CiD4UTpNB4UÇFJVNO}MPOKMp ÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ G ?$Ê Ã$ÊE? &7C KMPOKM}¶JVF!CTLeJ!DGLGLGK&PNOLGRUTp©U ³ C$CEBKMFHB4UÇNFM~$®°LGWF!KMK¶ÆeF!K4KM}¶JVNCTL ÊxÊrÊxÊ A ?$Ê Ã$Ê A ="D4UÇLiJ!DGÁ BCENLqJH}&CELqJ UT}¶J°UÇp½UÅpCEDGF!}&KxC LGCTNp!K ÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ q¿ ?$Ê H DLGLGKMPONLGR}&DGF!FKMLqJ½J!GF!CTDGRElU² ¨ À>ÈDGLG}&J!NCEL ÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊx/ Ê ?FG Ô ?$Ê HÊ@G lC$WGK&P UTÁNP JVCELNUTL¡ÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ/?FG ?$Ê HÊ Ã ¸SKMLKMFVUÇ)P hCEF!ÁÂDP:U hCEFQJ!GKSBGGCTJ!CÇ´bUÇp!p!NOpJ!KMWl®°LGWGFKMK¶Æe}MDGFF!KMLqJÊrÊrÊrÊxÊrÊx/ Ê ?Tà ?$Ê HÊ H ®½BGBGPON}jU3JVNCTL JVCUÅDGUTLqJVDGÁ BCENLqJ}&CELqJ UT}¶J ÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊx/ Ê ?A ?$Ê< ³ CELG}&PDGpNCELÊxÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ/?D ?$EÊ ? ®°BGBKMLWGN ÊxÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ ATÄ ?$Ê [email protected]Ê G ®½BGBKMLGWGN e® ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ ATÄ ?$Ê ?$Ê Ã ®½BGBKMLGWGN lÎ ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A7G ?$Ê ?$Ê H ®½BGBKMLGWGN ³ ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ AEà ?$Ê ?$<Ê ®½BGBKMLGWGN ¨ ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ AEà ?$Ê ?$EÊ ? ®½BGBKMLGWGN l» ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ AH ?$Ê ?$Ê A ®½BGBKMLGWGN lË ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A ,*, '!- # % 23!, #). .! * 1!, # % ,)'!- 1! d \ ^ Y+X ] ZT\ f Y d \1Y\ = A ] mGd \ ] { lm4_|_ÐAÐA ZT\ A Ê@G Ô UTPOP+K% KM}¶J ÊxÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê GEÊ@G Kx}&PUTpp!N}MUTP Ô UTPP K% KM}&JQÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê GEÊ Ã ±µLqJ!KMREK&F½iD4UTLqJ!DGÁ Ô UÇPP+K9K&}&J ÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê GEÊ H Ë4F!UT}&J!NCEL4UÇP9iD4UTLqJVDÁ Ô UTPOP9K% KM}¶JÅÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê Ã »,WGREKpJVUÇJVK&p ÊrÊrÊxÊrÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê Ã$Ê@G KxNOLiJ!KMREK&FiD4UTLqJVDGÁ Ô UTPP KMWGRTKxp¬J UÇJ!KMp ÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê Ã$Ê Ã K(hF!UT}&J!NCEL4UÇP9iD4UTLqJVDÁ Ô UTPOP9KMWREKpJ U3JVKMp·ÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A Ê H Í[DJ!J!NLGREK&FPNODNWvJVKMCEF¬ ÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ AÊ HÊ@G ²©CEL$´ } GNF!UTP[Í[DJ!J!NLGRTKMFPNOiDGNW J!GKMCTF ÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ ÆNNON 4 7 D A I A I A I A D ¿FG ¿Tà ¿H ¿F ¿F AÊ HÊ Ã AÊ< F!UTLGpB AÊ G Ê@G A Ê G Ê Ã A Ê G Ê H A Ê G Ê< ; 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A ±µL WG}z}MDGFF!K&LiJ p!K¶JVDGBGp% HGK&L U}&CELGp¬J UTLqJ ÆTCEPOJVUTREKlNpUTBGBPNK&W J!CU}MCELGWDG}&J!CEF7Up¬J UÇJ!NCEL4UÇF M} DGFF!K&LiJvNp JµBGNO}jUTPOPO2KMp¬J UTIPNpGKMW9Ê Ô C7K&ÆTKMF4HN JV UÁCEFKÕp!CTBGGNp¬JVNO}jUÇJ!KMW ÁKjUTpDGF!K&ÁK&LiJ 7K WGNp}MC1ÆEK&F;JVGUÇJQN J 4D}&JVDGUÇJVK&pQNLJVNÁK½UTF!CEDLGW JVK"UjÆEK&FVUTREK½ÆTUÇPDGK |ËNOREDGF!K½Ã$Ê G ÊGÌ"LGK½C[JVGK>QUjp JVC} 4UTF!UT}&J!KMF!N MKlJVGA K 4DG}¶JVD4UÇJ!NCELpC °JVGNOpp¬J UÇJ!NCEL4UÇF }&DGF!FKMLqJvNOpJ!C}MCTÁBGD$JVKeJVGKl}MDGFF!K&LiJ#* }MDGFF!K&LiJ}&CEF!FKMPU3JVNCTL hDGLG}&J!NCEL UTLW2J!C}jUTPO}MDGPUÇJVKÉN JVpË4CEDGFNKMFxJVFVUÇLG/p hCEF$ Á HGN} NpÅ}MUTPPOKMW JVK LGCENOp!KTÊ GK hDGLW4UTÁKMLqJ UTPF!KMUTp!CT: L hCE&F 4DG}&J!D4UÇJ!NCELGp JVC½C$}&}MDGFNpJVG=K UT}&JJV4UÇJKMPOKM}¶JVF!CTLGN}ÐJVF!UTLGp!BCEF¬J/Np Urp¬JVC} 4UTp¬JVN}©BGFC$}&KMp!p&~NOLG}MCEÁNLR CEDJ!RECENOLGR"pJVUÇJVK&p7C [KMPOKM}&J!F!CELp;UTFK½pB K&}MN 4KMWvIqUTLC$}&}MDGB4U3JVNCTL BGF!CTI4UTIGNOPNOJµ UÇLGWlJ!GKSJVF!UTLGp!ÁNpp!NOCELvJVFCEDGREÉJVKrp!UTÁBGPKSNOpHUTPpCÅBGF!CEIGUTIGNPONp¬JVN}ÇÊ !, :1!" ,).# !, GKÅ}MDF!F!K&LqJ I(t) 4CHNOLGRvJVGFCEDGREUvWK&ÆN}&KÅK$GNINOJVp"4DG}&J!D4UÇJ!NCELGp ∆I = I(t) − hIi NOLÕJVNOÁK UTF!CTDGLGW`JVKUjÆTKMFVUÇREK Ê+KLCENpKÂNpSWGK4LGK&WUTp"J!GKÅÁKjUTLp!iD4UÇF!K 4DG}¶JVD4UÇJ!NCELpSC ∆I B K&F DGLGN J hF!K&DKMLG}¶5IGUTLGWhIi HNOWJV),N|Ê KÇÊJVKlpB K&}&J!FVUTPWGK&LGp!N Jµ2C©JVGK 4DG}¶JVD4U3JVNCTLGpMÊл$B K&F!NOÁK&LiJVUTPP JVGK 4D}&JVDGUÇJVNOCELGp7UTF!KHÁKjUTpDGF!K& W HNOJ!GNLU 4LGNOJ!K hF!KMiDGK&LG}&I4UÇLGW HNW$JV WK&JVK&F!ÁNLGK&WIqUI4UTLGW´ B4UTp( p 4P JVK&FrFKMp¬JVF!NO}&J!NLGR hF!K&iDGKMLG}&NK&pxJVCzUTLNLqJVK&FÆÇUTP [ω − ∆ω/2, ω + ∆ω/2] Ê ÕUÇJ!GKMÁUÇJVNO}jUTPOPO ;KK BF!KMpp½J!G<K 4DG}&J!D4UÇJ!NCELGpNOLÉJVGNOpNLqJVK&FÆÇUTP[UTp hCEPOPC Hp ' HG (|~ HGKMFK 1 ∆Iband (t) = 2π Z ω+∆ω/2 dω[∆I(ω)e−iωt + ∆I ? (ω)eiωt ] , ÃÊ@G ω−∆ω/2 hCEF!ÁC Ê Ë4CEF ω JVK,ÁKMUTLp!iD4UTFKMWGDG}&J!D4UÇJVNOCELGp EHNW$JVeC∆I(t) JVGKEhFKMiDG∆ω KML}&eNLqJ!KMFÆÇUTP ∆f = ∆ω/2π Ê$GKMFK%hCEFK ∆I(ω) Np+J!GKË CTDGF!NOKMF9J!FVUTLGp h(∆I)2 i UTF!KSBGFCEBCEFJ!NCEL4UÇP JVCÂJ!GK ;KCEI$J UTNOLAhCTFQJVGKp!BKM}¶JVF!UTP[WGKMLp!NOJµ |Ã$Ê Ã ClWGK&F!NOÆTKJ!GKhFKMiDGK&LG}&q´µWKMBKMLGWGK&LG}MKÉC», Ê|Ã$Ê Ã )7K LGK&KMWJ!C$iLGC CEL0HGN} JVNOÁKÅp}jUTPOK JVGK 4DG}¶JVD4UÇJ!NCELpxJ UTKvBGPUT}MKÇÊNpÂNOp>WGK&p!}MFNIKMW IqJVGK}MCEFF!KMPUÇJVNOCEL hDLG}&J!NCELJ!4UÇJ>}MCELGLKM}&J!p JVGK 4D}&JVDGUÇJVNOCELGprUÇJJ*;ClWGN 9K&F!KMLqJrNLpJ UÇLiJ!p t UTLGW t + t0 ~ C(t) = h∆I(t + t0)∆I(t0 )i ÊË4F!CEÁ JVGK(K&NLGK&F©´ Ó NLqJV} GNOLGKxJ!GKMCTF!KMÁ 7K iLGC J!4UÇJ½JVGKÂpB K&}&J!FVUTPWGKMLGpNOJµzNp½KUT}&J!POe*J HN}&K>JVK Ë4CEDGF!NOKMF;J!FVUTLp/hCEFÁ C JVGK}&CEF!FKMPU3JVNCTLhDGLG}&J!NCEL ' H ? ( SI (ω) = (∆Iband )2 /∆f . SI (ω) = 2 Z ∞ −∞ dth∆I(t + t0 )∆I(t0 )ieiωt . ¿ ÃÊ H "!#$%& ' '(!)&+*,- './*%0'01'2 43-5 16-879%-$& ':( '3 -:5$ 5(<;=8<3>?(+!#[email protected]( :$(C%DE F113HG&A4?B'I 0 ¹½pD4UTPP BGqp!N}MUTPppJ!KMÁp4UjÆEKSUr}MKMF¬J UTNOLvF!K&PU UÇJ!NCELJVNOÁK τ GU #J!KM3F HGN} eUTPOP }MCEFF!KMPUÇJVNOCELGpUTFK P CEp¬JjÊ GK&F!K%hCTF!KJVGK>}MCEFF!KMPUÇJVNOCELAhDGLG}&J!NCELzJVKMLWGp°JVC MKMFC hCEF t τ Ê9²½CEFÁ UÇPPO NL:UÇL`KMPOKM}&J!F!NO} JVF!UTLGp!BCEF¬J;K BKMFNÁKMLqJQJ!GK°p!UTÁBPNLRxF!UÇJVK½Np;Á>DG} vp!POC;K&F;JV4UÇLÉUTLq } GUTFVUT}¶JVK&F!Np¬JVNO}°F!K&PU UÇJ!NCEL JVNOÁKÇÊÇGNOpÐWGC$K&p,LGCTJÐÁKjUTLÅJ!4UÇJ,JV4K 4DG}&J!D4UÇJ!NCELGp/ÆÇUTLNp!CEF,}MUTLG}&KMP4CEDJMÊEGK¶UTF!Kp¬JVNPOP$BF!KMpKMLqJ UTpHU HGNOJ!KSI4UT} iREF!CTDGLGWÕLCENpKTÊ ±µLJVK©}&PUTpp!NO}jUTP4PONÁNO!J JVK½}&DGF!FKMLqJ *Å}MDGFF!K&LiJ7}MCTF!F!K&PUÇJ!NCELNOp,F!KjUÇP UÇLGWp¬ÁÁK&J!F!N}Ç~ C(−t) = C ;K°}jUTLÉWGK 4LGK"J!GK C(t) iJVKMLvJVK"LGCENOp!K°pB K&}&J!FVUTP WGKMLGpNOJµNp7pÁÁK&J!F!NO}T~ SI (−ω) = SI (ω) ÊGÀ: pÁÁK¶JVF!N MK&WpB K&}&J!F!DGÁ SIsym(ω) = (SI (ω) + SI (−ω))/2 hCTF,B CEpNOJ!NOÆEK hFKMiDGKML}MNK&p ω HGN} Np WGK&J!KM}¶JVKMWeNOLvpJVUTLGW4UTFW)POC hF!KMiDGK&LG}&vLGCENpK°ÁKjUTpDGF!K&ÁK&LqJVpMÊ Ô C;K¶ÆEK&F4NLJVGK"iD4UÇLiJ!DGÁ PONÁNOJ JVGK½p!BKM}¶JVF!DÁ NpLGCÂPOCELGREK&F7p¬ÁÁK&J!F!N} SI (−ω) 6= SI (ω) UTLW JVGNOp7}&PUTpp!NO}jUTPWGKMp}MFNBJ!NCELvNOp7LCTJ ÆÇUTPNOWÉUTLqÁCEFKTÊ G: K hDGLGWGUTÁK&LqJ UTPp!CEDGF}MK&pSC ÐLGCENOp!K #JVGK&F!ÁUTP LGCENpK p!GCÇJSLGCENOp!K UTLGWUÁN $J!DGF!KrC ÐICTJV WGKMBKMLWÉCELJ!GK"F!K&PUÇJ!NCELIK&*J 7KMKMLvJ!GF!K&K"KMLKMF!RÇ p}jUTPOKMpM~iJVK°JVGK&F!ÁUTP KMLGK&F!RT kB θ J iJVGK½KMLGK&F!RT UTp!pC$}&NUÇJ!KM0 W HNOJVJVGK hFKMiDGK&LG}&C NLqJ!KMF!K&pJ ω UTLGWJVGKKMLGK&F!RT eV BGFC1Æ$NOWGKMW IqJVGKWGK&ÆNO}MK ÆECEP J UTRTKTÊqGFKMK°PONÁNOJ,}MUTp!K&p;C 9LGCENOp!K½UTF!K©WGNp}MDGpp!K&WÉNÁÁK&WGNUÇJ!KMP >NOLJ!GNpÐp!K&}&J!NCEL+ÊÌ"LJ!GK½CÇJVGK&F 4UTLGW $JVKMF!KSK $NpJ!Ep G LGCTNp!K FHN} eNp7}jUTDGpKMWlIqvp!POC } GUTLGREK&pNL J!GK"WGK¶Æ$NO}MKSFKMp!NOpJVUTLG}MKUÇLGW JVGK¶vUÇF!K hCEDLGWÉNLvÁCEp¬JQ}MCTLGWGDG}¶JVNLRÂÁ U3JVKMFNUTPOpM6Ê G LCENpK°WGCEÁNLGUÇJVK&pQUÇJ7ÆEKMF¬vPOC hFKMiDGKML}MNK&p &K G1Ä*#G1ÄTÄ' Ô ÂUTLW NpÂp¬JVF!CTLGREPOpDGBGBGFKMp!pKMW UTp hF!K&DKMLG}¶ NpÂNOLG}MFKjUTpKMW)p!C`JVGUÇJÅNLJVGNOp JVGK&p!NOp NOJxNprLGCTJ>UTWGWGFKMpp!KMW pNLG}&KLCENpK ÁKjUTpDGF!K&ÁK&LiJ!pÂUTFK}&CELGpNWGK&F!KMWUÇJ>NRE hF!K&DKMLG}&NKMp&Ê GKM L hF!KMiDGK&LG}&NLJVGK½FVUTLRE>K hFCEÁ G1ÄE( Ä Ô ½J!C GH4´ G1Ä> Ô $J!G>K HNOJVK½LGCENOp!K©K $Np¬JVp hp!DG} eUÇp JVGK&F!ÁUTPGLGCTNp!Kqp!GCÇJ7LCENpKiUÇLGWvUÇJ hFKMiDGK&LG}MNOKMpQUÇI C1ÆEK(G½¸ Ô 7K>4LW 4LNOJVK hFKMiDGKML}&LGCENOp!KTÊ L * ,! # . ®QJLGCEL´ MK&F!CÅJVK&ÁBKMF!UÇJVDF!KJVGK&F!ÁUTP GDG}&J!D4UÇJVNOCELGpQ}&CELqJVFNIGDJ!K"}MCTLGp!NOWGKMF!UTIGPOJVCÂJ!GKSLGCENOp!KK&ÆTKML NLJ!GKeUTIGpKMLG}&KzC HJ!GKeIGNUTp>}MDGFF!KMLqJ N|Ê KÇÊ/NL K&iDGNPONIGFNDGÁ¶ÊGK&p!KeJVGK&F!ÁUTP 4D}&JVDGUÇJVNOCELGpUTFK }jUTPOPK&WJVGK&F!ÁUTP;LGCTNp!KeUÇLGW UÇPp!C iLGC HL UTpÒTCEGLGpCEL$´µ²©iDGNp¬JLCENpKÉI K&}jUTDGpKlJ!GK& ;K&F!A K 4F!p¬J Ô F!K&B CEF¬JVK&W`K$BKMF!NOÁK&LqJ UTPOPOÉIq`ÒÊ+ΰÊ9ÒTCEGLGpCEL ' H A (UTLGWUTLGUTPO MK&WÕJVKMCEFK&JVNO}jUTPOPOÉIq Ê ²©iDGNp¬J MONQPSR%T$PUWVSXZY[ \0P$]^`_<aZUbP$]HP^cXZ[&d eI[&fDR1X#_fZ[4ahgdijP$kmlnUbXZYoXZfDR]<aZd U+a#aZUbP$]pefZP$i<Ri<Ubk+U=XgpaZgdqiIPkCr9ls[8_jah[ X#P θ V[?]PX#[)XZ[&d eI[&fDR1X#_fZ[$t I ' HE¿ (|ʱµLJVGKPNÁNOJ k θ eV, ω [J!GKMFÁ TU PÐLGCTNp!KWGCEÁNLGUÇJVK&prC1ÆTKMF>CÇJVGK&FxJµB K&p>CQLGCENpKTÊGK Á UÇRELGNOJ!DGWGK"CJ!GKLGCENpKSB C7KMF©NpBGFCEBCEFJ!NCEL4UÇPJ!CJ!GK}MCELGWDG}&JVUTLG}MK G CJ!GKppJ!KMÁC6HN} NpHUTLeNOPPDpJVF!UÇJVNOCEL CJ!GK<4DG}&J!D4UÇJ!NCEL$´ WGNpp!NOB4UÇJVNOCEL J!GKMCEFKMÁl~ |Ã$<Ê SI = 4kB θG , HGKMFK G = 1/R HNOJ! R NpF!K&p!NOpJ UÇLG}MKÇ6Ê KrLGCTJVKSJ!4UÇJHJVGKrK $BGFKMpp!NCTLÕNLl», Ê Ã$Ê QNpÆÇUTPNOWÉNL I CÇJVÉJVGKr}MPUTp!pN}MUTPUTLGWeJVGKiD4UTLqJ!DGÁ F!KMRTNÁKTÊG±µLeJVGKrPU3J!JVK&F}jUTpK 7Kx4UjÆTK>p!NOÁBPOJVCF!KMBPUT}&K G Iq JVK}&CELGWGD}&J UÇLG}MKiD4UTLqJVDGÁ NOL Í UÇLGW4UTDGK&F 1 p hCEF!ÁÂDPU G = 2e2 T /h HGK&F!K T NpÉJ!GK JVF!UTLGp!ÁNpp!NOCELÉBGF!CTI4UTIGNOPNOJµ hCEFp!NOLGREPKS} GUTLGLGK&P[}jUTpK ¶Ê GKMFÁ UTP9LGCENpKNpHUTPOp!CÅ}jUÇPPK&W HGNOJ!KSLGCENpK * J!GKp!BKM}¶JVFVUÇP[WGKMLGpNOJµeNOpNLGWGK&B K&LGWGK&LiJ©C f Ê B ' .: ! / 23!, *,#)+!,# . $À GCTJ7LGCENOp!K½NLUTLvKMPOKM}¶JVF!NO}jUTP }&CELGWGDG}¶JVCEF7Np7UxLGCTL$´µK&DNPNOIGF!NODGÁ hIGNUTpÐÆECTPOJ UÇREK V 6= 0LCENpK½CEFNRÇ´ NL4U3JVKMWhF!CEÁ JVGKSWNp!}&F!K¶JVKMLKMp!p©CJVK"} 4UTFREKMpHC KMPOKM}¶JVF!NO}jUTP}MDF!F!K&LqJjÊÀ$GCÇJLGCENOp!K"NOpUÂWGCEÁNL4UÇLiJ }MCELqJ!F!NIDJVNOCELzNOLzJ!GK>LGCTNp!( K HGKML eV kB θ, ω Ê À$GCTJ°LGCENpK;UTp"GF!pJ°WGKMp}MFNIKMW:Iq:À} GCTJ!J q ' HI ( HGCpJ!DGWGNK&WlJ!GK} 4UTFRE<K 4DG}&J!D4UÇJ!NCELeBGGK&LGCEÁKML4UÅNOLzUÂÆÇUT}&DGDGÁ JVDI KWGNOC$WGKÇÊ ± 7KÕUTpp!DGÁKÉJV4U3JJ!GKeKMPOKM}&J!F!CELpBGUTp!p}MCEÁBGPOK&JVK&PONLGWGK&B K&LGWGK&LiJJVF!CEDGRT U:}MCTLGWGDG}¶JVCE%F JVGK&LJ!GKLiDGÁ>I K&FxC ;iD4UTLqJ U N NOLUÉJVNOÁKÅNOLiJ!KMF¬ÆTUÇP T0 4D}&JVDGUÇJVK&pÂUTLGW}jUTLI KWGK&p!}MFNIKMWIi CENOp!pCELGNUÇLp¬J UÇJ!Np¬JVN}&pMÊGKUjÆEKMF!UTREKLDÁÂI K&FNprRENOÆTKMLIiJ!GKÁKjUÇL}&DGF!FKMLqJ hN i = hIiT /e UTLGWvJ!GK"ÁKjUTLÉpDGUTF!KSWGK&F!N ÆTU3JVNCTL Np h(∆N )2 i = hN i 7HN} ÉNp;DGp!K&WeJ!CÅ}jUTPO}MDGPUÇJVKHJVKS}MDGFF!0KMLqJ 4DG}¶JVD4UÇJ!NCELpHUÇJ MK&F!;C hF!KMiDGK&LG}& e2 h(∆N )2 i T02 ehIi = . T0 h(∆I)2 i = ÃÊE? 4Ë F!CEÁ»,Ê Ã$Ê H %;K,}MUTLWGK&J!KMF!ÁNLK/JVGK,DGLNOÆEK&F!p!UTPEp!CTJLGCENOp!K,K$BGFKMpp!NCTLxNOL"JVGK &KMF!C3´ JVK&ÁBKMF!UÇJVDF!K PNOÁN J |Ã$<Ê A SIP oisson = 2ehIi . GKlp!GCTJ LGCENOp!KlB C 7KMF NOpÅ*J HN}MKlJVGKlBGFC$WGD}&JC "JVKz} 4UÇF!REKzDGUTLqJVDGÁ UTLGW JVGKzÁKMUTL }MDGFb´ F!K&Li"J 4C HNLGRJVGFCEDGRE:UWGK&ÆNO}MKTÊ GNp½%K KM}¶JS}jUTLÕI KÂCTIGp!K&FÆEK&WNLzÆÇUT}&DGDGÁ JVDGIKMp½CEF°NOLzJ!DGLGLGK&P ÈDGLG}¶JVNCTLGp HGK&F!KÅJVGK} 4UÇF!REKÅiD4UTLqJVUzUTFKÂJVF!UTLG/p hK&F!FKMWNLGWKMBKMLGWGK&LqJxC ;KMUT} CÇJVGK&FMÊ+GNOpSp!GCÇJ LGCENOp!KÂNOp°}jUÇPPK&WCENOp!pCELGNUÇL`LGCENOp!KÅUTLW», Ê |Ã$Ê A ½N"p LC HLUTpÀ} GCTJ!J q' hCEF!Á>DGPUÊGNOp"F!K&p!DGP J NprÁCWGN 4KMWNLiD4UÇLiJ!DGÁ·J!FVUTLp!BCEFJ>WGDGKJ!CÕJVK UT}&JxJV4U3J>K&PKM}¶JVFCELGpÅUTF K hK&F!ÁNCELGNO}B4UTFJ!N}&PKMp HGN} eCEIK&JVGKrUTDGPN9BGF!NOLG}MNOBGPKÇÊ$±µLÉJVNp}jUÇp!K 4N &;K}MCTLGp!NOWGKMFNOLG}MNOWGKMLqJHKMPOKM}¶JVF!CTLGpCELlUÂB CÇJVKML´ JVNUTP/I4UTFF!NOKMF HGN} UTF!KFVUTLWGCEÁPO:K&NOJVKMFJVFVUÇLGp!ÁNOJJVKM W HNOJ!BF!CEI4UÇIGNPONOJµ T CTFxF%K 4K&}&JVK&W HNOJ! BGF!CTI4UTIGNOPNOJµ #JVGNOp;NpJVGK°F!KjUÇp!CELvJ!4UÇJQpGCTJLCENpK"Np7UTPpCÂ}jUÇPPK&W B4UÇFJVN JVNOCEL LGCTNp!K p!C JVG<K hFKMiDGKML}&Re=NLWG1KM−BKMTLGWKMLqJ°pGCTJ©LGCENOp!KSpB K&}&JVF!UTP[WGK&LGp!N JµeNp ' HD ( Ã$Ê ¿ SI = 2ehIi(1 − T ) . ± JQNOp;DGp%K hDGP9J!CÂWGK 4LGK"JVKSË UTLG C UÇ}&JVCTFQUTp;J!GK"F!UÇJVNOC>IK&*J 7KMKMLeJVK"NLGWKMBKMLGWGK&Lq4J hF!K&DKMLG}¶Ép!GCÇJ LGCENOp!KSWGN Æ$NOWGKMWeIqvJ!GKrCENOp!p!CTLzLGCTNp!K F ≡ SI , 2ehIi D ÃÊ I HGN} NpKMiD4UTP JVC 1 − T NLJVGK½BGF!K&p!K&LiJ;}jUTpKTÊ$±µLJVK°PNOÁN J/C9ÆEKMF¬PCJVF!UTLGpÁNOp!p!NOCEL (T 1) ;KFKM}MC1ÆTKMFHJVGKrCTNp!pCELGNUTLeLGCENpKTÊ ±µLiJ!KMFKMpJ!NLGRTPO [}&CEF!FKMPUÇJVNOCELBGGKMLCEÁK&L4UePN ÇKÅJVGKUTDPN,BGFNLG}&NBGPOKÅCEF ³ CEDGPCTÁÂINLqJVK&FVUT}¶JVNCTL }jUTLp!DGIpJ UÇLiJ!NUTPOPO:pDGBGBGFKMppÂp!GCÇJ>LGCTNp!KNLÁKMp!CTp!}MCTBGN}ppJ!KMÁpMÊË CTFrp¬pJVK&ÁpxN0 L HGN} }MDGFb´ F!K&LiJ>NpLGCTJ>}jUTFF!NOKMWNLDLGNOJ!pxC QKMPK&}&J!F!CEL} 4UTF!RTK JVGKREKMLGK&FVUT3P hCEFÁÂDGP U hCEFJVK p!CTJ>LCENpK Np SI = F 2qhIi HGK&F!KJVGKKMPOKM}¶JVF!CTLz} 4UÇF!REK e NOpF!K&BGPUT}&KMWlÔ IqlUTLzK 9K&}&J!NOÆEK} 4UÇF!REK q Ê4À$GCTJ©LGCENOp!K ÁKMUTp!DF!KMÁKMLqJ!p"B K&F/hCEFÁK&W`NLlJVGKhFVUT}¶JVNCTL4UTPiD4UTLqJVDGÁ UTPOPF!KMRTNÁK>UTPOPC7KMWÕJ!GKÂCEIGpKMF¬ÆTU3JVNCTL C +J!GK hFVUÇ}&JVNOCEL4UTP4} GUTF!REK°}MCEFF!KMpB CTLGWGNLR>JVCrJVGK"iD4UÇp!N ´ B4UTF¬JVN}&PK&p ' qÄ Già (8$UTLGWvpGCTJLCENpK°Np KMLGGUTLG}MK&WIq U UT}¶JVCEF [NLUl²"Ài´ ÈDGLG}&J!NCEL:IKM}MUTDGpKC JVK®½LGWGFKMK&Æq´ F!%K GKM}&J!NCEL ' H)( Ê9GKMFK¶´ hCEF!K qUSp!GCÇJ7LCENpKÁKMUTp!2DGFKMÁKMLqJRENOÆTKMpÐUTWWGNOJ!NCEL4UÇPN7L hCEF!ÁUÇJVNOCELÂUÇI CED$J,JVGKHKMPOKM}¶JVF!NO}jUTPJVF!UTLGp!BCEF¬J HGN} lNpLCTJ½UT}M}&KMp!pNIGPOKrÆNU}&CELGWGD}&J UÇLG}MKrÁKMUTp!DF!KMÁKMLqJ!pMÊ À$GCTJ©LGCTNp!KÁKjUTpDGF!K&ÁK&LqJ°UÇPp!C}MUTL I KDGpKMWÕJ!CWNpJ!NLGRTDGNplIK&*J 7KMK&L:}MPUTp!pN}jUÇPUTLGWÕiD4UÇLiJ!DGÁ p!}MUÇJ!J!KMFNLGRNLzU} 4UÇCTJVNO}r}MUjÆ$N Jµ ' G (8 NOJ7RENOÆTKMpQNOL hCEFÁ UÇJ!NCELUTICEDJ7JVGK"K&PK&}&JVFCELÉp!}MUÇJ!J!KMFNLGRÂBF!C}MKMpp!K&p©NOLÉU>WGN 9Dp!NOÆTK HNFK ' ? (9CEF;NOLÉU ppJ!KMÁ¤NL ³ CEDGPCTÁÂI IGPOC$}3UTWK°F!K&RENÁKTÊÌ"L J!GK°CÇJVGK&F;4UTLW)$pGCTJLCENpK½ÁKjUTpDGF!K&ÁK&LqJVpQ}MUTLvIK DGp!K&WÅJVC"BGFCEIKQB4UTF¬JVN}&PK7pJ U3JVNp¬JVNO}MpMÊÇÎ;CEpCELGp,K&ÁN J!JVK&WÂIqÅU°J!GKMFÁ UTPp!CTDGF!}&KQJVKMLWJ!C"IGDGL} FKMp!DPOJ´ NLGRNOL`Up!DGBKMFb´bCTNp!pCELGNUTLÕp¬J UÇJ!NpJ!N}& p ' A)(9 HGNPOKr; U hK&F!ÁNCELGNO}"JVGK&F!ÁUTP[p!CTDGF!}&K>KMÁNOJ!pHB4UTFJ!N}&PKMp p!K&B4UTFVU3JVKMP |UÇLiJ!N ´ IGDGLG} GNOLGR QPKMUTWGNOLGRÅJVCp!DGI$´µCENOp!pCELGNUÇLÕpJVUÇJVNOpJ!N}M p ' i¿ (|Ê GN?p UT}¶J½}&CELGp¬JVNOJ!DJVK&p UÂÆEKMF¬eNÁB CTFJ UÇLiJ7JVCCEP hCEFQJ!KMp¬JVNLRK&LqJ UTLGRTPKMÁKMLqJ©NLÉJ!GK}MCELqJVK J½C iD4UTLqJ!DGÁ }MCEÁBGDJVUÇJVNOCEL+Ê ±µL Á UT}&F!CEp}MCEBN}Õp¬pJVK&Á%p p!GCÇJÉLGCENOp!KÕNOpvLGCTJvBGF!K&p!KMLqJeIKM}MUTDGp!K:}&DGF!FKMLqJ 4DG}¶JVD4U3JVNCTLGpÉUTFK UjÆEKMF!UTREK&WzCED$J½IqÉK&PK&}&JVFCELGpJVF!UTLGp hKMF!FKMWeJVGFCEDGRElÁÂDGP JVNOBGPK©JVFVUÇLGp!BCEFJH} GUTLGLGK&PpMÊ , !" !,# .()$*.-*/%012 ,!,:. ±µLJ!GKHGNRT;hFKMiDGK&LG}&PONÁNOJ ω k θ, eV & KMFCÇ´µBCENLqJ 4D}&JVDGUÇJVNOCELGpÐNLÅJVK©WK&ÆN}&K©NOLqJVF!CWGDG}&K B UTLUÇpÁÁK&J!F NL5JVGKzp!BKM}&J!F!DGÁ S(ω) 6= S(−ω) Ê7±µL5JVGKzWGK%4LNOJVNOCEL CLGCENpK »Ð Ê Ã$Ê H Ô ˆ = I(t) NOpF!KMBPUT}&KMWlIi JVGKSJVNOÁK°WGKMBKMLGWKMLqJ"}&DGF!FKMLqJ©CEB K&FVUÇJ!CEFNLÉJ!GK KMNOp!K&LIKMFRBN}&J!DGF!K I(t) Ô K $B HNOJ! IKMNOLGRSJVGKJVNOÁKNOLGWGK&B K&LGWGKMLqJ UTÁNP JVCELNUTLxC J!GKHppJ!KMÁlÊiGK ˆK $B p!BKM}¶(iJVĤt) FVUÇP[WGIKMLGpNO(−i JµÉNOĤt) pLGC WGK 4ĤLGKMWzUTp SI (ω) = 2 ± 7K>LC5JVGKHNOLGNOJ!NUTP REFCEDGLGW pJ U3JVKMp SI (ω) = 4π X i,f Z ∞ ÃÊ D iωt ˆ ˆ dth∆I(t)∆ I(0)ie . −∞ |ii UTLGWJVGK 4L4UÇP NOLiJ!KMFÁK&WGNUÇJ!K pJ U3JVKMp ˆ 2 P (i)δ(Ef − Ei − ω) , |hf |I|ii| |f i 7K©CTIJ UTNOL ÃÊ@1G Ä hCEFQNOLGNOJ!NUTP p¬J UÇJ!KMpMÊ4±µLeCEFWGKMFQJ!CÅNLqJVK&F!BGFK&JHBGqp!NO}jUTPOPO NpH}&CEDGBGPOKMWeJVCvUWGK¶JVKM}¶JVCEF&Ê K 4LGWlJV4UÇJ S (ω) NOp BGF!CTB CEF¬JVNOCEL4UTP4JVCxJVGKSK&LGKMFRT J!FVUTLGphKMFQF!UÇJVK°I K¶J*;K&KMLeJVGKSp¬$p¬JVK&Á UTLGWvJ!GK"WGK¶JVKM}¶JVCEF&ÊG± E > E ω = E − E > 0 9KMLGK&F!RTNOp"JVFVUÇLGp/hK&F!F!K&W0h FCEÁ J!GKWGK&J!KM}&J!CEFSJVCÉJVKp¬pJVK&ÁzÊ4U3JxÁKjUÇLGp B CTp!NOJ!NOÆTKhF!K&iDGKMLG}&NK&pÅ}MCEFF!KMpB CTLGWJVC:UÇL2UTIGp!CTF!BJ!NCELK&LGKMFRTBGF!C}MK&p!p h F!CTÁ«J!GKvKMLqÆNFCELGÁKMLqJ! HGNPOK/LGK&RqUÇJVN ÆEK hFKMiDGKML}MNK&p}&CEF!FKMpB CELWrJ!C½UÇLrK&ÁNOp!pNCELSBGFC$}&KMppMÊ GKÐF!KMUTp!CELrCi JVK7UÇpÁÁK&J!F NLvJ!GKp!BKM}&J!F!DGÁ NOpQJVGKBGFKMpKMLG}&K>C MK&F!CÇ´ B CENOLqJ44 DG}¶JVD4UÇJ!NCELpMÊ4± JVGKpp¬JVKMÁ NOpNLeKMiDGNOPNOIGF!NODGÁ UÇJ MK&F!CJ!KMÁB K&FVUÇJ!DGF!KÐLGCK&LGKMFRT NpUjÆÇUTNPUTIGPKh CEFKMÁNp!pNCEL p!CJV4U3J S (−ω) = 0 IDJJ!GK ppJ!KMÁ }jUTL`[email protected];Uj$p°UTIGpCEF!I`K&LGKMFRTzUTLGWÕJ!GKMFK%h CEFK S (ω) 6= 0 Ê GKÂUÇpÁÁK&J!FlNp©NÁBCEFJVUTLqJ UTPpCUÇJ 4LGN JVKSÆECTPOJ UÇREK V UTLGWÉJVK&ÁBKMF!UÇJVDGFK θ [email protected] JVGK}&CELGWGN JVNCTL ω eV, k θ pJ!NPOP ÆÇUTPNOW+Ê ±µLÕKMiDGNOPNOIGF!NODGÁ$ UÇJ>4 LGNOJ!KrJ!KMÁB K&FVUÇJ!DGF!K θ 4 J!GK>BC;K&F°WGK&LGp!N JµzCTI K¶$p©JVGKrWGK¶J UTNOPKMWlI4UTPUTLG}&K F!K&PUÇJ!NCEL ' )I ( ÃÊ@GG S (ω) = e S (−ω) . G1Ä HNOJ! P (i) NOp;JVKSBGF!CEIGUTIGNPONOJµWGNOpJVFNIGD$JVNCTL SI (ω) 7KÂUÇp!p!DÁKJVGKrLGCENOp!Kp!CTDGF!}&K>pp¬JVKMÁ I f f i I I B I ω/kB θ I i (aL ) (aR ) L R SAMPLE PSfrag replacements (bR ) (bL ) : q *'I3;<'(! 1 '3%D' 1'1%D13>D' n'"#h*:b(; <3B0-5(;j?CD5(`0,$<; ! - ; -! jE q; $3p$ $<; - 6 (15$ 3b* % ' NL/R (aL/R ) (bL/R ) NL/R ±µLÉJVGKPNOÁN J;CPOC hF!K&DKMLG}¶ ω kB θ ;KFKM}MC1ÆTKMFHJVGK}MPUTp!pN}MUTP+}jUÇp!K SI (ω) = SI (−ω) Ê ±µLÕBGFNLG}&NBGPOK N J©NpHB CTp!p!NOIGPK"J!CÁKjUÇp!DGFKxpKMB4UTF!UÇJVK&POÉJ!GKSJ*;Cp!NWKMp©CJ!GKxpB K&}&J!F!DGÁC IGDJ½U p!BKM}&NUTP WGK&J!KM}&J!CEF 7HGN} e}jUÇLlWNp!}&KMFLlIK&J*7KMK&LzK&ÁNOp!pNCELÉUÇLGWeUTIGpCEF!BJ!NCELÉBF!C}MKMpp!K&p Np7LGKMK&WGKMW ' G!I7 GDFÃT7Ä 7à G hCTFQD }M(|Ê,DGF F!CK&LiIJHKeCTB }MCEK&FVLqUÇÆTJ!KMCELGFMNOFÊ KMKLq!J ÐGUTK&POp!F!CÂKjU NO#LqJVJVKMF!%F C,WGNDGL }&K"JVCTNJ!pÂGJVKMFKMWGp!KNOp 4,LG±xN JVNDGCTp!LvKÉC JV KlDGLGLp¬CT$J ÁUÇJ!ÁNCEK&L JVFNI(t) NLGp¬JVKMUTWÉC I(t) &KMW ˆ LGCENOp!KSp!BKM}¶JVF!UTP[WGKMLp!NOJµ Z ∞ |Ã$Ê G à + S (ω) = 2 dteiωt h∆I(0)∆I(t)i , S − (ω) = 2 Z −∞ ∞ |Ã$Ê GH dteiωt h∆I(t)∆I(0)i , HGN} F!K&PUÇJ!KMpÐJVCJVGK SI (ω) Iq S +(−ω) = SI (ω) S −(ω) ≡ SI (ω) Ê ON J!J!GKMpK°WGK4LGN JVNCTLGpqN J Np+iDGN JVK,pNÁBGPK JVCF!K&Á UÇF ½J!4UÇJ[NOL S +(ω) S −(ω)&B CEpNOJ!NOÆEK LGK&RqUÇJVN ÆEK h F!KMiDGK&LG}MNOKMp}MCTF!F!K&p!BCELGW JVC UTL KMÁNpp!NCTL F!UÇJVK hFCEÁ JVKÁKMpCEp!}&CEBGNO}WK&ÆN}&K HGNPOK:LGKMREUÇJVN ÆEK B CEp!N JVN ÆEK hFKMiDGKML}MNK&p }MCEFF!K&p!BCELGWeJVCUTLzUTIGp!CTF!BJ!NCELÉF!UÇJVKÇÊ ! −∞ ! GKNWGKMUÂC JVGKSp}jUÇJJVKMFNLGRUTBBGF!CqUÇ} UTPpCÅF!K%hK&F!FKMWzUTpQJVGKÍUTLW4UTDGK&F½UÇBGBGF!CEUT} ;NpQJ!CF!KMPUÇJVK JVF!UTLGp!BCEF¬J>BGFCEBKMFJ!NK&pxCJ!GK p¬pJVK&Á hNLB4UTFJ!N}&DGPUTF%[}MDGFF!K&LiJ4DG}¶JVD4UÇJ!NCELpJ!CÕN JVprp!}jU3J!JVK&F!NOLGR BGF!CTB K&FJVNOKM%p BHN} ÉUTFK"UTpp!DGÁKMWJVCÂIK iLGCHLhF!CTÁ UxiD4UTLqJVDÁ ÁKM} 4UÇLGN}MUTP }MUTP}&DGPU3JVNCTL+ÊEGNOp UTBGBGFCqUT} ;UTp½RTKMLGK&FVUTPON MK&WÉJVCÁÂDGP JVN ´ } 4UTLGLKMP8ÁÂDGP JVN8´ JVK&F!ÁNL4UÇP}&CELGWGD}&JVCTF!p' ?ÇÄ)( Ê ±µLJVGNOpxpKM}¶JVNCT)L 7Kv}&CELGp!NOWGKMFxUÕ}&CELGWGDG}¶JVCEFx}MCELLGKM}¶JVKMWJ!CÕ*J 7CÕJVK&F!ÁNL4UTPOprUTpxPOK%#J LxUÇLGW F!NOREqJ R HKMF!K½KjUT} PKjUÇW 4UTp NL/R } 4UÇLGLGKMPOp hp!KMK°ËNRTDGF!K½Ã$Ê Ã ¶ÊiGK½F!K&p!KMF¬ÆECENOF!p;UTF!K½UTp!pDGÁKMW p!CÉPUTF!REKJVGUÇJ"JVK&`}MUTL:IKÂ} 4UTF!UT}&J!KMF!N MK&W:Ii`U JVK&ÁBKMF!UÇJVDGFK θL/R UTLGWU } GK&ÁNO}jUTPB CÇJVKMLqJ!NUTP µL/R Ê 1 - 1! GNpNpÐU"pJVUTLGW4UÇF!WF!K&p!DGP J HGN} }jUÇLIK4hCEDGLWÅNL¯HK%hpMÊ '<?7G ?Eà (|ÊKNLqJVFC$WDG}MKLGC5CEBKMFVU3JVCEFp a†L/R,α (E) UTLW aL/R,α (E) HGNO} }&F!KjU3JVKUTLGW¼UÇLGLGNNPU3JVK`K&PKM}¶JVFCELGpAHNOJV KMLKMF!RÇ E NOL J!GK } 4UTLGLKMP α NOLÂJVGK;P%K #J/PKjUÇW LCEF/FNRTiJPKMUTW R HGNO} UÇF!KNL}MNWKMLqJ/DGBCELÅJVKQpVUTÁBGPOKTÊT±µLÅJ!GK pVUTÁK ;Uj 4JVGKS}&F!KMUÇJVNOCEL b† (E) UTLWzUÇLGLGNNPU3JVNCTL bL/R,α (E) CEB K&FVUÇJ!CEF!pWKMp!}&F!NOI KKMPOKM}¶JVF!CTLGp NLrJVGK7CEDJVRTCENLGR½pJVUÇJVK&pMÊEÎ7KML/R,α }jUÇDGp!K 7KUTpp!DGÁK;J!4UÇJJ!GK;p}jUÇJJVK&F!NLR"Á U3JVF!N NpNLGWKMBKMLGWGK&LqJ,C 4JVK p!BGNOLepJ U3JVKMp©C JVGKKMPOKM}¶JVF!CTLGp 7KxNORELGCEFK"JVGKp!BNLlNLGWGK eNLeJVKMp!KrCEBKMF!UÇJVCEFpMÊ4GK&p!KrCEBKMFVU3JVCEFp GG CEIK&eUTLqJ!N}MCTÁÁÂD$J UÇJ!NCELF!KMPUÇJVNOCELGpQpDG} `UÇp {a†L,α (E), aL,α0 (E 0 )}+ = δαα0 δ(E − E 0 ) , {aL,α (E), aL,α0 (E 0 )}+ = 0 , GKCEBKMFVU3JVCEFp a UTLGW {a†L,α (E), a†L,α0 (E 0 )}+ = 0 . b UTFKSF!K&PUÇJ!KMWÉÆNUÂJVGKp}jUÇJJVKMFNLGRÁUÇJVFN bL1 ··· bLNL bR1 ··· bRNR = s aL1 ··· aLNL aR1 ··· aRNR s UTp Ã$Ê GB . GKrp!}jU3J!JVK&F!NOLGRÁUÇJ!F!N s 4UTp©WGNÁKMLp!NCTLGp (NL + NR ) × (NL + NR ) Ê ± J½FKMPUÇJVK&p°UTPOP+F!K4KM}¶JVNOCEL BGF!C}MK&p!pKMpxUÇJJVGKPK #JrF!KMpKMF¬ÆECENOFUTPOPJVF!UTLGp!ÁNpp!NOCEL:BGF!C}MK&p!pKMphFCEÁ J!GKF!NRTiJrFKMp!K&FÆTCENFSJVCeJVK PK #JCELGK UTLGWeÆ$NO}MKSÆEK&F!p!UÊ4± J©4UTpJVGKIGPOC$} pJ!F!DG}¶JVDGFK r t0 t r0 Ã$Ê G ? HGKMFKJVGKHp!iD4UÇF!KHWGNUÇRECEL4UTP$IGPC}ip r pN MK NL × NL /UTLGW r0 p!NMK NR × NR WGKMp}MF!NOI KHK&PKM}¶JVFCEL F!K 4KM}¶JVNCTLÅBGF!C}MK&p!p!K&pHGNOPK;J!GKC ´µWGNUTRECELGUTP81FKM}&JVUTLGREDPUTFIGPC}ip t hp!NMK NR ×NL /UTLGW t0 pN &K NL × NR QWGK&p!}&F!NIKSJVGKK&PKM}¶JVFCELÉJVFVUÇLGp!ÁNpp!NCTLvJVGFCEDGREeJVGKp!UTÁBPKTÊ GK}MDGFF!KMLqJ½CEBKMFVU3JVCEF}MUTLzIKS}MCTLGp!NOWGKMFKMWÕNOLvJVGKPO%K #JHPKMUTWlUTp e X I(x, t) = 2mi σ Z ∂ψL (r, t) ∂ψL† (r, t) dydz ψL† (r, t) − ψL (r, t) ∂x ∂x ! , Ã$Ê [email protected] HGKMFK x y UTLW z UTF!K©JVGK½}MCCEF!WNL4UÇJ!KMp7NLJ!GK°POK%#J;PKjUÇW) ψL† UÇLGW ψL UTF!K©JVGK>hKMFÁNOCEL }&F!KjU3JVNCTL UTLGWUTLGLGNOGNPUÇJVNOCEL,4K&PWCEBKMFVU3JVCEF&Ê[GK UÇLGLGNNPU3JVNCTL,4KMPOWCTB K&FVUÇJ!CEFNpK$BGFKMp!pKMW NOLJ!KMF!ÁprC JVGKp}jUÇJJVKMFNLGRBGFCEBKMFJ!NK&pC J!GKpVUTÁBGPOKUTp 1 ψL (r, t) = √ 2π Z NL (E) dEe −iEt X χLα (y, z) p aLα (E)eikLα x + bLα (E)e−ikLα x . υLα (E) α=1 Ã$Ê G ¿ µ± L », Ê Ã$Ê G ¿ χLα(y, z) Np,JVGKHJVF!UTLGp¬ÆEKMFp!K>QUjÆEK>hDGLG}¶JVNCTLNOLJ!GK©PK%#JÐPOKjUTW} 4UTLLGKMP α HN} NOp LGCEFÁ UTPON &KMWvUÇp R dydzχLα(y, z)χ∗Lα (y, z) = δα,α J!GK"ÆEK&PC}MN Jµ C }MUTF!FNKMFp υLα (E) = kLα/m UTLGW kLα = p2m(E − ELα ) !HNOJ!rJ!GKLGCTJVNO}MK,JVGUÇJ&;Kp!K&B4UTFVU3JVK7J!GKÐKMLGK&F!RT E CKMPOKM}&J!F!CELpNOLiJ!C JVGKJ!FVUTLGp¬ÆEK&F!p!K½KMLGK&F!RT ELα }MCEFF!KMpB CTLGWGNLRrJ!CJ!GK©ÁCTJ!NCELÅC+KMPOKM}¶JVF!CTLGpÐNLJVGK α´ JVJVF!UTLGpÆTKMFp!K } 4UTLGLKMP |UT}&F!CEppJ!GKePKMUTW ÐUÇLGW J!GKePCELRENOJ!DGWGNLGUTP7K&LGKMFRT2}MCTF!F!K&p!BCELGWGNOLGRJVCJVGKlÁCÇJVNCTL C KMPOKM}&J!F!CELpxUÇPCELGRÉJ!GKPKjUÇW+Ê[GKp!DGÁÁ U3JVNCTL:NL»,Ê |Ã[email protected]Ê G1¿ "CELGPOÕNLG}&PDGWKMp} 4UTLGLKMPp HNOJ!FKjUTP K hKMF!ÁNCTLÅCEB K&FVUÇJ!CEF!pÐFKMPU3JVNCTLÅNL»,Ê |Ã[email protected]Ê G> hCEF,JVGKH}&DGF!FKMLqJQCEBKMF!UÇJVCTF 7K kLα ÊÀ$DIGpJ!NOJVD$JVNLRJ!G? 4UjÆEK 0 0 e I(t) = 2π Z dE Z dE 0 XX 0 0 0 0 ei(E−E )t a†nα (E)Aαα nn0 (L; E, E )an0 α0 (E ) , nn0 αα0 Gà Ã$Ê GI HGKMFK 0 0 Aαα nn0 (L; E, E ) Np,JVK©}&DGF!FKMLqJQÁUÇJ!F!N >KMPOKMÁKMLqJ!HGNO} WKMBKMLGWGp7CEL BCEpNOJVNOCELÅNLREK&LGKMF!UTP8 X 1 0 0 p (L; E, E ) = Aαα {[kLβ (E) + kLβ (E 0 )] 0 nn 0 kLβ (E)kLβ (E ) β h i 0 0 × e−i[kLβ (E)−kLβ (E )]x δβα δβα0 δnL δn0 L − ei[kLβ (E)−kLβ (E )]x s†Ln;αβ (E)sLn0 ;βα0 (E 0 ) +[kLβ (E) − kLβ (E 0 )] h io 0 0 × e−i[kLβ (E)+kLβ (E )]x δβα δnL sLn0 ;βα0 (E 0 ) − ei[kLβ (E)+kLβ (E )]x s†Ln;αβ (E)δβα0 δn0 L . Ã$Ê GD Ô C7K&ÆEK&F GK&F!K ;KWGNOp!}MDp!p©p!CEÁKUTp!pDGÁBJVNOCELGp©UTLGWlUTLqJVNO}MNOB4UÇJVKU hK¼F!K&p!DGP JVp&Ê • ®°pCEDF;ÁC$WGK&P WC$K&pQLGCTJQJVUTK°NLqJVCÅUÇ}M}MCTDGLqJNLGK&PUTp¬JVNO}½BGFC$}&KMpp!KMp%;KSRTK&JUÂWKMPOJVUhDGLG}¶JVNOCEL C/K&LGKMFRTA HGKM$ L ;KÂ}&CEÁBGDJVKJVK>UjÆEK&FVUTRTK>}MDF!F!K&LqJSUTLGWÕLGCENpKrUÇJ &KMFC hF!K&DKMLG}¶EÊ9®½p°UF!K&p!DGP J JVGKxUjÆEK&FVUTREKxpJVUÇJVNOCEL4UTF¬É}MDGFF!K&LiJ°Np©}MCELpJ UÇLiJ"UTLGWlJVGK MK&F!C hFKMiDGKML}&zLGCENpKxWC$K&p½LGCÇJ°WGK&B K&LGW CEL HGKMFKxN JNpÁKjUÇp!DGFKMW+Ê • ±µLBGFVUÇ}&JVNO}jUTPp!NOJ!D4UÇJ!NCELG%p TJVK½IGNUTp eV NpÐUTpp!DGÁKMWJVCI K©ÁÂDG} p!ÁUTPOPKMFJV4UTLJVK©} KMÁN}jUÇP B CÇJVKMLqJ!NUTP½C xJ!GK:PKMUTWGp&ÊÎ;KM}MUTDGpKÁCEpJvF!KMPOK&ÆÇUTLqJeÁCEÁK&LqJ U4UTBB K&LNOLJVGK:ÆN}&NLGN Jµ C rJVK } GKMÁN}MUTPiB CÇJVKMLqJ!NUTP HNOJ!GNL>>U hK eV 1JVGNOpNÁBGPONKMp+J!4UÇJJVGK7ÁCEÁKMLqJVU kLβ (E) UTLW kLβ (E 0) UTFK FVUÇJ!GKMF}&PCEpKTÊ ±µLxJ!GNp}jUÇp!K JVK7pKM}MCTLGW>IGNOR©J!KMFÁ NL hCEF!ÁÂDPUQC Aαα 0 CEp}MNPOPUÇJ!KMp[F!UTBGNWPO nn (L; E, E ) HNOJ!C U ;UjÆEK&PKMLRTJV π/kF ÊKMp!K kF CEp!}&NPOPUÇJ!NCELGp°}jUTLJ!iDGpÂIK LGK&REPK&}&J!KMWNOLJVNpr}MCELGWNOJVNOCEL+Ê GNpHUÇp!p!DÁBJ!NCEL HNOPP9IKrUTBGBGPONK&WvJVC}jUTPO}MDGPUÇJVK"ICTJ!zUjÆTKMFVUÇREKx}&DGF!FKMLqJ½UTLGWlLGCENOp!KTÊ Ô KMFKjU#J!KMFHNLÉJ!GNpQJ!GKMpNp7K HNPP.;CTF vNOLeJ!GNpHUÇp!p!DÁBJ!NCELpJ!GKMLC;K<HNPOP94UjÆTK 0 0 0 0 Aαα nn0 (L; E, E ) = δαα0 δnL δn0 L − X Ã$Ê ÃTÄ s†Ln;αβ (E)sLn0 ;βα0 (E 0 ) . β µ± L CEFWGKMFJVC}&CEÁBGDJVKeJVKÕUjÆEK&FVUTRTKÕ}MDF!F!K&LqJ!;N JNpLGKM}&KMp!p!UTF JVC}MCELp!NWKMFJ!GKlpJ U3JVNp¬JVNO}jUTP UjÆEKMF!UTREK<hCEFHUÅpp¬JVKMÁ UÇJJ!GKMFÁ UÇP+KMiDGNOPNIF!NDÁ Ã$Ê 7à G ha†nα (E)an α (E 0 )i = fn (E)δ(E − E 0 )δn,n δαα , HNOJ! fn(E) NOp7J!GK"Ë4KMFÁN8´ ¨ NOFVUT} Î;CEpK¶´µ»ÐNOLGpJ!KMNOLWNpJ!F!NOIGDJVNOCE; L hDGLG}&J!NCEL UTp!pC$}&NUÇJ!KMW HN JVvPOKjUTW HGCEpK°} GK&ÁNO}jUTP BCTJVK&LiJ!NUTP4NOp µn ~ fn(E) = 1/[exp((E − µn )/kB θn ) ± 1] ÊU iNLGRxNLqJVCxUT}M}&CEDGLqnJ JVGKDGLNOJ UÇFvC J!GKp!}jU3J!JVK&F!NOLGRÁ UÇJ!F!N FhF!CEÁ »,Ê |Ã[email protected]Ê G!I F;KrCEIJVUTNL 0 0 e hIi = 2π Z 0 0 ÃÊ ÃEà dE F [t† (E)t(E)][fL (E) − fR (E)] , HNOJ! t N pJVGK½C´ WGNUÇRECEL4UTP4IPC}C[JVGK"p}jUÇJJVK&F!NLRÂÁ U3JVF!N Ã$Ê G ? tαα = sRL;αα ÊGKSÁ U3JVF!N t† t }jUÇLzIKxWNUTRECTL4UTPNMK&W)UÇLGWÕ4UÇp½UF!KMUTPp!K¶J°CKMNOREKMLqÆÇUTPODGKMp hJ!FVUTLp!ÁNp!pNCELeBGFCEI4UTINPN JVNOKMp Tα Ê À$CÅJV4U3JJVGKrUjÆEK&FVUTREKr}MDF!F!K&LqJ½}MUTLlI <K HF!N J!J!KMLlUTp 0 Z e X hIi = dETα (E)[fL (E) − fR (E)] . 2π α 0 ÃÊ ÃH GNp,K&iD4UÇJVNOCELC+UjÆTKMFVUÇREKH}MDGFF!KMLqJ=;CTDGPW4UTpÐIKMKMLREK&LGKMF!UTPNMKMW hCEF,ÁUTLqÅ} 4UÇLGLGKMPOpUTLGWÁ UTLq JVK&F!ÁNL4UTPOp '<?Eà (|Ê GH ,1'# ! % # !,# . GKLGCENOp!KSNpWK%4 LGK&WÕNOLeJ!KMFÁpHC}MDGFF!K&LiJ©CEBKMF!UÇJVCTF!p©UTp 2 S (ω) = lim T →+∞ T + Z T /2 dt Z ∞ −∞ −T /2 |Ã$Ê Ã 0 dt0 eiωt [hI(t)I(t + t0 )i − hIihIi] . GKe}jUTPO}MDGPUÇJVNOCELC ©LGCTNp!K NLqÆECEP ÆEK&pBF!CWGDG}&J!p I(t)I(t + t0) C©J*7C}MDGFF!KMLqJCEBKMF!UÇJVCTF!pMʱ J JVGK&F!KhCEF!KSNOLiÆTCEPOÆTKMpREF!UTLGWÉ}MUTLGCELGNO}jUTP9UjÆEKMF!UTREK&p©C hCTDGF hKMFÁNOCELvCEBKMF!UÇJVCTF!p7HGN} e}jUÇLÉI KS}&CEÁ´ BGDJ!KMA W HNOJV NO} .1 pJVGK&CEF!K&Á ha†n1 α1 (E1 , t)an2 α2 (E2 , t)a†n3 α3 (E3 , t + t0 )an4 α4 (E4 , t + t0 )i = fn1 (E1 )fn3 (E3 )δn1 n2 δα1 α2 δn3 n4 δα3 α4 δ(E1 − E2 )δ(E3 − E4 ) 0 +fn1 (E1 )[1 ∓ fn2 (E2 )]δn1 n4 δα1 α4 δn2 n3 δα2 α3 δ(E1 − E4 )δ(E2 − E3 )e−i(E1 −E2 )t . $Ã Ê Ã ? GKA4F!p¬JJ!KMF!Á BGFKMpKMLqJVp J!GKlBGF!CWGDG}¶J C½JVGKzUjÆEK&FVUTREKz}MDGFF!K&LiJ!pMʱµL5J!GKzK $BGF!K&p!p!NOCEL hCEFJVK LGCENOp!K CELPOeJVGKxNFF!KMWDG}MNOIGPKr}MDGFF!K&LiJ"CEB K&FVUÇJ!CEF©}MCELqJVFNIGD$JVKMp%+UTLWÕJVGKxNLqJVK&REFVUÇP C1ÆTKMF°J!NÁKxREN ÆEK&p U WGK&POJ U hDGLG}&J!NCELzNL`K&LGKMFRT hCELGKxC,JVGK>NLqJVK&REFVUTPOp°WGFCEBGp"CTDJ Ê GK>LGCENpK }&CELGpNWGK&F!KMW:NOLÕJVK PK#JPKMUTW;NpCTIJ UTNOLGKMWlUTp 2e2 S (ω) = π + Z dE Xn + + αα0 αα0 ALR (L; E, E 0 Aαα RL (L; E, E + 0 Aαα RR (L; E, E 0 0 αα Aαα LL (L; E, E + ω)ALL (L; E + ω, E)fL (E)[1 ∓ fL (E + ω)] 0 + ω)AαRLα (L; E + ω, E)fL (E)[1 ∓ fR (E + ω)] 0 + ω)AαLRα (L; E + ω, E)fR (E)[1 ∓ fL (E + ω)] + 0 ω)AαRRα (L; E + ω, E)fR (E)[1 ∓ fR (E + ω)] HGKMFK Aαα 0 NOpK$BGF!K&p!pKMW:NLe», Ê Ã$Ê ÃTÄ ÊG®°pp!DGÁNLRJ!4UÇJ C KMLGK&F!RTnn;K(L; r4E, UjÆEK E ) 0 0 2e2 S + (ω) = π + Z X X α Ã$Ê ÃGA . NpNOLGWGKMBKMLWGKMLqJ Tα2 (fL (E)[1 ∓ fL (E + ω)] + fR (E)[1 ∓ fR (E + ω)]) ) Tα (1 − Tα ) (fL (E)[1 ∓ fR (E + ω)] + fR (E)[1 ∓ fL (E + ω)]) . $Ã Ê ÃE¿ GKxLCENpKxU3J ω = 0 NOpHCEIJ UÇNLGK&WCHN JVGCED$J°UÇp!p!DÁNOLGR snn ;αβ (E) NpHNLWGKMBKMLGWKMLqJ°C/KMLKMF!RÇ α UTp dE ( snn0 ;αβ (E) o 0 NL Z 2e2 X S (ω = 0) = dE {Tα (E)[fL (1 ∓ fL ) + fR (1 ∓ fR )] π α=1 + Ã$Ê ÃI GNp hCTF!ÁÂDGPU`NpUTPOp!C:CEI$J UTNOLGKMW hCEF hK&F!ÁNCELGp>IiDGp!NOLGRJ!GK;U1ÆTKÕB4UÇ}TK&J UTBGBF!CqUT} '<?H (|Ê,±µL JVGKvUTIGp!K&LG}MKeCHINUTpÂCTFÅUÇJÂNREJ!KMÁB K&FVUÇJ!DGF!K θ |µL − µR| << kB θ JVGKvJ*7C 4F!p¬JÂJVK&F!Áp WGCEÁNL4U3JVKTÊG¹½p!NOLGRJ!GKrF!KMPUÇJVNOCEL fi(1 − fi ) = −kB θ∂fi /∂E .7K>FKM}&C ÆTKMF°J!GKÅÒTCTGLGp!CTL`²©iDGNOpJ hCEF!Á>DGPU ' H AHq¿ ()hCEFQJ!GKMFÁ UÇP[KMiDGNPONIGFNDGÁ LGCTNp!#K '<? ( ±Tα (E)(1 − Tα (E))(fL − fR )2 S + (ω = 0) = 2 2e2 ( P π GB α Tα ) . kB θ = 4GkB θ , Ã$Ê ÃD HGKMFK G = e2 Pα Tα /π NOpÂJVGKlÍUTLGWGUTDGKMF}&CELGWGDG}¶J UTL}MKlC½J!GKeÁK&p!CEp}MCEBN}e}MNOF!}MDNOJjʱµL J!GK CEBGBCEp!N JVKxPNÁNOJHJ!GKÅIGNUÇp°PUÇF!REK&F½JVGUTL:JVGK>J!KMÁB K&FVUÇJ!DGF!K L − µR| >> kB θ .;KREK¶JUvp!GCÇJ LGCENOp!K HN} lNpHUTPOp!C}jUTPOPK&WÉF!KMWDG}MK&WÕp!CTJ©LGCENpKSCEFiD4UTLqJV|µ DÁ p!GCTJ©LGCTNp!K Ã$Ê HEÄ S + (ω = 0) = 2eF hIi , HNOJ! F NpÐJVGK°ËUÇLG C UT}¶JVCE=F HNOJ!JVF!UTLGpÁNOp!p!NOCEL NOp7K&LGKMFRTNLGWGK&B K&LGWGK&LiJQCEF;NOLJVK°PNOLGKjUÇFFKMRENOÁK F = P α Ã$Ê HG Tα (1 − Tα ) P . α Tα GK:pVUÇÁK`FKMpDGPOJ!p;K&F!KWGNp}MDGpp!K&WIq ÍUTLW4UTDGK&FeUTLGW ÕUTF¬JVNOL ' ?HF ? ? (UTBGBKjUTPONLGRJVC ;UjÆEK B4UT} ÇK&JVp&Ê ±µL`JVGKÂBFVUT}¶JVN}MUTPP ÉNÁBCEFJVUTLqJ°}MUTp!K HGK&LJ!GKÂp}jUTPOKÂC,JVKÂKMLGK&F!RTzWGK&B K&LGWGKML}MKC,JVF!UTLGpb´ ÁNOp!pNCEL}MCK }MNOKMLqJVp NpÁÂDG} ÉPUTF!REK&F,JV4UTL I CTJ! JVK½JVK&ÁBKMF!UÇJVDF!K"UÇLGWÉUTBGBPNK&WÆTCEPOJVUTREK JVGKÅiD4UÇLiJ!NOJ!NKMpSNOL»Ð TÊ α|Ã(E) $Ê Ã I °Á UjÕI KF!K&BGPUÇ}MKMWIq`J!GKMNOF"ÆTUÇPDGK&p"J U ÇKMLU3JSJVGKË4KMFÁNKMLGK&F!RTTÊ GKMA L 7KxCEI$J UTNOL " # X X 2e2 eV 2 2kB θ Tα + eV coth Tα (1 − Tα ) , S (ω = 0) = π 2kB θ α α + Ã$Ê Hqà µ± LÕJVK>}jUÇp!KÅUTPOP[JVGKrJVFVUÇLGp!ÁNpp!NCTLÕ}MCK }&NKMLqJ!pSUTF!Kxp!ÁUTPP}&CEÁBGUTF!K&WzJVC,GJ!KMFÁp½BGF!CEBCEF¬JVNCTL4UTP JVC Tα2 UTF!KLGK&REPK&}&JVK&W)GJVKML + S (ω = 0) = 2ehIi coth eV 2kB θ GNp4hCTF!ÁÂDGPUÂWGKMp}MFNIKMpJVGKpGCTJ©LGCENOp!K¶´ JVGK&F!ÁUTP+LGCTNp!KS}&F!CEpp!C1ÆEK&FMÊ " , !" # .!"+# !" Ã$Ê HH . ® B CENOLqJ½}&CELqJ UT}¶J"NOp©Dp!D4UTPOPOÉWK%4LGK&W:UTp½U}&CELGpJ!F!NO}&JVNOCELeI K¶J*;K&KML`J*7CvÁK&J UÇPPNO}°FKMp!K&FÆTCENFpMÊGK }MCELWGDG}&JVUTLG}&KC7iD4UTLqJVDÁ B CENOLqJ}MCELqJ UÇ}&JrWGNpBGPUjprUÉpJVK&BHNp!KNLG}&F!KMUTp!KUTprUhDGLG}¶JVNCTLC;JVK RqUÇJ!K"ÆECEP J UTRE# K ' GH ( Ê GKMFKQUTFK7WGN 9K&F!K&LiJ QUjp/C 7 UTIGFN}jU3JVNLR½U©B CTNLqJ}&CELqJ UT}¶JjÊ ± J/}MUTL>IKF!KjUÇPN &KM( W hCEFNLGp¬J UTLG}&K;NOL U"IGF!KMU E´OÈDGL}&JVNOCELÅIqÂBGDPPNOLGR"UÇB4UTFJ,U"BGNK&}MKC }MCTLGWGDG}¶JVCEF,DGLqJ!NPN J/IGFKjU ip&Êq±µLUÁCEFK;}MCTLiJ!F!CEPOPK&W QUj ;BCENLqJv}&CELqJ UT}¶JVpÉUÇF!, K hCEFÁK&W NOL à ´µWNÁKMLGpNCEL4UÇPKMPK&}&J!F!CEL RqUÇp!KM%p ;KÇÊ RGÊ7NL¸rUT®½p ®½P#¸UT®°p GK&J!KMFCÇ´µp¬JVF!D}&JVDF!KMp&Ê9ÎÕUÇBGBGPONOLGR UÆECEP J UTRTKJ!C pDGNOJVUTIGPOq´ p!4UTBKMWlRqUÇJ!KxK&PK&}&JVFC$WKMp JVKxK&PKM}¶JVFCEL RqUTp}MUTLeI KPC}jUÇPPOWGK&BGPK¶JVK&WzUÇLGWÕUÅBCENLqJ}&CELqJ UT}¶J©}jUTLlI KSW%K 4LGK&WÕPOC$}MUTPP EÊi®°LGCÇJVGK&F©ÁKjUÇLGpHC }MFKjUÇJ!NLGR>U>B CTNLqJ;}&CELqJ UT}¶JNp7Iq BCEp!N JVNOCELGNLR>UTLÉÀ$©e´ J!NBv}&PCEpK½J!C>JVGK°p!DGF UT}MK"C U>}MCTLGWGDG}¶JVCEF&Ê ®°PP/J!GK pN &KMprC QJVK}&CELGpJ!F!NO}&JVNOCELUTF!KUTp!pDGÁKMWJ!CÕIKpGCEFJ!KMFrJV4UTLJVGKÁKjUT0 L hF!KMKB4UÇJ! WGDGKÂJ!CeUTLqÕJµB KC p}jUÇJJVKMFNLGR [UTLGW`J!DpJ!FVUTLp!BCEFJ"J!GF!CTDGRE:JVGKÅBCENOLiJ"}MCELqJ UÇ}&JNp°I4UTPPONp¬JVN}ÇÊ ±µL UiD4UÇLiJ!DGÁ BCENLqJÅ}&CELqJ UT}¶!J /J!G K HNWJ!2C ½J!GKÉ}MCTLGpJ!F!N}¶JVNOCEL NpÅ}&CEÁBGUTFVUTIPK J!C`JVGKlË4KMFÁN QUjÆTKMPK&LGRTJ!+7Ê ="D4UTLqJVDÁ BCENLqJQ}&CELqJ UT}¶JHNpUÂpNÁBGPK©}MCTLGWGDG}¶JVCE4F HGNO} eNp;Dp!KMWeJVCxJVKMp¬JHCEDGFQLCENpK ÁKMUTp!DF!KMÁKMLqJHpK&JVDB+Ê G? .!, %0123!, !,# . ³ CELG}&KMF!LNLGRrJVGK GLGNOJ!K>hFKMiDGKML}&vLCENpKB K&F/hCEFÁNOLGRSJVGK½NLqJVK&REFVUTPC E NL »Ð Ê Ã$Ê ÃE¿ HNOJ! JVK LGCTJ!N}MKJVGUÇJ R ∞ dEf (E)[1 − f (E + x)] = x/(1 − e−βx) 7K CTIJ UTNOLJ!GK LCELGpÁÁK¶JVFN MK&W LGCENOp!KSp!BKM}¶JVFDGÁ−∞hCEFKMPOKM}¶JVF!CTLlp¬pJVK&Á UTp ' ÃÇÄ)( NL 2e2 X 2ω S (ω) = − Tα2 π α 1 − eβω + Ã$Ê HG NL 2e2 X ω − eV eV + ω − + , Tα (1 − Tα ) π α 1 − eβ(eV +ω) 1 − eβ(ω−eV ) HGKMFK NL NpJVGK"LiDGÁ>I K&FQC} 4UTLLGKMP9 V NOp7JVK"UTBGBPNK&W ÆECTPOJ UÇREKTÊ$»,Ê Ã$Ê HG ÐNp7LGCTJQp¬ÁÁK&J!F!N} hCEFBCEp!N JVNOÆTKSUTLGWlLGK&RqUÇJVN ÆEK<hF!K&iDGKMLG}&NK&pMÊ ±µLKMiDGNOPNOIGF!NODGÁ V = 0)7K FKM}&C ÆTKMFrJVGK 4DG}&J!D4UÇJ!NCEL$´ WGNpp!NOB4UÇJVNOCELÕJ!GKMCEFKMÁ«UÇJ 4LGN JVK hF!K ´ iDGKMLG}&NK& p '<?GA)( 2(−ω) Ã$Ê H ? S + (ω) = 2G , 1 − eβω ±µLxJVK MK&F!C©JVKMÁBKMFVU3JVDGFK7PONÁNOJVNOLGR;}MUTp!K 7K;FKM}&C ÆTKMFJ!GKDGUTLqJVDGÁ LGCENOp!K HN} ÂNOp WNp!}&DGp!pKMW NL¯H%K bÊ ' ÃÇÄ (7UTLGWp!GC HLNOL», Ê 9HÊ Ã I "NOLJ!GKLGK Jx} 4UTBJ!KMF>C ;JVNpSJVGK&p!NOpMʱ ;J!GKFKMpKMFÆTCENFp 4UjÆEKCELGK} 4UÇLGLGKM9P GJVGKFKMp!DPOJVp©UTFKNPPODGpJ!FVUÇJ!KMWÉUÇ p '<?EB¿ ?I)( 2 (2e /π)T (1 − T )(eV − ω)Θ(eV − ω) , S (ω) = (2e2 /π)[−2T 2 ω − T (1 − T )(eV + ω)Θ(−eV − ω) + T (1 − T )(eV − ω)] , + N N ω ≥ 0, ω < 0, Ã$Ê H A HGKMFK Θ(x) NpSJVGK Ô KMUjÆ$NOp!NOWGK hDGLG}&J!NCELUTLGW T NOpJ!GKÅJVF!UTLGp!ÁNpp!NOCELBGF!CEIGUTIGNPONOJµTÊ9±µL UÇ}&J!)7K CEIJVUTNLz»,Ê Ã$Ê H A IqÕDp!NLR JVGKxF!K&PUÇJ!NCEL S +(ω) = SI (−ω) ÊGKÂBPCTJ½C S +(ω) NOp°p!CHL:NOL JVGKDGBB K&F©BGUTLGKMP[NOLeËNRTDGF!KrÃ$Ê HÊ !#. zKjUÇp!DGFNLGRJVGKrLGCEL$´ pÁÁK¶JVF!NMK&WzLGCTNp!KrÁKjUTLGp©IKMNLRvUTIGPOKSJVC WNpJ!NLGRTDGNplIK&J*7KMK&L:KMÁNpp!NCTL ω > 0 7K&LGKMFRT 4C HpvJ!CJVGKÕWGK&J!KM}&J!CEF UTLGW UTIGpCEF!B$JVNCTL ω < 0 ;KMLKMF!RÇ#4CHphFCEÁ J!GK WGK&J!KM}¶JVCEF C;JVGKWGK&ÆN}&K DGLWGKMFrJVKMp¬JjÊ Ô C7K&ÆTKMFK$B K&F!NOÁK&LiJVUTPP :NOJrNOprWN}MDGP JxJ!CzWNpJ!NLGRTDGNp DGL4UTÁ>IGNREDCEDGp!P ÅIK&*J 7KMKMLep¬$ÁÁK&JVFN &KMWvUÇLGWvLGCEL´µpÁÁK¶JVFN MK&W LGCTNp!K $BGUTFJ!POIKM}jUÇDGp!K HGUÇJNp C#JVK&LeÁKMUTp!DGFKMWlNpQJ!GKK $}MK&p!p°LCENpKTÊ EÌ #J!KM)L *J ;CpDG} zÁKjUÇp!DGFKMÁKMLqJVpHUTFK"BKM/F hCTF!ÁKMWÉCELvJ!GKSpVUTÁK"p¬$p¬JVK&Áz~J!GEK 4FpJ HGNOPK°N JNp WGF!N ÆEK&L:CEDJ"C 7K&iDGNPONIGFNDGÁ hKTÊ [email protected]Ê Iq`UTBBGPONLRÉU WG}ÂÆTCEPOJVUTREK ©UTLGW`J!GKÂp!K&}MCELWNOL`KMiDGNOPNOIGF!NODGÁ hJ!GKrÆTCEPOJVUTREKrp!CEDF!}MKxNpHJVDGFLÕC ¶Ê GK>K $}MKMppSLGCENOp!KrNp©WG%K 4LKMW:UTp©JVGKrWGN 9K&F!KML}MK>NOLzJ!GKxLCENpK I K¶*J ;K&KMLlJVG<K 4FpJ½UTLGWeJVGKpKM}MCTLGWzÁKjUÇp!DGFKMÁKMLqJj~ ÃÊ Hq¿ SM,excess(ω) = SM,noneq (ω) − SM,eq (ω). ±µLvÁCEpJ;}jUTpKMpQÁKMpCEp!}&CEBGN}½pVUTÁBGPOKMpQUÇF!K°WF!NOÆTKMLÉCED$JQC K&DNPNOIGF!NODGÁ¦Iq UTLvK J!KMF!LGUTP9WG}½ÆECEP J UTREK V 4pCJ!4UÇJ |Ã$Ê HI SM,excess(ω) = SM (ω, V 6= 0) − SM (ω, V = 0). GK©K $}MK&p!pLGCENOp!KHNpDGp%K hDG6P HGKM L ;K°UTF!K©NLqJVK&F!K&pJVK&WvNLPCC iNLRSNLqJVCrJVGK©} 4UTLGRTK°NLJVK½p¬$p¬JVK&Á HGN} UTF!KlWGDGKlJVCWF!NOÆNOLGRCED$J C "KMiDGNPONIGFNDGÁlÊ/± JNpUTPpCDGp%K hDG?P HGK&L UB4UTF¬JVNO}MDGPUTFp!K&J!DGB [email protected] PSfrag replacements 2 S+ 1.5 1 0.5 0 -2 -1 0 1 2 1 2 h̄ω/eV 0.5 + Sexcess 0.4 0.3 0.2 0.1 0 -2 -1 0 h̄ω/eV [email protected]$<-* 'I 0 121%-;?3% A '([email protected](-* 2'Ib91'O% + n!D%? ')'(!j!D [email protected]$-A S b>< #'(! 3 - ,2<; 'D13pb8?B22 2(;+ $:(1%O:'I 0 + ~ω/eV 4e V /h T = 0.5 Sexcess(ω) +B Z '(! 3 6 2<; 'D13/b:- ; ' nm' 4e V /h T = 0.5 U KM}&J!p°JVKÅÁKMUTp!DGFKMÁKMLqJIq`NOLiJ!F!CWGDG}&NLGRÉUÇLUTWGWGN JVNCTL4UTPLGCENOp!K:HGNO} Np"NOLGWGKMBKMLWGKMLqJCÐJVK pVUTÁBGPOKSpJ U3JVK pC IqeJ UiNLRJ![email protected] KMFKMLG}&K>IK&J*7KMKMLzJVGKJ*;CLGCENOp!KrB C7KMF!p>;Kr}jUÇLÕREK¶J°FNWlC JVGKNOLGpJ!F!DGÁKMLqJ U3JVNCTL$´µWGK&B K&LGWGK&LiJHLCENpKSB C7KMFMÊ ²½C ;KUTBGBGP :», Ê Ã$Ê HI ½J!Cz}MUTP}&DGPUÇJ!KÂJVGKK }&KMpp>LGCENOp!KC ;DGUTLqJVDGÁ BCENLqJ}&CELqJ UT}¶J>U3J MK&F!CÅJVK&ÁBKMF!UÇJVDGFK}MCEFF!KMpB CTLGWGNLRJ!CNOJVppB K&}&J!F!DGÁ C LGCENOp!K S + NLl»,Ê |Ã$Ê H A ¶~ |Ã$Ê HD + Sexcess (ω) = (2e2 /π)T (1 − T )(eV − |ω|)Θ(eV − |ω|) . GKp!BKM}¶JVF!UTP4WGKMLp!NOJµÂC K $}MK&p!pLGCENpKI KMUTF!p,ÁCEp¬J,C NOJVp ;K&NREqJÐLGKMUTF MK&F!<C hF!K&DKMLG}&NKM%p iIGDJ,JVK K hFK¶´ LGCENOp!K>WKM}MFKjUTpKMpPNOLGKjUÇF!PO JVC MK&F!CÉC1ÆEK&FSUvFVUÇLGREK [0, ±eV ] hCEF½I CTJ!`BCEp!N JVNOÆTKÂUTLW:LGKMRqU3JVNOÆT iDGKMLG}&NK&p $UTLGWÆÇUTLNp!KMp;IK&TCELGW J!GK©B CENOLqJVp ω = eV hp!KMK©JVGK½PC 7KMFB4UTLGK&PNOLvËNOREDGFK©Ã$Ê H ¶ÊiGK K $}MK&p!pHLGCENOp!K°J!GKMF%K hCEFKS}MCELqJVUTNLGpQUÂp!NLREDGPUÇF!NOJµ~NOJ!p;WGK&F!NOÆÇUÇJ!NOÆTK"WGNOÆTKMFREKMpHUÇJ;JVGNOpQB CTNLqJjBÊ KEGLGW JV4U3J J!GKÐp!BKM}¶JVFVUÇPWKMLGpNOJµC $JVGKÐK $}&KMp!pLGCEL$´ pÁÁK&J!F!N MKMWLGCTNp!K Sexcess L hDGLG}¶JVNOCEL + (ω) Np UTLrK&ÆEK&: ³ C,JVK:hF!K&DKMLG}¶ Sexcess + + (ω) = Sexcess (−ω) Ê CELp!KMiDGK&LqJVPO+JVGKÅK $}MKMpppÁÁK&J!F!NMKMWÕLGCENpK [email protected] KMFp°CELGP eIq` U UÇ}&JVCTF½*J 7C ÁKjUTpDGF!K&W`NLÕ¯©K bÊ ' 7à G( sym + + Sexcess Sexcess hF!CEÁ JV(ω) GKK =$}MSK&p!excess pÂLCE(ω) L$´µp¬+ÁÁ K&J!F!N &(−ω) KMWLGCENOp!KTÊiDGpxK $}MK&p!pÅLGCENOp!KK $BKMFNÁKMLqJVpxNLJVGKiD4UTLqJVDGÁ F!K&RENÁKx}MUTL`DGpD4UTPOPO[email protected] KMF!K&LqJVPOeIKÂK BPUTNOLGKMW`IqzDGp!NOLGRvLGCTL$´µp¬$ÁÁK&JVFN &KMWÕCTF°pÁÁK¶JVFN MK&W LGCENOp!KK $BGF!K&p!pNCEL+Ê,À$C JV' C iLGC ¤BGF!K&}MNOp!KMP H4UÇJÅiD4UTLqJ!NOJµNpxÁKMUTp!DGFKMW NOLp!D} K $B K&F!NÁKMLqJ!p HGKMFKxJ!GK>K }&KMppSLGCEL$´ pÁÁK&J!F!N MKMWzLGCENOp!KrNp©pÁÁK&J!F!NO} ;K>LGKMK&W:URECCWÕDGLGWKMF!p¬J UTLWGNLGR C JVGKWGK¶JVK&}&JVNOCELeBGF!C}MK&p!p&Ê G¿ ËNLNOJVKh FKMiDGK&LG}&5LGCENOp!KlNpJVGKlp!DGIiÈKM}&JvCS UWGKMI4U3JVKTÊ3 4UÇJNpUT}¶JVD4UTPOPOÁKMUTp!DGFKMWNOL GLGNOJ!K hF!K&DKMLG}¶zLCENpKrÁKjUÇp!DGFKMÁKMLqJVp %Å®½p>7Kr4UjÆEK>WNp!}&DGp!pKMW`NOLlJ!GKrp!KM}¶JVNOCELzC/iD4UTLqJVDÁ LCENpK4NOJ Np/NOÁBCEF¬J UTLqJJ!Cp!BKM}MN # USÁKMUTp!DF!KMÁKMLqJÐBGFC$}&KMWGDF!KHNLCEFWGKMF/J!CrWKM}MNOWGK HGN} LGCTNp!K}&CEF!FKMPUÇJVCEF NpHÁKjUTpDGF!K&W+ʯ©K&}MKMLqJ!PO JVKMCEFK&JVNO}jUTP[K9 CTFJVp½4UjÆEKxI K&KMLÕÁUTWGKJVCWGK&p!}MFNIKxJ!GKxNREAhF!K&DKMLG}¶ LGCENOp!KSÁKjUTpDGF!K&ÁK&LiJ©BGFC$}&KMpp©IGUTp!K&WÕCELÉJVKrW$$LGUTÁNO}jUTP ³ CEDGPCTÁÂIÉIGPC}3 UTWGKSJVKMCEF¬EÊ ±µL5JVNp} 4UTBJ!KMF ±:4 FpJWGNp}MDGppJ!GKÕpNLGREPOKeKMPK&}&J!F!CEL2JVDGLLGKMPONLGRNL5UDGP JVF!U3´µpÁ UTPOP7J!DGLGLGK&P ÈDGLG}¶JVNCTL) HGNO} UÇF!KAi LGCHL UTpWL4UÇÁNO}jUTP ³ CEDGPCTÁÂI2IGPC}3 UTWGKTÊGNpÂJ!GKMCEF¬ NpU:I4UÇp!Np h CEF WGK&J!KM}¶JVNLR5LGCTNp!KNOL U2ÁK&p!CEp}MCEBN}WGK&ÆNO}MKIi }&CEDGBGPONLGR2N Jz}MUTB4UT}&NOJ!NOÆEK&PO5JVC5U`ÈDGL}&JVNOCEL NL U LGKjUÇF!IqWGK&J!KM}¶JVCEF}MNOF!}MDNOJ!HGN} NpUTPpCWGNp}MDGpp!KMW NLJVGNOp} GUTBJVK&FMÊ$À$CEÁKHK$BKMF!NOÁK&LqJVpÐC+ LCENpK WGK&J!KM}¶JVNCTLzUÇF!KB CTNLqJVK&WlCTDJHNLÉJVKrPUTpJHpKM}¶JVNCTL+Ê $ " ³ CEDGPOCEÁÂI:IGPOC$}3UTWGKNOp"JVGKNL}MF!KMUTp!K&WFKMpNpJVUTLG}&K UÇJpÁ UÇPPINUTpSÆECTPOJ UÇREKMpSC;UTLKMPOKM}&J!F!CELN}ÅWGK ´ ÆN}MK½}MCEÁBGFNp!NOLGRÂU3JQPKMUTpJQCELGK"POCQ´ }jUTBGUT}MN J UTLG}&K©J!DGLGLGK&PÈDGLG}¶JVNOCEL+Ê ³ CEDGPCTÁÂIvIGPOC$}3UTWGK>QUÇp44F!p¬J CEIGpKMFÆTKMWUTLGWDGLGWKMF!p¬JVCC$W HN JVGNOLÕJVGK hF!UTÁK ;CTF lC p!NLREPKÂK&PK&}&JVFCEL:JVDGLLGKMPONLGRvNOL:p!ÁUTPP }jU ´ B4UT}&NOJ UÇLG}MK©ÁK¶J UTPOPNO}QJVDGLGLKMPiÈDLG}&J!NCELG=p HNOJ!ÉUxPUTF!REK©LiDGÁÂIKMF7C ;KMU iPOJVF!UTLGp!ÁNOJJVNOLGR} 4UTLGLGK&Pp&Ê .7# !/,!!,..!,-*1# -*/ ,!!, ,! .# ! ® VJ DGLGLKMPGÈDGLG}&J!NCEL4NLlNOJVp©p!NOÁBPKMp¬J hCEF!Á }MUTLÕIKrWGKMp}MF!NOI K&W$hCEF©KUTÁBGPOK UTp½UJ!GNLlNLGpDGPU3JVCEF I4UTFF!NK&FQIK&J*7KMKMLzJ*;CLGCEFÁ UTP9}MCELGWDG}&J!NLGRKMPOKM}¶JVF!CWGK&php!K&K>ËNOREDGFK<HÊ G ¶Ê4®}&}MCEFWGNLGRÅJ!CJ!GKPU! C[}&PUTpp!N}MUTP4KMPOKM}¶JVF!CWL4UTÁN}&p iLGCx}MDGFF!KMLqJ}MUTL 4C J!GF!CEDREvUTLNLGpDGPU3JVNLRrIGUTF!FNKMF&Êi®}M}MCTF!WGNOLGR JVCÕJ!GK P!U C HiD4UTLqJVDGÁ·ÁK&} 4UTLGNO}M%p GC 7K&ÆTKMF /J!GKMFKvNp>UÕLGCEL´ ÆÇUTLGNOp!GNOLGRÕBGFCEI4UTINPN Jµ' hCEF>UTL KMPOKM}&J!F!CELCEL CTLGK½pNWGK½C 9J!GK°IGUTF!FNKMFÐJVCrF!KMUT} J!GK°CTJ!GKMF7p!NOWGKTÊ GK&LÉUxIGNUTpÐÆECEP J UTREKHNOp;UTBBGPNOKM)W JVGNOpÁKMUTLGpQJ!4UÇJHJVGK&F!<K HNPOP9IKrU}&DGF!FKMLq>J 4C Ê ¨ DGKJVCeJVGKWGNp}MFK&JVK&LGKMpp>C;KMPOKM}&J!F!NO}jUTP,} 4UÇF!REK }&DGF!FKMLqJ:GCHpJVF!CEDGRTUÉJVDGLLGKMP+ÈDGLG}¶JVNCTL NprUzpKMFNKMpxC K¶ÆEKMLqJ!pÂN0 L HGN} K UT}&J!POCELGKKMPOKM}&J!F!CELB4UTpp!KMp hJVDLGLGKMPOp "JVGFCEDGREJVGKI4UTFF!NK&FMÊ ®QJ &KMF!CvJ!KMÁB K&FVUÇJ!DGF!K>UJVDLGLGKMPONLGR BGF!C}MK&p!p"POKjUTWGNOLG R hFCEÁ J!C NOp°CELPOlB [email protected] ,J!GK [email protected] KMFKMLG}&KC } GUTF!RENOLGRKMLGK&F!RENOKMpHI%K hCEFKUTLGWzU #JVK&FQJVGKSJVDLGLGQKMPONLGRÅQBF!−C}MeKMpp½NOpB CTp!NOJ!NOÆTK ∆E = Q2 (Q − e)2 − >0. 2C 2C HÊ@G = e/2C Ê GNp}MCTLGWGNOJ!NCELNOp7p!UÇJVNOp/4K&W [email protected] Q > e/2 CTFÐJ!GK½ÆTCEPOJVUTREK©UT}&F!CEpp7JVKÐÈDGL}&JVNOCEL U > Uc ± ;KvUTp!pDGÁKJV4UÇJ>pJVUTFJ!NLGRÕUÇJÅUÕ} GUTF!REK |Q| < e/2 J!GKÈDGL}&JVNOCELNOp>} 4UTFREKMW IqJVGK NWGKMUTP GI Insulator Normal metal Normal metal * & 3 + '(!)p* %D<;?<;?%? '><3p-4DA<* G';I!E'c<p; -1* <;=;9'h* <; * % D<;-;&<%& '< K JVK&F!L4UÇP}&DGF!FKMLqJ I Ê KML |Q| > e/2 UTLKMPOKM}¶JVF!CTLÁUjÂJVDGLLGKMP4JVKMF!K&IqWKM}MFKjUTpNLGRJVK©} GUTF!REK CEL:JVGK½ÈDLG}&J!NCELIKMPC JVGKÂJ!GF!K&p!GCTPW e/2 Ê9GNpBGFC$}&KMp!pC}M}&DGF!p<HNOJ!U hF!KMiDGK&LG}& f = I/e UTLGWÂNOp/}jUÇPPK&W>pNLGREPOK;K&PKM}¶JVFCEL>JVDLGLGKMPONLGR°CEp!}&NPPUÇJVNOCEL ' ?D AEÄ ( Ê K;Á UjxUTPpC"DGpKHUTLÅNWGKMUTPiÆECEP J UTREK p!CEDF!}MKQUTLGW hKMK&WU°}MDF!F!K&LqJ,JVC°J!GK ÈDGLG}¶JVNOCEL>JVF!CEDGRTU°PUTFREKF!KMpNp¬JVCEF&ÊT± JVpF!KMpNp¬J UTLG}&KQNpUTp!pDGÁKMW JVCIKÐp!ÁUTPPOKMF J!4UTLSJVK/JVDGLGLKMPNOLGRF!K&p!NOpJ UÇLG}MKÐC JVGK ÈDLG}&J!NCELIGDJPUTF!RTK/KMLGCTDGREJVCNLGNIGN J[4U UTp¬J F!K&} 4UTF!RTNLGR"C 4JVGK}MUTB4UT}&NOJVCTF/U #J!KMF/U°J!DGLGLGK&PNOLGR"K&ÆTKMLqJjÊTGKMF?K HNOPPiIKQLGCS}MDF!F!K&LqJÐ@N 4JVKQK JVK&F!L4UÇP ÆECEP J UTRTK°Np7p!ÁUTPPOKMF,JVGUTL e/2C ÊGÀ$: C 7KE4LGWvJVGUÇJUÇJ MKMFCÂJVK&ÁBKMF!UÇJVDGFK iJVGK"U1ÆTKMF!UTREKS}MDF!F!K&LqJHNL JVGKS}&DGF!FKMLqJ´ ÆECEP J UTRTKS} 4UTFVUÇ}&JVK&F!NOpJVNO}SNpp[email protected] #J!KMWeNLvÆTCEPOJVUTREKSIq e/2C ÊGNpp[email protected] #JHNOLvJVGKS}&DGF!FKMLqJ´ ÆECEP J UTRTKÅ} 4UTFVUÇ}&JVK&F!NOpJVNO}NpS}jUÇPPK&W:JVGK ³ CEDGPOCEÁÂIRqUTBUTLGWJVKBGKMLGCTÁK&LGCELC;pDGBGBGFKMp!pNCELC JVGK}&DGF!FKMLqJ©I K&PC Uc NpF!K hKMFF!KMWeJVCUTp ³ CEDPCEÁÂIeIGPOC$} 3UTWKTÊ GK ³ CEDGPOCEÁÂIIGPC} 3UTWGK½Np7UTPpC>CEIGpKMF¬ÆTUÇIGP>K HGKMLvJ!GK°JVK&ÁBKMF!UÇJVDF!K©Np;POC KMLGCTDGREvpC>JV4U3J JVGK°} 4UTF!RTNLGRÂK&LGKMFRT hJVKSKMLGK&F!RT Ec = e2 /2C iJVGUÇJNp7F!K&DNF!K&WvJVCÅ} 4UÇF!REK°J!GKÈDGL}&JVNOCEL HN JV CELGKÅK&PKMÁKMLqJVUTFz} 4UTFREK °NOpSPUTFREKMF©JV4UTL:J!GKÂJVGK&F!ÁUTPK&LGKMFRT:C ÐJ!GKÅ} 4UTFREK}jUTFF!NK&F kB θ Ê[Ë4CEF }jUTBGUT}MN J UTLG}&KMpHIKMPC 10−15 Ë JVGKSJ!KMÁB K&FVUÇJ!DGF!KÁÂDGp¬JHI KIKMPOC UTICEDJ G Ó Ê ±µL UT}¶!J TJ!GKHp!NOLGREPKÈDGLG}&J!NCEL}MUTLGLGCTJIKWGKM}&CEDGBGPOKM; W hF!CEÁ JVGKHFKMpJC JV?K 7CEFPWCEF,F!K&BGPUÇ}MNLR NOJ!pÂp!DGFF!CEDLGWGNLREpIiNWGKMUTPQ}MDF!F!K&LqJCTFÂÆECEP J UTREKÉpCEDGF}MKMp&Ê Ke4UjÆEKÉJ!C}&CELGpNWGK&FJ!GK>ÈDGLG}¶JVNCTL KMÁÂIKMWWGKMWNLÅJVK©K&PK&}&JVFN}MUTPG}MNOF!}&DGNOJ' AG AEÃF AH)(|Ê ¨ L4UÇÁNO}jUTP ³ CTDGPCEÁ>IIPC}ÇUÇWGKNpÐUiD4UTLqJVDGÁ %K KM}¶J HGNO} UTBGBKjUTFp HGKMLUDGUTLqJVDGÁ }MCEKMF!K&LqJS}MCELWGDG}&J!CEF"NOp°}MCTLGLGKM}¶JVK&W:NL`pKMF!NOKM>p HNOJV`UTL KMPOKM}&J!F!CEÁUTRELK&JVNO}°NOÁBKMW4UÇLG}M# K ' AG (|Ê # !,. ! # % !,!, !, .# ! , .! ! '71# - !. ! 1# !,!" K hCEPPOC¦±µLGRECTPWUÇLGW ²°U j UTFC Æ ' AG)(7WGKMFNOÆNLRzJVKvK$BGFKMp!pNCEL CJVDGLGLKMPNOLGRlFVUÇJ!KJVF!CEDGRT U ÈDGLG}¶JVNCTL+Ê GKQiD4UTpN ´µBGUTFJ!N}MPOKMp NL>J!GK7J*7CSÁK&J UÇPK&PK&}&JVFC$WKMp,UTFK;WGK&p!}&F!NIKMWÅIqrJVGK Ô UTÁNP JVCELGNUTL X † X 9H$Ê Ã Hqp = k ckσ ckσ + q c†qσ cqσ , qσ kσ HGKMFK k UÇLGW q UÇF!KÕJ!GK`KMLKMF!RTNKMp CiD4UTp!N8´µB4UÇFJVNO}MPOKMp;HNOJV#QUjÆEK`ÆTKM}¶JVCEF k UÇLGW q HGNPOK σ WGKMLCTJVK&pJVGK&NFp!BGNOL+ÊÇK 4F!p¬JÐUTLGWxJVGKQpKM}&CELGWp!DGÁ }MCTF!F!K&p!BCELGWÂJ!C°JVGK;PK%#JUTLGWÅFNREqJKMPOKM}&J!F!CWGK F!K&p!BKM}&J!NOÆTKMPOTÊ GD DGLLGKMPONLGRNpNOLqJVF!CWGDG}&KMWeIqÉJVGK Ô UTÁNOPOJ!CELGNUÇL ' AqÃF7A B? 7A HT = X A( HÊ H Tkq c†qσ ckσ e−iφ + h.c. , kqσ HNOJ! φ Np J!GKBG4UTpK φ(t) = e R t dt0U (t0 ) HGK&F!K U = Q/C NpJVGK:ÆECTPOJ UÇREK:UT}&F!CEppeJ!GK ÈDGLG}¶JVNCTL+Ê NOJ! J!GKBGDGFB CEpKCÂ−∞ pJ!DGWNLR J!GKK9K&}&JzCxKMLqÆNF!CTLGÁK&LqJzJ!C2J!GK} 4UTLGRENOLGR C } 4UTFREKCELlJ!GK;ÈDGLG}¶JVNCTLlK&PK&}&JVFC$WKMp©IiÉJ!GKCEBKMFVU3JVCEF e−iφ 7Kx}&CELGpNWGK&F©CTLGPOvJ!GK<4DG}¶JVD4UÇJ!NCELp UTF!CTDGLGWÅJVGKÁKjUÇLÆÇUTPODGKQWGK¶JVK&F!ÁNLGK&WIqÂJ!GKK J!KMF!LGUTPGÆECEP J UTRTK V ÊE4U3J7NOLGWGDG}&KMp,DGp,J!Cr}&CELGpNWGK&F φ̃(t) = φ(t) − eV t UTLGW Q̃ = Q − CV Êq±µLiJ!F!CWGDG}&NLGR φ̃ NLqJ!C HT 7K½BKMF hCEF!Á¦UJ!NÁK¶´µWKMBKMLGWGK&LqJ DGLGN J UTF¬ JVF!UTLGp hCEF!ÁUÇJVNOCEL H̃ = U † HU − iU † ∂U/∂t HN JV U= Y h i exp ieV tc†kσ ckσ . X Tkq c†qσ ckσ e−iφ̃ + h.c. kσ 9H$Ê< GKLGK%JVDGLLGKMPONLGR Ô ÇU ÁNOPOJVCTLGNUTLJVGK&LzFKjUTWp H̃T = HÊE? kqσ UTLGWÉJVKrLK% Ô UTÁNOPOJ!CELGNUÇLC JVKrK&PK&}&JVFC$WKMpHNp H̃qp = X (k + eV )c†kσ ckσ + X 9H$Ê A q c†qσ cqσ , qσ kσ p[email protected]#J!KMWeKMLKMF!RÇePK&ÆTKMPOp©IK&J*7KMK&LzJ!GKPKjUÇWGpMÊ ±µLJVK>hCEPOPC HNLR 7K"HNPOPGDGpK½JVK©J!DGLGLGK&PNLR Ô UTÁNOPOJ!CELGNUÇLÂNLJVGK?hCEF!Á HÊ ? iUTLW LGC VJ K JVCTJVUTP Ô UTÁNOPOJ!CELGNUÇLC JVKrp¬pJVK&Á Np 9H$Ê ¿ H = H̃qp + Henv + H̃T , HGKMFK Henv WGK&p!}MFNIGNOLGReJVGKKMLqÆNFCELGÁKMLqJjʱµLCTDGFx}MUTp!K [JVGKKMLqÆNF!CTLGÁK&LqJ>NOprFKMBGFKMpKMLqJVK&W Ii JVGKWGK¶ÆN}M<K HGK&F!K ;K<7CEDGPOWzPON TK½JVCÁKMUTp!DGFKSLGCENOp!KrUTLGWÉJVKQÈDLG}&J!NCELeNpDp!KMWÕUTp©UÅWGK&J!KM}&J!CEFMÊ HNOJ! eV .: .).# ! # % !,!,..!,- $.! * ,!,! ,! .#! Î;K%hCTF!K©}jUTPO}MDGPUÇJVNOLGRrJ!GK½J!DGLGLGK&PNOLGRxF!UÇJVK&p7J!GF!CEDRE JVKÐÈDGLG}¶JVNOCEL IqDGp!NOLGRÂBKMF¬JVDGFI4UÇJVNOCELJVGK&CEF ;KzÁ UT KeJ*;CNOÁBCEF¬J UTLqJUTpp!DGÁBJVNOCELGp&Ê7ËNF!p¬J!/ J!GKzJ!DGLGLGK&PNLRFKMp!NOpJVUTLG}MK R NOpPUTF!REKe}&CEÁ´ B4UTFKMWJ!CÕJVGKF!K&p!Np¬J UTL}MKviD4UTLqJ!DGÁ R = 2π/e ÊGNpxNÁBGPNOKMp"JVGUÇJ>J!GKvpJVUÇJVK&pÂCELJ!GKJ*7C KMPOKM}&J!F!CWGKMpÐCELPOÂÁN xÆEK&F;7KjUi POÂp!CJV4UÇJÐJ!GK Ô UTÁNP JVCELGNUTL H Ê<A NpÐURECC$WÅWGK&p!}&F!NB$JVNCTLC9JVK iD4UTp!N8´µB4UÇFJVNO}MPOKMprNLJVKvKMPOKM}&J!F!CWGKMp&ÊKvJVGK&L ÁUj}&CELGpNWGK&F>JVGKvJ!DGLGLGK&PNOLGR Ô UÇÁNOPOJVCTLGNUTL H̃ UTp©UÅB K&FJVDF!I4UÇJ!NCEL9Ê4GKp!KM}&CELGW:UÇp!p!DÁBJ!NCELeNOp} 4UTF!RTKxK&DNPNOIGF!NODGÁ¤I K&NLGRKMp¬J UTIPNpGKMWlIK%h CEFK T 2 K T U J!DGLGLGK&PNOLGRK&ÆEK&LqJvC}M}MDF!pp!CJ!4UÇJJVKzp¬J UÇJ!KMpJVCI KlDGp!K&W NOL J!GKzBKMF¬JVDGFI4UÇJ!NCEL JVKMCEFK&JVNO}jUTP }jUTPO}MDGPUÇJVNOCELÉUTFKSKMiDGNOPNIF!NDÁ p¬J UÇJ!KMpMÊ GKSJVDGLGLKMPNOLGRFVUÇJ!KSNpREN ÆEK&LzIqvJ!GKrË K&F!ÁN+RECEPOWGKMLeFDGPK 9H$Ê I Γi→f = 2π|hf |H̃T |ii|2 δ(Ei − Ef ) . GNpHNOpHJ!GKrFVUÇJ!KrC JVF!UTLGpNOJVNOCELGpIK&*J 7KMK&L`JVGKNLNOJVNUTP9p¬J UÇJ!K |ii UTLGWeJV( K 4LGUTPpJ U3JVK ÊÀ$B K&}MN ´ N}MUTPP 7K>pK&J |ii = |Ei|Ri UTLGW |f i = |E 0i|R0i HGK&F!K |Ei |E 0i UTF!KriD4UTpN ´µBGUTFJ!N}M|fPOKipJ U3JVKMp°C ÃTÄ F!K&p!BKM}&J!NOÆTKK&LGKMFRT E E 0 UTLGW |Ri |R0i UTF!KF!K&p!K&FÆECTNFrpJ U3JVKMp } GUTF!REKpJVUÇJVK&pEHNOJ!K&LGKMFRENK&p ER ER0 Ê4GKÁUÇJVFN KMPK&ÁK&LqJHNL 9HÊ I 7J!GKMLlI K&}MCEÁKMp H D Ê hf |H̃T |ii = hE 0 |HTe |EihR0 |e−iφ̃ |Ri + hE 0 |HTe † |EihR0 |eiφ̃ |Ri , P UT}&J!pvNL J!GK`iD4UTpNB4UÇFJVNO}MPOKzpB4UT}MKÇÊQGKÕJ!KMFÁ hE 0|Tkq c†qσ ckσ |Ei † = kqσ Tkq cqσ ckσ HNOJ! HTe RENOÆTKMpJVGK`LCEL$´ &KMFC }MCELqJVFNIGD$JVNCTL CELGP HGKMLJ!GK`NLNOJVNUTPHUTLGW#4L4UTP½pJ U3JVKMpvUTF!K`C"JVGK$hCEF!Á |Ei = |..., 1kσ , ..., 0qσ , ...i UTLGW |E 0 i = |..., 0kσ , ..., 1qσ , ...i F!K&p!BKM}¶JVNOÆTKMP EÊ GNprÁKMUTLGpJV4U3J>NOL JVGK½NLGN JVNUTPGpJVUÇJVK°UTLvK&PKM}¶JVFCELvNpC}M}MDBiNOLGRÂJVGK½pJVUÇJVK (k, σ) NOLJVK°PK#J;K&PK&}&JVFC$WKBHGKMFKjUTpQJ!GK pJVUÇJVK (q, σ) Np>DGLGC}M}MDBGNK&W NOLJVGKÉFNREqJ>KMPK&}&J!F!CWGK /POKjUTWNLGRÕJ!C Pβ (E) UTpU`}MCEÁ>IGNL4U3JVNCTLC f (k )[1 − f (q )] Ê ± SJVGK:UÇBGBGPNOKMW2ÆECEP J UTRTK eV NOp ÁÂD} p!ÁUTPPOKMFÅJVGUTLJVGKÕË4KMF!ÁNHKMLGK&F!RT 7K:Á Uj5UTpp!DGÁK JV4U3JÉUTPOP©iD4UÇp!N ´ B4UTF¬JVN}&PKepJVUÇJVK&pÉNLqÆECTPOÆEK&W4UjÆEK:K&LGKMFRENK&pv}MPOCEp!KlJVCJ!GK:Ë4KMFÁNHK&LGKMFRTEÊ7U iNOLGR JVGKSJ!DGLGLGK&PNLRÁUÇJVFN vKMPK&ÁK&LqJ©J!CIKxUTBGBGFC $NOÁ UÇJ!KMP NOLGWGKMBKMLWGKMLqJ"C k q 7K>ÁU1vF!KMBPUT}&K P p hCEFJVGK°WGKMLGpNOJµC pJVUÇJVK&p |T |2 IqvUÇLeU1ÆTKMF!UTREKMWÉÁUÇJ!F!N K&PK&ÁK&LiJ |T |2 7HGNO} eUT}M}&CEDGLqJV4 UÇJ;k,q,σ J!GKË4KMkqFÁNKMLGK&F!RTTÊG®°POP }&CELGp¬J UTLqJJVK&F!ÁpUTFK"}MCEPOPK&}&JVK&WvNLvJ!GK°J!DGLGLGK&PNLRÂF!K&p!Np¬J UTL}MK RT Ê$GK JVCTJVUTP+F!UÇJVEK hCEFK&PK&}&JVFCELÉJVDGLLGKMPONLGR hF!CTÁ PK #JJVCF!NOREqJHNp Z ∞ 1 dEdE 0 f (E)[1 − f (E 0 )] Γ→ (V ) = 2 e RT −∞ X × |hR0 |e−iφ̃ |Ri|2 Pβ (R)δ(k + eV + ER − q − ER0 ) . 9HÊ G1Ä Z ∞ Z ∞ 1 dt 0 exp (i(E − E 0 + eV )t) f (E)[1 − f (E 0 )] Γ→ (V ) = 2 dEdE e RT −∞ 2π −∞ X iφ̃(t) × Pβ (R)hR|e |R0 ihR0 |e−iφ̃(0) |Ri . 9H$Ê@GG R,R0 ½² C 7K,JVFVUÇ}MKÐCEDJ[J!GKÐKMLqÆNF!CTLGÁK&LqJp¬J UÇJ!KMp&Ê1GK7BF!CEI4UÇIGNPONOJµ©CF4LGWGNOLGRJVGKÐNLNOJVNUTP3F!KMpKMF¬ÆECENOF pJVUÇJVK |Ri Np Pβ (R) = hR|ρβ |Ri HNOJVSJ!GK,KMiDGNOPNIF!NDÁ WGKMLGpNOJµSÁUÇJ!F!N ρβ = Zβ−1 exp(−βHenv ) Ê Ô K&F!K Z = F {exp(−βH )} NOpQJVGKB4UÇFJVN JVNOCEL hDGLG}¶JVNOCELeCJ!GKKMLqÆNFCELGÁKMLqJjÊ ¯©K HFβ!NOJ!NLGRJVKÂWGKMP J ; U henv DGLG}&J!NCELzNL HÊ G1Ä NLzJ!KMFÁ C ÐNOJVp½Ë4CEDGFNKMFHJVF!UTLGp hCEF!Á UTLGWÕDGp!NOLGR JVK Ô K&Np!K&LiI K&F!RBGF!K&p!KMLqJVUÇJVNOCEL)7KrCTIJ UTNOL R,R0 *, @ *, :# 1.)'# ! ! , . , .# ! % ,! .#! KSWK%4LGKJVKrK&iDGNPONIGFNDGÁ¤}MCEFF!K&PUÇJ!NCEL h DLG}&J!NCEL heiφ̃(t) e−iφ̃(0) i = = p!CÅJVGUÇJ?;KREK¶J X R Pβ (R)hR|eiφ̃(t) e−iφ̃(0) |Ri 1 X hR|eiφ̃(t) e−iφ̃(0) e−βHenv |Ri , Zβ R Z ∞ 1 Γ→ (V ) = 2 dEdE 0 f (E)[1 − f (E 0 )] e RT −∞ Z ∞ dt exp (i(E − E 0 + eV )t) heiφ̃(t) e−iφ̃(0) i . × −∞ 2π Ã7G 9H$Ê@G à 9HÊ GH ± QJVKvLGCENOp!KNpŸUTDGpp!NUTL)[JVGK}MCEFF!K&PUÇJ!NCELhDGLG}¶JVNCTLWGK%GLGKMW NOL 9H$Ê@G à }MUTLIK !p NOÁ[email protected] UTBGBGP NLGR>J!GKSREK&LGKMF!UTPN MKM W N} J!GKMCTF!KMÁ UTp |UÇPp!CÅpKMKS®½BGB K&LGWGN É® ' »Ðip&Ê ?$Ê ? TU LGW ?$Ê ?E¿ ( [email protected]Ê G> heiφ̃(t) e−iφ̃(0) i = exp(h[φ̃(t) − φ̃(0)]φ̃(0)i) . Ë4CEF,PUÇJJVK&F,}MCELqÆEK&LGNK&LG}M"K ;K©NLqJVFC$WDG}MKJVK°UTIIGF!K¶Æ$NUÇJVNOCEL}jUTPOPK&WB4UTp!&K *BG4UÇp!K°}&CEF!FKMPUÇJVNOCEL hDGLG} ´ JVNOCEL+~ [email protected]Ê G ? J(t) = h[φ̃(t) − φ̃(0)]φ̃(0)i , UTLGWÉJVKxË4CEDF!NK&F;JVF!UTLGp hCEF!Á C J!GK}MCEFF!KMPUÇJVNOCE L hDGLG}¶JVNOCEL 9HÊ GB ~ 1 P (E) = 2π Z ∞ −∞ HGN} lNp}MUTPPOKMWvJ!GKWGNp¬JVF!NOIGDJ!NCEL hDLG}&J!NCEL+Ê GKNLqJVK&REFVUTP9C ÆTKMF©KMLGK&F!RTvC P (E) NpLCEF!ÁUTPNMK&W JVCAGHN} l}MCELGF!Áp UTIGNOPNOJµWGK&LGp!N Jµ Z ∞ ®°LGCÇJVGK&FBGF!CEBKMF¬JµeC 9HÊ [email protected] dt exp [J(t) + iEt] , P (E) UTpHUBGF!CTI$´ P (E)dE = eJ(0) = 1 . P (E) NOpQJVGKp!C3´µ}jUÇPPK&WÉWGK&JVUTNPOKMWÉI4UÇPUTLG}&KSpÁÁK¶JVF 9H$Ê@G ¿ −∞ P (−E) = e−βE P (E) , 9H$Ê@GI HGN} ÁKMUTLGpÐJV4UÇJ7JVGK½BGF!CTI4UTIGNOPNOJµxJVCxK$}MN JVK½J!GK°K&LqÆ$NOF!CELÁK&LiJ;NpPUTFREKMFÐJVGUTLJ!GK°BGFCEI4UTINPN Jµ JVCUTIGpCEF!IzKMLGK&F!RTAhF!CTÁ JVGKrKMLqÆNFCELGÁKMLqJ"IqlU Î7CEPOJ MÁUTLGLA UT}&J!CEFMÊ ³ CELp!KMiDGK&LqJVPO9LGCKMLGK&F!RT }jUTLlIK>UTIp!CEFI K&$ W hF!CTÁ JVGKrKMLqÆNFCELGÁKMLqJ°UÇJ MKMFCJ!KMÁB K&FVUÇJ!DGF!K UTLGW P (E) JVGK&LzÆÇUTLGNOp!GK&p>hCTF LGKMREUÇJVN ÆEKSKMLKMF!RTNKMp&Ê L ,!,!.'!- 1) ¹½pNLGRÅJVKrWK%4LGN JVNOCELÉC P (E) ;Kr}jUÇLlFK%HFNOJVKSJ!GK<hCEF/;UTF!WeJVDLGLGKMPONLGRÅF!UÇJVKSNOL 9HÊ GH QUTp 1 Γ→ (V ) = 2 e RT Z ∞ −∞ dEdE 0 f (E)[1 − f (E 0 + eV )]P (E − E 0 ) . 9HÊ GD GKÂK BF!KMpp!NOCEL 9HÊ GD QJ UÇKMpSNOLiJ!CÉUT}&}MCEDGLqJ"J!GKÂBCEp!pNIGNOPN JµÉCÐKMLKMF!RÇ`K$} 4UÇLGREKÅIK&J*7KMKMLJVK JVDGLLGKMPONLGRKMPOKM}&J!F!CELUTLGW J!GKÕK&LiÆNOF!CELGÁKMLqJMÊ=K`ÁU12NOLqJVKMFBGF!K¶J P (E) UÇpJVK`B CTp!p!NOIGNPONOJµJVC KMÁNOJJVKvKMLGK&F!RT E JVCÕJ!GK K $J!KMFL4UTP;}&NF}MDGN JjÊ ³ CEF!FKMpB CELWGNLGRTPO P (E) hCTFxLGK&RqUÇJ!NOÆEK KMLGK&F!RENOKMp WGKMp}MFNIKMpÐJVGK½UTIGpCEF!BJ!NCELC 9KMLGK&F!RTIqJVGKHJVDLGLGKMPONLGRK&PKM}¶JVFCEL+Êi±µLqJVKMRTFVUÇJ!NLGRC1ÆEK&F7ÆÇUTF!NUTIGPOK E 0 ;KCEI$J UTNOLeJ!G<K 4L4UTP hCEF!ÁÂDP: U hCE4F hCEF QUÇF!WÉJVDGLLGKMPONLGRFVU3JVKSIKMNLR H Ê ÃÇÄ À$NÁNPUTF!P 7K }jUÇP}MDPUÇJ!KÂJVGKI4UT};UTF!WJ!DGLGLGK&PNOLGRzF!UÇJVKÇÊ Ô C7K&ÆEK&FNOJrNpFVU3JVGK&FxCEIqÆNCTDGp<hF!CEÁ JVGKp¬$ÁÁK&JVF¬vC UÅÆECEP J UTREK°IGNUTpKMWlp!NOLGREPKÐÈDGLG}&J!NCELlUTp 9H$Ê Ã7G Γ (V ) = Γ (−V ) . ±µLz}&CELG}&PDGpNCEL)4 J!GKx}&CELGpNWGK&FVUÇJ!NCELeC }MDGFF!KMLqJ *iÆECEP J UTRTKx} 4UÇFVUT}¶JVKMFNp¬JVN}rC/ UpNLGREPOKSJVDGLGLKMPÈDGLG} ´ JVNOCELzFKMWGD}MKMp½JVCJVGKxWGK&J!KMF!ÁNLGUÇJVNOCELeC P (E) CTF©J!GK>BGGUTp!Kx}MCEFF!KMPUÇJVNOCEL h DGLG}¶JVNCTL J(t) Ê ±µLÕJ!GK LGKJHpKM}&J!NCEL;K<HNOPPBGF!K&p!KMLqJ°UTLeKUTÁBGPK"h CEFQp!BKM}&NUTP9NÁB K&W4UTLG}&K HGN} eFKMPU3JVKMpQJ!GKE4 DG}¶JVD4U ´ JVNOCELGpC J!GKÐÆECEP J UTREK7UT}&F!CEpp J!GK[ÈDLG}&J!NCEL>UTLGWxJ!GK}MDGFF!KMLqJ 4 DG}&J!D4UÇJ!NCELGpNOLÂUHLGKjUTFIqxÁKMpCEp!}&CEBGNO} 1 Γ→ (V ) = 2 e RT Z ∞ −∞ dE E P (eV − E) . 1 − exp(−βE) ← → WGK&ÆNO}MK ' ÇÃ Ä |( Ê ÃEà $ +! ! " " ±µL ¯HK%bÊ ' ÃTÄ)9( EJVK°UTD$JVGCEFpÐBGF!CEBCEpK°UÁKMUTp!DF!KMÁKMLqJp!K¶JVDGBhCEF,WGK&J!KM}¶JVNLR>iD4UTLqJVDÁ¡LGCENpKHC ÆTKMF7U HNWGK=hF!K&DKMLG}¶ÅFVUTLREKQDGpNLGRSU°J!DGL4UTIPK7J*7CÇ´ PK&ÆTKMPGp¬pJVK&Á¡UTp,U½WGK&J!KM}&J!CEFMÊTGKWGK&J!KM}¶JVCEF/}&CELGpNpJ!p C;UeWCEDGIGPOKÅDGUTLqJVDGÁ WGCÇJ ¨ = ¨ EHGN} NpS}MUTB4UT}&NOJ!NOÆEK&POz}MCEDBGPK&WJ!CeJ!GKPKMUTWGpC;UeLKjUTFIi ÁK&p!CEp}MCEBN}S}MCTLGWGDG}¶JVCEF&Ê4GKxp} GKMÁKrCJ!GKppJ!KMÁ NOpp!GCHLÕNOLlËNREDF!K<HÊ Ã$Ê a) ΓL ΤC ΓR EL ε eV det ER b) DETECTOR CIRCUIT DEVICE CIRCUIT Zs C 1/3 V det C g Cc C 1/3 V det Zs Cg C s V dev Mesoscopic Device Cc Vg C 1/3 V det 1'2<; b"? L `#A,'B3 011<'* 3H'(*!m%0'01' 2b -, %+*'(!n G c5$'; O G 3-5 1 = Z + m!E'Q<2$$( ? 5$; A *(.!,. .(:!" ®WGCEDIGPK7DGUTLqJVDGÁ¦WCTJ,NpU hDGPP }MCELqJ!F!CEPOPUTIGPOKJ*7CPK¶ÆEKMPppJ!KMÁ HN JV>J!GKp!K&B4UTFVU3JVNCTLÂI K¶J*;K&KML PK¶ÆEKMPOp = EL − ER }MCTLiJ!F!CEPOPK&WlIqlRqUÇJ!KrÆTCEPOJVUTREK UWG}rNLGK&PUTp¬JVNO}S}MDGFF!KMLqJ°}jUTL`}&NF}MDGPUÇJVKSNOLzJVK WGK&J!KM}¶JVNCTL}&NF}MDGN JSCELGP `@N JV; K hFKMiDGK&LG}& NpSBGFC ÆNOWGKMWIq`JVGKÁKMpCEp!}&CEBGN}WGK¶ÆN}MKÇÊ[GK JVDGLLGKMP,F!UÇJVKIK&*J 7KMKMLJVGKWGCTJ!pSNprUTpp!DGÁKMωW=ÁÂDG} p!ÁUTPPOKMF°J!4UTLJVGKÅJ!DGLGLGK&P,FVUÇJ!KMprUT}MFCEp!pJVK PK #JHUTLGWlF!NOREqJHI4UTF!FNK&F!pp!CÅJ!4UÇJHJVGKNOLGKMPUTpJ!N}"}&DGF!FKMLqJ©NpREN ÆEKMLlIq 9H$Ê ÃEà Iinel () = eTc2 P () , HGKMFK Tc NOprJ!GKJ!DGLGLGK&P7}&CEDGBGPONLGReIK&*J 7KMK&LJVGKWGCTJVp&ÊG K hCEFÁÂDGPU HÊ G ? "}jUTLI KF!K HF!N J!JVK&L UTp HGK&F!KeLGC J!G K 4DG}¶JVD4UÇJ!NLGR:B4UTp!K φ̃(t) F!K&PUÇJ!KMpÂJVC:J!GK 4DG}¶J(t) JVD4UÇJ!N=LGReh[δÆECTφ̃(t) POJ UÇREK −UÇδ}Mφ̃(0)]δ F!CTp!pJVGφ̃(0)i K ¨ = ¨ ÈDGLG}¶JVNCTL δVDQD (t) = V (t) − hV δ(t)i IqJVGKF!K&PUÇJ!NCEL Rt Ê KÅ}MUTP}&DGPU3JVKÂJV; K GF!pJBGUTFJrNL`J!GKBG4UTpK}&CEF!FKMPUÇJVCEF"UTLGW:J U ÇK δ φ̃(t) = e −∞ dt0 δVDQD (t0 ) ) NLqJVC>UT}M}&CEDGLqJQJ!GK°WGK 4LGN JVNCTL hCEFÐJVGK"LCEL$´µp¬ÁÁK&J!F!N &KMWB C 7KMFQpB K&}&JVF!UTP9WKMLGpNOJµC [JVGK½ÆECEP J UTREK 4DG}¶JVD4UÇJ!NCELpHUT}MFCEp!pJVKQÈDLG}&J!NCEL SV (ω) = 2 Z ∞ −∞ dteiωt hδVDQD (t)δVDQD (0)i , ÃH 9H$Ê ÃH JVGK&LC7KCEIJVUTNL e2 hδ φ̃(t)δ φ̃(0)i = 2 Z t dt −∞ 0 Z 0 dt −∞ 00 Z ∞ 0 00 dωe−iω(t −t ) SV (ω) . 9H$Ê Ã −∞ K&/F hCEFÁNOLGRJ!GKÂNLqJ!KMREF!UTPp©C1ÆEKMF"J!GK>J!NÁK t0 UÇLGW t00 UTLGW:LGCÇJVNLRJVGUÇJ"JVK:4DG}¶JVD4UÇJ!NCELp°CÐJVK ÆECEP J UTRTKQUT}&F!CEppJVGK ¨ = ¨ ÈDGLG}&J!NCEL>F!KMPUÇJVKJVC°J!GKQ}&DGF!FKMLqJ 4DG}¶JVD4UÇJ!NCELpJVGFCEDGRExJVGKQÁKMpCEp!}&CEBGNO} WGK&ÆNO}MKvUÇp SV (ω) = |Z(ω)|2SI (ω) &HN JV Z(ω) NpJVGKJVFVUÇLGp!NOÁBKMW4UÇLG}MK}MCELLGKM}¶JVNLR`WGK&J!KM}¶JVCEF UTLGWlWGK&ÆNO}MKr}MNF}MDGN JVpHUTLW SI (ω) = 2 R ∞ dteiωt h∆I(t)∆I(0)i UTBGBKjUÇF!p©NLlULCEL$´µp¬ÁÁK&J!F!N &KMW −∞ hCEF!C Á 77K4UjÆEK Z ∞ |Z(ω)|2 π 9H$Ê Ã ? dω J(t) = SI (ω)(e−iωt − 1) , RK ω2 N pQJ!GKiD4UTLqJVDGÁ CFKMp!NOpJVUTLG}MKÇÊ ±µLÕJVGKxPNOÁN JHC,p!ÁUTPOP)4DG}&J!D4UÇJ!NCELGp½C,ÆECTPOJ UÇREK>UÇ}MF!CTp!p½J!GK©ÈDGL}&JVNOCEL) hCEFPCTLGRÂJVNOÁK&p7KK $B4UTLGW eJ(t) ' 1 + J(t) NLl»,Ê HÊ [email protected] QUTLGWeWGK&F!N ÆEK HNOJ! RK = 2π/e2 P () ' π 1− RK Z −∞ ∞ J(t) NOp½LCTJ"WGN ÆEK&F!RENOLGR |Z(ω)|2 π |Z()|2 dω S (ω) δ() + SI () . I ω2 R K 2 −∞ 9H$Ê ÃGA GK"GF!pJ;B4UTF¬J;FKMLGCEFÁ UÇPN &KMp,JVK½K&PUTp¬JVN}©}MDF!F!K&LqJ HGK&L = 0 ʱµL UT}¶J!iIq }&CELqJVF!CTPPNOLGR"JVK°RqUÇJ!K ÆECEP J UTRTKQNLÂJ!GKWGK&J!KM}¶JVCEFÐ}MNOF!}&DGNOJ/J!CS}MCELqJ!F!CEP 6= 0 ;KHWGCLGCÇJ,}MCELp!NWKMF hDGFJ!GKMF/J!GNpJVKMFÁzÊT± J!GK NÁB K&W4UTLG}&K Z NOp/p!ÁUTPOP$K&LGCEDGRE Z = 0.1R ÇJ!GK}MCTDGBGPNOLGR"C J!GKQLGCENOp!KQNOLqJVC"JVKWGK&J!KM}&J!CEF,Np p!D }&NKMLqJ!POÂK 9SK&}&J!NOÆEKHIGD$J,JVGKJVF!UTLGpSNÁB K&W4UTLG}&KKQNpÐUTBBGF!!C $NÁUÇJVN ÆEK&POrNLGWKMBKMLGWGK&LqJ7C hFKMiDGKML}&~ J hCEF!ÁÂDPU 7KÅCEIJVUTNC L hCEF½J!GKÂNOLGKMPUTpJ!N}r}MDGFF!KMLqJSJ!GF!CEDRE 2 Ê GKÂPUTp" |Z(ω)|2 ' |Z(0)|2 ≡ κ2 RK ¨ ¨ JVGK = Np T 2 SI () 9HÊ ÃE¿ Iinel () ' 2π 2 κ2 c 2 . e K 4LW5JVGUÇJJVGKl}MDGFF!K&LiJ 4DG}¶JVD4U3JVNCTLGp U3J hFKMiDGK&LG}& ω FKMpDGPOJNL2JVKlNOLGKMPUTpJ!N}É}&DGF!FKMLqJvU3J PK¶ÆEKMPWGN 9K&F!KML}MK = ω Ê GUÇJÂÁKjUTL( p 7Kv}jUÇLpJ!DGWJVGKBGF!CEBKMF¬JVNOKMprC H}MDGFF!K&LiJ>LGCENOp!K NOLJ!GK ÁK&p!CEp}MCEBN}SWGK¶Æ$NO}MKIqeNLqÆEK&pJ!NRqUÇJ!NLGRÂJ!GKNLGK&PUTp¬JVNO}"}MDGFF!K&LiJ©C WGK¶JVKM}¶JVCEF&Ê GNp"NOLqJVKMFKMp¬JVNLRvJVGK&CEF!K¶JVN}MUTP NWGKMUv4UTp"p C UTF"KMPODGWGK&W:K $B K&F!NOÁK&LiJVUTPÆ[email protected] 4}MUÇJVNOCE)L 9BCEpp!NIPO I K&}jUTDp!K NOLJ!GK WGCTDGIGPKWGCTJxpp¬JVKMC Á UTWGWGN JVNCTL4UTP DBL ;UTLqJVK&W xpCEDGF!}&KMpÂC NLGK&PUTp¬JVNO}Åp!}jU3J!JVK&F!NOLGR F!K&LGWGKMFU`BF!KM}&NpKÉLGCENpKvÁKjUTpDGF!K&ÁK&LiJÂDNOJVKÉWN }MDGP JjÊ ± JÅNpxJVGK&F!%K hCTF!KÉLGK&}MKMppVUTF¬ JVC:POC$C hCEF WGK&J!KM}¶JVNCTLx}&NF!}&DGNOJ!p HGN} >UTF!KPK&p!pÆ$DPLGK&FVUTIGPOK,JVC°WNp!pNB4U3JVNCT)L jUTp NpJVK7}MUTp!KC GpDGBKMF!}&CELGWGD}&JVNOLGR }MNOF!}MDNOJV%p GIKM}jUÇDGp!KC JVGKBGFKMpKMLG}&K>C JVGKpDGB K&F!}&CELGWGDG}¶JVNOLGRREUTB+Ê , !" # .!"+# !" # 1:(# % !,# . ®°pUTLUTBBGPNO}jUÇJ!NCEL) 7KpJVDW`JVKÅ}MDGFF!KMLqJrLGCENOp!KÅpB K&}&JVFDGÁ C;UviD4UTLqJVDÁ B CTNLqJS}MCTLiJVUT}&JMÊ[±µL ¯©K bÊ ' ÃÇ)Ä (84LCENpKxC iD4UTLqJVDGÁ BCENLqJH}&CELqJ UT}¶J pKMKx»,Ê ÃÊ HG ;NLl} 4UTBJ!KMFSà NOpH}MCELGpNWGK&F!K&W:UTp©U hDGLG}¶JVNCTLvC F!K&LGCEFÁ UTPON &KMW hFKMiDGK&LG}& ν = ω/|eVdev | hCEFQp!NOÁBPN}&NOJµ;K°JVUTK eVdev > 0 FHN JV P N = 2 UTLGWeWGN 9K&F!K&LiJHÆÇUTPODGKMpC JVKJ!CTJ UÇP9JVF!UTLGp!ÁNpp!NOCEL T = N α Tα N ν>1, 2T ν P P 2e3 Vdev α Tα (1 − Tα )(1 + ν) + α Tα2 2ν N 0 < ν < 1 , 9HÊ Ã I P SI (ω) = @ N T (1 − T )(1 + ν) −1 < ν < 0 , π α α α N ν < −1 , 0 K HÊ H$ÊqGK©UTpÁÁK¶JVF Iinel (ν) HNOJVÆÇUTPDKMp,C JVGKJ!CTJ UTP$JVFVUÇLGp!ÁNpp!NCTLUTF!KHBGPOCTJ!J!KMWNLËNOREDGF!4 C LGCENOp!K"WK&JVK&F!ÁNLGK&p;JVKSÁ UTNOL hKMUÇJVDGFKSC Iinel (ν) ʱµLvJVGKUTIp!CEFBJVNOCELvp!NOWGK FhCEFQCEBKMLÉ} GUTLGLGK&Pp à b #"'(! !E'0'* .5(;b%"'(! -/D'D<; -1*: O% ' 2 2 2 1'h I%inel 12'(ν) 3b:#':? 1'&16π +'κT; b$c /hV 6(-dev 3j 113.; +<3:-,;=b ?qOD(` W0%Z-<1T*: = 1, 1.5, 2 % '4323<1 '(! !E'n (13S5(<;=4'(! -:!D [email protected]$-A !E' G0'2? 'B<3 Iinel ν = −0.25 ν = 0.25 !E' *8 O% ' b Z HGNO} 4UjÆTK Tα }MPOCEp!KJVC 1 [NLJVGNOp}jUTpK T = 1 UTLGW T = 2 J!GKLGCEL$´ KMiDGNPONIGFNDGÁ B4UTF¬J>C JVGKLGCENOp!KNp MK&F!C`UÇLGWLGCÕKMLGK&F!RT}jUÇLIK UTIp!CEFI K&WIqWK&JVK&}&J!CEF&HGNPOK hCEFxLCEL$´µCTB K&L} 4UTL´ LGKMPOp°JVKLCEL$´µK&iDGNPONIGFNDGÁ LGCTNp!KÅNOp 4LGN JVKUTLWJ!GKÅWGK¶JVKM}¶JVCEF}MUTLUTIGp!CTF!IKMLGK&F!RT`K&ÆEK&LU3J MKMFC JVK&ÁBKMF!UÇJVDGFKTÊ»,ÁNOp!pNCELNpB CTp!p!NOIGPK?hCEFI CÇJV CTB K&LÉUTLGWLGCEL$´ CEB K&L } 4UÇLGLGKMPOpQWGDGK©JVC MKMFCÇ´µBCENOLiJ 4DG}¶JVD4UÇJ!NCELpp!CJVGKNLKMPUÇpJVNO}"}MDF!F!K&Lq>J hCTF ν > 0 NpHUTP QUjp 4LGNOJ!KTÊ GKQNLKMPUÇpJVNO}7}&DGF!FKMLqJÐWGK&B K&LGWGNLRSCELÂJVGK7JVFVUÇLGp!ÁNpp!NCTL>Npp!GC HLNLxJVGKQNOLGp!K¶JVpJ!CËNRTDGF!K HÊ H HNOJ! U $KMW ÆÇUTPODGK½C 9JV"K hF!K&iDGKMLG}¶EÊ$±µLJVK"UTIGpCEF!B$JVNCTL}MUTp!K Iinel (ν) CEp!}&NPOPUÇJ!KMpUTp7( U hDGLG}¶JVNOCEL C T PO%K #JvNOLGp!K¶J HKMF!KMUTpÉNOJNpvUTLNL}MF!KMUTp!NOLGR hDGL}&JVNOCEL HN JV BGPUÇJVKMUTD$´µPON ÇA K hKMUÇJVDF!KMpÉNOL JVK KMÁNpp!NCTLÉ}jUTp K hF!NOREqJHNLGpK&J Ê ±µL }MCTLG}MPODGp!NOCEL)QJVKNOLGKMPUTpJ!N}:}MDF!F!K&LqJzJ!GF!CEDRE¼U ¨ = ¨ UÇJlPOC JVK&ÁBKMF!UÇJVDGFK}MUTL IKU BGF!CTI KeC °LGCENpKzNOL ULGKMUTF!Iq ÁK&p!CEp}MCEBN}l}MCELGWDG}&J!CEFMÊÐGKÕUTpÁÁK¶JVFI K¶*J ;K&KML UTIGpCEF!B$JVNCTL UTLGWeKMÁNpp!NOCELvBGFC$}&KMp!pKMp©RENOÆTKMpHUÅ}MPOKjUTF;ÁKMUTp!DGFKMÁKMLqJ©C JVGKLGCTL$´µK&DNPNOIGF!NODGÁ¡iD4UTLqJVDÁ LGCENOp!KTÊ $ K:LC J!4UÇJSLGCTNp!KxNp"LCTJ"CELPOzUTL:DGL;UTLqJVKMWp!NOREL4UTP|Ê ± J"}MCTLiJVUTNLpSU7KjUTP JV:CÐNOLhCEF!ÁUÇJ!NCEL LGCTJHBGFKMpKMLqJVK&W`NLeUjÆEK&FVUTRTKMWÕCTIGp!K&FÆÇUTIGPOKriD4UÇLiJ!NOJ!NKMp&ÊGGKÁCEpJHNOÁBCEFJVUTLqJiDGKMp¬JVNOCELzNOpGC 7K UT}&J!D4UTPOPOxWGK&J!KM}&JÐJ!GKHLGCENpK %H±µLBGFNLG}&NBGPOK ;KH}jUÇLÁKjUTpDGF!KQJ!NÁK¶´ F!KMpCEPOÆTKM)W ÇJV4UÇJÐNOp,}MCELqJ!NLiDGCEDGpPO CEIJVUTNLÕÆÇUTPODGKMp°C ÐJVGK 4D}&JVDGUÇJVNOLGRÉiD4UTLqJVN Jµ`UTLWPUÇJVK&F°WGCep¬J UÇJ!NpJ!N}&p°CEL:JVKÂW4UÇJVU +}jUÇP}MDPUÇJ!NLGR hCEFNLGp¬J UTL}MKJ!GKLGCENpKBC;K&FMÊ ± JÅUTPOp!ClNpJVGKI4UTpN}WGK%GLGNOJ!NCELCQJ!GK LGCTNp!KÇÊ Ô C;K¶ÆEK&FLCENpK ÁKMUTp!DF!KMÁKMLqJ!p©NOLeK $B K&F!NOÁK&LiJ©UÇF!KWGCELGKrUTP JVK&F!L4UÇJ!NOÆTKMPO hCEPOPC HNLGRxJVGKp!CTDGF!}&KMp©C LCENpKTÊ # . , .#! 1 !" GKÅp!CTJSLGCENOp!KÂNOp"DGpD4UTPP eJVKMp¬JVK&WhCTPPCHNOLGRJ!GKÀ$} GCÇJ!J q'hCTF!ÁÂDGPU »Ð Ê|Ã$Ê<A °U3J MK&F!CvJVK&Á´ B K&FVUÇJ!DGF!K CE: F hCEPOPC HNLGRlJVGK }MF!CTp!p!C1ÆTKMFUTpÅNL»Ð Ê |Ã$Ê HH xUÇJÂLGCEL$´ MKMFC`JVK&ÁBKMF!UÇJVDGFKTÊGK 4F!p¬J K$BKMF!NOÁK&LqJ UTPHp!CTJvLGCTNp!KlÁKjUTpDGF!K&ÁK&LiJ!pvNL p!K&ÁNO}MCELWGDG}&J!CEF iD4UÇLiJ!DGÁ B CENOLqJ}&CELqJ UT}¶JVp 7KMFK WGCELGK½IqɯHK MLN TC1Æ <AE¿ +UÇLGW Ó DGÁ UÇF AI |ÊË4CEF;U>p!NOLGREPOK½} 4UÇLGLGKMP9pVUTÁBGPKqJVGK"LCENpK 4UTp©UB KMU hCEFQJVF!UTLGpÁNOp!p!NOCEL 1/2 UTLGWlp!DGIp!KMiDGK&LqJSCEp!}&NPOPUÇJ!NCELGp;UTF!KrCEIGpKMF¬ÆEKMW`UTpJVGKrLiDGÁÂIKMF ' ( à ' )( ? D'*: q!E'1 *: '2<:'(!s- &* 2; 1'% O&+"'(! @$<-* 3<')1'$1%#13>#' n' 1'O$%Z /3 <3>B([email protected](-* 2'I+c1'O$%-c6mI6 $3o( 4 ; D134<D%;b; ' b - -+H'(! ?B?<;91'<2; +HD' 96 -B1'2;=bo#' 6s3o-B1'3$%D<1S'(! [email protected] $?* 2'Ib 1'O$%-c:< 3 m>( :; 0'/013"D' ?<,- * Gm'(! ; 1%? '+:-,@$<-* 3<'- DA<* *%-E G B5$'; O .2<2; 13 G% n1 <3 G +*/-$1H'(!:-Hh * 113 - ' - @(-* 2'Ib)1'D$% ! 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ËNFpJ67KÂ}&CELGp!NOWGKMF½JVGKÅÎ7CERECEPODGI C1Æq´µWK¸SKMLGLKMp°J!GKMCEF¬`UÇBGBGPNOKMWChCEF½J!GK>REK&LGKMF!UTPNMK&W`p!K&ÁNO}MCEL$´ WGDG}¶JVCEFÁC$WKMP ' EI Ä I7G ( Ê K;UTF!K7LGC5}&CELGpNWGK&F!NLR½JVKQREKMLKMFVUÇP$K&PKM}¶JVFCELÂpp¬JVKMÁCHN JV>UTLÅUTBGBPNK&WÂUTFIGNOJ!FVUTF¬KJVK&F!L4UÇP B CÇJVKMLqJ!NUTP V (x) Ê4GKK% KM}¶JVNOÆTK Ô UTÁNPOJ!CELGNUTL C JVNppp¬JVKMÁ NOpp!NOÁBGP HFNOJJVKMLzUTp Hef f ( X −∇2 = dx − µS + V (x) Ψσ (x) 2m σ o + ∆(x)Ψ†↑ (x)Ψ†↓ (x) + ∆∗ (x)Ψ↓ (x)Ψ↑ (x) , Z Ψ†σ (x) Ê@G HGKMFKJ!GKCEB K&FVUÇJ!CEF!p Ψ Ψ† pVU3JVNp#NLGRJVK;UTLqJ!N ´µ}&CEÁÁÂDJ U3JVNCTL°F!DPKMp&Ê GK&}MUTL>IK7WKM}MCTÁBCEp!K&W NLvJ!KMF!ÁpHC JVGK&NFHË4CEDGFNKMF;}MCEÁB CTLGKMLqJVp©UTp Ψσ (x) = X eikx ckσ , Ê Ã k Ψ†σ (x) = X e−ikx c†kσ , k HNOJ! c†kσ ckσ NpJVK;}MFKjUÇJ!NCEL |UTLGLNGNOPUÇJ!NCEL CEBKMFVU3JVCEF&hCEF/UÇL>KMPOKM}¶JVF!CTL) HN} Â4UÇp/ÁCTÁK&LiJ!DGÁ ³ k UTLGW2pBGNL σ NLJ!GKeÎ ÀJVGK&CEF ' IG ( ÊGK 4F!p¬JÅJVKMFÁNOL GÊ G xWGK&pJ!F!C1p UTLW }&F!KMUÇJVK&pCTLGK KMPOKM}&J!F!CELeUÇLGWlJ!GKMFK%hCEFKS}MCELp!KMF¬ÆEK&p½JVKSLDÁÂI K&FCB4UTFJ!N}&PKMp&ÊGÎ;D$JJVGKSJ*7CPUTpJJ!KMFÁpNOLG}MFKjUTpK CEFWGKM}&F!KjUÇp!K`*J 7CB4UTF¬JVN}&PK&pMÊÐGK`ÁKjUTL C "JVKÕBGFC$WGD}&J Ψ† Ψ† ΨΨ UTFKÕLGCTL$´ ÆÇUTLGNOp!GNOLGRUTLGW JVGK&p!KJVK&F!Áp HNOPP/BPUjUTLNÁB CTFJ UÇLiJ"F!CEPOKTÊ ∆(x) NOpr}MUTPPOKMW: JVKBGUTNFSBCTJVK&LqJVNUÇP Ê+GK%K KM}¶JVN ÆEK Ô UTÁNP JVCELNUTL GÊ G ;NOpWGNUTRTCEL4UTPON MK&WvIqÉJVGKrÎ;CTRECEPDI C1ÆJVFVUÇLGp/hCEFÁ U3JVNCTL Ψ↑ (x) = Ψ↓ (x) = Xh k uk (x)γk↑ − Xh † vk∗ (x)γk↓ i † uk (x)γk↓ + vk∗ (x)γk↑ k i , , Ê H HGKMFK γ γ † UTF!KLGK% CTB K&FVUÇJ!CEF!prpJ!NPP,p!UÇJVNOp/#NLRzJVK hK&F!ÁNCEL}MCEÁÁÂDJVUÇJVNOCELF!K&PUÇJ!NCELGp&Ê[GK& UTF!K }jUTPOPKMW hK&F!ÁNCELDGUTp!N8´µB4UTF¬JVNO}MPKCEBKMFVU3JVCEFpMÊ/GKep¬J UÇJ!K u (x) v (x)>}MCEFF!K&p!BCELGWGpÅJVC:J!GK QUjÆT"K hDGLG}¶JVNOCELC+UrKMPK&}&J!F!CEL$´ [email protected] CEPK ´µ[email protected]iD4UTp!N8´µB4UÇFJVNO}MPOKHUÇkJB CEpNOJ!kNCEL x ÊiGK©}MCEFF!K&p!BCELGWGNOLGR HqÃ Ô UTÁNP JVCELNUTL4UTp©UÅWGNUÇRECEL4UTP hCEF!Á Hef f = Eg + X Ê< † Ek γkσ γkσ , HGKMFK Eg Np/JVK©RTF!CEDGLWp¬J UÇJ!KHKMLGK&F!RTÅC Hef f UTLGW Ek NOp/JVGKHKMLKMF!RÇÅCJVGKHK$}&NOJ U3JVNCTL n ÊqGNOp Ô UTÁNP JVCELNUTLBGF!C1ÆNWKMpÅJV4U3JÅJVGKeKMPOKM}¶JVF!CTL5UÇLGW5CEPK ;UjÆEKChDGLG}&J!NCELpÅpVUÇJ!Np/#JVGKlÎ;CTRECEPDI C1Æ KMiD4UÇJ!NCELp kσ ∇2 − µS + V (x)]u(x) + ∆(x)v(x) , 2m −∇2 Ev(x) = −[ − µS + V (x)]v(x) + ∆∗ (x)u(x) . 2m &JVKMp!KKMiD4U3JVNCTLGp[LGKMK&WxJ!C©IK,p!CTPOÆEK&WxpKMP ´µ}MCTLGp!NOpJVK&LqJVPOTÊjGK uk UÇF!K,KMNOREKML Eu(x) = [− ±µLBGF!NOLG}MNOBGPK CUPONLGKMUTFQp¬$p¬JVK&Á HNOJVe}&CEF!FKMp!BCELGWNLGRKMNOREKMLqÆÇUTPDKMp Ek ~ vk u u E = Ω̂ . v v GÊE? 7hDGLG}&J!NCELp Ê A GKCEBKMFVU3JVCEF Ω̂ Np Ô K&F!ÁNOJ!NUTLvpCJ!4UÇJJ![email protected] KMFKMLqJ©KMNRTKMLhDGL}&JVNOCELGp u UTFKSCEFJ!GCERECTL4UTP Ê ± u Np;J!GKSp!CEPODJVNOCEL hCEF7JVGKSK&NREK&LiÆÇUTPODGK E J!GKML −v Np7JVGKSpCEPDvJ!NCELhCEF;JVKSKMNRTKMLqÆÇUTPDGK v u −E Ê ±µ0 L H4U3<J hCEPPOC Hp";K;HNPOP/UTpp!DGÁK V (x) = 0 Ê)K GLGWJ!GKKMNOREKMLhDLG}&J!NCELGpSNOL:JVGKREK&LGKMF!UTP hCEF!Á u(x) = u0 eikx Ê9± 37KCTLGPOl}MCTLGp!NOWGKMFSK&LGKMFRENK&p E > ∆ J 9JVKMF!K HNOPP I KÂUvB4UTNF°C Á UÇRELGNOJ!DGv(x) WGK&pQC k UÇp!vp!0C}MNUÇJVK&W HNOJVeKMUT} ÕK&LGKMFRT √ 1/2 Ê ¿ k ± = 2m µS ± (Ek2 − ∆2 )1/2 , HNOJ! Ek = (∆2 + 2k )1/2 k = k − µS Ê zCTF!KMC1ÆTKMF IKM}jUÇDGp!KC ;J!GKÎ ³ ÀBGUTNFNLGRÉC k UTLGW 2m −k 7KÁÂDGpJ"}MCELGpNWGK&F"ICTJV:pNRELGp°C k 9pCÉJV4UÇJ°JVGK&F!KÂNOpSU hCEDGF hCEPW`WKMREK&LGKMF!UT}&ÕC F!KMPOK&ÆÇUTLqJ pJVUÇJVK&"p hCEF½KjUT} E pKMKÅËNOREDGFK GÊ Ã HNOJVlJVGKxLGCTJ!N}MKrJV4U3J°JVKx*J 7ChCEPWzp!BGNOL`WGK&REKMLGK&FVUT}¶`CELPO U KM}&J!p LGCTF!ÁUTPNjUÇJ!NCEL pNLG}&KÕJVGK&F!KzNp LCp!BGNOL$´ GNB BGF!C}MK&p!p!K&pMÊ ³ CTLGp!NOWGKMFNLGRJ!GKÕFKMPUÇJVNOCELGpC CEBKMFVU3JVCEFp7NOLvÎ;CERTCEPDGIC1ÆÅJVF!UTLG/p hCTF!ÁUÇJVNOCE)L 7K 4LGW J!4UÇJ7JVGK½K $}MN J UÇJ!NCELGp;UÇJ ±k+ UÇLGW ±k− UTFK BGF!K&WGCEÁNL4UÇLiJ!POKMPOKM}&J!F!CEL´µ[email protected] TKSUÇLGWzCEPK ´µ[email protected] TK $}&CEF!FKMp!BCELGWNLGREP EÊ ±µLREKMLKMFVUÇ8P &;K}jUTL NOLiJ!F!CWGDG}&KJ!GKvKMPOKM}¶JVF!CTL GCEPOKK $}MNOJVUÇJVNOCELCTB K&FVUÇJ!CEF!p HNOJ!JVGKKMLGK&F!RT F!K&DNF!K&WJVC`ÁU ÇKÉUTL K $}&NOJ U3JVNCTL HNOJ! } 4UTF!RTK e Np Eek = µ + Ek HGNOPKJV4UÇJ>JVC`ÁU TKvUTL K $}MN J UÇJ!NCE' L HN JV} 4UTFREK WGN 9K&F!pSIq UTLGWNOp ÊË4F!CTÁ JVK }MCELp!KMF¬ÆÇUÇJVNOCEL C °} 4UTFREK`−e UTLW JVGKl}MCTLGp!K&2µFÆÇUÇJVNOCEL C SEKMhkLGK&=F!RT −µ =7K+,4ELkW =JVG−(µ K&F!KÕUT−F!A K EhkCE)DGFB CTp!p!NOIGPK BGF!C}MK&p!pKMpSNLqÆTCEPOÆNLRJ!FVUTLGp hKMF"C ÐU p!NOLGREPOKxK&PKM}¶JVFCELGN}x} 4UTFREK>NOL`p!DGIppJ!KMÁp GÅUTLGWÃ~ K&PKM}¶JVFCEL JVF!UTLG/p hK&F!>p hF!CEÁ GrJVCeà CEF©F!K¶ÆEKMFp!K +GCTPKJVFVUÇLG/p hK&F!"p hFCEÁ G>J!CÉà hCEF©F!K&ÆTKMFp!K +}MFKjUÇJ!K>K&PKM}¶JVFCEL NL GrUTLGWlGCEPOK"NLz à CTFWGKMp¬JVF!C1eICTJV G}&F!KjU3JVKGCEPOKSNL GrUTLGWlKMPOKM}&J!F!CELeNOLÕ Ã hCEFWGKMp¬JVFC vI CTJ! ¶Ê K 4LWJ!4UÇJJVGK&p!KKMLGK&F!REK¶JVNO}jUTPOPO:UÇPPC 7KMW:JVF!UTLGpNOJVNOCELGpSC}M}&[email protected] ICTJV ±Ek,i UTFKp¬ÁÁK&J!F!N} HNOJ!lKMUT} µi p!K&KrËNREDF!K GÊ H ¶Ê ? ? 2 L 1 !,..#! ! 1 , .#! # %,, . . ) * .!" 1% K G F!pJ}&CELGpNWGK&FJ!GKÅp!} KMÁUÇJVNO}ÅWGNUTRTFVUTÁ CKMLGK&F!RTÕÆ$pÁCEÁKMLqJVDGÁ UÇJ²°ÀNOLiJ!KMF UT}MK HGNO} NpvpGCHLNOL ËNRTDGF!K Ê<GÊÌSLK:KMPK&}&J!F!CEL NL}MNWKMLqJÉCEL JVK:NLqJVK&F/ UT}&K'hF!CEÁ JVGK:LGCTF!ÁUTP°p¬J UÇJ!K $ReoUb] M [ ]PXZUW\0[ XZY<RX Y<[&f#[?R1^X#[&f V[&]<PXZ[-aXZY[ ah_<ej[?f#\&P]<V_j\OXZU+] Rd e<kbUbXZ_<V[ P^QejRU+fnejPXZ[&](X#U+Rk ∆ n l Y[?] U=X T1Rf#U+[?a ahejR1XZUWRk+k+gt ∆(x) HH /XZY<[&Pf#gr9lnYUW\DYHUWa Ub]H^ R\OX XZY<[ *8 1'3$%#'< 5$`% ' '(!q Qq @ $% 02$(`? ; BG($%4b4?.; ' <;=; ' c<;=;-,-<% ? ' 3 0O013/b8#$ Z $<; !2<; < $* & 3 C* '(! A"5 * '*-* +#`!E$16Q3%0bG+/<"; 1%- ' -<1*: ZD13 ' -71%D13 $.'2/(b ; %83'D '; %16 ->; '013o(+ Z:3$'# 6 6 0 2 4 5 6 ; 1%? '16<3>? ( ' : 2'Ibqb:-,3+ 1%& ' '(!s-,'2:5$; 'j( A Z HNOJ!K&LGKMFRT E > ∆ UTprNLGWN}jU3JVKMWIq:JVK UTFF!C¡U3JrJ!GKpJ U3JVKPUTIKMPOKMW 0 NLËNRTDGF!K GÊ l}jUTL JVF!UTLGp/hK&F7J!GF!CEDREvJVGK°NLqJVK&F/ UT}&K"HNOJ! JVK QUjÆTKÆTKM}¶JVCEFQCTLvJVGK"p!UTÁK°p!NOWGK"C[J!GKSË4KMFÁN pDGF/ UT}&K q + → k + +}&CEF!FKMpB CELWGNLGRÉJ!CeJ!GKB CTNLqJ 4 HNOJ!J!GKÅBGFCEI4UTIGNOPN Jµ C(E) °CEFEHN JV`J!GK QUjÆEK ÆEK&}&JVCTF}&F!CEpp!NOLGRSJVGFCEDGREJVGK©Ë4KMFÁN$p!DGF UT}MK q+ → −k− E}&CEF!FKMpB CELWGNLGRSJ!CJ!GKHB CTNLqJ 2 HNOJ! JVGKBGFCEI4UTINPN Jµ D(E)¶ÊGNp©KMPOKM}&J!F!CELe}MUTLÕIKF!%K GKM}&J!KMWÕUTp©UTLzK&PK&}&JVFCEA L HNOJVeJVKrBF!CEI4UÇIGNPONOJµ B(E) }MCTF!F!K&p!BCELGWGNOLGRJ!CJ!GKxBCENOLiJ 5 CEF°UÇp°UGCEPKrCELlJVGKxCTJVGK&F½pNWGKrC /J!GKÂË4K&F!ÁN[p!DGF UT}MK IqÕ®½LGWGF!K&K&ÆÕF!%K GKM}&J!NCEL HNOJ!:BGF!CTI4UTIGNOPNOJµ A(E) h}MCEFF!KMpB CTLGWGNLRvJVC J!GKÂBCENLqJ 6¶ÊGKÅPU3J!JVK&F BGF!C}MK&p!pQNOLqÆECEP ÆEKMpQUxJVF!UTLG/p hK&FQC U>BGUTNF}jUTFFNLRÂ} 4UTFREK 2e FHGNPOK©J!GK"CTJ!GKMF7BGF!C}MK&p!pKMpJVF!UTLG/p hK&F CELGP ÉUÅp!NOLGREPK°KMPK&}&J!F!CELGNO}S} 4UTFREKTÊ4GKr}MCTLGp!K&FÆÇUÇJVNOCELeC BGF!CEIGUTIGNPONOJµF!K&DNF!K&pJV4UÇJ GÊ I A(E) + B(E) + C(E) + D(E) = 1 . 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I K&}jUTDp!KÂC K&LGKMFRTÕ}&CELGpKMFÆÇUÇJ!NCEL JVKÂpVUTÁK>K&LGKMFRT`FKMiDGNOF!KMÁKMLqJ!pSUTp"NOL:ËNRTDGF!K ?$Ê ÃÇI`4UjÆEKÂJ!C I Kp!UÇJVNO/p 4K&, W hCEFJVGK 4L4UÇP p¬J UÇJ!KM p K&PKM}¶JVFCELG?p HNOJVzCEBGBCEp!N JVKKMLKMF!RTNKM%p p!KMK>ËNRTDGF!K ?$Ê qU Ê4±µLzJVK ;´µÁUÇJVFN J!KMFÁNOLGCEPCTRT hCTFJ!GNpJVF!UTLGp!N JVNOCEL J!CC}M}&DGF ÐÆNFJ!D4UTPHpJVUÇJVK&pv}MCTF!F!K&p!BCELGWGNOLGRJ!CJVK KMLGK&F!RTvC J!GKWGCTJ½UTF!KSFKMiDGNOF!KMW 6HN} lp!DGBGBF!KMpp½J!GK®°LGWGFKMK¶ÆlJ!DGLGLGK&PNOLGRÅ}MDGFF!K&LiJ½IKM}jUÇDGp!KxC PUTFREKÂK&LGKMFRTWGKMLGCTÁNOL4UÇJVCTF!p"NOLJ!GKÅJVFVUÇLGp!N JVNCTL:FVUÇJ!KTÊ ËNRTDGF!K&p ?$<Ê l)I [} [)W WGKMp}MF!NOI KÅJ!GK}MUTp!K&p HGKMFKUTLÂK&LqÆ$NOF!CELÁK&LiJ/Np}&CEDGBGPOKMW>J!C°JVKQpVUTÁKQ² ¨ À>}MNOF!}&DGNOJMÊT,F!C1ÆNWKMWÂJVGUÇJJ!GNpKMLqÆNF!CTLGÁK&LqJ }jUTLNK&PWCEF,REN ÆEKHp!CEÁKC 9NOJVp,K&LGKMFRTÅJVCSJ!GK©² ¨ ÀWGK&J!KM}&J!CEF qK&PK&}&JVFCELGNO}QJVFVUÇLGp!N JVNCTLGp/ÆNU"JVGKHWGCÇJ HD j} UTL:IKM}&CEÁKÂÁÂDG} ÁCEF!KxPN ÇKMP zIKM}MUTDGpKÅKMPK&}&J!F!CEL:K&LGKMFRENK&pSCEL`JVGK>LGCEF!ÁUTP p!NOWGKÂ}jUÇLIKÂ}&PCEpK JVClJV4UÇJ>CJVKvWGCTJ>PK&ÆTKMP|Ê/À$D} JVFVUÇLGp!N JVNCTLGpx}MUTLJViDGp>C$}&}MDGFxK&ÆEK&[email protected]!GKv} GK&ÁNO}jUTPB CÇJVKMLqJ!NUTP C JVGKrLGCEFÁ UTP9PKjUÇWzK }&KMK&WGp½J!4UÇJ©C J!GKrp!DGBKMF}MCELGWDG}&J!CEFMÊ ®°?p 7Kxp4UTPP[pKMKrPUÇJ!KMFHCE)L GJ!GKrIGNUTp ÆECEP J UTRTK;}jUÇLUT}&JÐUÇp,U½ÆÇUTPOÆTK hCEFBGGCTJ!CÇ´bUTpp!NOpJVK&WÅKMPOKM}&J!F!CELÂJ!FVUTLp!NOJ!NCELpMÊ3± J,NpBGF!K&}MNpKMP xJ!GKMpKPUÇJJVKMF p!N JVD4UÇJ!NCEL4p HGN} , HNOPP[IKxK $BGPCEN JVK&WÕNLlCEFWGKMFHJVCÁKMUTp!DGFKrJ!GK>LCENpKxC JVGKrÁKMUTp!DF!NLR}&NF!}&DGNOJ hJ!GK bK&LiÆNOF!CELGÁKMLqJ ¶Ê a) µL D ∆ µS µL ∆ c) d) D h̄ω ∆ µS ∆ D h̄ω ∆ D h̄ω ∆ µ L µS µS µL PSfrag replacements b) ∆ ∆ 3 1-5 -71%& ':+q- ?%? 'j< 3 1-5 -71%& '8+q-; O? %+*G $3 m'D' ZO% O#13 $3 1-5/ -71%? '<6 q >2<'#'. ,2( '(5 313.#'"'< 2( '(5 313HG?A ?815b'*% 3 G0'2? '.'(! 'I'2 2$+ - -5(`0 2'D' ZO% O#13 3 1-5o -71%? '6 q$ B 2<'#'H 2( '(5 313 G&A?)15+ '* -(' 0%)G 6 63 6 , [email protected] b 2$O%b - ' - 3'-6$-)?<; + '(! ; 1%- ' [email protected]$& ;+ # , ! ! ! #!.! GK Ô UÇÁNOPOJVCTLGNUTL?HGNO} rWKMp!}&F!NOI K&pJ!GK,WGK&}MCEDGBPKMWLGCTF!ÁUTPTÁK&JVUTPTPOKjUTW i* p!DGBKMF}MCELGWDG}&J!CEF *K&LqÆ$NOF!CELÁK&LiJ ÁKMpCEp!}&CEBGN}S}&NF}MDGN J7p¬$p¬JVK&Á FKjUTWp [email protected]Ê G H0 = H0 + H0 + Henv , HGKMFK X † ?Ê Ã H0 = k ck,σ ck,σ , L S L k,σ WGKMp}MFNIKMpSJVGKK&LGKMFRT:pJVUÇJVK&pNL:JVGKPOKjUTW).HNOJ! c†k,σ UÇLK&PK&}&JVFCEL}MFKjUÇJ!NCELCEBKMF!UÇJVCEF&Ê+GKp!D´ B K&F!}&CELGWGDG}¶JVCEF Ô UTÁNOPOJ!CELGNUÇL 4UTpJ!GKSWGNUÇRECEL4UTP hCEF!Á X ?Ê H † H0 − µ S N S = Eq γq,σ γq,σ , S qÄ q,σ HGKMFK γq,σ , γq,σ UTFKÅDGUTp!NOB4UTFJ!N}&PKÂCEBKMF!UÇJVCTF!p HGN} F!K&PUÇJ!KÂJVCÉJVGKË4K&F!ÁN/CEBKMFVU3JVCEFp † IqvJVGKrÎ;CTRECEPNODGIC ÆJVF!UTLGphCEF!ÁUÇJVNOCEL † c−q,↓ = uq γ−q,↓ − vq γq,↑ , † c†q,↑ = uq γq,↑ + vq γ−q,↓ , cq,σ , c†q,σ ?Ê< TU LGW Eq = p∆2 + ζq2 Np©JVGK>DGUTp!NOB4UTFJ!N}&PKrKMLGK&F!RT ζq = q − µS Np©JVGKÂLCEF!ÁUTP pJ U3JVKÂpNLGREPOK¶´ KMPOKM}&J!F!CELKMLGK&F!RT}MCTDGLqJVKMW hFCEÁ¦JVGK"Ë K&F!ÁN PK&ÆTKMP S UTLGW ∆ NOp7JVK°p!DB K&F!}MCTLGWGDG}¶JVNLRÅRqUTB HGNO} HNPOP,I KÉUÇp!p!DÁK&WJVC`IKJ!GKvPUÇF!REK&pJxKMLGK&F!RTp!}MUTµPKv NOLJ!GKMpKÉ}jUTPO}MDGPUÇJVNOCELGpMÊ Ô KMFKjU#J!KMF ;KvUTPpC WG%K GLGK eV = µL − µS UTLGWlUTp!pDGÁK µS = 0 Ê Ô KMFKF;KSWGC>LGCTJQpB K&}MN #J!GK Ô UÇÁNOPOJVCTLGNUTLC[J!GK°K&LqÆ$NOF!CELÁK&LiJQI K&}jUTDGpK"JVGK°KMLqÆNF!CTLGÁK&LqJ F!K&BGF!K&p!KMLqJ!p"UTLzCEB K&Lzp¬pJVK&Áz~ JVKrÁKMpCEp!}&CEBGNO}x}&NF!}&DGNOJ HN} ÕFKMBGFKMpKMLqJVp°J!GKrKMLqÆNF!CTLGÁK&Lq>J HNOPP CELGP Á [email protected] hKMp¬J>N [email protected] ;Æ$NUlJ!GKvBG4UÇp!K 4D}&JVDGUÇJVNOCELGp hφ(t)φ(0)i )HGNO} 2UTF!KNLGWGD}MKMW2U3JxJ!GKɲ"À ÈDGLG}¶JVNCTL CTFxPUÇJVK&FrCTL hCEFJVGK ² ¨ À}&NF}MDGN J! UÇJrJVGKWGCTJ *ip!DGBKMF}MCELGWDG}&J!CEF©ÈDGLG}¶JVNOCELSI K&}jUTDGpK C JVGKPC}jU3JVNCTLÉC J!GKr}jUTB4UÇ}MNOJ!CEFQBGPUÇJVK&pMÊ4±µA L H4UÇ>J hCEPPOC Hp77Krp!4UTPOPUTp!pDGÁKJ!4UÇJHJVKrDLGpÁÅ´ ÁK¶JVFN MK&W LGCTNp!K½p!BKM}¶JVFVUÇP9WGK&LGp!N Jµ + UTpQWGK 4LGKMWÉNOLv», Ê |Ã[email protected]Ê G1à ÐNOLv} 4UTB$JVKMFÃr}MCTF!F!K&p!BCELGWGNOLGR JVCBGGCTJ!CELÕK&ÁNOp!pNCEL hCEFHBCEp!N JVN ÆE<K ShF!(ω) K&DKMLG}¶ 9CE%F UTPOJ!KMF!LGUÇJVN ÆEKMP SI (ω) GJ!GKrp!BKM}&J!FVUTP WGK&LGp!N Jµ C QLCENpK}&CEF!FKMp!BCELGWNLGReJVCzBGGCTJ!CELUÇIGp!CEFBJVNOCE)L [NOpxpB K&}[email protected] GKMWIqJVKJ!FVUTLGpB CTFJxBGF!CEBKMF¬JVNOKMprC JVGK°ÁK&p!CEp}MCEBGNO}"}MNOF!}&DGNOJ ' G!IGDFIq¿ ( Ê Ô KMFK hh· · · ii p¬J UTLWGp4hCEFQUÇLÉNF!FKMWGD}MNIPK°LCENpK°}MCTF!F!K&PUÇJ!CEF HGKMFKSJVGKBGFC$WGD}&J©C U1ÆTKMF!UTREKr}MDGFF!KMLqJ!p½4UÇp©IKMK&LzpDGIJ!FVUT}¶JVKMWzCEDJMÊ GK©J!DGLGLGK&PNOLGR Ô UTÁNP JVCELNUTLÂWKMp!}&F!NOIGNLGRSJ!GKHKMPK&}&J!F!CELJVF!UTLG/p hK&F!FNLGRIK&*J 7KMKMLJVGKHp!DB K&F!}MCTL$´ WGDG}¶JVCEFHUTLWlJ!GKLGCEF!ÁUTP ÁK¶J UTP+POKjUTWeNOLeJ!GKr²"ÀxÈDGLG}¶JVNCTLlNOp HT = X Tk,q c†k,σ cq,σ e−iφ , ?ÊE? k,q,σ HGKMFKJVGKNOLGWGN}&KMp k UTLW q F!%K hK&F/JVCSJVKLGCEF!ÁUTP$ÁK¶J UTP$PKjUÇWUÇLGWp!DGBKMF}MCELGWDG}&J!CEFMÊ KH}MCELGpNWGK&F hCEFp!NOÁBPN}&NOJµÅJ!4UÇJ Tk,q = T0 Ê ²½CTJ!K"JV4UÇJHJ!GK"JVDGLLGKMPONLGR Ô UÇÁNOPOJVCTLGNUTL}MCELqJVUTNLGpHU 4DG}¶JVD4U3JVNLR BG4UTp<K UT}&J!CEF HGNO} zFKMBGFKMpKMLqJVp°J!GK}MCEDBGPNOLGRÅJVCÅJVGKrÁKMp!CTp!}MCTBGN}}MNOF!}&DGNOJMÊG±µLGWGK&KMW) IKM}jUÇDGp!KxC JVGKH}MUTB4UT}&NOJVN ÆEK}&CEDGBGPONLGRSIK&*J 7KMK&LJVK©pNWGK&p7C J!GK©²"À©ÈDGLG}¶JVNCTL UTLWJ!GKHÁK&p!CEp}MCEBN}H}MNOF!}MDNOJ!TU }MDGFF!K&Li?J 4D}&JVDGUÇJVNOCELÉJVFVUÇLGp!PUÇJVK&pNLqJVCUÂÆECTPOJ UÇREEK 4DG}¶JVD4UÇJ!NCELlUT}&F!CEppJVGKr²"ÀrÈDGL}&JVNOCEL+Ê4Î7CTJVlUTFK F!K&PUÇJ!KMW:ÆNUvJVGKÅJ!FVUTLGpb´µNOÁBKMW4UÇLG}MKÅC JVK}&NF}MDGN # J ' ÃÇ)Ä ( V (ω) = Z(ω)I(ω) Ê+²½K J![JVGKÅÆTCEPOJVUTREK 4DG}¶JVD4UÇJ!NCELpvJVFVUÇLGp!PUÇJVK:NOLiJ!C2BG4UTp0 K 4D}&JVDGUÇJVNOCELGplUT}MFCEp!pÉJ!GK ÈDGLG}&J!NCEL HUTpÉJVGKBG4UÇp!KNpÉJVK }jUTLCELGN}MUTP+}&CEL3ÈDGRqU3JVKC J!GK} 4UTFREK>U3J©J!GK;ÈDGLG}¶JVNCTLG p '<AG)( ~GJ!GKxB4UTp!KNOpHJ!iDGp½}&CELGpNWGK&F!KMWÕUTp©U iD4UTLqJVDGÁ ÁK&} 4UTLGNO}jUTP/CTB K&FVUÇJ!CEFMÊ+¸N ÆEKMLUÉp!BKM}&@N 4}}&NF!}&DGNOJ }jUÇB4UT}MN JVCEFp F!KMpNp¬J UTLG}&KM%p K¶JV}ÇÊ °JVK BG4UTpKS}MCEFF!K&PUÇJ!CEF7NOp;JVKMF!K hCEF!KSK $BGF!K&p!p!K&WzNOLvJVK&F!ÁpQC [JVK"JVFVUÇLGp¬´ NÁB K&W4UTLG}&K°C JVK"}MNOF!}MDNOJUÇLGW JVGKpB K&}&JVF!UTP[WGK&LGp!N JµÉC LGCENpK ' ÃTÄ (8HN} lNpp!C HLzNLl», Ê 9HÊ Ã ? Ê GKvBGFKMp!K&LqJp¬pJVK&ÁIKjUTFpÂp!NOÁNOPUTFNOJ!NKM"p HNOJVJVGKvp¬JVDGW$C HNOLGKMPUTpJ!N}®°LGWGFKMK¶ÆF!K 4KM}¶JVNOCEL NL JVKl}MUTp!C K HGKMFKlJ!GKep!DGBKMF}MCELWGDG}&J!CEF }&CELqJ UTNOLGpBG4UTpC K GDG}&J!D4UÇJVNOCELGp ' II3ID)(|ÊÀ$DG} 5BG4UTpK 4DG}¶JVD4UÇJ!NCELpÅWGKMp¬JVF!C1 J!GKepÁÁK&J!F IK&*J 7KMK&LKMPOKM}¶JVF!CTLGpUTLGW GCEPK&p ,UÇLGW U 9K&}&JJVGKe}&DGF!FKMLqJ ÆECEP J UTRTK"} 4UTF!UT}&J!KMF!NOpJ!N}MpHC JVKx²°À>ÈDLG}&J!NCEL+Ê L ,!,!.'!- 1!" GKÉJVDGLLGKMPONLGR:}&DGF!FKMLqJUTp!pC$}&NUÇJ!KMW HNOJ!J*;CKMPOKM}&J!F!CELpNOpÂREN ÆEKML2IqJ!GKlË K&F!ÁN;RECTPWGK&L FDGPK I = 2eΓi→f 7HNOJVÉJ!GKSJVDGLGLKMPNOLGRFVUÇJ!K Γi→f = 2π X f |hf |T |ii|2 δ(i − f ) , G ?Ê A HGKMFK i UTLW f UÇF!KvJVK JVDGLLGKMPONLGR`K&LGKMFRENK&pÅCJVKÉNLGN JVNUÇPÐUTLGW 4L4UÇP;pJVUÇJVK&p/NOLG}MPODGWGNOLGRÕJVK KMLqÆNFCELGÁKMLqJ!4UÇLGW T NpQJ!GKSJVFVUÇLGp!N JVNCTLÉCEB K&FVUÇJ!CEF7HGN} lNpK BF!KMpp!K&W:UTp T = HT + HT ∞ X n=1 1 HT iη − H0 + i n , ?Ê ¿ HGKMFK η NpBCEp!N JVNOÆTK"NL74LGNOJ!KMpNÁUTP Ê GF!CEDREGCEDJÅJ!GNp>} 4UTBJ!KMF,CTLGKÉ}MCELp!NWKMF!pÂJ!GKÉBGGCÇJVCÇ´µUTp!pNpJ!KMW JVDLGLGKMPONLGR ®, >}&DGF!FKMLqJ WGDGK"J!CÂJVGKSGNORE hF!K&iDGKMLG}¶É}MDF!F!K&Lq>J GDG}&J!D4UÇJVNOCELGpC J!GKSÁK&p!CEp}MCEBN}"WGK¶ÆN}MK UTp;J![email protected] KMFKMLG}&K '<A )(|~ ?$Ê I IP AT = I(K&LqÆ$NOF!CELÁK&LiJ ) − I(LCK&LqÆ$NOF!CELÁK&LiJ ) , HGKMFK 7NLREK&LGKMF!UT8P ÐJVKÕJVCTJVUTP©}MDGFF!KMLqJ hCEFJVDGLLGKMPONLGRC KMPK&}&J!F!CELGpJVGFCEDGRE J!GKÈDGL}&JVNOCEL Np I = I → − I← Ê Ô C;K¶ÆEK&F+K BKMFNÁKMLqJ UÇPPO4NOJ©Np©WGN}&DGPOJ©JVC}MCEDBGPKx}jUTB4UÇ}MNOJ!NOÆTKMPOÉUTLGWÕJ!GKMLÕJ!C FKMÁC ÆTK>JVK ÁK&p!CEp}MCEBN}WGK&ÆNO}MKr}MNF}MDGN J hF!CEÁ JVGKrWGK&J!KM}¶JVCEF©}MNOF!}MDNO J ' ÃEà (|Ê 4UÇJ©N%p 4NL UT}&J 4C #JVK&LÕÁKjUTpDGF!K&W NpJVGKK $}MK&p!pLGCENOp!K N|Ê KÇ@Ê JVK WGN 9K&F!KML}MKvIK&*J 7KMK&L }&DGF!FKMLqJ 4D}&JVDGUÇJVNOCELGpxUÇJÅUÕRTNOÆEK&LIGNUTp>UÇLGW MK&F!CIGNUÇpNLvJVKrÁKMpCEp!}&CEBGNO}S}MNF}MDGN JjÊÍ U3JVKMFCT)L 7K HNOPP }jUTPO}MDGPUÇJVK°J!GKSK $}&KMp!p½LGCENOp!K Sexcess + C S +(ω) ÊG±µLeJVGNO4p ;CE/F 6;KJViDGpHÁKjUTpDGF!KSJV[email protected] KMF!K&LG}MKIK&*J 7KMK&LzJVKr}&DGF!FKMLqJVpHJVF!CEDGRTl(ω) J!GK WGK&J!KM}¶JVCEF HKMLU°IGNUTpÆTCEPOJVUTREKQUÇLGWU MK&F!CSIGNUTp/UÇF!KUTBBGPNOKMWxJ!C°JVKWGK&ÆN}&KQ}MNOF!}&DGNOJ ÇUÇp,U hDGLG}¶JVNCTL C WGK&J!KM}&J!CEFHIGNUÇpQÆECEP J UTREK HGNO} zNOpWG%K 4LKMW`UÇp ?$Ê D ∆IP AT (eV ) = I(eVd 6= 0, eV ) − I(eVd = 0, eV ) . GNprUTPOp!Cl}MCEFF!K&p!BCELGWGprJVCeJVGK[email protected] KMFKMLG}&KIK&*J 7KMK&LJVGK ®,}MDGFF!KMLqJ!pxJVF!CEDGRTJ!GKWK&JVK&}&J!CEF HGKML U IGNUTp ÆECTPOJ UÇREK:UTLGW U &KMF!C2IGNUTpÉUTFKUTBGBGPONK&WJVCJVK:WGK&ÆN}&K:}MNOF!}MDNO!J ∆IP AT (eV ) = p HNOJVLGClKMLqÆNFCELGÁKMLqJ IP AT (eVd 6= 0, eV ) − IP AT (eVd = 0, eV ) IKM}MUTDGpKJVK}&CELqJVFNIGDJ!NCELE }jUTL}MKMPHCED$JjÊÐGKÕ[email protected] KMF!K&LG}MKzI K¶*J ;K&KML ®, }MDGFF!K&LiJ!pJ!Dp BGFC ÆNOWGKMp}MF!D}MNUÇP©NOL hCEFÁ UÇJ!NCELCEL JVGKpB K&}&JVF!UTPWGKMLp!NOJµlC K $}&KMp!p°LGCENpKC J!GKxÁKMpCEp!}&CEBGNO}rWK&ÆN}&KTʲ½CTJ!N}&KJ!4UÇJ©CEDGF©}jUÇP}MDPUÇJ!NCEL UTBGBGPONK&p°JVCvJ!GK MKMFCÉJVKMÁBKMFVU3JVDGFKÅ}jUTpK hCEFS}MCELqÆTKMLGNOKMLG}&K IGDJN JS}jUTLIKÅREKMLKMFVUÇPN &KMW:JV C GLGNOJ!K JVK&ÁBKMF!UÇJVDGFKMp&Ê .!,- . . 1# ! !,!'.!- ®°P JVGCEDRE CEDF hC}MDGpCNLqJ!KMF!K&pJ }MCELG}&KMFLGpvBGGCÇJVCÇ´µUTp!pNpJ!KMW®½LGWGFKMK&Æ F!K4KM}¶JVNCTL)=7K`LGK&KMW J!C }MCEÁBGDJ!KSUTPOP9BCEp!pNIGPOK°}MCTLiJ!F!NOIGDJVNOCELGp&ÊGK}MDF!F!K&LqJ©UTp!pC$}&NUÇJ!KMW HNOJ!ÉCELGKSKMPOKM}¶JVF!CTLvJVDGLGLKMPNOLGRÅNp RENOÆTKMLeIqÉJVGKrË4KMFÁN RECEPOWGKMLeF!DPK I = 2πe X f |hf |HT |ii|2 δ(i − f ) . ?$Ê G1Ä GK;}MUTP}&DGPU3JVNCTLrCGJVGK7}MDGFF!K&LiJ/BGF!C}MK&KMWGpNLxJ!GK;p!UTÁK=;UjÂUTp JV4U3JC4U½LGCEFÁ UÇPÁK&JVUTPMÈDGLG}¶JVNCTL '<A )(9 K $}MKMB$J>JVGUÇJrCELGK4UTpJVCeJ U ÇK NLqJ!CzUÇ}M}MCTDGLqJ>JVKpDGB K&F!}&CELGWGDG}¶JVNOLGRÕWGK&LGp!N JµCQp¬J UÇJ!KMprCEL JVGK©F!NOREqJ;pNWGK©C +J!GK/ÈDGLG}&J!NCEL BHN} vNOpWCELGK°IqK $BGPOCENOJ!NLGRSJ!GK"Î;CTRECEPNODGIC ÆxJVF!UTLG/p hCTF!ÁUÇJVNOCEL+Ê Ë4CEF>JVKz}MUTp!KeC °KMPK&}&J!F!CELGpÅJ!DGLGLGK&PNL' R hF!CEÁJ!GKeLGCEFÁ UTP;ÁK¶J UTPQPOKjUTW J!CJ!GKlp!DGBKMF}MCELGWDG}&J!CEF ià > % (eV > ∆) GJVKr}&DGF!FKMLqJ hF!CEÁ PK #JJ!CF!NRTiJHFKjUTWp Z ∞ I→ = e dte−i(µS −µL )t hHT (t)HT† (0)i −∞ Z ∞ X 2 dte−i(µS −µL )t = eT0 hc†k,σ1 (t)ck0 ,σ2 (0)ihcq,σ1 (t)c†q0 ,σ2 (0)ihe−iφ(t) eiφ(0) i −∞ = 2eT02 Z k,k 0 ,q,q 0 ,σ1 ,σ2 ∞ dt −∞ X k,q |uq |2 e−iEq t eik t eJ(t) Z 2π ∞ |Z(ω)|2 E SI (ω) δ(E − ) 1− dω = d dE √ RK −∞ ω2 E 2 − ∆2 −∞ ∆ Z ∞ Z |Z( − E)|2 E 8π 2 eT02 NN NS eV $Ê dE √ SI ( − E) . d + 2 − ∆2 ( − E)2 RK E ∆ −∞ 4πeT02 NN NS Z eV ∞ Z ? @GG HNOJ! N UTLW N J!GKÂWGK&LGp!N JµÕC/pJ U3JVKMp°C,JVGKxJ*;CvPOKjUTWp NOLÕJVK>LGCEFÁ UÇP p¬J UÇJ!K Ê K>LGCTJ!N}&K JV4U3JJ!GNNp}jUTPO}MDGPUÇSJVNOCEL2Np pNÁNPUTFÂJVCJVGK`}MUTP}&DGPU3JVNCTL CSJVGKlJVDGLGLKMPNOLGRF!UÇJVKzCSK&PKM}¶JVFCELGpvNOL JVGK:BF!KMpKMLG}&KC ÂUTL K&LiÆNOF!CELGÁKMLqJAHGNO} NOpÉp!GCHL NL p!K&}&J!NCELGpAH$Ê@GEÊ H2UTLGW HÊ GEÊ hCEPPOCHNLGR JVGKxpVUTÁK>p¬JVK&BGp Ê+GNp½}MDGFF!K&LiJ"NLG}&PDGWGK&p°ICTJV`UTL`KMPUTpJ!N}>UTLGW:UTLÕNLGK&PUTp¬JVN}r}MCTLiJ!F!NOIGDJVNOCE)L 4JVK hCEF!ÁKMFIKMNOLGRF!K&LGCEF!ÁUTPON MK&WÉIqÉJVGKBGFKMp!K&LG}MKxCJ!GKKMLqÆNFCELGÁKMLqJ' ÃTÄ ( Ê Ô KMF!K NOpHJ!GKKMLGK&F!RT C °UTL K&PKM}¶JVFCEL5NOL J!GKeLGCEFÁ UTPÁK&JVUTPQPOKjUTW5UTLGW E NOpÅJVGKlKMLGK&F!RT2C SUDGUTp!NOB4UTFJ!N}&PKÉNL2JVK p!DGBKMF}MCELWGDG}&J!CEF;POKjUTW9Ê ³ 4UTLGRENOLGRÆTUÇF!NUÇIGPK&p,NLJVGK©NLGK&PUTp¬JVNO}QJVKMFÁ¦JVC Ω = − E δ = + E $UÇL DGp!NOLGR R dx(x + a)/p(x + a)2 − b2 = p(x + a)2 − b2 /2 U #JVK&F}MCEÁBGDJ!NLGRxJVGKS}&DGF!FKMLqJ hF!CEÁ F!NOREqJQJVCPK #JHNLlUÅp!NOÁNOPUT3F QUj 67KrCEIJ UÇNL ∆IP AT ∞ |Z(ω)|2 + el Sexcess (−ω) K1e (eV ) = −C1e dω 2 ω −∞ Z eV −∆ |Z(Ω)|2 + inel +C1e dΩ Sexcess (−Ω)K1e (Ω, eV ) , 2 Ω −∞ Z ?Ê@G à HGKMFKvJVGKJVFVUÇLGp!ÁNpp!NCTL}MCK }&NKMLqJCJ!GKɲ°ÀzÈDLG}&J!NCELNOLJVKvLGCEFÁ UTPpJVUÇJVK NpÂWGK4LGK&W5UÇp K 7KMNOREq>J hDLG}&J!NCELGpHUÇF!KWG%K GLGKMWÕUTp T = 4π 2 NN NS T02 C1e = eT /RK Ê4G< p ?$Ê GH el K1e (eV ) = (eV )2 − ∆2 , p ?$Ê GB inel K1e (Ω, eV ) = (Ω − eV )2 − ∆2 . À$NÁNPUTF!P 7KxCEI$J UTNOLvJVG<K hCEFÁÂDGPUÂC ∆IP AT hCTFQJVGK}jUÇp!K eV ≤ −∆ ∆IP AT ∞ |Z(ω)|2 + el = −C1e dω Sexcess(−ω) K1e (eV ) 2 ω −∞ Z ∞ |Z(−Ω)|2 + inel +C1e dΩ Sexcess(Ω)K1e (Ω, eV ) . 2 Ω eV +∆ Z ?Ê@G ? Ë4CEF J!GK;}MUTp!K −∆ ≤ eV ≤ ∆ JVGK&F!KUTFK;LGC°KMPUÇpJVNO}J!FVUTLp!NOJ!NCELpIKM}MUTDGp!K;KMPK&}&J!F!CELGp}jUTLGLCTJ/K&LiJ!KMF JVGK°p!DGBKMF}MCELGWDG}&J!NLGRÅRqUÇB+ʲ©K&ÆTKMFJ!GKMPOKMpp4WGDGK"J!C>JVK"BGF!K&p!K&LG}MKC[J!GK"KMLqÆNFCELGÁKMLqJ!$KMPOKM}&J!F!CELp }jUTLUÇIGp!CEFI:CEF"K&ÁN J"KMLKMF!RÇ, hF!CTÁ CEF©JVCvJ!GKÅKMLqÆNFCELGÁKMLqJSp!CvJ!4UÇJUÇL:NLGK&PUTp¬JVN}riD4UTpNB4UTF¬JVNO}MPK }MDGFF!K&Li?J 4C Hp% eV −∆ |Z(Ω)|2 + inel Sexcess(−Ω)K1e (Ω, eV ) 2 Ω −∞ Z ∞ |Z(−Ω)|2 + inel Sexcess(Ω)K1e (Ω, eV ) . −C1e dΩ 2 Ω eV +∆ ∆IP AT = C1e Z dΩ H ?Ê@[email protected] # . #! !,!'.!- # 2 , , ' . L GK:p!NOLGREPOK`KMPOKM}&J!F!CEL hDGUTp!NOB4UTFJ!N}&PKÅJVDGLLGKMPONLGR }&DGF!FKMLqJlNpvC(4FpJeCEFWGKMFÉNOLJVGK`J!DGLGLGK&PNLR UTÁBGPN JVDGWGKÇÊ K½LC 5JVDGFLJ!CxBGFC$}&KMpp!KMp=HGN} NLqÆECTKHJVGKHJVDGLLGKMPONLGRC J*;CxKMPK&}&J!F!CELGp,J!GF!CEDRE JVGK"²"À NLqJ!KM/F UÇ}MKTÊ$±µLGWGK&KM)W 4IKM}MUTDGp!KSCTDGFUTNOÁ¤NOp7J!CÅp!GC JV4UÇJQ®°LGWGFKMK¶ÆvF!K4KM}¶JVNOCELÉ}jUTLeIK"DGpKMW JVC"ÁKjUÇp!DGFKLGCENpK ;KLGK&KMWÅJVCSK UTÁNOLGKUTPOP*J 7CKMPOKM}&J!F!CELÅBF!C}MKMpp!K&p 7KpJ UÇFJ HN JVÂJVKQJVFVUÇLG/p hK&F CQJ*7C`KMPOKM}¶JVF!CTLGpÅUTpxDGUTp!NOB4UTFJ!N}&PKMpxUTIC1ÆEKJ!GKvRqUÇB+Ê ³ UTP}&DGPUÇJ!NCELprCJVGKÁ UÇJ!F!N KMPOKMÁKMLqJÂNOL »,Ê ?Ê A HUTFKrJ!GKML`}MUTF!FNK&WÕCEDJ©JVCp!K&}MCELGW:CTF!WGK&F½NOLzJVKxJVDLGLGKMPONLGR Ô UÇÁNOPOJVCTLGNUTLÉDp!NLRJVK T Á U3JVF!N 2 X 1 ?$Ê G ¿ HT |ii| δ(i − f ) . I = 4πe |hf |HT GKNLGN JVNUTP p¬J UÇJ!KNpHUÅBGF!CWGDG}¶J! iη − H0 + i f ?$Ê@G!I |ii = |GL i ⊗ |GS i ⊗ |Ri , HGKMFK |GLi WGKMLCTJVK&p;UREF!CTDGLGWpJ U3JVKHN} }&CEF!FKMpB CELWGpJ!CrU 4POPK&WË4K&F!ÁNGp!KMU hCEF,J!GKHLGCEF!ÁUTP KMPOKM}&J!F!CWGKTÊ |GS i NpJVGKÎ ³ À>REF!CEDLGWÅpJVUÇJVKQNOL>JVKQp!DGBKMF}MCELWGDG}&J!CEF,PKMUTW+Ê |Ri WKMLGCTJ!KMp/J!GKQNLNOJVNUTP pJVUÇJVKSC JVGKK&LiÆNOF!CELGÁKMLqJMÊ Ì"LlJ!GKCTJVGK&FH4UTLGW)CEDGFHREDGK&p!p?hCEFQJVGK<4LGUTP+pJVUÇJVKpGCEDGPOWzFKjUTW ?$Ê GD † |f i = c†k,σ c†k ,σ γq,σ γq† ,σ |GL i ⊗ |GS i ⊗ |R0 i , HGKMLe*J 7C KMPOKM}¶JVF!CTLGp©UTF!KSK&ÁN J!JVK&W hF!CEÁ¤JVGKrp!DGBKMF}MCELWGDG}&J!CEFMÊ GK µREDGKMpp C ,», Ê ?$Ê GD ;NOpHUTL NL hCTF!ÁKMWÉCELGKÇ~ UTLeKMPOKM}&J!F!CELlB4UTNOFQNpHIGFC TK&LzNOLÉJVGKp!DB K&F!}MCTLGWGDG}¶JVCE%F UÇLGWzCTLGKKMPK&}&J!F!CELeJVDGLLGKMPOp JVC>J!GK"LGCTF!ÁUTP ÁK&JVUTPPKjUÇ)W FHNPK©JVGK°CTJVGK&FQIKM}MCTÁK&pUÂiD4UTp!NOB4UTF¬JVN}&PK©NL J!GKSp!DB K&F!}MCTLGWGDG}¶JVCEF JVGKp!UTÁKrBGFC$}&KMp!p©NpJ!F!DGK hCEFJVGKrp!K&}MCELWÕKMPOKM}¶JVF!CT$ L HGNO} zJVDLGLGKMPOpJVCJ!GKLGCEFÁ UTP[ÁK&JVUTP[PKMUTW+Ê GKMLeJVKrpDGBKMF!}&CELGWGD}&JVCTF½POKjUTWlUTIGpCEF!IpH*J 7CKMPK&}&J!F!CELG%p ?$Ê ÃTÄ † |f i = ck,σ ck ,σ γq,σ γq† ,σ |GL i ⊗ |GS i ⊗ |R0 i . GK bREDGK&p!p ©C [», Ê ?$Ê ÃTÄ /Np,J!4UÇJÐ*J ;CxKMPOKM}&J!F!CELp7}MUTLJ!DGLGLGK&6P hFCEÁJ!GK½LCEF!ÁUTPGÁK&JVUTP4PKMUTW UÇLGW I K&}MCEÁKrDGUTp!NOB4UTFJ!N}&PKMpHNOLzJVKxpDGB K&F!}&CELGWGDG}¶JVCEF&Ê Ô KMF!K |R0i NpHJVG( K 4L4UTPpJVUÇJVKxC /J!GKxK&LqÆ$NOF!CEL´ ÁK&LqJjÊ KlUTFA K GF!pJ}&CELGp!NOWGKMFNLGR:J!GKe}jUTpKzC ©*J 7CK&PKM}¶JVFCELGpJVDGLGLKMPNOLG' R hF!CTÁJVGKepDGB K&F!}&CELGWGDG} ´ JVCEFÅPOKjUTW2JVC:J!GKzLCEF!ÁUTPQÁK&JVUTPQPKMUTW+Ê,±µLqJVFC$WDG}MNOLGRJ!GKe}MPOCEp!DGFKeF!KMPUÇJVNOCEL hCEFÅJVKlK&NREK&LGpJVUÇJVK&p C JVRGK:LCELG}MCTLGLGKM}¶JVK&W pp¬JVKMÁ {|υii} ;UTLGW DGp!NOLGR JVG' K UÇ}&JÉJVGUÇJ hυ|(i − H0 ± iη)−1 |υi = ∞ G E C G L K M } T U z L K $B E C L M K q L V J N Ç U V J r K T U O P P V J r K & K LGKMFRTeWGK&LGCEÁNL4UÇJ!CEF!p&7Ê K4UjÆEK dtei( − ±iη)t ∓i 0 i 0 0 0 0 0 0 0 0 υ I← = 2e X Z f,υ1 ,υ2 ∞ dt −∞ Z ∞ dt 0 0 Z ∞ 0 00 dt00 e−η(t +t ) eii (t−t 0 00 +t0 ) hi|HT† |υ1 i ×eiυ1 hυ1 |HT† |f ie−if t hf |HT |υ2 ie−iυ2 t hυ2 |HT |ii . 0 t0 ?$Ê Ã7G GKMLTIq>J!FVUTLGphCEF!ÁNLR©J!GK;JVNOÁK7WGKMBKMLWGKMLqJÐBG4UÇp!KMp,NOLqJVCSU½J!NÁK7WKMBKMLGWGK&LG}MKHC4J!GK;J!DGLGLGK&PNLR Ô UTÁNP JVCELNUT)L 7K}jUTLNLqJ!KMREF!UÇJVKÅCED$J>UÇPP GL4UTP,UTLWÆNF¬JVD4UÇP/pJVUÇJVK&pMÊ[GNOpxUÇPPC Hp"DpJ!CeF!%K HFNOJ!K JVGK½JVDGLGLKMPNOLGRÂ}&DGF!FKMLqJHNLvJ!KMF!ÁpQC [J!DGLGLGK&PNOLGRÂCEBKMF!UÇJVCEFpQNLJVGKSNOLiJ!KMF!UT}&J!NCELvFKMBGFKMpKMLqJ UÇJ!NCELvJ!C PC 7KMpJ©CEFWGKMF O(T04) ÊGGKr}MDGFF!KMLqJ©F!KMUTWGp I← = 2e Z ∞ dt −∞ ×hHT† (t Z ∞ dt 0 −t 00 0 Z ∞ 0 00 0 dt00 e−η(t +t ) ei(µS −µL )(2t−t −t 0 † )HT (t)HT (t0 )HT (0)i , 00 ) ?Ê ÃEà HGKMFKSJVGKSJVNOÁK°WGKMBKMLGWKMLG}&K>C JVGKCEBKMF!UÇJVCTF!pHNpREC1ÆTKMF!LKMWÕIq HT (t) = ei(KL +KS +Henv )t HT e−i(KL +KS +Henv )t , HNOJ! KL = H0 − µLNL UTLGW KS = H0 − µS NS ÊEGKHNLqJVK&FVUT}¶JVNOCELÅBGF!K&p!K&LiJVUÇJVNOCELNL», Ê ?$Ê ÃEà 4UTp;J!GKSUTW$ÆTUÇLiJVUTREK©JV4UÇJ ;KSCTLGPO4UjÆEK"J!CÂ}jUÇP}MDPUÇJ!K°pJVUÇJVNOpJ!N}jUÇP UjÆTKMFVUÇREKMpC #JVNOÁK ´µWGK&B K&LGWGKMLqJ }MCEFF!K&PUÇJ!NCEL hDGLG}&J!NCELpMÊ KSK$BGFKMp!pHJVKJ!DGLGLGK&PNOLGR Ô UÇÁNOPOJVCTLGNUTLNLvJ!GK}MFKjUÇJ!NCEL UTLGLGNOGNPUÇJVNOCEL CEBKMFVU3JVCEFp,NLÅJVGK©LGCEFÁ UÇP ´UÇLGWpDGBKMF¬´ }MCELWGDG}&J!NLGRrPKMUTWGp&ÊiÎ;KM}MUTDGp!K©JVGK&F!KHNpÐLGCÇJ;UÇLiÂNLqJVK&FVUT}¶JVNCTL I K¶*J ;K&KMLzB4UTFJ!N}&PKMpNOLÉ*J ;CPKMUTWGp 7;KrCEIJVUTNL L I← = S 2eT02 × Z ∞ dt −∞ X Z ∞ dt 0 0 Z ∞ 0 00 0 dt00 e−η(t +t ) ei(µS −µL )(2t−t −t 00 ) 0 hck1 σ1 (t − t00 )ck2 σ2 (t)c†k3 σ3 (t0 )c†k4 σ4 (0)i k1 ..k4 ,q1 ..q4 ,σ1 ..σ4 00 0 ×hc†q1 σ1 (t − t00 )c†q2 σ2 (t)cq3 σ3 (t0 )cq4 σ4 (0)iheiφ(t−t ) eiφ(t) e−iφ(t ) e−iφ(0) i . ?$Ê ÃH »,Ê ?$Ê ÃH 7NOpQJVF!DKEhCEFQJ*;CKMPOKM}¶JVF!CTLGpJVDGLGLKMPNOLGRUTpiD4UTpNB4UÇFJVNO}MPOKMpUTLWÕUTp®½LGWGFKMK&ÆeFK%4K&}&J!NCEL+Ê Ë4CEF©J!GK>}&CEF!FKMPUÇJVNOCELzC,CEBKMFVU3JVCEFp°NLlJVGKxLGCEFÁ UTPÁK&JVUTPPKjUÇW) Dp!NLRJVK: N}61 p½JVKMCEFKMÁC.7K CEIJVUTNL hck1 σ1 (t − t00 )ck2 σ2 (t)c†k3 σ3 (t0 )c†k4 σ4 (0)i = −e−i(k1 −µL )(t−t 00 −t0 ) δk1 ,k3 e−i(k2 −µL )t δk2 ,k4 00 0 +e−i(k1 −µL )(t−t ) δk1 ,k4 e−i(k2 −µL )(t−t ) δk2 ,k4 . Ô K&F!K7K}MCELp!NWKMFHiD4UTp!NOB4UTF¬JVN}&PK°J!DGLGLGK&PNOLGRGpC hc†q1 σ1 (t − t00 )c†q2 σ2 (t)cq3 σ3 (t0 )cq4 σ4 (0)i = −|vq1 |2 |vq2 |2 e−iEq1 (t−t 00 −t0 ) ?$Ê Ã δq1 ,q3 e−iEq2 t δq2 ,q4 00 0 +|vq1 |2 |vq2 |2 e−iEq1 (t−t ) δq1 ,q4 e−iEq0 (t−t ) δq2 ,q3 . ?$Ê Ã ? GKrK$B CTLGKMLqJVNUTPp©CBG4UTpKxCEBKMF!UÇJVCTF!p½UTF!K}jUÇP}MDPUÇJ!KMWzUTp½NOLlJ!GKr®°BGBKMLGWN l® pKMK», Ê?$ÊE? I UTLGWÉJVKrB4UTp!K ´µBGGUTp!Kr}MCEFF!K&PUÇJ!NCELhDGL}&JVNOCELeNpWGK4LGK&WÕNLe», Ê 9HÊ G ? Gp!CÅJ!4UÇJ' DEÃF6DH)( $? Ê ÃGA he e e e i = e . Ë4CEPPOCHNOLGR», Ê 9HÊ Ã ? 7K hDGF¬JVGK&FUÇp!p!DÁK JVGUÇJ J(t) 1 EHGN} ÁKMUTLGpU PC·J!FVUTLp¬´ iφ(t−t00 ) iφ(t) −iφ(t0 ) −iφ(0) J(t−t00 −t0 )+J(t−t00 )+J(t−t0 )+J(t)−J(−t00 )−J(t0 ) NÁB K&W4UTLG}&KHUTBGBGFC!NOÁ U3JVNCTL)ÇJ!CEREK&J!GKMF3HNOJ!ÅJVGK? UT}&JÐJ!4UÇJ J(t) NOp ;K&PP IKM4UjÆEK&WÉUÇJPUÇF!REKQJ!NÁKMpMÊ GNpQUTPPOCHp7DGpQJVC>K$B4UTLGWeJVK"K$BCELGKMLqJ!NUTP+CBG4UTpKr}&CEF!FKMPUÇJVCEFp eJ(t) ≈ 1 + J(t) ÊGKSFKMp!DPOJ hCEFxJ!GKv}&DGF!FKMLqJÅ}MCTLiJVUTNLpÂI CÇJV UTL K&PUTp¬JVNO} UTLW2UTLNLKMPUÇpJVNO}}&CELqJVFNIGDJ!NCEL I← = I←el + I←inel HGKMFK Z eT 2 2π ∞ |Z(ω)|2 ?$Ê ÃE¿ el el I← ' Ψ0← − dω SI (ω)K2e← (ω, eV, η) , HNOJ! Ψ0← UTLGW 16π 3 RK el K2e← (ω, eV, η) inel I← ' RK ω2 WGK4LGK&W`UTpNOLz»,ipMÊ $? ÊE?D QUTLGW ? Ê AEÄ ;NLe®°BGBKMLWGN lÎ" UTLGW eT 2 16π 2 RK Z ∞ dΩ 2∆+2eV −∞ |Z(−Ω)|2 inel SI (−Ω)K2e← (Ω, eV, η) , Ω2 ?$Ê ÃI HGKMFK K2e← inel (Ω, eV, η) NOpWGK%4LKMW NOL », Ê ?$Ê<AG >NOL5®½BGBKMLGWGN ΰÊÐGKlKMPUTpJ!N}É}MCTLiJ!F!NOIGDJVNOCEL K $Np¬JVpHCELGP vN eV < −∆ ÊË4CEF −∆ < eV CELGP J!GKNLGK&PUTp¬JVN}½B4UTF¬J½}&CELqJVFNIGDJ!KMpJVC I← Ê ? $À NÁNPUTF!P ;K}MUTP}&DGPUÇJ!K:hCEF I→ Ê[GKMFKNpSUÉpÁÁK¶JVF¬:I K¶J*;K&KMLJVGKÅÁUTRELGN JVDGWKÂI K¶J*;K&KML JVGK°F!NRTiJ;UTLGWÉPOK%#J;ÁC1ÆNLGRx}MDGFF!K&LiJHDB CEL IGNUTp7F!K¶ÆEKMFpVUTP|~$J!GK"K$BGFKMpp!NCTLAhCTF I← NOp7J!GK"pVUÇÁKSUÇp I→ 4N &;KrF!K&BGPUT}&K −eV Iq eV Ê À$C GNOLeJ!GKNLqJVK&FÆÇUTP |eV | ≤ ∆ 7KxCEI$J UTNOL ∞ |Z(−Ω)|2 + inel Sexcess(Ω)K2e→ (Ω, eV, η) 2 Ω 2∆−2eV Z ∞ |Z(−Ω)|2 + inel Sexcess(Ω)K2e← (Ω, eV, η) , −C2e dΩ 2 Ω 2∆+2eV ∆IP AT (eV ) = C2e HNOJ! inel inel K2e→ (eV ) = K2e← (−eV ) Z UÇLGW dΩ ?$Ê ÃD C2e = eT 2 /16π 2 RK Ê # '71# ! ,!,!.'!- & # # , . ! 1 .# ! ±µLzJ!GNpH}jUÇp!K67KÂUTPOp!CLGKMK&WÕJVC}jUTFFeCEDJ½}jUÇP}MDPUÇJ!NCELGpC/JVGKrÁUÇJVFN ÉK&PK&ÁK&LiJ½NLz», Ê $? Ê A QJ!C p!K&}MCELGW CEF!WGK&F;NOLJ!GK½J!DGLGLGK&PNLR Ô UTÁNOPOJ!CELGNUÇL+ÊTÐBGN}MUTPP EJVK°NLNOJVNUTPp¬J UÇJ!K"HNOPP4IK"UTpp!GCHLÉNOL »,Ê $? Ê GI ¶Ê4ÌSLÉJ!GKCTJVGK&FH4UTLGW)4CTDGFREDGK&p!p?hCEFQJVGK<4LGUTP[pJVUÇJVKF!KMUTWGp $? Ê HEÄ |f i = 2 (c c −c c )|G i ⊗ |G i ⊗ |R i , L −1/2 † † k,σ k 0 ,−σ † † k 0 ,σ k,−σ L HGKMLzU ³ C$CTB K&F©BGUTNFQNOpKMÁNOJJVKMW hFCEÁ J!GKp!DGBKMF}MCELGWDG}&J!CEF CEF 0 S |f i = 2−1/2 [ck,σ ck0 ,−σ − ck0 ,σ ck,−σ ]|GL i ⊗ |GS i ⊗ |R0 i , ?$Ê HG HGKML`J!GK>pDGBKMF!}&CELGWGD}&JVCTFSPKjUÇW`UTIGpCEF!IpSU ³ CCEB K&F°B4UÇNFMÊ Ô KMF!K |R0i NOp½JVK4L4UÇPp¬J UÇJ!KÂC,JVK KMLqÆNFCELGÁKMLqJjÊ7GK bRTDGKMpp ÕCr»,p&Ê ?Ê HEÄ UTLGW ?$Ê HG NOpvUTREUTNL UTL NLhCTF!ÁKMW CTLGKT~7NLGWGK&KMW) JVGK s8´ QUjÆEKÅp¬ÁÁK&J!F`C J!GKÅp!DB K&F!}MCTLGWGDG}¶JVCEFNÁBCEp!K&p°JV4U3JSCELGP ÕpNLGREPOK&JSBGUTNFp"CKMPOKM}&J!F!CELp }jUTL:IKÅKMÁNOJJVK&W:CEFSUTIGpCEF!IKMW9Ê[GNp"BGKMLGCTÁK&LGCEL4UTpSIKMKMLWGK&p!}&F!NIKMWNOL`JVGKÅKMUTF!P , 7CEF zCEL KMLqJ UÇLGREPK&ÁK&LqJÐNLÅÁKMp!CTp!}MCTBGN}BGqp!NO}Mp ' DT7Ä DG(8qUÇLGWÅJVGKHF!K&p!DGP JVNOLGER 4L4UTPpJ U3JVKH}jUTL ENLBGFNLG}&NBGPOK I KSWK&JVK&}&J!KMWzJ!GF!CTDGREzUÅÆNOCEPUÇJ!NCEL C /Î7KMPP9NLGK&DGUTPN JVNK&p ' GHG (|Ê Î;KM}MUTDGp!K ;K UTF!KUTPOp!Cl}MCELGpNWGK&F!NOLGRlJ!GKJVDGLGLKMPNOLGRlBGF!C}MK&p!p>C ;*J 7C`KMPOKM}&J!F!CEL( p hFCEÁ JVGKp!D´ B K&F!}&CELGWGDG}¶JVCEFHJVCJVGKxLGCEFÁ UTP[ÁK&JVUTP[PKMUT)W 4JVKx}&DGF!FKMLqJ"NOp½K BF!KMpp!K&WUÇp½NOL`», Ê ?$Ê ÃTà ¶Ê4±µLlJVGNOp }jUTpK JVGKxBGqp!N}MUTP NLqJVK&F!BGFK&J U3JVNCTLÕC л, Ê ?$Ê ÃEà Np½U GCEBBGNLGRBGFC$}&KMp!p°C /*J 7CÉKMPK&}&J!F!CELG?p HNOJ! CEBGBCEp!N JVKp!BGNOLGp:hFCEÁ JVGKp!DGBKMF}MCELGWDG}&J!CEFJ!GKMFKMIqFKMÁC1Æ$NOLGR`U ³ C$CEBKMFxB4UTNOF>NLJVGK p!DGBKMFb´ }MCELWGDG}&J!CEF 9UTLGWI4UT} ÕUTRqUTNOL+ÊGKÅWGKMPUjzJ!NÁKMp°IK&*J 7KMK&LJ!GKÂ*J 7CeJ!DGLGLGK&PNOLGRvBGFC$}&KMp!pKMpC ÐJVK KMPOKM}&J!F!CEL( p HN JVGNOLUlB4UTNOFSNpRENOÆTKMLIi 0 UTLGW 00 FKMpB K&}&JVN ÆEK&PO &HGK&F!KjUÇpxJVKJ!NÁKÅI K¶*J ;K&KML WGK ´ pJ!F!C1NLGRlUTLGW}MFKjUÇJ!NLGRlU ³ CCEB K&FB4UTNF°t NpRENOÆTtKMLIq t ÊË4CEPPOCHNOLGRÉ», Ê ?$Ê ÃH "UÇLGW}&CELGpNWGK&F!NLR CELGP `JVGK®°LWGF!K&K&ÆBGF!C}MK&p!p &7K }jUÇLLGC HF!N JVKJVGKJVDLGLGKMPONLGRl}MDF!F!K&LqJÅUTpxC U hDGL}&JVNOCELC QJVK LGCEFÁ UTP UTLGW:UTLCEÁ UÇPCEDGp ½¸SF!K&KM)L 1 p hDGL}&JVNOCELGp"C ,JVGKÂLCEF!ÁUTPÁK¶J UTP PKMUT)W GLσ JVKÂiD4UTLqJVDGÁ WGCTJ G UTLGWJVGK©p!DB K&F!}MCTLGWGDG}¶JVCEF% F p!K&K°®½BGB K&LGWGN ³ HGNO} Np,J!GK½p!UTÁK½UÇpNOL ¯HK%bÊ ' Dqà ( Dσ σ I← = 2eT04 Z ∞ dt −∞ X Z ∞ dt 0 0 × −G> Lσ (k, t 0 0 k,k ,q,q ,σ Z ∞ 0 00 0 dt00 e−η(t +t ) ei(µS −µL )(2t−t −t 00 ) 0 0 ∗ 0 00 0 − t00 − t0 )G> L−σ (k , t)Fσ (q , −t )F−σ (q, t ) 00 > 0 0 ∗ 0 00 0 + G> Lσ (k, t − t )GL−σ (k , t − t )Fσ (q , −t )Fσ (q, t ) ×eJ(t−t 00 −t0 )+J(t−t00 )+J(t−t0 )+J(t)−J(−t00 )−J(t0 ) . GKF!K&p!DGP J hCEFQJVGK}&DGF!FKMLqJ°}&CELqJ UTNOLGpHI CÇJVlUTLeKMPUTpJ!N}UTLGWlUTLlNLKMPUÇpJVNO}"}MCTLiJ!F!NOIGDJVNOCEL+~ el inel I← = I ← + I← , A ?$Ê Hqà ?$Ê HH HGKMFKSJVGKKMPUTpJ!N}S}&CELqJVF!NOIGDJ!NCELÉFKjUTWGp el I← Z ∞ Z ∞ Z ∆2 eT 2 −eV 0 p dE dE d ' √ 2π 3 eV ∆ ∆ E 2 − ∆2 E 0 2 − ∆2 Z Z 1 2π +∞ |Z(ω)|2 SI (ω) 4π +∞ |Z(ω)|2 dω SI (ω) dω − . × 1− 0 el RK −∞ ω2 D← RK −∞ ω2 D← ?$Ê HG GKNLGK&PUTp¬JVNO}"}MCELqJ!F!NIDJVNOCELvJVC I← Np Z ∞ Z ∞ Z ∞ Z ∞ eT 2 inel dE 0 dE d0 d I← ' 2 π RK eV ∆ ∆ eV ∆2 |Z(−( + 0 ))|2 SI (−( + 0 )) p . ×√ inel ( + 0 )2 D← E 2 − ∆2 E 0 2 − ∆2 ?$Ê H ? HGKMFK,JVGK,WGK&LGCEÁNL4U3JVCEFp[UTF!K,pB K&}MN 4KMWrNLS®½BGB K&LGWGN ¨ Ê I→ NOp[WGKMFNOÆTKMWrNLUQp!NOÁNOPUTFÁUTLGLGK&FjIGD$J NOJ!p+K$BGFKMp!pNCELxNp[CEÁNOJJVK&WSGKMFKTÊ ²½K¶ÆEK&FJVKMPK&p!p%ÇN JVp+K9K&}&JVp&HNPOPTIKÐWGNpBGPUjEK&WNLSJVGKÐÁKjUTpDGF!K&ÁK&LqJ C JVGKLGCENOp!KSC UÅB CTNLqJH}MCELqJVUT}&JMÊ GKUTIC ÆTKÂK $BGFKMpp!NCTLGp}MCELGp¬JVN JVDJ!K>JVGK>}MKMLqJ!FVUTP/FKMpDGPOJSC ,JVGNOp ;CTF ~ 7KÅDGLGWGK&F!p¬J UTLGWLGC GC ¦JVK}&DGF!FKMLq: J 4D}&JVDGUÇJVNOCELGprNLJVGKLGKMNOREiI CEFNLGReÁKMpCEp!}&CEBGN}}MNOF!}&DGNOJrREN ÆEKF!NpKÅJVCÕNOLGKMPUTpJ!N} UTLGWKMPUTpJ!N}Å}MCTLiJ!F!NOIGDJVNOCELGpSNOLJ!GK}MDGFF!KMLqJxI K¶*J ;K&KML UeLGCEFÁ UÇP/ÁK¶J UTPÐUÇLGWUep!DB K&F!}MCTLGWGDG}¶JVCEF&Ê K?4LGWJV4U3J;ICTJV}MDGFF!K&LiJ;}MCELqJ!F!NIDJVNOCELGpÐ4UjÆEK©JVGK©pVUTÁ?K hCEF!C Á iUTLGWJVGK©}MDGFF!K&LiJ 4DG}¶JVD4UÇJ!NCELp C +J!GK°ÁKMpCEp!}&CEBGN}½WGK&ÆNO}MKSU KM}¶J;J!GK"}MDF!F!K&LqJNLJVGK½WGK&J!KM}&J!CEFUÇJJVK"KMLGK&F!RT}MCTF!F!K&p!BCELGWGNOLGRxJ!C JVGKSJ!CTJ UTP[K&LGKMFRTeC *J ;CKMPK&}&J!F!CELGpHNOLlJ!GKLGCEF!ÁUTP[POKjUTW+Ê®°LGWGFKMK¶ÆzF%K 4K&}&J!NCELeJVGK&F!K hCEF!KxUT}&J!p°UÇp UTLeKMLGK&F!RT GPOJVK&FMÊ ²½K J;K}MUTLÉ} 4UTLGRTKÆÇUTFNUTIPKMp;UTpQNOLvJVGKSBGFK&ÆNCTDGpp!K&}&JVNOCELGp&Ê NOJ!eUTLeUTF!INOJVF!UTFIGNUTp eV ;KCEI$J UTNOL HNOJ! ¨ Ê 4 +∞ |Z(ω)|2 + Sexcess(−ω)KNelS (ω, eV, η) ∆IP AT (eV ) = −CN S dω 2 ω −∞ Z 2eV |Z(Ω)|2 + CN S dΩ Sexcess(−Ω)KNinel − S (Ω, eV, η) 2 2 Ω −∞ Z CN S ∞ |Z(−Ω)|2 + − dΩ Sexcess(Ω)KNinel S (Ω, eV, η) , 2 2 Ω 2eV Z CN S = eT 2 ∆2 /π 2 RK Ê9GK ÇKMFLGKMPhDLG}&J!NCELGp KNelS UTLGW KNinel S ?$Ê HGA UTFKÂp!CHLNOL`®°BGBKMLWGN , !" # .!"+# !" # 1:(# % !,# . µ± L:JVGNOp"p!K&}&JVNOCEL)7KNPOPDGp¬JVFVU3JVKxJ!GKÅBGFKMp!K&LqJ>FKMp!DPOJVpEHN JVUvp!NOÁBPKÂK GUÇÁBGPOKTÊ.KÅ}&CELGp!NOWGKMF<hCEF JVGNOpBGDGF!BCEpK`UDGUTLqJVDGÁ B CTNLqJ}MCELqJ UÇ}&J! HGNO} NpUWGK&ÆN}&K$HGCTp!KÕLCENpKlpB K&}&J!FVUTP©WGKMLp!NOJµ Np ;K&PP} 4UTF!UT}&J!KMF!N MK&W IiDGpNLGRÕJ!GK p}jUÇJJVK&F!NLRÕJVGK&CEF '<?I)?E¿ (|Ê Ô KMFK C 7K&ÆTKM%F ;Kv}&CELGpNWGK&F DGLGp¬$ÁÁK&JVFN &KMWeLGCENOp!KS}MCTF!F!K&PUÇJ!CEF!%p HGNO} z4UjÆTKxIKMK&LzpGC HLÕNLl», Ê Ã$Ê H A Ê ®°pB CTNLqJVK&W CTDJUTIC ÆTK 7KlUTF!KÉ}&CEÁBDJVNOLGR`JVGKeWGN 9K&F!K&LG}MKeC ©JVGKe®, }&DGF!FKMLqJVpNL JVK BGF!K&p!K&LG}MK°UTLGW NOLJ!GK½UTIGp!K&LG}MK½C 9J!GK©WG}½INUTp&ÊqGNpÁKMUTLGp,JVGUÇJ 7K©NLGpKMFJJVK½pB K&}&JVF!UTP WGKMLp!NOJµ C K $}MK&p!pLGCENOp!KQC J!GKÁK&p!CEp}MCEBN}QWGK¶ÆN}MK HGN} hCEF/U"B CENOLqJ,}MCELqJVUT}&JÐIKjUTFp,ÁCTpJ,C N JVp 7KMNOREqJ i¿ 1000 1000 ∆IP AT 300 eVd = 0.3 eVd = 0.5 eVd = 0.8 200 500 500 100 0 PSfrag replacements 0 0.2 0.4 0.6 0.8 1 0 0.985 0.99 0.995 1 0 0.2 0.4 0.6 0.8 1 eV eV eV 2; '#D13>:!#<%& '8'(! ?,3< G 5$'; O6! '<mB* %0'01'2 3-5< 1 $5 '; D G ∆I 1'?bP<AT ':;=b 6 3j113H; + $3 3'##13H; + ; -!C-61D6Q<3 E 2$$; eVd 0.3 0.5 0.8 32 % %b; @$%b2((`& ; q?<; +6< n' @ $%+2$(`? ; ?<; +63 3 1-5" -71%? 'j + ∆IP AT # '(! 6 q - 01 D $ C2e T = 0.6 LGKjUÇF MK&F!C'hFKMiDGKML}MNK&pMÊ,»$}MK&p!pLGCENOp!KWGKM}&F!KjUÇp!KMpPONLGKMUTF!P `JVC&KMFC:C ÆTKMFUÕF!UTLGREK [0, ±eVd ] hCEF B CTp!NOJ!NOÆTKUÇLGWLGK&RqUÇJVN ÆEK;hF!K&DKMLG}&NKMp GK&F!K7KvFK%HF!N JVK»,Ê ÃÊ HD <HN JVJ!GKLGCTJVNO}MKGKMFKjU#J!KMF% ;KDGpKrJ!GKLGCTJVUÇJVNOCEL Vd NLGp¬JVKMUTWeC V J!CpGC¼JVKÆTCEPOJVUTREKSCÁKMp!CTp!}MCTBGN}WGK¶Æ$NO}MK + Sexcess (ω) = 2e2 T (1 − T )(eVd − |ω|)Θ(eVd − |ω|) . π ?Ê Hq¿ KÅ} GCCEpKUvREKMLGK&F!NO} hCEF!Á hCEF°J!GK>JVF!UTLGpNÁB K&W4UTLG}&K+pNÁNPUÇF©NOL`p!BNF!N J"JVCvJ!4UÇJS} CEp!K&LNOL ¯©KbÊ ' ÃÇÄ)( Ê ³ CELp!NWKMF!NOLGRJVGK©}MNOF!}MDNOJÐNLËNOREDGF!K&p"?$Ê@G°UTLGW&?$Ê HUÇJ ω = 0 EJVGK©WGK¶Æ$NO}MK°UTLGWWGK&J!KM}&J!CEF UTF!KSLCTJ}MCTDGBGPK&WÕUÇLGWÉJVGK"J!FVUTLGpNÁB K&WGKML}MKSp!CEDGPWÉJ!GKMFK%hCEFKÆÇUTLNp!9Ê Ì"LeJ!GKSCTJ!GKMF4UÇLGW)GJVK JVF!UTLGp!NOÁBKMWGUTLG}MKNp BGFKMWGNO}&JVK&W>JVC°4UjÆEKU©}MCELGp¬J UTLqJ,IKMGU1ÆNOCEFU3J/PUÇF!REK hF!K&iDGKMLG}&NK&pMÊK;J!GKMFK%hCEFK } GCCEp!KSJVK hCTPPC HNOLGRÂREK&LGKMFNE} hCEFÁ hCEFQJ!GKSJVFVUÇLGp!NOÁBKMW4UÇLG}MK |Z(ω)|2 = (Rω)2 , ω02 + ω 2 ?$Ê HI HGKMFK R Np©JVGKrJµBGN}MUTPGNORE$hFKMiDGK&LG}&ÕNÁB K&W4UTLG}&KÂUTLWÕJVGKx}MFCEp!pC ÆTKMF<hF!K&iDGKMLG}¶ ω0 Np½KMp¬JVN ´ Á U3JVKMWhF!CEÁ JVGKK $B K&F!NÁKMLqJVUTPW4U3J U"C¯HK%bÊ ' ÃEà ( E} C$CEpNLGR"U 4LGNOJ!KQ}MD$JVChF!KMiDGK&LG}& ω0 ÁKjUTLp JV4U3JUÇJ hF!K&iDGKMLG}&NK&p ω ω0 TJ!GKÁKMp!CTp!}MCTBGN}H}MNOF!}&DGNOJ,GUTp,LGCNLGDGKMLG}&KHCELJ!GKWGK¶JVKM}¶JVCEFÐ}&NF!}&DGNOJ I K&}jUTDp!KPC hF!KMiDGK&LG}MNOKMp½WGCLGCTJHBGFCEB4UTRqU3JVK"JVF!CEDGRTÕUÅ}jUÇB4UT}MN JVCEF&Ê K}jUÇP}MDPUÇJ!KÅLDÁK&F!N}MUTPP zJ!GK ®,¦}&DGF!FKMLqJVprNL:J!GK UÇI C1ÆEKJVF!KMK}MUTp!K&pM~ p!NLREPKÅUÇLGW*J 7C iD4UTp!NOB4UTF¬JVN}&PKHJVDGLLGKMPONLGR>UTLGWv®½LGWGF!K&K&Æ F!%K GKM}&J!NCEL9Ê®°POP K&LGKMFRENK&pQUTFK"ÁKjUTpDGF!K& W HNOJV F!KMpB K&}&JQJ!C JVGKHpDGB K&F!}&CELGWGDG}¶JVNOLGRxREUTB ∆ Êq±µLJVKMp!K©DGLGN JVp%;K©} GCEpK ω0 = 0.3 UTLW η = 0.001 Ê ³ DGF!FKMLqJVp7UTFK JµBGN}MUTPP BGPCTJJVK&WeUTp U hDGL}&JVNOCELÉC [J!GKSWG}"IGNUTp;ÆTCEPOJVUTREK eV C [JVGK°WGK&J!KM}&J!CE4F hCEFQpK&ÆTKMFVUÇP9ÆÇUTPDGK&p C /J!GKÂÁKMp!CTp!}MCTBGN}xIGNUTp©ÆECEP J UTRTK eVd hJVK>KMLqÆNFCELGÁKMLqJ Ê[Ì"DGF½ÁCTJ!NOÆÇUÇJ!NCELeNpHJVC }MCELGpNWGK&F°JVK ®,2}MDF!F!K&LqJVp HNOJ!ÂJVGK}&CELGWGN JVNCTL |eV | < 1 |eV | < ∆HGK&F!KQJ!GK%K KM}¶JÐC JVGKK&LiÆNOF!CELGÁKMLqJ CELÅJVGKH®, }&DGF!FKMLqJ;NOp,ÁCTpJ,BGFCELGCEDLG}MK&)W UTLGW ;KHp4UTPPBGF!K&WGN}¶J,JV4UÇJÐ*J 7CKMPK&}&J!F!CELGp,J!DGLGLGK&PNLR UTpxU ³ CCEBKMFrB4UTNF h®°LGWGFKMK¶ÆBF!C}MKMpp!K&px}&CELqJVF!NOIGDJ!KMpJVGKÁCEp¬JJVCzJ!GK ®,¦}&DGF!FKMLqJ! K$}MK&BJ }MPOCEp!KvJ!C eV = ∆ ÊÐÎ;KM}MUTDGpKzC ©JVKÉpÁÁK&J!F IK&*J 7KMK&L LGK&RqUÇJ!NOÆEK eV UTLW BCEpNOJVN ÆEK eV ;K I GW NpBGPUjrJVGKHFKMp!DPOJVp hCTF eV > 0 ÊEK½®, }&DGF!FKMLqJVp hCEF/J!GK©UTI C1ÆTKHJ!GF!K&KBGF!C}MK&p!pKMpQUÇF!KBGPOCTJ!J!KMW LGKJHJVCCELKxUÇLGCTJVKMFHNLeËNOREDGFK ?$ÊE?:hCEF}&CEÁB4UTF!NOp!CEL9Ê K(4LGWÕNOL`ËNOREDGFK ?EÊ ?ÅJV4UÇJ½[email protected]GKx} GKMÁN}MUTPB CÇJVKMLqJ!NUTP[CJVGKxLGCEF!ÁUTP[ÁK&JVUTPPKjUÇWzNOp½}&PCEpK JVC`J!GKvBCTJVK&LiJ!NUTP7C HJVGKÉpDGBKMF!}&CELGWGD}&JVCTFÅPKjUÇW eV ∆ JVKe ®,¤}MDGFF!KMLqJ!pNOLJVGKJVGFKMK }jUTpKMpSUÇF!KÂpDGBGBGFKMp!pKMW+ÊË4CEF©JVGKx*J ;CÉ}MUTp!K&pSC ÐiD4UTp!NOB4UTF¬JVN}&PKJVDGLLGKMPONLGR JVKÅ ®, }MDF!F!K&LqJVpUTFK KMiD4UTPSJ!C &KMFC IKMPC U }&KMFJVUTNL JVF!KMpGCEPOW+ÊGKp!NLREPK:iD4UTpNB4UÇFJVNO}MPOK:}[email protected] KMFp hF!CEÁ MK&F!C UÇJ UJ!GF!K&p!GCEPO)W 3HGN} NOp[email protected] 4K&W UTp ∆ − eV ÐJV4UÇJN%p 7iD4UÇp!NBGUTFJ!N}MPOKMpUTFK`UTIGPOKlJ!C JVDGLLGKMP9UTI C1ÆTK"JV4UÇJ;p!DGBKMF}MCELWGDG}&J!CEFRqUTBÉCTLGPO@N [J!GK&}jdUÇLlICEFF!C JVGK°LGKM}&KMp!p!UTFeK&LGKMFRT hF!CEÁ JVGKÁKMp!CTp!}MCTBGN}WGK&ÆNO}MKTÊ GNOpxK BPUTNOLG<p HiJVK}&DGFÆTKMpÅUTpp!C}MNUÇJVK&W HNOJ!WGN 9K&F!K&LiJrÆÇUTPDKMprC JVGK:ÁKMpCEp!}&CEBGNO}:WGK&ÆN}&KIGNUTpvÆTCEPOJVUTREK:UÇF!Kp!GN #JVK&W J!C JVGK:FNREqJlUTp eVd WGK&}MFKjUTpKMpMÊ©Ë4CEF *J 7C iD4UTp!NOB4UTF¬JVN}&PK°J!DGLGLGK&PNOLGR 7K>CEIp!KMF¬ÆEKrJV4UÇJJ!GKx ®,¼}&DGF!FKMLqJ°GUTp°UÅpNÁNPUÇF7JVF!KMpGCEPO)W HGN} }MCEÁB4UTFKMW J!CÅËNOREDGF!K ?$Ê ?TU Np;BDGp!GK&WlJ!C QUTFWvJVGK°F!NRTiJ;NLÉËNREDGFK ?EÊ ?ÇI IKM}jUÇDGp!KSÁCEF!K½KMLGK&F!RT NpLGK&KMWGK&W JVCJVFVUÇLG/p hK&F*J 7CK&PKM}¶JVFCELGpvUÇI C1ÆEKeJVKzRqUÇ)B Ð}MCEÁB4UTFKMW5J!CUp!NOLGREPKÉK&PK&}&JVFCEL+ʲ½CÇJ p!DGFBGF!NOp!NOLGREPOJVGKr}MCTF!F!K&p!BCELGWGNOLGR}MDGF¬ÆEKMp°UTF!KrCELG}&K>UÇRqUTNLep[email protected] #J!KMWeJVCJVGKrF!NOREq>J HN JVlWGKM}&F!KMUTp!NOLGR eVd Ê4KMp!Kr}MDFÆEK&p"UTPOP94UjÆTK>UÅp!GUTF!BlÁ U NOÁÂDGÁ UÇJ eV = ∆ Ê KSJVDF!LeLGC J!CJ!GK®°LGWF!KMK¶Æl ®,[email protected] KMFKMLG}&Kr}&DGF!FKMLq!J HGNO} zWCEÁNOL4UÇJ!KMp;J!GKrUTI C1ÆTK*J 7C BGF!C}MK&p!pKMpvU3Jp!ÁUTPPUÇLGW5ÁCWGKMF!UÇJVKeIGNUTp!K&pMÊв½CÇJVKÉJV4U3JJ!GKeJVCTJVUTPQ®°LWGF!K&K&Æ }MDGFF!K&LiJ}MCTLiJVUTNLp UTL K&PUTp¬JVNO}l}&CELqJVFNIGDJ!NCEL5UT; p ;K&PP©UTpUTL NOLGKMPUTpJ!N}e}MCTLiJ!F!NOIGDJVNOCEL I K&PC J!GKÕRqUÇ)B Ð}MCELqJVF!UTF J!C iD4UTp!NOB4UTF¬JVN}&PK JVDGLGLKMPNOLG4R HGNO} x4UÇp[}MCELqJVFNIGD$JVNCTLGp[I K&PC JVGK,RqUÇBrCTLGPO½I K&}jUTDp!KÐJVGK&p!KÐBGF!C}MK&p!pKMp UTF!KrBGGCÇJVCÇ´µUTp!pNpJ!KMW+ÊÎ;K&}jUTDp!: K 7KÂUTFK>}&CEÁBDJVNOLGRJ![email protected] KMFKMLG}&K>IK&*J 7KMKML®, }MDGFF!KMLqJ!"p HNOJ! 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UTL K BKMFNÁKMLqJ UÇP7ÁKjUTpDGF!K&ÁK&LqJÅCJVKÉ ®,¤}MDGFF!K&LiJÂI K ´ PC JVKÅRqUTB:}MCTDGPWÕJ!iDGpp!KMF¬ÆEKUTpSUÇL:K% KM}&J!NOÆTKÅLGCENpKÂÁKjUTpDGF!K&ÁK&LiJ[UTp°J!GK 7KMNRTiJ hDGLG}&J!NCELp UTF!<K iLGCHL+Ê KNelS (ω, eV, η) UTLGW KNinel S (Ω, eV, η) ²½CTJ!N}MKzJV4UÇJeNOL UTPP©CEDGFÉLiDGÁKMFN}jUÇP"F!K&p!DGP JVp 7JVGK ®, }MDF!F!K&LqJVpzUTF!K:BGPOCTJ!J!KMW NOL DLGNOJ!pÉC Ê KÅBGDJSpCEÁKÂJVKMLqJVUÇJVN ÆEKÅLiDGÁÂIKMFpSNL`JVKMp!KiD4UTLqJ!NOJVNOKMp&Ê Ô K&F!K e3 R2 T 2 T (1 − T )∆/8π 3 ~2 RK T = 0.6 Np/JVK©K 9K&}&J!NOÆEKHJVF!UTLGpÁNOp!p!NOCELÅ}MCK }&NKMLqJ7C JVGK©²"ÀÅNLqJ!KM/F UÇ}MK ∆ = 240 µeV T = 0.5 NpJ!GKJVF!UTLGp!ÁNpp!NOCELÂC JVGKiD4UÇLiJ!DGÁ¦BCENLqJÐ}&CELqJ UT}¶J,JVCI KÁKjUÇp!DGFKM)W iUTLW R = 0.03RK RK Np JVGKSFKMpNpJVUTLG}&KriD4UÇLiJ!DG Á ;NpQJVKrFKMpNpJVUTLG}&K HN} eKMLqJVK&F!pHJVGK"J!FVUTLp!NÁBKMW4UTL}MKTÊGNpNOÁBGPONK&p KTÊ RGÊ hCEF;J!GK ®, }MDGFF!K&LiJ©NOLeËNREDF!K ?$<Ê AÅUÇJJ!GK"JVCEBeC J!GKSB KMU J!4UÇJ ∆I ' 10−10 A HNOJ! JVGKrWGK¶Æ$NO}MKxIGNUTp Vd = 0.8∆/e = 48 µV HGN} `pKMK&Áp°UTL`UT}&}MK&BJ UTIPKxÆÇUTPODGP KAT}MCEÁB4UÇJ!NIGPOEK HNOJ! }MDGFF!K&LiJ©ÁKjUTpDGF!K&ÁK&LqJ©J!KM} GLNiDGKMp&Ê ! ! KLGCJVDGFLJ!CzUeWGN 9K&F!KMLqJpK&JVDB hCEFSLGCTNp!KWGK¶JVK&}&JVNOCELHGKMFKK&PKM}¶JVFCELGpNLUÉLGCEFÁ UTPÁK¶J UTP PKMUTWÂJVF!UTLGp!N J,JVGFCEDGREUSiD4UTLqJVDGÁ¡WCTJÐNLÅJVK ³ CEDGPCTÁÂIIGPC}3UTWGKF!K&RENÁKQIK%hCEFKHRECENLRSNLqJVCSJVK p!DGBKMF}MCELWGDG}&J!CEFMÊ4KSKMp!pKMLqJVNUTP[NLREF!K&WGNK&LiJ!pHUTF!K½JVGKpVUÇÁKSUÇpNLvJVKSBGF!K¶Æ$NOCEDGpHp!K&}&J!NCEL)$K$}MK&BJ JV4U3JSUTWGWNOJVNOCEL4UTP[K&LGKMFRTC 4POJ!KMF!NOLGR C}M}&DGF!p°I K&}jUTDGpKÂC ,J!GKÂWGCTJMÊ ±µL`J!GNp½p!K&}&JVNOCE)L 6;KÅ} C$CEpK>JVK B4UTF!UTÁK¶JVK&F!pC JVGKWGK¶ÆN}MKpCJ!4UÇJHCELGP v®½LGWGF!K&K&ÆeBGFC$}&KMpp!KMp°UTF!KFKMPK¶ÆÇUTLqJjÊ # , #!.! GK Ô UTÁNOPOJ!CELGNUÇL;HN} ÉWGK&p!}MFNIKMpQJ!GK"WGK&}MCEDGBPKMWÉLCEF!ÁUTPÁK¶J UTPPKMUTW * WGCÇJ *pDGB K&F!}&CELGWGDG}¶JVCEF * KMLqÆNFCELGÁKMLqJ ÁKMpCEp!}&CEBGN}}&NF!}&DGNOJ7pp¬JVKMÁ FKjUTWGp ?$Ê HD H0 = H0 + H0 + H0 + Henv , HGKMFK,JVGK Ô UTÁNP JVCELNUTL©C J!GKÐLGCEFÁ UTPÇÁK¶J UTPqPOKjUTWrUTLWJ!GKÐp!DGBKMF}MCELWGDG}&J!CEFPOKjUTWrUTFK,WGKMp}MFNIKMW UTpHUTIC1ÆEKTÊ GK Ô UTÁNP JVCELGNUTLhCTFQJVGKiD4UTLqJVDÁ WGCTJHF!KMUTWGp L H0 D = D X σ S D c†D,σ cD,σ + U n↑ n↓ , ?$Ê qÄ HGKMFK U HNOPPIKÕUTpp!DGÁKMW JVCI KlNLGLGNOJ!KÐUTpp!DGÁNLGRUpÁ UÇPPQ}MUTB4UT}&NOJ UÇLG}MKlC"J!GKzWCTJjÊ K }MCELp!NWKMFQJVGUÇJHJVGKWGCTJHBCEpp!KMpp!K&p°CELGP ÉUp!NOLGREPK°KMLGK&F!RTePOK&ÆEK&P)hCEFp!NOÁBPN}&NOJµTÊ GK"JVDGLLGKMPONLGR Ô UTÁNPOJ!CELGNUTLNOLG}MPODGWGK&p;JVGK°KMPK&}&J!F!CELÉJVDLGLGKMPONLGR>I K¶*J ;K&KMLeJVGKSpDGB K&F!}&CELGWGDG} ´ JVCEFHUÇLGWÉJVGKWGCTJ UT4p 7KMPPUÇpHJ!GKSJVDGLLGKMPONLGRÅIK&*J 7KMK&LÕJVGKWGCÇJ©UTLGWeJVGKLGCEFÁ UÇP9ÁK&J UÇP+PKMUT)W Ô HT = (HT 1 + HT 2 ) + Ê } , HT 1 = X TD,q c†D,σ cq,σ e−iφ , q,σ HT 2 = X Tk,D c†k,σ cD,σ , k,σ ?$Ê G HGKMFKÅJVGKÅNOLGWGN}&KMp k D [UTLW q F!K%hK&F"JVCÉJVKÅLGCEF!ÁUTPÁK¶J UTP/POKjUTW9iD4UTLqJVDÁ WGCTJ!+UTLGWp!DGBKMFb´ }MCELWGDG}&J!CEFMÊK}MCELp!NWKMFJVGKSpNÁBGPK°}jUTpK TD,q = T1 UTLGW Tk,D = T2 Ê Ë CTF7BGCTJVCÇ´µUTp!pNp¬JVKMWÉ®½LGWGFKMK&Æ BGF!C}MK&p!pKMp7;K"LKMKMWÉJ!CÂ}jUTFFCEDJ;}jUTPO}MDGPUÇJVNOCELGpÐC[JVK"Á U3JVF!N KMPOKMÁKMLqJ>NOL», Ê ?$<Ê A SJ!, C hCEDGF¬JVCEF!WGK&F>NOLJ!GKJ!DGLGLGK&PNLR Ô UÇÁNOPOJVCTLGNUTL9Ê9±µL H4U3: J hCEPOPC Hp)7K ?7G TU p!pDGÁK>JV4U3J°JVKÂWGCTJSNOp°NOLGNOJ!NUTPOPOvK&ÁB$Jµ9CHNOLGR JVCJVGKÅUÇpÁÁK&J!FzI K¶J*;K&KMLJVGKxJ*;CvJ!DGLGLGK&P I4UTFF!NK&F!p&ÊTGK©I4UTFF!NK&F/IK&J*7KMKMLJVGKHLGCTF!ÁUTPÁK&JVUTPPOKjUTWUTLGWÅJ!GKHWGCTJÐNp,pDGBGBCEp!K&WJ!CI KCEBGUTiDGK }MCEÁB4UTFKMWJVCJV4U3J7IK&*J 7KMK&LvJVGKHWGCÇJQUTLWJ!GK½pDGBKMF!}&CELGWGD}&JVCTFMÊ®°p7UrFKMp!DPO!J qJ!GK½F!UÇJVKHC9KMp}jUTBK C[K&PKM}¶JVFCELGp hFCEÁ¡JVGK"WCTJ;J!C>JVGK½p!DGBKMF}MCELWGDG}&J!CEFNp7p!DGIpJ UÇLiJ!NUTPOPOPUTF!REK&FJ!4UTL J!GK°J!DGLGLGK&PNLR FVUÇJ!KSC KMPK&}&J!F!CELGp hFCEÁ J!GKLGCEF!ÁUTP+POKjUTWÉJ!CJ!GKWGCTJ hp!KMKrIKMPC hCEFHUT}¶JVD4UTP[LiDGÁ>I K&F!p Ê GKMFKUTF!K>J*7CeB CTp!p!NOIGNPONOJ!NKMp hCEF°} 4UTF!RTKÂJVFVUÇLGp/hK&F"BGF!C}MK&p!pKMpM~ U ³ CCEBKMF"B4UÇNF°NOLÕJVKÅp!DGBKMFb´ }MCELWGDG}&J!CEF°NOp½J!FVUTLGpÁN J!J!KMWzJ!CJ!GKÂLGCEFÁ UÇP POKjUTWzCEF½ÆNO}MKxÆTKMFpVUÊ G: K 4FpJSBGFC$}&KMpp"NLqÆECTPOÆEK&p°JVK KMPOKM}&J!F!CEL hFCEÁ U ³ C$CTB K&F/B4UTNOFJVDLGLGKMPONLGR°CELqJVC"JVKWGCTJ ÇLGK JTJVGNOp/KMPOKM}¶JVF!CTLÅKMp!}MUTBKMpÐNL>J!GKPKMUTW JVGKCÇJVGK&F,KMPK&}&J!F!CEL hFCEÁJ!GKpVUTÁK ³ CCEB K&F,B4UTNOFJVKMLDGLGWGK&F!RECKMp,JVKpVUTÁK;J*7CrJ!DGLGLGK&PNLRSBGF!CÇ´ }MK&p!p!K&pMÊ À$NOÁNOPUTF,JVF!UTLGpNOJVNOCELG%p NLvJ!GKCEBGBCEp!N JVK"WNF!K&}&J!NCE)L UTF!KSLKM}MK&p!p!UTF hCEF7*J ;CKMPOKM}&J!F!CEL4p hF!CEÁ JVGKLGCEFÁ UTPPKMUTWJ!C`KMLGW DB UTpÅU ³ CCEBKMFxB4UTNFxNLJ!GK pDGBKMF!}&CELGWGD}&JVCTFMÊ,²©CTJVK J!4UÇJxJVGNOp>WGK ´ p!}&F!NB$JVNCTL C K¶ÆEKMLqJ!p½UÇp!p!DÁK&pQNÁBGPNO}MN Jµ>JVGUÇJQJVK°p!DB K&F!}MCTLGWGDG}¶JVCEFPOKjUTWvFKMÁUTNLp;NLJVGK°REF!CEDLGW pJVUÇJVKÂNOLÕJVKÂNLGN JVNUTPUTLG' W 4LGUTPp¬J UÇJ!KMp ®½LGWGFKMK&ÆÕBGF!C}MK&p!p ¶ÊÌ"L`JVKÂCTJVKMF"4UÇLG)W 9N ,JVGK>LGCEF!ÁUTP ÁK¶J UTP PKMUTW`NOp°NLNOJVNUTPP vNOLÕJVGK>REF!CEDLGW:pJVUÇJV K 4PPOKMW:Ë4K&F!ÁNpKjU 9NOJ°Np½P%K #J°NL:UÇL:K $}MN JVKMWp¬J UÇJ!K HNOJ!*J 7ClKMPK&}&J!F!CELGp4UjÆNLRlK&LGKMFRENK&prUTI C1ÆTKË4K&F!ÁN/KMLGK&F!RT EF NL:JVK 4L4UTP,p¬J UÇJ!KTÊ[GKK JVF!U KMLGK&F!RTv4UTpHI K&KMLlBGF!C1ÆNWKMWlIiÉJ!GKKMLqÆNFCELGÁKMLqJjÊ4ÐBGN}MUTPP ?$<Ê qà |ii = |GL i ⊗ |GS i ⊗ |0QD i ⊗ |Ri , HGKMFK |GL,S i WKMLGCTJ!KMpU©REFCEDGLGWxpJVUÇJVK HN} >}&CEF!FKMpB CELWGpJ!C°U 4PPOKMWrË4KMFÁNqpKj>U hCTFJ!GKLGCEF!ÁUTP KMPOKM}&J!F!CWGKTÊ |0QD i NpJVGKÆÇUT}MDGDÁ«C ;JVK iD4UTLqJVDÁ«WCTJÂUÇLGW |Ri WKMLGCTJ!KMp>J!GKNLGN JVNUÇPpJVUÇJVKC JVGKK&LiÆNOF!CELGÁKMLqJMÊ Ì"LlJ!GKCTJVGK&FH4UTLGWeCEDGFHREDKMp!p hCEFQJ!G<K 4L4UTP[p¬J UÇJ!Kp!GCEDPWeF!KMUTW ?$Ê H |f i = 2−1/2 [c†k,σ c†k ,−σ − c†k ,σ c†k,−σ ]|GL i ⊗ |GS i ⊗ |0QD i ⊗ |R0 i , HGKMLzU ³ C$CTB K&F©BGUTNFQNOpKMÁNOJJVKMW hFCEÁ pDGB K&F!}&CELGWGDG}¶JVCE%F CEF ?$Ê |f i = 2−1/2 [ck,σ ck ,−σ − ck ,σ ck,−σ ]|GL i ⊗ |GS i ⊗ |0QD i ⊗ |R0 i , HGKML`J!GK>pDGBKMF!}&CELGWGD}&JVCTFSPKjUÇW`UTIGpCEF!IpSU ³ CCEB K&F°B4UÇNFMÊ Ô KMF!K |R0i NOp½JV K 4L4UÇPp¬J UÇJ!KÂC ,JVK KMLqÆNFCELGÁKMLqJjÊ GK°È[email protected] 4}MUÇJVNOCE, L hCEF°J!GKÅ} GCENO}MKÂC ;»,ipMÊ ?$Ê H °UTLW ?$Ê ½NOp°JVKÅpVUTÁKÅUTpSNOL p!K&}&JVNOCEL ?$Ê Ã$EÊ ?Ê 0 0 0 0 !, % # 1 . % # $*, ,*,#)# F . ! 1 1!" GK>J!KM} GLGNOiDGK ;KÅDp!K>J!Cv}jUTPO}MDGPUÇJVKrJVGK>}MDGFF!KMLqJ"J!GF!CTDGRE`JVKŲ ¨ ÀÈDLG}&J!NCEL`NOp½J!GKÂpVUÇÁKÅUÇp JVGK>QUj ;K°CEIJ UÇNLJVGKS», Ê ?$Ê ÃEà ʮ½POJ!GCEDGRE ;K"UTPpC>}MCTLGp!NOWGKMFJVGK©JVDGLGLKMPNOLGR>BF!C}MKMppQC+J*7C KMPOKM}&J!F!CELp7IGD$J;JVK°KMPOKM}¶JVF!CTLGp;ÁÂDpJ74UjÆEK°ÆNF¬JVD4UÇPp¬J UÇJ!KMp7CELviD4UTLqJVDÁ¤WCTJ!iJVGK&L JVK°}jUÇP}MDPUÇJ!NCEL ?Eà C JVGKÁUÇJVFN KMPK&ÁK&LqJHNLl»,Ê $? Ê<A 7JVC hCEDGF¬JVlCEF!WKMFNLÉJ!GKSJVDGLGLKMPNOLGR Ô UÇÁNOPOJVCTLGNUTLRENOÆTKMp I← = 2e XZ f,υi ∞ dte i(i −f )t −∞ hi|HT† |υ1 i ∞ Z Z ∞ 0 0 dt01 e−i(i −υ1 −iη)t1 hυ1 |HT† |υ2 i Z ∞ Z ∞ 0 dt02 e−i(i −υ2 −iη)t2 0 dt3 ei(i −υ4 +iη)t3 dt03 e−i(i −υ3 −iη)t3 hυ3 |HT† |f ihf |HT |υ4 i 0 Z0 ∞ Z ∞ ×hυ4 |HT |υ5 i dt2 ei(i −υ5 +iη)t2 hυ5 |HT |υ6 i dt1 ei(i −υ6 +iη)t1 hυ6 |HT |ii ×hυ2 |HT† |υ3 i 0 0 = 2e XZ f,υi ∞ dt −∞ Z 0 ∞ dt1 0 Z ∞ dt2 0 Z ∞ dt3 0 Z ∞ dt01 0 Z ∞ Z dt02 0 ∞ 0 0 0 dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 ) 0 hυ2 |HT† |υ3 ieiυ3 t3 hυ3 |HT† |f i hυ1 |HT† |υ2 ie e ×hi|HT† |υ1 ie ×e−if t hf |HT |υ4 ieii (t1 +t2 +t3 ) e−iυ4 t3 hυ4 |HT |υ5 ie−iυ5 t2 hυ5 |HT |υ6 ie−iυ6 t1 hυ6 |HT |ii . ii (t−t01 −t02 −t03 ) 0 iυ2 t02 iυ1 t01 ?$Ê ? ,» Ê ?$Ê< ? ;WGKMp}MFNIKMp°K¶ÆNWGK&LiJ!POÉJ!GKrJ!DGLGLGK&PNOLGRCJ*7C K&PKM}¶JVFCELGp?hF!CEÁ J!GKrp!DGBKMF}MCELGWDG}&J!CEF©J!C JVGKSLCEF!ÁUTP+ÁK&JVUTP9POKjUTWÉÆNUUÅiD4UTLqJVDGÁ WGCTJ! UÇLGWeI4UT}ÉUTREUTNL)FHNOJ!eJ!GKWGKMPUjvJVNOÁK&pQI K¶J*;K&KML J*;CÅCEBGBGNOLGRÅpJ!KMBGpHUTFK"REN ÆEKMLeIq t1 t2 t3 t03 t02 t01 F!K&p!BKM}¶JVNOÆTKMP 4UTLWvJVGK"J!NÁK½IK&J*7KMK&LzJVK WGKMp¬JVFC NOLGR:UTLGW }&F!KMUÇJVNOLGR:U ³ CCEBKMF>BGUTNFxNp>RTNOÆEK&LIqJMÊGNOp>JVDLGLGKMPONLGRzBGF!C}MK&p!pÅÁUTK&p>JVK }MCEFF!K&p!BCELGWGNOLGR}MDGFF!KMLqJ½I K&NLGRUÇp I← = 2e Z ∞ dt Z ∞ dt1 Z ∞ dt2 Z ∞ dt3 Z ∞ dt01 Z ∞ dt02 ∞ Z −∞ 0 0 0 0 0 0 iµS (2t−t01 −t02 −2t03 −t1 −t2 ) −iµL (2t−t02 −t03 −2t1 −t2 −t3 ) 0 0 0 dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 ) e ×e † ×hHT 1 (t − t01 − t02 − t03 )HT† 2 (t − t02 − t03 )HT† 1 (t − t03 )HT† 2 (t) ×HT 2 (t1 + t2 + t3 )HT 1 (t1 + t2 )HT 2 (t1 )HT 1 (0)i . ?Ê< A GKBGF!CTIGPK&Á Np½JViDGprF!KMWDG}MK&WJ!CeJ!GK}jUTPO}MDGPUÇJVNOCEL`C7}MCTF!F!K&PUÇJ!CEF!p°C7J!GKÂJVDLGLGKMPONLGR Ô UTÁNOPOJ!CÇ´ LGNUÇL NLJVKlRTF!CEDGLW p¬J UÇJ!KTÊ,¹©p!NOLGR N}61 pÅJ!GKMCEFKMÁCJVKMp!Ke}MUTL2I KeK$BGFKMpp!KMW5NL JVK&F!ÁpÅC½U p!NOLGREPKB4UTF¬JVNO}MPKv¸FKMK&L)1 phDGLG}&J!NCEL IKM}MUTDGpKvJVGK Ô UTÁNPOJ!CELGNUTLCHJVGKvNOp!CEPUÇJVK&W}&CEÁBCELGK&LqJVpÅNp iD4UTWGF!UÇJVNO} hK$}MK&BJÅÁU1IK hCTFrJ!GK K&LiÆNOF!CELGÁKMLqJ HGNO} NprWGKMUTPOJxp!K&B4UTFVU3JVKMP ¶ÊGKvWGK¶J UTNOPÐC JVGNOpx}MUTP}&DGPU3JVNCTLNOpxpGC HL NL®°BGBKMLGWN »H&Ê K }jUÇLLGC HF!N JVKJ!GKJ!DGLGLGK&PNOLGRÕ}&DGF!FKMLqJÅUTp>U hDGLG}¶JVNCTLC JVGK½LGCEFÁ UTP UTLGW UTLGCEÁUTPCTDGp ,¸SF!K&KM)L 1 p hDGL}&JVNOCELGpC 9J!GK©LGCEF!ÁUTP4ÁK&JVUTP PKjUÇ)W GLσ JVGKiD4UTLqJ!DGÁ WGCTJ G UTLWlJ!GKp!DGBKMF}MCELGWDG}&J!CEF F HGNO} ÕUTFKp!GCHLÕNOLÉ®°BGBKMLWGN ³ Dσ I← = 2eT14 T24 Z ∞ dt σ ∞ Z dt1 ∞ Z dt2 Z ∞ dt3 Z ∞ dt01 Z −∞ 0 0 0 0 0 iµS (2t−t01 −t02 −2t03 −t1 −t2 ) −iµL (2t−t02 −t03 −2t1 −t2 −t3 ) ∞ dt02 Z ∞ 0 0 0 0 dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 ) e ×e X ∗ 0 0 0 0 > 0 × −Fσ (q , −t1 − t02 )F−σ (q, t1 + t2 )G> Lσ (k, t − t2 − t3 − t1 − t2 − t3 )GL−σ (k , t − t1 ) k,k 0 ,q,q 0 ,σ × Gt̃Dσ (−t01 )Gt̃D−σ (−t03 )GtDσ (t3 )GtD−σ (t1 ) + Fσ∗ (q 0 , −t01 − t02 )Fσ (q, t1 + t2 ) 0 0 > 0 t̃ 0 t̃ 0 t t × G> Lσ (k, t − t2 − t3 − t1 )GL−σ (k , t − t1 − t2 − t3 )GDσ (−t1 )GD−σ (−t3 )GD−σ (t3 )GDσ (t1 ) 0 0 0 0 0 0 0 0 0 0 ×eJ(t−t1 −t2 −t3 )+J(t−t3 )+J(t−t1 −t2 −t3 −t1 −t2 )+J(t−t3 −t1 −t2 )−J(−t1 −t2 )−J(t1 +t2 ) . ?H i ?$Ê i¿ GKÕFKMpDGPOJ hCEFÅJVGKz}MDGFF!K&LiJ }MCELqJVUTNLGpICTJV UTL K&PUTp¬JVN}lUTLWUTL5NLGK&PUTp¬JVN}É}&CELqJVFNIGDJ!NCEL9Ê,GK KMPUTpJ!N}"}&CELqJVF!NOIGDJ!NCELÉFKjUTWGp Z Z Z ∞ eγ12 γ22 −eV ∞ ∆2 0 p d ' dE dE √ 2π 3 eV ∆ ∆ E 2 − ∆2 E 0 2 − ∆2 Z Z 1 2π +∞ |Z(ω)|2 SI (ω) 4π +∞ |Z(ω)|2 SI (ω) − , × 1− dω dω 0 el RK −∞ ω2 D← RK −∞ ω2 D← el I← ?$Ê I HGKMFK D←0 Np7J!GKSCEFNRENOL4UTP4WGK&LGCEÁNL4UÇJ!CEF3HGN} eNp7LGCTJHU 9K&}&JVK&WÉIq JVK"KMLqÆNFCELGÁKMLqJ!FHGN} eNp WG%K GLGKMWIq», Ê ?$Ê ¿ ? +UTLW D←el Np½JVGKÅWGK&LGCEÁNL4U3JVCEF½BGF!CWGDG}¶JU KM}&J!KMWIq`JVKÅKMLqÆNF!CTLGÁK&LqJ pKMKr»,Ê ?$Ê ¿ A 7C®°BGBKMLWGN lË ¶Ê4GKNLKMPUÇpJVNO}"}MCTLiJ!F!NOIGDJVNOCELÉF!KMUTWGp inel I← eγ 2 γ 2 ' 21 2 π RK ×√ Z ∞ d eV Z ∞ d eV 0 Z ∞ dE ∆ Z ∞ dE 0 ∆ |Z(−( + 0 ))|2 SI (−( + 0 )) ∆ p , inel ( + 0 )2 D← E 2 − ∆2 E 0 2 − ∆2 2 ?$Ê D HGKMFK D←inel Np[J!GKÐWGKMLGCTÁNOL4UÇJVCTF+BGF!CWGDG}¶J/U3J!JVFNIGD$JVKMWSJ!C©J!GKÐNLGK&PUTp¬JVN}}MDGFF!KMLqJ HGNO} xNOpWGK%4LKMW NLe», Ê ?$Ê ¿E¿ ;C®°BB K&LGWGN eË;Ê Ô KMFK γ = 2πN T 2 UÇLGW γ = 2πN T 2 WGK4LGKrJVGKrJVDGLGLKMPNOLGR F!UÇJVK&p½IK&J*7KMK&L:JVGKxp!DGBKMFb´ }MCELWGDG}&J!CEF7UT1 LGW J!GK½WSCTJQ1 UÇLGWvIK&2J*7KMKMLvJ!GK°N W2CTJQUÇLGWJ!GK°LGCTF!ÁUTP4ÁK¶J UTP POKjUTW)qFKMpB K&}&JVN ÆEK&POBHNOJ! NS UTLW NN UTpxJ!GKvWGK&LGp!N JµC Hp¬J UÇJ!KMpÂBKMF>p!BGNOLC JVK *J ;C:ÁK&JVUTPprNLJVGKvLCEF!ÁUTPÐpJVUÇJVKÉU3J JVGKr} GK&ÁNO}jUTPBCTJVK&LiJ!NUTPOp µS UTLGW µL FKMp!BKM}¶JVN ÆEKMP EÊ®°POP[}MCELqJVFNIGD$JVNCTLGp©J!CJ!GKx}&DGF!FKMLqJS}&CELqJ UTNOL WGKMLCEÁNOL4UÇJ!CEF!p HGK&F!KÕJ!GKÕNOL 4LGN JVKMpNÁUTP η UTWGNUÇI4UÇJVNO}e/p HNOJ!} GNLRB4UTF!UTÁK&JVK&F ÅNpNLG}&PDGWKMW NOL CEF!WKMF;J!CU1ÆTCENWÉWNOÆEK&F!REK&LG}MK&pMÊ ±µ L UT}¶!J GN JQ4UTpIKMK&Lep!GC HLlNLɯH%K hp&Ê ' DT)Ä ([UTLW ' D ? (+JV4U3JHUÂBGF!CTB K&F F!K&p!DGÁÁ U3JVNCTLC +JVGK½B K&FJ!DGF!I4U3JVNCTLpKMF!NOKM%p $NOLG}MPODGWGNLRxUÇPP F!CEDLGWJ!F!NB=p hFCEÁ¦JVGK½WGCTJJVCrJVGK½LGCEF¬´ Á UÇPGPKMUTWGp qPOKjUTWpJ!C>UIGF!CEUTWGKMLNLGRrC JVGK©WGCTJ7PK¶ÆEKMP|Ê KJVU TK½NLqJVCrUT}&}MCEDGLqJJVGNOpÐIGF!CqUÇWGKMLGNOLGRrIi F!K&BGPUT}&NLGR η Iq γ/2 6HNOJV γ = γ1 + γ2 NLqJ!CvCEDGF"}MUTP}&DGPU3JVNCTLGp hNLG}&PDGWNLGRCELGPOzUvIGFCqUTWGK&LGNLR WGDGKQJ!CJ!GKp!DB K&F!}MCTLGWGDG}¶JVCEF ¶Êq®°pÐÁKMLqJVNOCELGK&WUTI C1ÆTK 7K©GU1ÆTK©UTp!pDGÁK&WÅJV4UÇJ γ1 γ2 ÊT±µLCTF!WGK&F JVCUjÆECTNWvJVKSK $}MN J UÇJ!NCELÉC iD4UTpNB4UÇFJVNO}MPOKMpUTIC1ÆEK"J!GKSRqUT)B $JVGK&p!KFVUÇJ!KMpHUTPOp!CÅLGK&KMWÉJVC hDGP 4PP JVK L H4UÇ=J hCEPOPC Hp7K ÇKMKMBvJ!GK©LGCTJ U3JVNCTL η NOL CEDGFÐK BF!KMpp!NOCELGp $I KMUTF!NOLGR }MCELWGNOJ!NCEL D + γ < ∆ ʱµ NLxÁNOLGWxJ!4UÇJ/N JFKMBGFKMpKMLqJVp,JVKQPNOLG%K HNOWJV>UTp!pC$}&NUÇJ!KM: W HN JV>J!GK;POKjUTWGp&ÊEË4CEF LiDGÁK&F!NO}jUTPBGDGFB CEpKM%p NO4J HNOPPI KSpD }MNOKMLqJHJVCUTp!pDGÁK"JV4UÇJ η N4p ÇKMBJHÆTKMFepÁ UÇPP9}&CEÁB4UTF!K&WvJVCÅJVKSp!DGBKMF}MCELGWDG}&J!NLGR RqUT)B GUTp ;K&PP[UTp©UTPOP9JVKrFKMPOK&ÆÇUTLqJHPK¶ÆEKMP[K&LGKMFRENK&p WGCTJHPOK&ÆTKMPB CTp!NOJ!NCEL $IGNUTpQÆECTPOJ UÇREKM%p 4K&JV}ÇÊ Ê GK>UÇI C1ÆEKrK $BGFKMpp!NCTLGp°}&CELGpJ!NOJ!DJVKSJVKrpKM}&CELGW`ÁUTNLÉFKMpDGPOJ©C JVGNO?p 7CE/F ~ 7KxDLGWGKMFpJVUTLGW LGC GC JVGK}MDGFF!K&Li&J 4D}&JVDGUÇJVNOCELGpNLSJVKLKMNRTICEFNLGRHÁKMp!CTp!}MCTBGN},}&NF!}&DGNOJRTNOÆEKÐFNp!K/J!C½NOLGKMPUTpJ!N} UTLGWeKMPUTpJ!N}S}&CELqJVF!NOIGDJ!NCELGpNOLvJVGK}&DGF!FKMLqJ°NOLÉJVGKr² ¨ ÀeWGK¶ÆN}MKÇÊ K 4LGWrJV4U3JICTJ!rFNREqJ UTLW>PK #J}MDGFF!K&LiJ}MCTLiJ!F!NOIGDJVNOCELGp[4UjÆTKJ!GK7p!UTÁ3K hCTF!ÁlÊjGKÐ}MDGFF!K&LiJ¬´ }MDGFF!K&Li; J 4D}&JVDGUÇJVNOCELGpC ©JVGKeÁKMp!CTp!}MCTBGN}eWGK¶Æ$NO}MKlU KM}¶JJ!GKeWGK&J!KM}¶JVCEF}MDF!F!K&LqJvUÇJÅJ!GKeKMLGK&F!RT }MCEFF!K&p!BCELGWGNOLGR"JVC"J!GKQJVCTJVUTPK&LGKMFRTÂC *J 7CrKMPK&}&J!F!CELGpK $NOJ!NLGRSNO L CEF,K&LqJVKMFNLGER hFCE Á JVKLGCEF!ÁUTP PKMUTW+ÊKMF!K hCEF!K ;KrBGF!C}MK&KMWeJVCÅJVKrp!UTÁKS} 4UTLGREKC ÆÇUTF!NUTIGPOKMpU hCEFQJVKSp!NLREPK²°À>ÈDGL}&JVNOCEL+Ê Ë CTF eV > 0 KMPUTpJ!N}}MDGFF!K&LiJ}&CELqJVFNIGDJ!NCELpxNOL I→ UTF!KBGF!K&p!K&LiJIDJÂJVKvpVUTÁK }&CELqJVFNIGD$´ JVNOCELGp NOL I← ÆÇUTLNp! #JVGKÕCEBGBCEp!N JVKÕNOp JVFDGK,hCEFJVGKÕ}jUTpK:C eV < 0 Ê ³ 4UÇLGRENLRÆÇUTF!NUTIGPK&p NL JVKzNOLGKMPUTpJ!N}É}MCTLiJ!F!NOIGDJVNOCELGpUTLGW5WG%K GLGNLGRJV$ K ÇKMF!LKM?P hDGLG}&J!NCELp KNelDS (ω, eV, D , η) UTLGW ? PSfrag replacements 400 4e+05 eV = 0.1, D = 0.5 eV = 0.3, D = 0.3 eV = 0.2, D = 0.5 eV = 0.5, D = 0.3 3e+05 eV = 0.1, D = 0.3 300 eV = 0.5, D = 0.5 eV = 0.7, D = 0.5 el KN DS eV = 0.2, D = 0.3 2e+05 200 1e+05 100 0 0 -1 -0.5 0 ω 0.5 1 -1 -0.5 0 ω q n( !D%? ' B!D%? '.'(!)!D [email protected] el KN ω DS 2$<; |eV | < D KNinel DS (Ω, eV, D , η) UTpHNOLÉ®°BGBKMLGWN lË hCTF ∆IP AT (eV ) = −CN DS CN DS 2 CN DS − 2 − HNOJ! CN DS = eγ12 γ22 ∆2 /π 2 RK Ê eV > 0 UTLGW eV < 0 0.5 42$$;=6 1 |eV | ≥ D 7KCEIJVUTNL +∞ |Z(ω)|2 + Sexcess(−ω)KNelDS (ω, eV, D , η) 2 ω −∞ Z 2eV |Z(Ω)|2 + Sexcess(−Ω)KNinel dΩ DS (Ω, eV, D , η) 2 Ω −∞ Z ∞ |Z(−Ω)|2 + Sexcess(Ω)KNinel dΩ DS (Ω, eV, D , η) , 2 Ω 2eV Z n; -! dω ?ÊE?TÄ G$ K 4FpJJVK&F!Á NL5», Ê ?$EÊ ?ÇÄ ÂWGKMp}MF!NOI K&pJ!GKzK&PUTp¬JVNO}l}&CELqJVFNIGDJ!NCEL NL2JVKÕ®, }MDGFF!KMLqJMÊ ®°P JVGCEDRE<;K7UTF!K,POKMppNOLiJ!KMFKMpJ!KMWxNLSJVGNOp[}MCELqJ!F!NIDJVNOCEL)7K7}MUTLGLGCÇJNORELGCEFK,NOJNOLrBFVUT}¶JVN}&KÐI K&}jUTDGpK NOJ;}MCELqJVFNIGD$JVKMpQJ!CÂJVGK"J!CTJ UÇP ∆IP AT ÊGGKSKMLqÆNFCELGÁKMLqJ©U KM}¶JVpQJVNp}MDF!F!K&LqJH}MCELqJVFNIGD$JVNCTL)IGD$J UÇJÂJ!GKÉKMLGW2C ©JVKvJVDGLGLKMPNOLGR`BGFC$}&KMpp!KM%p ,JVGK&F!KeNpÂLC:KMLGK&F!RT K $} 4UTLGREKeIK&*J 7KMK&L5JVKÉWGK&ÆN}&K UTLGW JVKWGK¶JVK&}&JVCTFl}&NF}MDGN JjÊQGKp!K&}MCELWJVK&F!Á NOL »Ð Ê ?EÊ ?TÄ WGKMp}MF!NOI K&plJ!GK:JVDGLLGKMPONLGR2C >U ³ C$CTB K&FB4UÇNF hF!CEÁ J!GK;LGCTF!ÁUTPqPKMUTWxJ!C°JVGK7p!DGBKMF}MCELWGDG}&J!CEFÆNUJVK;iD4UTLqJVDÁWGCÇJ!HNOJVxKMLGK&F!RT K$} 4UTLREKTÊ4GKK&PKM}¶JVFCELGp}jUÇLzUÇIGp!CEFIeKMLGK&F!RT hNLÉ}MUTp!KSJVKMNF7JVCTJVUTP9K&LGKMFRTÉNp;p!ÁUTPPOKMF7J!4UTLÉJVK p!DGBKMF}MCELWGDG}&J!CEF;} KMÁN}jUÇP BCTJVK&LqJVNUÇP µS ÐCEFKMÁNOJK&LGKMFRT [email protected] JVGK&NFÐJVCTJVUTP KMLGK&F!RTNp7IGNRTREKMF,JVGUTL ³ CCEB K&FB4UTNF µS ¶ÊGKPUÇpJJVK&F!Á NL», Ê ?$Ê ?TÄ "WGKMp}MFNIKMpJVGKNLqÆEK&F!pKJVDLGLGKMPONLGRlBGFC$}&KMp!p&~ U UTIGpCEF!IGp°KMLGK&F!RTC hF!CEÁ J!GKÂLGK&NREiICEF!NOLGRWK&ÆN}&K 9N JVp½}MCELGp¬JVN JVDGK&LiJ"KMPK&}&J!F!CELGp½JVGK&L`JVDGLGLKMPJ!C JVK LGCEFÁ UTPPKjUÇW+Êq±µLJ!GNpK&ÆTKMLq!J iJVGK©JVCTJVUTP4KMLKMF!RÇC +JVK½CED$JVRECENOLGRKMPOKM}&J!F!CELp7NOpBCEpNOJVN ÆEKÇÊq± CTLJVK }MCELqJ!FVUTF¬[JVGNOpxJ!CTJ UTP,K&LGKMFRTNprLGKMRqU3JVNOÆTK[JVKMLJ!GK ³ CCEBKMF>BGUTNF4UTpxKMÁNOJJVKMWKMLKMF!RÇJ!CzJVK WGK&ÆNO}MKTÊ ±µLÉCEF!WGK&F7J!CÂDGLGWGK&F!p¬J UTLGWeGC JVK"WGK&J!KM}¶JVCEFQ}&NF}MDGN JQU KM}¶JVp;J!GK"IKM4UjÆNCTF;C [J!GK°}&DGF!FKMLqJ NOL JVGKSBF!KMpKMLG}&K>C JVKrK&LqÆ$NOF!CELÁK&LiJ 7KNLqÆEK&pJVNORqUÇJ!K"JVG<K 7KMNRTi?J hDGL}&JVNOCELGp KNelDS (ω, eV, D , η) UTLGW KNinelDS (Ω, eV, D , η) pKMB4UTF!UÇJVK&POTÊ[GK 7KMNOREqJ hDGLG}&J!NCEL KNelDS (ω, eV, D , η) NOpBGPCTJJVK&WNOL ËNRTDGF!K ?$Ê IUTp½; U hDGL}&JVNOCELeC hFKMiDGK&LG}& hCTFQ*J ;CÆTUÇPDGK&p©C JVKxIGNUTpQÆECTPOJ UÇREKrUTLGWe*J ;CÆTUÇPDGK&p C QJ!GKvWGCTJxPK¶ÆEK&P;BCEp!N JVNCTL+Ê[GNp>KMPUÇpJVNO} ωÇKMFLGKMP7Np>p¬ÁÁK&J!F!N}DGLGWKMFÅUÕINUTpÆECEP J UTRTKFK&ÆEK&F!p!UTP ' KNelDS(−eV ) = −KNelDS (eV )(|ÊQË4F!CEÁ J!GK`F!NOREqJvB4UÇLGKMP½C SJVGNOp 4RTDGF!K HGK&F!'K 7K:}MCELGpNWGK&F |eV | ≥ D 47K4LGW JV4UÇJeJ!GKMFKNOpeU pÁ UÇPP"p¬JVKMBUÇJ ω = ∆ − D UTLGWU p4UTF!B BKjU U3J ω = −∆ + D ÊGKB KMU NprUTp¬$ÁÁK&JVFN} UTLGWN JVpGK&NREqJxNprÁÂDG} PUTF!RTKMFSJV4UÇLJVGUÇJrC QJVK ?? 0 80000 80000 eV = −0.5, D = 0.5 eV = −0.3, D = 0.5 eV = 0.1, D = 0.3 eV = 0.1, D = 0.5 -1e+06 eV = 0.3, D = 0.5 eV = −0.3, D = 0.3 60000 inel KN DS 60000 40000 40000 20000 20000 -2e+06 -3e+06 -4e+06 PSfrag replacements eV = 0.5, D = 0.5 eV = 0.3, D = 0.3 -5e+06 0 -1 -0.5 0 0.5 1 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Ω Ω Ω j- n( n!D%? ' D ! % ? ' ( ' ! # ! 1 @ A ! $ 2 < b ; 6 I1#m2$<; inel KN Ω eV < 0 DS E q2$<; eV ≥ - &< *4 )- !E' eV ≥ 0 < eV < D D D pJ!KMB+Ê) GK&L ω < −∆ + D KNelDS } 4UÇLGREKMprp!NORELUTLGWIKM}MCTÁK&prLKMRqUÇJ!NOÆTKTÊ[GKÆECEP J UTRTKÅIGNUTp eV ÁUTNLPOeU KM}¶JVp½JVGKÅUÇÁBGPONOJ!DGWGKC,JVK>BKjUzUTLGW`C,JVGKxpJVK&B:NL KNelDS Ê9KÂPK#J°B4UÇLGKMPC ËNRTDGF!& K ?$Ê IÕWGKMp}MFNIKMp KNelDS HGKML |eV | < D ÊGKvBKjUGK&NREqJÂWKM}MFKjUTpKMpÅiDGN JVK UTp¬JÅUTp>U hDGLG}¶JVNCTLlC eV UTLGWÕN JVpPOC$}MUÇJVNOCELÉNpp[email protected] #J!KMWzUÇJ ω = −∆ + eV Ê4GKrB KMU vNpHp¬$ÁÁK&JVFN}EhCEFHU PUTFREK"IGNUTpMÊ K°J!DGF!LÉLC JVCxJVGK"J!F!DGP BGCTJVC3´bUTpp!Np¬JVK&WÉBGF!C}MK&p!p!K&p HGNO} ÉNLqÆECTPOÆEKSK&NOJ!GKMF;UTIGp!CTF!BJ!NCELvCTF KMÁNpp!NCTL:C ;K&LGKMFRTEÊG; K ÇKMF!LKMP inel eV, D , η) NpSBPCTJJVKMWNOLËNREDGFK ?$Ê DlUÇpxU hDGLG}¶JVNOCEL C hF!K&iDGKMLG}¶ Ω 7HN} É}MCTF!F!K&p!BCELGKWGpQNJ!DSC>(Ω, JVK°JVCTJVUTP9K&LGKMFRTC *J ;CÅK&PKM}¶JVFCELGp FhCEF D > 0 ʱµLvJ!GK PK #J7BGUTLGKM9P eV NOpÐLGKMRqU3JVNOÆTK $UTLGWNLJVGK©}MK&LiJ!KMFQBGUTLGKM9P eV NOpBCEpNOJVN ÆEK©IGDJ eV < UTLGW 4L4UTPOPO JVGKxF!NOREqJ"B4UTLKMPC ËNREDF!K ?$Ê DWGKMp}MF!NOI K&p eV ≥ D Ê KGLGW`JV4U3J HGK&L eV < DD J!GKMFKÂNp°U pJ!KMBUÇJ Ω = D + eV Ê GK&L 7KNOLG}MFKjUTpK eV }MPCTp!KÂJVC D 9J!GKpJ!KMBpJVNOPPWGCEÁNL4UÇJ!KMp KNinelDS IGDJ©JVGK&F!KxNp½Up!ÁUTPOP BKjU eUÇJ Ω = −∆ + eV Ê GKML eV ≥ JVNp hNLqÆEK&FJVK&W ½BKjU eNOp©ÆTKMF p!4UÇF!B+ÊiGNOp;NpK $BGPON}&NOJNLJVGK½F!NOREqJ;B4UÇLGKMP|ÊK°NLqÆTKMFJ!KMWvBKjU .DBHGNO} v4UTp;UxPUÇF!REK©UTÁBGPN JVDGWK Á U TKMpÐN J,LGC WGN }MDGP J/JVCrCEIGpKMFÆTK©J!GKHpJ!KMB+ÊqGK hNLqÆEK&FJVK&W ,B KMU >NOp,PC}jUÇJ!KMWUÇJ Ω = −∆ + D Ê ®°RqUÇN)L eV ÁCTpJ!POÕU 9K&}&J!p"JVGKUTÁBGPONOJVDWGKxC KNinelDS Ê[Ë4CEF D < 0 LGCÇJSp!GC HL +JVGKÅFKMpDGPOJNOp p!NOÁNOPUTF"J!C`JV4UÇJÂC D > 0 HN JV CEBB CEpNOJ!K eV IGDJÂJ!GKÉUTÁBGPONOJVDWGKC ©J!GKvBKjU NpxWGCEDGIGPOKMW }MCEÁB4UTFKMWÉJVCÅJ!4UÇJHC D > 0 HGKML |eV | ≥ |D | Ê ²½CTJ!K½JVGUÇJ;DLGWGKMFpJVUTLGWGNOLGR>JVK°IKM4UjÆNCEFC +J!GK½*J 7C:7KMNOREqJ hDGLG}¶JVNOCELGpQUÇp; U hDLG}&J!NCELC +JVK [email protected] KMFKMLqJÅB4UÇFVUTÁK&J!KMF!p eV UTLW D xC HJVGKÉWK&JVK&}&J!CEFÅNpÂ}&F!DG}&NUTP|ʱ JÅUTPPOC HpxDGpÂJVC:}&CELqJVFCEP;JVK %K KM}¶J°C JVGKrWGK¶Æ$NO}MKÆECEP J UTRTKrINUTp CELeJVGKrWG}r}MDGFF!KMLqJ°C J!GKxWK&JVK&}&J!CEF UTLGWzNOJHNOp©J!GKMF%K hCEFK JVGK ÇK& hCEFK $J!FVUT}¶JVNOLGRÅJVGKLGCENOp!EK heVFCEdÁ J!GKÁKMUTp!DGFKMÁKMLqJ©C JVGNOpWG}x}&DGF!FKMLqJjÊ , ' .: .# ! # 23!, #.!" :#! 7 KLGC¡}MUTP}&DGPUÇJ!K ∆I hF!CEÁ », Ê $? Ê T? Ä HNOJ!J!GKpB K&}&J!FVUTPWGKMLp!NOJµC;K$}MK&p!pÂLCENpKCQU iD4UTLqJVDGÁ BCENLqJ½}MCELqJ UÇ}&J! RENOÆTKML`IqÕ»,Ê ? Ê Hq¿ ¶Ê6Kx}MCELGpNWGK&F½J!GKÅ ®, }&DGF!FKMLqJUTp°UhDGLG}¶JVNCTL C9J!GK©WGK&J!KM}&J!CEFÐÆECEP J UTREK eV hCEFÐp!K¶ÆEKMF!UTP ÆTUÇPDGK&p7C JVGK©WGK¶Æ$NO}MK©ÆECEP J UTREK eV BHN} vUÇF!K©p!GCHLvNOL ËNRTDGF!K,$? Ê G1ÄÊBK GLGWvJV4U3JQJVGK&F!KSUÇF!K°J*7CÂÆÇUTPDGK&pQC eV UÇJ4HGNO} ∆I } 4UTLREKMpWGF!UTpJ!N}MUTPP EÊ P AT d ËNFpJ J!GKMFKÂNp°UvpJ!KMB:PC}jU3JVKMW`UÇJ eV = D UÇLGW`p!K&}MCELW)[UvË UTLGC3´µ[email protected] AT B KMUzUTBGBKjUTFpSUÇJ ?GA eV = 40000 eVd eVd eVd eVd 20000 80000 = 0.8 = 0.6 = 0.4 = 0.2 ∆IP AT ∆IP AT 40000 0 0 -20000 PSfrag replacements -0.5 0 -40000 0.5 0 -0.5 eV 0.5 eV 2 ; 'ZD13Bq:!D%? '8'(! ?$,3%#1%D' G )5$'; O ! '< 3' `A.; -5$;b D = 0.4 ; -!C2(; <∆I 3 P AT E q2(; <3"!E' 0-5(;<5(<;=%q'(! 3 -5 1"G 0.6 eVd GK hLGKMRqU3JVNOÆTK [WGKMFNOÆÇUÇJ!NOÆEKUÇJ = −D pKMKMÁpJ!C°WGN ÆEK&F!REKÇÊ3K;GK&NREqJ C 4ICTJVJVK;BKjU TU LGW:JVKp¬JVK&BNOLG}MFKjUTpKMpNLUÉÁCTLGCTeV JVCTLGCEDGp"ÁUTLGLKMFUTprUhDGLG}¶JVNCTL:C7J!GKFVUÇJ!NCÉCJVKÅWGK&ÆN}&K ÆECEP J UTRTK eVd WGNOÆNOWGKMWIq JVGKzWGCTJ PK&ÆTKMP D Ê3 GKML eVd Npp!ÁUTPP9/JVGKzB KMU NpÁÂDG} GNOREGKMF JV4UÇLJVGKpJ!KMB+ʱµLG}MFKjUTpNLGR eVd J!GKvBKjU hDGF¬JVGK&F>NOLG}MFKjUTpKMp IGDJ>J!GK p¬JVK&B GKMNOREqJÂNL}MF!KMUTp!K&p UTpJ!KM%F 9pJVUTFJ!NLGR hF!CEÁ JVK>JVGFKMpGCEPW:WK&ÆN}&K>ÆECEP J UTRTK eVd = ∆ − D Ê9±µLËNOREDGFK ?$Ê G1Ä 6;K GLGW JV4U3EJ hCEF D = 0.4 UTLGW eVd = 0.8 JVGKBKjU zGKMNOREqJNpSpJ!NPOPGNOREGKMF½JV4UTL:J!GKpJVK&)B [IGD<J HN JV Ô D = 0.6 UÇLGW eVd = 0.6 4JVK>pJ!KMB`IKM}&CEÁKMp°NREGK&F©J!4UTLlJVGKxB KMU Ê KMFK hCTF©pB K&}MN 4}MN Jµ6;K CELGP É}MCELp!NWKMF½J!GK>}MUTp!( K HGKMFK > 0 IGD$J°FKMp!DPOJV>p hCEF D < 0 }jUÇL`I KrCEIJVUTNLKMWÕNOLÕUp!NOÁNOPUTF Á UÇLGLGKM%F GK $BGPOCENOJ!NLGR>KMPK&}&J!F!CELlGCEDPOKSpÁÁK&J!FTÊ ±µLCTF!WGK&FJVEC hDGF¬JVGK&F,DGLGWGK&F!p¬J UTLGWJVKI K&4UjÆ$NOCEF/C ∆IP AT 7KH}MCELGpNWGK&F/JVGK;[email protected] KMFKMLqJÐ}MCELqJ!F!N ´ IGDJ!NCELGp C GJ!GNpBGGCTJ!CÇ´bUÇp!p!NOpJ!KMW>}&DGF!FKMLq!J HGN} ÂUTF!K7p!GC HLÅNL>ËNRTDGF!K ?$Ê GGEÊÇÀ$BKM}MN 4}jUÇPPO !;K;BGPCTJ JVGK©KMPUTpJ!N}©}MDGFF!K&LiJ;F!K&LGCEF!ÁUTPON MK&WIqJVK½K&LiÆNOF!CELGÁKMLqJ UT=p ;K&PPUTpJVGK©F!NOREqJQUTLW PK #J;NOLGKMPUTpJ!N} }MDGFF!K&LiJ!pMÊ K;4LGWJ!4UÇJrJVGKKMPUTpJ!N}B4UTF¬J>NOpxp¬$ÁÁK&JVFN}IK&*J 7KMKML eV B CTp!NOJ!NOÆTKUÇLGWLGK&RqUÇJVN ÆEKÇÊ ± JÂNOpÂUTPOÁCEp¬J>K&iD4UTP7J!C MK&F!' C HGK&L ʱ JÅp!C HpUzpJVK&B2UÇJ Ê GNpÂ}MUTL IK DGLGWGK&F!p¬JVCC$W hF!CTÁJV4K UT}&J,J!4UÇJÐUÇJ,J!|eV GKQJV| <F!KMpGDCEPOW eV = D EKMPOKM}&J!F!CELp/|eV JVDGLG| L=KMP hF!DCTÁJVKLGCEF!ÁUTP ÁK¶J UTPPKMUTWvJVCÂJ!GK"pDGB K&F!}&CELGWGDG}¶JVCEFHBGFKMWGCTÁNOL4UTLqJVP Iq Á U iNOLGRÂF!K&p!CELGUTLqJJVFVUÇLGp!N JVNCTLGp7J!GF!CEDRE JVGKrWGCTJMÊ»ÐPOKM}¶JVF!CTLGp°KMUTp!NOPOvJ!DGLGLGK&P J!CJVKxiD4UTLqJ!DGÁ WGCT!J NOL`UpKMiDGK&LiJ!NUTP ÁUTLGLGK&F½IKM}&CEÁNOLGRU ³ C$CTB K&F½B4UÇNF©NLzJ!GK>pDGBKMF!}&CELGWGD}&JVCTFMÊ[Ë4CEF eV = − JVGKxpVUTÁK>FKjUTpCELGNLR }jUÇL`I KxÁ UÇWGKhCEF NLG}&CEÁNLGRvCEPK&prCTF [KMiDGN ÆÇUTPK&LiJ!PO hCEFKMPOKM}&J!F!CELprK $NOJVDNOLGRÉJVGKp!DB K&F!}MCTLGWGDG}¶JVCEF&~U ³ CCEB K&FB4UTNF NL`J!GKp!DGBKMF}MCELGWDG}&J!CEFxNOpSp!BGPONOJSNOLqJVCÉ*J ;ClKMPOKM}¶JVF!CTLGp )HGNO} J!DGLGLGK&P/JVCÉJVKiD4UÇLiJ!DGÁ WGCTJrUÇLGW JVGK&LJ!CÉJVGKLGCEFÁ UÇP/ÁK¶J UTPPKMUTW+Ê[Î;K&}jUTDp!K C 7J!GKKMLGK&F!RT:}&CELGpKMFÆÇUÇJ!NCEL}MCTLGWGNOJ!NCEL 9NOJ"Np"JVKML LGKM}&KMppVUTF¬ÉJVC4UjÆEK eV < −D Ê DGFLGNLR:LGC J!CJ!GKÉNLKMPUÇpJVNO} }&DGF!FKMLq!J 7Ke}MCELGpNWGK&F>JVKl}&CELqJVFNIGDJ!NCELC H*J ;CKMPOKM}&J!F!CELp I K&NLGR"J!FVUTL/p hKMFF!K&WNOLGKMPUTpJ!N}jUÇPPO< hFCEÁJ!GKLGCEFÁ UÇPGÁK¶J UTP$PKMUTWÅJVCSJVGKpDGBKMF!}&CELGWGD}&JVCTF HGN} Np }jUTPOPK&WJVK IRinel NLvËNREDGFK ?$Ê GGEÊ GK&L eV < D KMPK&}&J!F!CELGp7JVDGLGLKMP J!GF!CTDGRE J!GK"iD4UTLqJVDÁ WGCÇJ JVCJVGKxp!DGBKMF}MCELGWDG}&J!CEF°IqÕUTIGp!CTF!IGNOLGR CEF©KMÁNOJJVNOLGRKMLGK&F!RTeJVCJVGKxKMLqÆNFCELGÁKMLqJjÊ Ô C7K&ÆEK&F[UÇp NLvJ!GKKMPUÇpJVNO}"}jUÇp!K GJ!GKSJVF!UTLG/p hK&4F hF!CEÁ¤JVGKLGCTF!ÁUTP9ÁK&JVUTP9JVCÅJ!GKWGCTJHNp;ÁCEFEK UjÆECEF!UTIGPEK HKML eV > D HGNO} zK $BGPUTNOLGpQJVKrBF!KMpKMLG}&KxC J!GKpJVK&BÕNOL IRinel UÇJHJVGNOpIGNUÇpQÆECEP J UTREKÇÊ −D Ê ?E¿ 40000 2e+05 20000 I el inel IR ILinel 20000 10000 0 0 -20000 -10000 1e+05 0 PSfrag replacements -1e+05 -20000 0 eV -0.5 0.5 -40000 0 eV -0.5 0.5 -0.5 0 eV 0.5 ; O? q<3.b; O? q1'?bG? ' #' 6I) !D%? 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±µLÉJVGNOp½UÇBGB K&LGWGN 7;Kr}MCELp!NWKMFJVGKrUjÆEK&FVUTRTKrCK BCELGK&LqJVp°CJ!GKBG[email protected] KMFKMLqJHJVNÁKTÊ Ë CTPPC HNOLGRJ!GKÕJ!GKMCEFKMÁ NL iD4UTLqJVDÁ ÁKM} 4UTLN},hCEFCEBKMFVU3JVCEFp;N C ≡ [A, B] pVUÇJ!Np4KMp [A, C] = [B, C] = 0 JVKML ?$Ê ?Eà eA eB = eA+B eC/2 , AEÄ [email protected] A NpPONLGKMUTFQNOLzÎ7CEp!K}MFKjUÇJ!NCELlUTLWÕUTLLGNGNOPUÇJ!NCELCEBKMF!UÇJVCEFp7K4UjÆEK Ë4F!CEÁ J!GKMpKxJ*7CvBGFCEB K&FJ!NKMp% [email protected] CEIJVUTNL heA i = ehA 2 i/2 NOp°U´µLiDGÁÂIKMFHJVGK&L ?$Ê ?H . C = hCi = hABi − hBAi A2 + B 2 A B he e i = exp AB + . 2 C 67K>KMUTp!NOPO ?$Ê ? À$NÁNPUTFhCEFHU hCEDF¬´µBCENOLiJQ}MCEFF!KMPUÇJVNOCEL hDGL}&JVNOCEL) ?$Ê ? ? heA1 eA2 eA3 eA4 i = heA1 +A2 eA3 +A4 ie[A1 ,A2 ]/2 e[A3 ,A4 ]/2 . $À NL}MK Ai + Aj i, j = 1, 2, 3, 4NOp©UTPpCPONLGKMUTFQNOLzÎ7CEp!K}MFKjUÇJ!NCELlUTLWÕUTLLGNGNOPUÇJ!NCELCEBKMFVU3JVCEFp ;K}MUTLzBKMFhCEF!Á¤JVGKrUjÆEK&FVUTRTKxC JVKrK $B CELKMLqJVp½UTp he A1 +A2 A3 +A4 e (A1 + A2 )2 + (A3 + A4 )2 i = exp (A1 + A2 )(A3 + A4 ) + 2 +# "* X 1X 2 Ai . Ai Aj + = exp 2 i i<j ?$Ê ?GA K>UTBGBGP ÉJVGK&p!K>FKMpDGPOJ!p½JVCCEDGF©}jUTPO}MDGPUÇJVNOCELGp4HN JVzJ!GK>B4UTp!K φ(t) = e R t dt0V (t0) V (t) ÆECEP J UTRTK,CJ!GKÐÁK&p!CEp}MCEBGNO}ÐWGK&ÆN}&K=HKMF!K37K=7CEDGPOWxPON TK/J!C½ÁKjUÇp!DGFKLCENpKTÊ1± GL−∞ CENpKNOp¸rUTDp!p!NUTL) Ê K"UÇPp!( C 4LGW J!4UÇJ [φ(t), φ(t0)] = 0 ÊiGKMpKSBGF!CTB K&FJVNOKMpC φ(t) pVUÇJ!Np # φ(t) NOp7PONLGKMUTFÐNLI CTp!CELGp&F JVGK}&CELGWGN JVNCTLGp;JVCUTBBGPOv»Ðip&Ê ?$Ê ? QUTLGW ?EÊ ?GA ¶ÊG±µLGWKMKMW hφ2(t)i = hφ2(0)i 7;Kr4UjÆEK ?$Ê ?E¿ heiφ(t) e−iφ(0) i = eJ(t) , UTLGW ?$Ê ?I heiφ(t3) eiφ(t2) e−iφ(t1) e−iφ(0) i = exp[J(t3 ) + J(t2 ) + J(t3 − t1 ) + J(t2 − t1 ) − J(t3 − t2 ) − J(t1 )] , HGKMFK<;KrF!K&}jUTPOP+GKMFK , ! J(t) = h[φ(t) − φ(0)]φ(0)i Ê ±µLJ!GNprUTBB K&LGWGN )7KWK%4LGK ?$Ê Ã I 7UTp Z Ψ0←= el inel Ψ0← K2e← (ω, eV, η) 9UTLGW K2e← (Ω, eV, η) −∆−eV ∆+eV p p dδ (δ − eV )2 − ∆2 (δ + eV )2 − ∆2 Z 1 δ + iη NOL»Ðip&Ê ?$Ê ÃT¿ "UTLGW 1 1 − δ + iη δ − iη , ?$Ê ?D −∆−eVp p (δ − eV )2 − ∆2 (δ + eV )2 − ∆2 ∆+eV 1 1 1 1 1 1 1 + − − + , × 2 δ + iη δ + ω + iη δ + iη δ − iη δ + iη δ − ω + iη δ + ω − iη el K2e← (ω, eV, η) = dδ ?$Ê ATÄ × Z Ω−2∆−2eV p p (Ω + δ − 2eV )2 − 4∆2 (Ω − δ − 2eV )2 − 4∆2 2∆+2eV −Ω ! ! 1 1 1 1 1 1 − Ω−δ + Ω+δ − Ω+δ − Ω−δ . $Ê Ω+δ Ω−δ + iη − iη − iη − iη + iη + iη 2 2 2 2 2 2 inel K2e← (Ω, eV, η) = dδ ? <AG AG , ! ±µLÉJVGNOp½UÇBGB K&LGWGN 7;KrF!K&}jUTPOP9JVGKWGK4LGN JVNCTLÉC Ó K&PWp!`¸FKMK&L)1 p hDGL}&JVNOCELGp' DD ( Ê ËNFpJ! 7KÉWGK%GLGK JVKÉUTLGCEÁUTPOCEDGpx¸FKMK&L)1 phDGLG}&J!NCELWGK&p!}&F!NINLGRÕJ!GKvB4UÇNF!NOLGRlCHKMPOKM}&J!F!CELp HNOJ!lCTBGB CTp!NOJ!KSp!BGNOLGpHNLvJ!GKp!DGBKMF}MCELGWDG}&J!CEF Fσ (q, t − t0 ) ≡ −hTK c−q,−σ (t)cq,σ (t0 )i , Fσ∗ (q, t − t0 ) ≡ hTK c†q,σ (t)c†−q,−σ (t0 )i . ± ©I CTJ! t UTLW t0 UTFKvNL J!GKÉDGBGBKMFÅIGF!UTLG} )ÐUÇLGW t > t0 CEFÂICTJV t UTLGW t0 UTF!K NL JVGKÉPOC;K&F IGFVUÇLG} ),UTLW t0 > t JVGK&L Fσ (q, t+ − t0+ ) = Fσ∗(q, t− − t0−) = pREL (σ)uq vq e−iE (t−t ) ÊGK&p!K ¸FKMK&)L 1 >p hDGLG}¶JVNCTLGp°K&LiJ!KMF©JVGKÂWKMp!}&F!NOBJVNOCELÕCJVGKx®°LGWGFKMK¶Æ`BGFC$}&KMp!p&Ê9± 7KÂ}&CELGp!NOWGKMF©JVGKxp!NOLGREPK iD4UTp!NOB4UTF¬JVN}&PK>J!DGLGLGK&PNOLGRÉNL:JVKpDGBKMF!}&CELGWGD}&JVCTF ;KDGpKJ!GK}MCELqÆTKMLqJVNOCEL4UTPÐWGK4LGNOJ!NCEL:C;JVK ¸FKMK&)L 1 p hDGL}&JVNOCELlUTp hCEFLGCEFÁ UÇP9ÁK&J UÇPpMÊ À$KM}&CELGWGP 7KrWK%4LGKJVK>¸SF!K&KML)1 p?hDGLG}&J!NCELeC JVKrCTLGKSPK¶ÆEKMP = ¨ 0 q ?Ê Aqà GDσ (t − t0 ) ≡ hTK cDσ (t)c†Dσ (t0 )i . À$NÁ[email protected] 4}MUÇJVNOCELGpC$}&}MDGFI K&}jUTDGpKJVGKiD4UTLqJ!DGÁ¦WGCTJ/4UTpÐUSp!NOLGREPOrC}M}&DGBGNK&WPK&ÆTKMPHN JVÅK&LGKMFRT Ê GK?4F!p¬JQKMPOKM}¶JVF!CTL Np,J!FVUTLGphKMFF!KMWJVCxJ!GK©PKjUÇWIK%hCEFK©J!GK°pKM}&CELGWvGCTBGp;CELJVK°iD4UTLqJVDÁ¤WCTJ;DpC JV4U3JQNL CEDGF 7CEF/.7;KSCELPO}MCELp!NWKMF;J!GK = ¨ ¸FKMK&L)1 p4hDLG}&J!NCEL HKMF!KSICTJ! JVNOÁK©iD4UTLqJVN JVNK&p t UTLGW t0 UÇF!K"NOL JVGKSDBGB K&FQCEFQPOC ;K&FQIGFVUÇLG} ) 4UTLWvJVGK¸SF!K&KML)1 p4hDGLG}¶JVNOCELvÆÇUTPDGK&pQCELGP ;HGKML t > t0 @N t t0 NLÉJVGKrDGBGBKMFHIGF!UTLG} GtDσ (t − t0) = e−i (t−t ) UTLW t0 > t N t t0 NLeJVKrPOC ;K&F©IFVUTLG} JVGK&L Gt̃Dσ (t − t0 ) = e−i (t−t ) Ê GKx¸FKMKML 1 ?p hDGLG}¶JVNCTLlNOLÉJVGKSLGCTF!ÁUTP+ÁK&J UÇP+PKMUTWeF!KjUÇWGp ?Ê AH GLσ (k, t − t0 ) ≡ hTK ckσ (t)c†kσ (t0 )i . ±µL`CEDG>F ;CTF 67KÂ}MCELp!NWKMF°J!GK>}MUTp!K&Ep HGKMFK>*J ;CÉK&PK&}&JVFCELGp°J!DGLGLGK&PNOLG R hFCEÁ CEF½J!C JVGK>LGCEF!ÁUTP ÁK¶J UTP POKjUT)W ipC>JV4U3J 7K"CELGP }MCELGpNWGK&FQLGCEFÁ UÇP4ÁK¶J UTP9¸FKMK&)L 1 =p hDGLG}¶JVNCTLGp HGK&F!K t UÇLGW t0 UTFK°NOL JVGK©WGN 9K&F!KMLqJ7IGFVUÇLG} GKMp&ÊGË4CEFÐJVGK©}jUÇp!K°C [KMPOKM}&J!F!CELp7JVDLGLGKMPONLGR hF!CTÁ¦JVGK©p!DGBKMF}MCELWGDG}&J!CEF;J!CxJVK LGCEFÁ UTPÁK¶J UT9P );KDGp!KJVGKREF!KMUÇJVK&F¸SF!K&KM)L 1 p hDGLG}&J!NCEL G>Lσ (k, t − t0) = e−i( −µ )(t−t ) )HNOJ! R hF!CEÁ JVGKLCEF!ÁUTPÁK&JVUTP$J!CSJVGKHp!DB K&F!}MCTLGWGDG}¶JVCE%F k > µL ÊqË4CEFJ!GK}jUÇp!KHC KMPK&}&J!F!CELGp/J!DGLGLGK&PNOLGE ;K;DGp!K;J!GKQPOKMp!pKMFиSF!K&KM)L 1 p hDGLG}¶JVNOCEL G<Lσ (k, t−t0) = −hc†kσ (t0 )ckσ (t)i = −e−i( −µ )(t−t ) HNOJ! k ≤ µ L Ê D D , ! 0 0 k L 0 k L 0 µ± LvJVGNOpQB4UTF¬J!F;KSBGFKMpKMLqJHJVGKSWGK&LGCEÁNL4U3JVCEFBGF!CWGDG}¶JVp4HGNO} lUTBGBKjUTFQNOL JVK"JVDGLGLKMPNOLGRÂ}&DGF!FKMLqJ JVGFCEDGRE JVGK`²"À:ÈDGL}&JVNOCEL UTp U ³ CCEBKMF BGUTNF&Ê;ÀDG} WGKMLCEÁNOL4UÇJ!CEF!p}MCEÁK$hFCEÁ JVGKzKMLGK&F!RT WGKMLCEÁNOL4UÇJ!CEF!pQC J!GKSJVFVUÇLGp!N JVNCTLÉCEB K&FVUÇJ!CEF T ~ 0 −1 (D← ) = el −1 (D← ) 1 0 E + + iη 1 1 + E − − iη E + − iη , 1 1 1 = + E 0 + + ω + iη E − − iη E + − iη 1 1 1 + + 0 , E + + iη E − + ω − iη E + + ω − iη Aqà ?$Ê<AG ?$Ê<A ? inel −1 (D← ) 1 1 = + E 0 − 0 + iη E 0 + + iη 1 1 1 1 + + + . × E + 0 − iη E + − iη E − − iη E − 0 − iη K} 4UTLREKSÆTUÇF!NUÇIGPK&pQNL inel inel I→ I← UTp KSWK%4LGK ?$Ê<A A Ω = + 0 , δ = − 0 . ?$Ê<Aq¿ π + 2 arcsin( x+iη ) 1 1 ∆ = p , dE √ χ(x, η) ≡ 2 2 2 E − ∆ E − x − iη 2 ∆ − (x + iη)2 ∆ Z ∞ JVGK&L)7KrWK%4LGKJVK 7KMNOREqJ?hDGLG}&J!NCELpHUTp el K (ω, eV, η) = K inel Z eV d {[2χ(−, −η) + χ(− − ω, −η)] [χ(, η) + χ(−, η)] −eV + χ(−, −η) [χ( − ω, η) + χ(− − ω, η)]} , ?$Ê<AI Ω−2eV Ω−δ Ω+δ dδ χ( , −η) + χ(− , −η) (Ω, eV, η) = 2 2 2eV −Ω Ω−δ Ω+δ Ω+δ Ω−δ × χ(− , η) + χ(− , η) + χ( , η) + χ( , η) . 2 2 2 2 Z ?$Ê<AD , ! ±µLJVGNOpvUTBB K&LGWGN =;K:}&CEÁBDJVKÕJ!GK`BGFC$WDG}&JÉCSJVDGLGLKMPNOLGR Ô UTÁNOPOJ!CELGNUÇL CTB K&FVUÇJ!CEF!pvNOL JVK NOLGNOJ!NUTP ,RTF!CEDGLWzp¬J UÇJ!KHGNO} zNOpp!GCHLÕNOLe»,Ê ?$Ê< A ¶Ê hHT† 1 (t − t01 − t02 − t03 )HT† 2 (t − t02 − t03 )HT† 1 (t − t03 )HT† 2 (t) ×HT 2 (t1 + t2 + t3 )HT 1 (t1 + t2 )HT 2 (t1 )HT 1 (0)i X = T14 T24 hc†q1 σ1 (t − t01 − t02 − t03 )c†q2 σ3 (t − t03 )cq3 σ6 (t1 + t2 )cq4 σ8 (0)i k1 ..k4 ,q1 ..q4 ,σ1 ..σ8 ×hck1 σ2 (t − t02 − t03 )ck2 σ4 (t)c†k3 σ5 (t1 + t2 + t3 )c†k4 σ7 (t1 )i ×hcDσ1 (t − t01 − t02 − t03 )c†Dσ2 (t − t02 − t03 )cDσ3 (t − t03 )c†Dσ4 (t) ×cDσ5 (t1 + t2 + t3 )c†Dσ6 (t1 + t2 )cDσ7 (t1 )c†Dσ8 (0)i 0 0 0 0 ×heiφ(t−t1 −t2 −t3 ) eiφ(t−t3 ) e−iφ(t1 +t2 ) e−iφ(0) i . ?$Ê ¿TÄ $À NÁBGPN 4}jU3JVNCTLGpC}M}&DGFÐI K&}jUTDp!KJVGKHiD4UTLqJ!DGÁ¡WGCTJÐ4UÇp7USpNLGREP >C}M}&DGBGNOKMWPK&ÆTKMPHN JVKMLGK&F!RT D Ê4®½p½NOLz¯HK%bÊ ' DEÄ (94JVGK<4FpJ½KMPOKM}&J!F!CELlNpJ!FVUTLGphKMFF!KMWlJVCJVGKPOKjUTWlI KhCEF!KSJ!GKxpKM}&CELGWÕCEBGp©CEL JVGKiD4UTLqJ!DGÁ WGCTJMÊ4GKMF%K hCEFK hcDσ1 (t − t01 − t02 − t03 )c†Dσ2 (t − t02 − t03 )cDσ3 (t − t03 )c†Dσ4 (t) ×cDσ5 (t1 + t2 + t3 )c†Dσ6 (t1 + t2 )cDσ7 (t1 )c†Dσ8 (0)i = hcDσ1 (t − t01 − t02 − t03 )c†Dσ2 (t − t02 − t03 )ihcDσ3 (t − t03 ))c†Dσ4 (t)i ×hcDσ5 (t1 + t2 + t3 )c†Dσ6 (t1 + t2 )ihcDσ7 (t1 )c†Dσ8 (0)i = Gt̃Dσ1 (−t01 )δσ1 σ2 Gt̃Dσ3 (−t03 )δσ3 σ4 GtDσ5 (t3 )δσ5 σ6 GtDσ7 (t1 )δσ7 σ8 , AH ?$Ê ¿7G ¨ KMp}MF!NOIGNLRJ!GK®°LGWF!KMK¶ÆeBGF!C}MK&p!p%67KxUTp!pDGÁK X q1 ..q4 = X q1 ..q4 = X q1 ,q4 hc†q1 σ1 (t − t01 − t02 − t03 )c†q2 σ3 (t − t03 )cq3 σ6 (t1 + t2 )cq4 σ8 (0)i hc†q1 σ1 (t − t01 − t02 − t03 )c†q2 σ3 (t − t03 )ihcq3 σ6 (t1 + t2 )cq4 σ8 (0)i Fσ∗1 (q1 , −t01 − t02 )δσ3 ,−σ1 Fσ8 (q4 , t1 + t2 )δσ6 ,−σ8 , Ë CTF°JVGKÅ}&CEF!FKMPUÇJVNOCEL`CCEBKMF!UÇJVCEFp"NL:LCEF!ÁUTPÁK&JVUTPPOKjUTW9DGpNLGRÉJVK CEIJVUTNL X k1 ..k4 = X k1 ..k4 ?$Ê ¿Eà N}61 p"J!GKMCTF!KMÁC7K hck1 σ2 (t − t02 − t03 )ck2 σ4 (t)c†k3 σ5 (t1 + t2 + t3 )c†k4 σ7 (t1 )i [−hck1 σ2 (t − t02 − t03 )c†k3 σ5 (t1 + t2 + t3 )ihck2 σ4 (t)c†k4 σ7 (t1 )i +hck1 σ2 (t − t02 − t03 )c†k4 σ7 (t1 )ihck2 σ4 (t)c†k3 σ5 (t1 + t2 + t3 )i] 0 0 > = −G> Lσ2 (t − t2 − t3 − t1 − t2 − t3 )δσ2 ,σ5 GLσ4 (t − t1 )δσ4 ,σ7 0 0 > +G> Lσ2 (t − t2 − t3 − t1 )δσ2 ,σ7 GLσ4 (t − t1 − t2 − t3 )δσ4 ,σ5 , ?$Ê ¿H ³ CELG}MK&F!LGNOLGRvJVKÅBG4UTpK 4DG}&J!D4UÇJ!NCELGp% J!GK:hCEDGFb´µBCENLqJ°}MCEFF!KMPUÇJVCTF6HGN} :NOp"NÁBGPNO}MN JHNL`JVK K$BGFKMp!pNCELCÐJVKÂJVDGLGLKMPNOLGRÉ}MDF!F!K&LqJxNOp HF!N J!J!KMLUTpSUvJVNOÁKxCEF!WKMF!K&WBF!CWGDG}&JMÊÌ"LG}MKÅCTF!WGK&F!KMW JVGKÅBF!CWGDG}&JrC JVKK $B CELKMLqJVNUTP,RENOÆTKMpSJVKK $B CELKMLqJVNUTP/CJVGKpDGÁ C;UTPPB4UTNFNLGRTp"I K¶J*;K&KML BG4UTpKÅCEBKMFVU3JVCEFpSUTp"pGC HLNOL`®°BGBKMLWGN `®>Ê9®½pUvF!K&p!DGP !J HN JV`J!GKÅWG%K GLGNOJ!NCEL`C ;», Ê [email protected]Ê [email protected]? hF!CEÁ »,Ê ?$Ê ?I 7;KrREK&J 0 4 0 0 0 heiφ(t−t1 −t2 −t3 ) eiφ(t−t3 ) e−iφ(t1 +t2 ) e−iφ(0) i 0 0 0 0 0 0 0 0 eJ(t−t1 −t2 −t3 )+J(t−t3 )+J(t−t1 −t2 −t3 −t1 −t2 )+J(t−t3 −t1 −t2 ) = . 0 0 eJ(−t1 −t2 )+J(t1 +t2 ) ?$Ê ¿ , ! ±µL JVGNOpÅUTBGBKMLGWN 7KlBF!KMpKMLqJJVGKÉWKMLGCEÁNLGUÇJVCEFxBGF!CWGDG}¶JVp HGNO} UÇBGB KMUTFÂNOLJVGKvJ!DGLGLGK&PNLR M} DGFF!K&LiJHJ!GF!CEDRElJ!GKr² ¨ ÀxÈDGLG}¶JVNOCEL+Ê D 0 NOpQJVGKCEFNRENOL4UTPWGKMLCEÁNOL4UÇJ!CEF HGNO} zNOpLGCTJ©U KM}¶JVKMWlIqÉJVGKK&LiÆNOF!CELGÁKMLqJ 0 −1 (D← ) = UTLGW el D← 1 ++ + D + iη)( + D + iη)(E + D − iη) 1 1 × , + (− + D − iη)(E − − iη) ( + D − iη)(E + − iη) (E 0 iη)(E 0 ?$Ê ¿ ? NOpQJVGKWGK&LGCEÁNL4U3JVCEFQBF!CWGDG}&J½U KM}¶JVK&WzIqÉJVKrK&LqÆ$NOF!CELÁK&LiJ 1 el −1 (D← ) = 0 0 (E + + ω + iη)(E + D + ω + iη)( + D + iη)(E + D − iη) 1 1 + × (− + D − iη)(E − − iη) ( + D − iη)(E + − iη) 1 + 0 0 (E + + iη)(E + D + iη)( + D + iη)(E + D + ω − iη) 1 1 × + , (− + D − iη)(E − + ω − iη) ( + D − iη)(E + + ω − iη) AG ?$Ê ¿GA HGKMFK Dinel NOprJ!GK WKMLGCEÁNLGUÇJVCEFSBF!CWGDG}&JÂUÇJ!J!F!NOIGDJVK&WJ!CzJVK NLKMPUÇpJVNO}}MDGFF!KMLqJ |U KM}¶JVKMWIi KMLqÆNFCELGÁKMLqJQUTLWzNOpWGK%4LKMW`UÇp 1 1 = + 0 0 0 0 0 (E + D − − + iη)(E − + iη) (E + D + iη)(E 0 + + iη) 1 1 × 0 0 (D − + iη)(D − − iη) (E + D − − − iη)(E − − iη) 1 1 + + 0 0 (E + D − iη)(E + − iη) (D − + iη)(D − 0 − iη) 1 1 + × . (E + D − − 0 − iη)(E − 0 − iη) (E + D − iη)(E + − iη) inel −1 (D← ) KWGK4LGK ± x1 6 x GJ!GKML = Z ∞ Π(x1 , x2 , η) = dE √ ∆ ?$Ê ¿E¿ ?$Ê ¿I 1 1 . E 2 − ∆2 (E − x1 − iη)(E − x2 − iη) 2 1 Π(x1 , x2 , η) = 2(x1 − x2 ) CEFKMPOp!K Π(x1 , x2 , η) = GKML;KrWGK%GLGK +iη +iη π + 2 arcsin( x1∆ ) π + 2 arcsin( x2∆ ) p − p ∆2 − (x1 + iη)2 ∆2 − (x2 + iη)2 ! , +iη )) (x1 + iη)(π + 2 arcsin( x1∆ 1 + . 2(∆2 − (x1 + iη)2 )3/2 ∆2 − (x1 + iη)2 Ψ0← (, D , η) Z ∞ ∞ = dE dE 0 √ 1 1 p 0 ∆ ∆ E 2 − ∆ 2 E 0 2 − ∆ 2 D← Π(, −D , η) Π(−, −D , η) + = Π(−, −D , −η) , ( + D + iη)(− + D − iη) ( + D + iη)( + D − iη) Z ?$Ê ¿D Ψel (, , ω, η) Z ∞← Z D∞ = dE dE 0 √ 1 1 p el ∆ ∆ E 2 − ∆ 2 E 0 2 − ∆ 2 D← Π(− − ω, −D − ω, −η)Π(, −D , η) + Π( − ω, −D − ω, η)Π(−, −D , −η) = ( + D + iη)(− + D − iη) Π(− − ω, −D − ω, −η)Π(−, −D , η) + Π(− − ω, −D − ω, η)Π(−, −D , −η) , + ( + D + iη)( + D − iη) ?$Ê IEÄ Ψinel (, 0 , , η) Z ∞← Z ∞ D 1 1 p = dE dE 0 √ inel ∆ ∆ E 2 − ∆ 2 E 0 2 − ∆ 2 D← = (Π( + 0 − D , 0 , −η) + Π(−D , −, −η)) Π( + 0 − D , , η) + Π(−D , −0 , η) Π( + 0 − D , 0 , η) + Π(−D , −, η) + . × (D − 0 + iη)(D − − iη) (D − 0 + iη)(D − 0 − iη) ?$Ê IG A? À$NL}MK 0 = D←0 (−) UTLGW D→el () = D←el (−) Ψ0→() = Ψ0←(−) UTLW Ψel→() = Ψel←(−) Ê Ô C7K&ÆEDK&F→ () NLA UT}&JN 7Kx} GUTLGREKrJVGKxL4UTÁKrCÆÇUTF!NUTIGPOK 0 J!GKMLÕ} 4UTLGREKÆÇUTF!NUTIGPOK 0 ;K;HNPOPCEI$J UTNOLJ!GKpVUTÁK hCEF!Á>DGPU hCEFICTJV}MUTp!K&p eV >→0 UÇLGW eV < 0 Ê À$NOLG}MK UTLGW =0 − UTFK NLGWKMBKMLGWGK&LqJVPOÂKMiDGNOÆÇUTPOKMLqJ!TNOJÐNp,K¶ÆNWGK&LiJÐJ!4UÇJ Ψinel Ê Ô K&F!KjU #JVKMF% 0 inel 0 → (, , D , η) = Ψ← (, , D , η) ;KLGK&REPK&}&JHJVK ← CEF → NLWGK eNOLÉJVGK&p!<K hDGLG}¶JVNCTLGpMÊ ± eV > 0 GJVGKrKMPUTpJ!N}S}&DGF!FKMLqJ"}&CELqJVF!NOIGDJ!NCELGpHNOL I→ K $Np¬J½IDJ½J!GKrKMPUÇpJVNO}S}MDGFF!K&LiJ°}MCELqJ!F!N ´ IGDJ!NCELGpNOL I← ÆTUÇLGNp hNLe}MCTLiJ!FVUTp¬JHJVCÅJVGK}MUTp!KrC eV < 0 Ê K} 4UTLREKSÆTUÇF!NUÇIGPK&pQNLeNLKMPUÇpJVNO}"}MCTLiJ!F!NOIGDJVNOCELGpHUTp UTLGWeWGK4LGK Ω = + 0 , δ = − 0 , Z KNelDS (ω, eV, D , η) = HNOJ! Z inel KN DS (Ω, eV, D , η) = eV dΨel.tot (, D , η) , −eV Ω−2eV dδΨinel ( 2eV −Ω Ω+δ Ω−δ , , D , η) , 2 2 Ψel.tot (, D , ω, η) = 2Ψ0 (, D , η) + Ψel (, D , ω, η) Ê AA ?Ê Iqà ?Ê IH Aq¿ $ 4 L *( . . 7 GK Ô UTPOPGK% KM}¶J!WGNp}MC1ÆEK&F!KMW Ii»,WHNOL Ô UTPP4NOLCG!Iq¿Dq4UÇBGB K&LGp3HGKMLÉUÇLK&PKM}¶JVFN}H}MDGFF!K&LiJ 4CHp JVGFCEDGREeUÂ}&CELGWGD}&JVNOLGRBGPU3JVK°NOLeUÅÁ UTRTLGK&J!N}"GKMPWÉBKMFB K&LGWGNO}MDGPUTF;JVCÂJ!GKSBGPUTLGKTÊ$GKSÁUTRELGK¶JVN} 4KMPOWK $KMF¬JVpÂUlÍ[CEF!K&LqJ ;hCEF!}&KCTLJ!GKÁC1ÆNLGRe} 4UÇF!REK}jUTFF!NOKMF!p<HGNO} J!KMLGWGpJVCeBGDp!J!GKMÁ J!C CELGK½p!NWK"C [JVK°}MCTLGWGDG}¶JVCEF&Ê®¼IGDNPWDGB C } GUTF!REK"UÇJ;J!GK"p!NOWGKMp;C [JVGK°}MCELGWDG}&J!CEF!p HNOPP I4UTPUTLG}&K JVGNOp ÁUTRELGK¶JVNO}zNOL 4DGK&LG}MK ;BGFC$WGD}MNLR UÁKjUTpDGFVUTIPKzÆTCEPOJVUTREK VH I K¶*J ;K&KML *J ;C2pNWGK&pvC SJVK }MCELWGDG}&J!CEF hp!K&KvËNOREDGF!K AÊ G Ê K Ô UTPOP,F!K&p!Np¬JVN Æ$N JµNprBGF!CTB CEF¬JVNOCEL4UTPJ!CÕJVK UTÁBGPONOJVDWGKCQJVK Á UÇRELGK&J!N} GKMPW9Ê !"- 23 "! 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JVGK 4KMPOW+Ê ± J>ÆÇUTF!NOKMWNLUzpJVK&BHNp!K UTp!GNOCEL hp!K&KÉËNRTDGF!K A Ê Ã ¶Ê± J:;UTpÅUÇPp!CChCEDGLGWJV4UÇJHGKMFK 4 JVGK Ô UTPP,FKMp!NOpJVUTLG}MK;;UTp(4UÇJ![JVKPOCELGREN JVDGWNL4UTPF!K&p!Np¬J UTL}MKWGNp!UTBGBKjUTFKMW+ÊGK;4KMPOWU3JHGNO} JVGK°BGPUÇJ!KjUTDpQUTBGBKjUÇF!KMWCEF=HGKMFK"JVGK"POCELGREN JVDGWNL4UTP4FKMpNpJVUTLG}&K°ÆÇUTLGNpGKMW7;UTpNLGWKMBKMLGWGK&LqJHC JVGKÁ U3JVKMFNUT9P +JVK&ÁBKMF!UÇJVDGFK CEF>CÇJVGK&F>ÆÇUTF!NUTIGPK&pxC JVGKvK $B K&F!NÁKMLqJ IGD$JÅCELGP WKMBKMLGWGK&W2CEL UÕ}MCTÁÂIGNLGUÇJVNOCELC ?hDGLW4UTÁKMLqJ UTP7}MCELpJ UÇLiJ!p h/e2 Ê/GNOpÂBGGK&LGCEÁKMLGCTL }jUTL IKvDGLGWKMF!p¬JVCC$W2NOL JVK&F!ÁpC J!GKrÍUTLGW4UTDlPK¶ÆEK&P4p hCEFÁK&WlNOLlUÁUTRELK&JVNO} 4K&PW+Ê GK Ô UTÁNPOJ!CELGNUTL#hCEFeU2B4UTF¬JVNO}MPKÕp!DGIÈK&}&J!KMW J!C5U2ÁUTRELGK¶JVN}'GKMPW BKMFB K&LGWGN}&DGPUÇFvJVC NOJVp WGNFKM}¶JVNCTLÉC ÁCTJ!NCELv}MUTLlI <K HF!N J!J!KMLeWGC HL`UTp H= HNOJ!eJ!GKSÆEK&}&JVCTF©BCTJ!KMLqJVNUTP N X pi − eA(ri )2 2m i A(r) , A$Ê@G } CEp!K&LÕNLÉJ!GK<hCEPPOCHNOLGR>REUTDGREK 1 A(ri ) = B(yi , −xi ) , B = B ẑ . 2 GKKMLGK&F!RTeK&NREK&LiÆÇUTPODGKMpCJ!GKrUTI C1ÆTK Ô UTÁNOPOJ!CELGNUÇL UTFK En,ky = (n + 1/2)ωc , AI AÊ Ã A$Ê H _ _ _ _ _ _ + + _ B I VH + + + + + m <;=;Q 1% 'j1 q S; 1%-E <; h 4 1 G #1%D13HD' .?CD5(`0/* %? ; 3 $, #' - 'ZF !E'16? < ;=;5('; O4 2( 'I3113oG% 1 4 '>% 3% '(! ?$,1'3%D'` HNOJ! ωc = eB/mc Np½J!GK>}¶}MPCÇJVF!CTL$hF!K&iDGKMLG}¶ n = 0, 1, 2, ... Ê »ÐUT} zÆTUÇPDGKxC n }&CEF!FKMpB CELWGp JVCÕUÕÍUTLGW4UÇDPK¶ÆEK&P ÊÀ$NL}MKJ!GKÍ UÇLGW4UTDPOK&ÆTKMPprWGC`LCTJ>WKMBKMLGWCTLJVGKiD4UTLqJVDÁ·LiDGÁÂIKMF ky JVGK¶2UTFKlNREGP WGK&REKMLKMFVU3JVKTÊÐGNOpWGKMREK&LGKMF!UT}&ÐWGK%4LKMWUÇpJ!GKlLDÁÂI K&FC°p¬J UÇJ!KMpB K&FDGLGNOJ UTF!KMU 9ÁUjÕIKÅiD4UTLqJVN 4KMWIq`J!GKÅF!K&PUÇJ!NCEL Ê+± JNp"Dp!%K hDP/JVCÉWGK&p!}MFNIKÂJVGKÅNOLqJVKMRTKMF iD4UTLqJVDGÁ Ô UTPOP+K% KM}¶J h±'= Ô » QIiÉJ!GK<4PPONLGρR(B UT=}&J!CEeB/hc F ν = ρ/ρB , A$Ê< HGN} :NOp°JVKÂLiDGÁÂIKMF"CK&PKM}¶JVFCELGp"BKMFÍUTLGWGUTD:PK¶ÆEKMPUTLGWUT}¶JVpUTp"UvÁKMUTp!DGFKÂCÐJVKÅUTBGBGPONK&W Á UÇRELGK&J!N} GKMPW9Ê ±µLeËNREDF!K AÊ 7à iJVGKSBPUÇJ!KjUTDGp;C$}&}MDGFU3JKjUT} eNOLiJ!KMREK&F;ÆÇUTPDK"C JVG"K 4PPONLG( R UT}&J!CEF ν Ê$GNp}MUTL I KÉDLGWGKMFpJ!C$CW JV4UÇJ HKML KjUT} 2C ©JVGKÉWKMREK&LGKMF!UÇJVKepJVUÇJVK&pC ½UÍUTLW4UTD2PK¶ÆEKMP;Np 4POPKMW JVK F!K&p!Np¬JVN Æ$N JµÅNL}MF!KMUTp!K&p7IKM}MUTDGp"K h%K 7KMF;UTLG; W h%K 7KMF7pJ U3JVKMpF!K&Á UÇNLDGLGC}M}&DGBGNK& W HN JVGNOLJ!4UÇJKMLGK&F!RT PK¶ÆEKMP|Ê GKMLJ!GKvÍUTLW4UTDPOK&ÆTKMPNpr}MCEÁBGPK¶JVK&PO' hDGPO8P UlRqUTBK $NpJ!p>FKMiDGNFNLGRÕC U 4LNOJVK"ÈDGÁBNOL KMLGK&F!RT JVCF!KMUT} zJ!GKLGK J©p!K¶J½C WGK&REKMLKMFVU3JVKrKMLGK&F!RTep¬J UÇJ!KMp N|Ê KÇ@Ê J!GKLGK J°ÍUTLGW4UTDePOK&ÆEK&P Ê Ô C;K¶ÆEK&F WGDGKÅJVCeNÁBGDGFNOJ!NKMp½NLUÉpVUÇÁBGPOK9J!GKWGKMLp!NOJµ`C;p¬J UÇJ!KMp<HNPOPK¶ÆECEP ÆEK hF!CEÁ p!4UTFB ÍUTLGW4UTDPK&ÆTKMPOp>JVC:UzIGF!CqUÇWGKMFÂpB K&}&J!F!DGÁ«C ©PK¶ÆEKMPOp hp!KMKeËNREDGF& K A$Ê H ÊKMF!KeUÇF!K*J 7CiNLGWGp>C PK¶ÆEKMPOp PC}jUTPON &KMWvUÇLGWÉK JVK&LGWGK&)W GNLJVGK"L%K p!BKM}&J!F!DGC Á GUÇLGWÉNOJ;Np;K $B K&}&JVK&WÉJV4UÇJ;JVGK"K JVKMLWGKMW pJVUÇJVK&pÂC$}&}MDGBqUÕ}MCEFKvLGKMUTF>JVKvCEF!NORENLGUTPÐÍUTLGW4UTD POK&ÆTKMP HNPKJVGK PC}jUTPON MK&Wp¬J UÇJ!KMpÂJ!F!NRTREKMFxU IGF!CEUTWGKMLNLGReC 7JVGKWGKMLp!NOJµC QpJVUÇJVK&pMÊÌ"LGPOzJVGKK JVK&LGWGK&W pJ U3JVKMpr}jUÇL}MUTF!F¬}MDGFF!KMLqJÂUÇJ MKMFC JVK&ÁBKMF!UÇJVDGFKTÊEK©K $NpJ!KMLG}&K½C J!GKHPC}jUTPON MK&WÂpJVUÇJVK&pÐ}jUTLK $BGPUTNLÅJVKHUTBGBKjUTF!UTLG}MKHC BGPUÇJVKMUTDGpMÊ ®°prJVGKWGKMLGpNOJµNpxNLG}&F!KjUÇp!KMWJVGKPC}jUTPON MK&Wp¬J UÇJ!KMpÂRTFVUTWGDGUTPP ' 4PPÐDGB HNOJVCEDJÅUÇLi} 4UÇLGREKvNOL C$}&}MDGBGUÇJVNOCEL>C4J!GKQK $J!KMLGWKMWpJVUÇJVK&pÇJ!iDGp HNOJ!GCEDJ/UTLq>} 4UÇLGREKQNOL>JVGK Ô UTPOP$FKMpNpJVUTLG}&KTÊT±µLÂJ!GKMpK }jUTpKM%p JVK Ô UTPOPÐF!KMpNp¬J UTLG}&KvNprCEL UzpJVK&B2UTLGWJ!GKvPOCELGREN JVDGWGNOL4UTP/FKMpNpJVUTLG}&K ÆÇUTLGNpGKMp |UÇJ MKMFC JVK&ÁBKMF!UÇJVDGFK Ê ±µL GDIqÃ7 ¨ Ê ³ ÊÇ p!DGN ' G1ÄG($F!K&B KMUÇJVK&W Ó PN J M NOLGR1 pKjUTFPNK&FK$B &K F!NOÁK&LiJ!p HN JV>NREGK&FÁCEIGNOPN Jµ p!K&ÁNO}MCELGWDG}&J!CEFGK&J!KMF!C3´µpJ!F!DG}¶JVDGFKMp,UÇLGW>GNOREGK&FÁUTRELGK¶JVNO} 4KMPOW+Ê GKMNOFRECEUTPQUÇpJVC½ÁNOLGNÁN & KJVK 4 7'# ! 32 !" F!CEPOKC, NÁBGDGFNOJ!NKMpHUÇLGWÕJVCKMLGGUTLG}MKxJVGKrK% KM}¶JVp"C/KMPK&}&J!F!CEL$´ KMPOKM}&J!F!CELzNLqJVK&FVUT}¶JVNOCELGpMÊ GK¶ÕGU1ÆTK hCEDGLGWJVGUÇJJVGKBGPUÇJVKMUTDGprUÇJrNLqJVK&REKMF<4POPNOLGR UT}¶JVCEFpxUÇF!KÁÂDG} L4UTF!FC;K&F UTLGWI K¶J*;K&KMLJ!GKMÁ ÁCTF!K©BGPUÇJ!KjUTDp;UTFK°pKMKMLlUÇJ=h FVUÇ}&JVNOCEL4UTP 4 POPNOLGR< UT}&J!CEFKMpB K&}MNUÇPPO 1/3 UTLGW 2/3 hp!KMKËNREDGFK AÊ< ¶Ê AD m>O#2 0 G $5 '<)'(! -B-<D5$`0p%% O? 5 A 3 - ; ' -3+$<; %% O& 5 A ρ > !#$%& 'H'(!4* $%& ; 3 n 2; #<B'(! ?$/?< Dxy5$ 0 %% O? 5 A 1'Ib( 3 - - ρ 3 Oxx %b2$? '; %OqG$5 '<m'(! ? ; ' -3+$<; %% O? 5< A8 $Bb#% 5;b:'(! ν GKMpKvCEIGpKMF¬ÆTU3JVNCTLGp>}&CEDGPWLGCTJÂIK K $BGPUTNOLGKMWIqJ!GKvLGCEL´µNLqJ!KMFVUÇ}&JVNOLGRlDGUTLqJVDGÁ«ÁK&} 4UTLGNO}jUTP JVGK&CEFTÊ$K& UTF!K½F!KjUÇPN &KMWJVCÂIK"U>FKMpDGPOJ;C+JVK°ÁUTLqi´ I CWK% KM}¶JVpQCTLvNLqJVK&FVUT}¶JVNOLGR>K&PKM}¶JVFCELGpMÊ 4UÇJ©ÁKjUTLGpHLGC 7KxLGK&KMWlJVCNLG}&PDGWKJ!GKK% KM}¶JVp°CNOLiJ!KMF!UT}&J!NCELGpHNOLeJ!GK Ô UTÁNP JVCELNUTL AÊ G JVGK&L N N X pi − eA(ri )2 X e2 A$EÊ ? H= + . 2m |ri − rj | ±µLJ!GNp Ô UTÁNP JVCELNUTL) JVKBCTJVK&LqJVNUÇPK&LGKMFRTNpLGCzPCELGRTKMFrUzpÁ UTPOP/JVK&F!Á }MCTÁB4UÇF!KMW0HNOJ!JVK iNLGK¶JVN}ÐJVK&F!ÁlÊDZ JNLGWDG}MK&pJVGK=hFVUT}¶JVNOCEL4UTPiiD4UTLqJVDÁ Ô UÇPPiK% KM}¶J |Ë = Ô » IKMNOLGR°CTLGKQCGJ!GKQp¬JVF!CTLGR }MCEFF!K&PUÇJ!NCELvBF!CEIGPOKMÁpMÊ i i<j < =A m4] = _ f|d5gmG^3f m \ f Y dm4_>mqg A ]d ZT\ f Y d > ±µL GDIH7¯HCEIKMFJÍ UÇDGREGPONL ' G1Äqà (+BGFCEBCEp!K&WlNpQUÇLGpVUÇJ EhCEFQU>ÆÇUTFNUÇJ!NCELGUTPQUjÆTK hDLG}&J!NCELHGNO} }MCELqJVUTNLGK&WeLGC hFKMKB4UTF!UTÁK&JVK&F!p&~ ψ= Y i<j (zi − zj ) 2p+1 exp − X |zi |2 i 4l2 ! , A$Ê A HGKMFK zi = xi + iyi NpSJVGK}MCTÁBGPOK}&C$CTF!WGNOL4UÇJVKÅC;JVK ith B4UTF¬JVN}&PKUTLW l2 = hc/eB NOpJ!GK Á UÇRELGK&J!N}ÅPOKMLGRTJ!+ÊGKvÍUTDREGPNOL0;UjÆEK hDLG}&J!NCELRqUjÆEK UTLUT}&}MDGF!UÇJVKWGKMp}MFNBJ!NCELCHUTPOP 4PPONLGR hFVUT}¶JVNOCELGp ν = 1/(2p + 1) UÇLGW QUTp©p!C HLlJVCUTPOÁCEp¬JKUT}¶JVP ÉÁ U3JV} eLiDGÁK&F!NO}jUTP+RTF!CEDGLWzp¬J UÇJ!K QUjÆTK hDLG}&J!NCELGp hCEDGLGWhCEF7p!ÁUTPP Ë = Ô »2ppJ!KMÁpMÊ Ô C7K&ÆTKMFGJ!GKÍUTDGREPNL QUjÆTK hDLG}&J!NCELÉ}MUTL LGCTJHIKDGp!K&WlJ!CK BPUTNOLvJVGKF!K&Á UÇNLGNOLG: R 4POPNL: R hFVUT}¶JVNOCELGp p!DG} ÕUTp 2/5 3/7 +ÊOÊ Ê GKÂÍUTDGRTGPNOLGRUTBGBGFCqUT} ÕUTPPOC 7KMWeJVCÁ U TKBGF!K&WGN}¶JVNOCELGp½UTI CTDJHJVGKrK $}MN J UÇJ!NCELp©C ,U hF!UT}¶´ JVNOCEL4UTP iD4UTLqJVDÁ Ô UÇPPppJ!KMÁlÊGGKMpKSK$}MN J UÇJ!NCELp ;K&F!KSpGCHLeJ!CÂ}jUÇF!F UTLGCTÁ UTPOCEDGp3hFVUT}¶JVNOCEL4UTP } 4UTFREK νe UTLWJ!CÉ4UjÆEK hFVUÇ}&JVNOCEL4UTPpJVUÇJVNOpJVNO}Mp&Ê[Ì"LREKMLGK&FVUTPREF!CEDLGWGp JVGKÍUTDGREPNL`UTBGBGFCqUT} ¿TÄ E Extended states Localized states ωc PSfrag replacements N (E) ) <* '(! 3jS; -5$; 3:D'.b* 2E & %q+:?$ &* 2; }jUTLIKÅDGLGWGK&F!p¬JVCC$WNL`J!KMFÁpSCK&PKM}¶JVFCELGpUTW3ÈDpJVNOLGRvJVCÉJ!GKÅÁ UÇRELGK&J!N}(4KMPOW)HGNO} NOpSF!K&BGF!K ´ p!K&LiJ!KMWeIq G7D J!DGIKMp;J!GF!KMUTWGNLR>JVGK½*J ;CÅWGNOÁK&LGp!NOCEL4UTP BGPUTLGKÇÊ»,PK&}&J!F!CELGpQUTW3ÈDGp¬JQJVGK&NFPC}jUÇJ!NCEL NLÉCEFWGKMFJ!CÁNLNÁN MKHJVKrRTF!CEDGLWzp¬J UÇJ!KrK&LGKMFRT HN JVÉJVGK&p!KrÁ UÇRELGK&J!N} GKMPWl}MCTLGpJ!FVUTNOLGpMÊ { ,6 Y [email protected] f \ A Ai^3{f Y d @Âm4d X = Ai^3d f|{ Y d @ \ = A Y ^@9 ±µL G!DIDÐÒEUÇNLGK&LGWGFVUÒEUTNOL ' G1ÄH (HNWGK&[email protected]ÉpK&JC½DGUTp!N8´µB4UTF¬JVNO}MPK&p hCEFÂJVKÕË = Ô » ppJ!KMC Á Q}jUÇPPNOLGRJ!GKMÁ b}MCTÁBCEp!N JVKChKMF!ÁNCTLGp V~7K&PKM}¶JVFCELGpHNOJ! 4DF iD4UTLqJ U2UÇJJ UT} GK&W JVC KjUT} lC JVGK&ÁzÊ KMLÕKMUT} ÕC J!GK}MCEÁB CTp!NOJ!EK hKMFÁNOCELeÍUTLGW4UTDlP2n K¶ÆEK&PpHNp44POPK&WeJ!GKMFKxK $NpJ!p©UTL KMLGK&F!RTRqUÇB p!K&B4UTF!UÇJVNOLGR>N J hFCEÁ¦JVGK½LGK J;}MCEÁB CTp!NOJ!K°ÍUTLW4UTD POK&ÆTKMP ÊiGK"Ë = Ô »2C ÒEUTNLpJVUÇJVK&p }jUTL:IKÂDGLWGKMFpJVCCWUTpSUTL:±'= Ô » C}MCTÁBCEp!N JVK(hKMF!ÁNCTLGp°U3JUTL:K% KM}¶JVN ÆEKÅNLqJ!KMREK&F"4POPNLRG~ p = LiDGÁÂIKMFC }MCEÁB CTp!NOJ!K hKMFÁNOCEL /LiDGÁÂIKMFHC DGL4UÇJJ UT} KM$ W G7D eiD4UTLqJ U = ν/(1−2nν) Ê1GKMLJVK F!K&PUÇJ!NCELxI K¶*J ;K&KMLJVG4K 4POPNOLG>R UÇ}&JVCTF ν UTLGWÅNOLiJ!KMREK&F!p n UÇLGW p hCEF,ÒEUTNOLÅpJVUÇJVK&p,Np ν = p/(2np+1) Ê GNp©}MCTÁBCEp!N JVEK hK&F!ÁNCELlBGN}¶JVDGFKNp½UTIGPKSJ!C K $BGPUTNLlICTJVeJVGKxÍUTDGREPNA L hFVUÇ}&JVNOCELGp p = 1HUTp ;K&PP[UÇp©ÁUTLqÉC JVKrFKMÁUTNLNLG: R hF!UT}&J!NCEL p p 6= 1¶Ê GKiD4UÇp!N ´ B4UTF¬JVN}&PK&pSNL:JVGK}MCTÁBCEp!N JVK hK&F!ÁNCEL:p¬$p¬JVK&Á·UÇF!K}MCEÁBCEp!K&WC ;I CÇJVB4UTFJ!N}&PKMp hJ!GKvKMPOKM}¶JVF!CTLGp >UÇLGW 4KMPOW hJ!GK 4FD iD4UTLqJ U ¶Ê± JÂF!KMiDGNOF!K&pUÕRqUTDGRTK JVF!UTLG/p hCTF!ÁUÇJVNOCELJ!C:ÁC1ÆTK B4UTF¬JVN}&PK&p;JVCiD4UTpN ´µBGUTFJ!N}MPOKMp%HGN} lNp iLGCHLÕUTpQJVGK ³ GK&F!L$´bÀ$NÁCELGpQJ!GKMCTF ' G1ÄG (|Ê xY $ ±µLJVGKzJVGK&F!ÁC$WL4UÇÁNO}ÕPNOÁN JULGCELGNOLqJVKMF!UT}&J!NLGRJ*;CÇ´ WGNÁKMLp!NCTL4UTPHKMPOKM}&J!F!CEL REUTpv4UTpvUTL NL$´ }MCEÁBGFKMp!pNIGPOKxRTF!CEDGLWÕpJVUÇJV( K HGKMLK&ÆEK&FUTL`NOLiJ!KMREF!UTP[LiDGÁÂIKMF½C ÐÍUTLGW4UÇD`PK¶ÆEKMPOp°NOp>hDGPOP8GN|Ê KÇÊ@ JVK } GKMÁN}MUTP/BCTJVK&LqJVNUÇP ÈDGÁBG<p hF!CEÁ J!C ω (n + 3/2) U #JVK&F"JVGK nth ÍUTLGWGUTDPK&ÆTKMPÐNp 4PPOKMW+Ê1± J hCEPPOC Hp J!4UÇJ hCTF"U GLGNOJ!K7p!ωUTcÁ(nBP+K7UT1/2) PPip!NOLGREPOK¶´µBGcUTFJ!N}MPOK,pJVUÇJVK&p HGN} >C$}&}MDGFp,UÇJKMLGK&F!RENOKMp/NOL JVGK&p!K½RqUTBGpÁÂDGp¬J;IK½POC$}MUTPNMK&WU3JJ!GK°K&WGREK©C+J!GK½p!UTÁBGPKÇÊqGK°iD4UTLqJ!DGÁ Ô UTPP4K&WGREK©pJVUÇJVK&pQUTFK } GNF!UTP hp!K&KËNREDGFK AÊ ? Ê+GK&`}MCELGp¬JVN JVDJ!KÂJVGKÅiD4UTLqJ!N MK&WUTL4UTPOCEREDGKÂC J!GK}MPUTp!pN}jUÇPp iNOBGBGNLR CEF!INOJVp;C } GUTF!REK&WÕB4UÇFJVNO}MPOKMpQNOLzUÁUTRELGK¶JVN"} 4KMPOWlpDGIÈK&}&J©JVCKMPOKM}&J!F!CEp¬J UÇJ!N}S}&CEL 4LGK&ÁK&LqJjÊ ¿7G $ OD2 q 0 G$5 '<j'(!-n-D5$`0 ;b; %% OD<1 < 3q?; ' ?3b$<; %% OD<1 R !#$%& ' '(!.* %? ; 3 q 2; #< '(!8- -< D5$H`0 %% O& 5 A 1'Ib( 3 - - R 3 O%b2$? '; %OqG$5 '<m'(! ? ; ' -3+$<; %% O? 5< A8 !D%? ' '(! ν B q<0 v F q,+;= A8'(! ?$:@$<-* 4 q=0 q>0 <;=;<13:OD#% 3 ,D' ? ('< #F ! '< 1 *(.!"- 23!, ,- ( ) Ô UTPOB K&F!NOL ' G1Ä ? ( GF!pJNOLqJVF!CWGDG}&KMWUÉp!NOÁBPK>BN}&J!DGF!K>CÐJVGKÅNOLiJ!KMREK&FSiD4UTLqJVDGÁ Ô TU POPK&WGREKÅpJVUÇJVK&p L CÐLGCEL´µNLqJ!KMFVUÇ}&JVNOLGRvK&PKM}¶JVFCELGpSNL`U GPPK&W:ÍUTLGW4UÇD:PK&ÆTKMP,UÇLGW'hCEDGLGW:J!4UÇJ"J!GKÅKMWGRTKÅK$}MN J UÇJ!NCELGp }jUTLÕI KxF!KMBF!KMpKMLqJVK&WIiÕU CELK>WGNOÁK&LGp!NOCEL4UTP[} GNOFVUTPË4KMFÁNPNOiDGNW+Ê4±µLzJVGNOp½BGNO}&J!DGF!KJ!GKÂ[email protected] ÍUTLGW4UTDePOK&ÆEK&Pp HNOJ!lUTLeKMLGK&F!RTepB4UT}MNOLGRC ωc UTFKSI K&LiJ½DGB ;UTF!WlLGKjUÇFHJ!GKKMWGRTKrC J!GKpVUTÁBGPK Iq JVKS}MCELGLGNLGRÅBCTJ!KMLqJVNUTP HN JVeUTLÉK&PKM}¶JVFN} 4K&PW w hp!KMKrËNOREDGFK AÊ<A ¶ÊË4CEFUÂÁUT}MFCEp!}&CEBGN}MUTPP PUTFREK;p!UTÁBGPK ÇJVKMF!KHUÇF!KÁUTLqÂKMWREKpJVUÇJVK&p,LGKjUTF/J!GKHË4KMFÁN$KMLGK&F!RT EF C KMPOKM}&J!F!CELp,K&ÆEK&L HKML JVGK&F!KrUTFKSLGCp¬J UÇJ!KMpHLGKMUTF EF NLÉJ!GKIGDGP Ê Ë CTPPCHNOLGRÅÀJ!CELGK ' G1Ä A)( 7K}MCELGpNWGK&FJVGK ν = 1 iD4UTLqJVDÁ Ô UTPOP+REF!CTDGLGWlpJ U3JVK<HNOJ!zUÅpNLGREPOK B4UTF¬JVN}&PK ;UjÆE K hDLG}&J!NCE L hCEFrKjUT} ky K&NREK&LGpJVUÇJVK ψk (x, y) = eikxe−eB(y−k/eB) /2 NOLJ!GKPNLKjUTF RqUTDGRTK Ax = −By ʱ 7U PNOLGKjUTF©}MCTL 4LGNOLGRvBCTJVK&LiJ!NUTP V (y) = Ey NOp°BGD$JSN)L JVK>WGK&REKMLGK&FVUT}¶:Np PN #JVKMW 9JVGKKMLKMF!RÇC UepJ U3JV; K HNOJV k = k IKM}&CEÁK&p (k) = Ek/B UTLGWKMUT} p¬J UÇJ!K}MUTF!FNK&p U Ô UTPP;}MDGFF!KMLqJMÊ7Ë4DGF¬JVGK&F,JVKzBGqp!NO}jxUTPHpDGF/ UT}&KzC°JVKeJ*7CWNÁKMLGpNCEL4UÇPQKMPOKM}&J!F!CEL RqUTpNpU3J K hDGPP UTLWJVCEp!K°UÇJÐJVGK½F!NRTiJ7UTFK½K&ÁB$Jµ p!K&KSËNOREDGF!K A$Ê A Ê y = 0 JVKMLJ!GK°p¬J U3JVKMpJVCJVGK½PK #J;UÇF!? GKrBGq$pN}MUTPK&WGREKrC J!GKrWGF!CEBPK&J©NOL y p!B4UT}&K>}MUTLÕ[email protected] 4K&C W HN JVeJVGK ¬Ë4KMFÁN9p!DGF UT}MK xUÇJ 2 ¿Eà E (5/2)ωc (3/2)ωc EF PSfrag replacements (1/2)ωc kx edge bulk 3j ; -5$; ,<3/?$413 OD#%q! '<cp'ObD$%&+/ 'p3+*% '$<;; 1%?'4<: + 81' 9+H2'D? <; ->(*:I`A <3.-A(; '- ',!D [email protected] ν=1 EF ωc "<<2; %O813>( O? '8("1!1 n5$% '(! 13.3 OD' & '16 + W6 /3 12; $1* 2'Ib <; 'B-,13B2$ '2(<&+ q -,5$; 'I( A h(x) x v HN JV:U¬Ë4KMFÁN[ÆEK&PC}MN Jµ xIKMNOLGR JVKÂWGF!N #J½ÆTKMPOC$}&NOJµ E/B Ê[À$NOLG}MK>UTPPJVKÂpJVUÇJVK&p"ÁC1ÆTKÂNL JVGK°pVUTÁK°WGNOF!K&}&JVNOCEL)iJVGK°K$}MN J UÇJ!NCELGp7}jUTLÉIK"WGK&p!}&F!NIKMWeIq U ¨ NF!UT}°K&DGUÇJVNOCEL+ÊGNpQK&WGREKSp¬J UÇJ!K WGKMp}MFNBJ!NCELz4UTpSUTPOPC 7KMWÕJ!CvK $BGPUTNLzNL:U F!UÇJVKMF"NOLiJ!DGNOJ!NOÆTKxÁUTLGLGK&F½J!GKÂBGqp!NO}MpSC,J!GK±'= Ô » ' GjÄqF¿ &G1Ä I (8 UÇp©J!GK&lBGFC ÆNOWGKrUWGNOF!KM}¶J©F!KMBF!KMpKMLqJ U3JVNCTLÕC JVGKxÍUTLGW4UTDKMF¬´µÎ JJVN ÇKM?F hCEF!Á>DGPUÂC /U ÁK&p!CEp}MCEBN}S}MCTLGWGDG}¶JVCEF&Ê k=0 4 L *(%01 7.# !, 32 , !", ,- ) ±µLJ!GK©Ë = Ô » EJVGK½ÍUTLGW4UTDPK¶ÆEK&P4Np 4PPOKMWB4UTF¬JVNUTPP EÊ3±µLJ!GK©UTIGp!K&LG}MK½C+K&PK&}&JVFCEL$´ KMPK&}&J!F!CELNLqJ!KMF¬´ UT}&J!NCELp°JVGK&F!K 7CEDGPWJVKMLI KUTLKMLCEF!ÁCEDGpSREFCEDGLGWp¬J UÇJ!KWKMREK&LGKMF!UT}& IDJSJVGNOpSWGKMRTKMLGK&FVUT}¶ NpPN #JVKMW2Iq JVGKzNLqJVK&FVUT}¶JVNCTLGpMÊЮQJp!BKM}MNUTP?4PPONLGR UT}¶JVCEFpÐp!DG} UÇp ν = 1/3 ,JVGKzppJ!KMÁ Np K$BKM}&J!KMWJ!C`}MCELWGKMLGpKÉNLqJVCÕUz}&CEF!FKMPU3JVKMWPNiDGNOWp¬J UÇJ!KTÊ GNpxPNODNWpJ U3JVKvNOpxNOLG}MCEÁBGFKMp!pNIGPOK UTLGW4UTpxUlREUT0 B hCTFxUÇPP/K }&NOJVUÇJVNOCELGpMÊ[±µLJVKBGFKMpKMLG}&KvC QK&WGREK&( p 7K UÇLiJ!N}&NB4UÇJ!KPC ¦P $NOLGRÉKMWREK Ô K$}MN J UÇJ!NCELp[UTpNOLJ!GK±'= »ÊGKMFK UTFKp!K&ÆTKMF!UTP QUjpxJ!CzWGK&p!}&F!NIKÅJVGK hF!UT}&J!NCEL4UÇP/iD4UTLqJVDGÁ Ô UTPOP KMWGREK°pJ U3JVKMp&Ê4Î;DJ7JVG"K 4F!p¬JHUTLGWvpNÁBGPK&pJ QUjJVCÅDGLGWKMF!p¬J UTLWvJVGK"W$$LGUTÁNO}Mp7C hFVUT}¶JVNOCEL4UTP iD4UTLqJVDGÁ Ô UTPOP;K&WGREKeK $}MN J UÇJ!NCELpÂNp>J!C:DGp!KÉJ!GKÉq$WF!CWL4UTÁN}vBGNO}&J!DGF!KÉNOLiJ!F!CWGDG}&KMW2Iq KML ¿H E E+ (k) E− (k) kvF −kvF EF PSfrag replacements 0 −kF kF k (<G'; G$$3>O-%- '(! *%O;b; + :$3 ; +(E F1? '< ' GjÄD ([NOL HGN} eJVGKLGK&DJVF!UTP[KMWREKK$}MN J UÇJ!NCELÉNOpHUWK%hCEFÁ UÇJ!NCELÉC J!GKKMWGRTKxUÇp©pGCHL`UTp4;UjÆi p!CEPONWzPNLK>NLzJVGKËNOREDGFK AÊ ¿$Ê9KÅKMPK&}&J!F!CEp¬J UÇJ!N}>BCTJ!KMLqJVNUTP}MUTLI K>K$BGF!K&p!pKMWNOLÕJVK&F!Áp"CÐJVK PC}jUTP;WGNpBGPUT}&KMÁKMLqJ CEFGK&NREqJ ÅC "J!GA K 4DGNW h UÇJUBCENLqJ x UTPOCELGR:JVGKzKMWGRTKTÊÐGK`UÇDJVGCTF p!GC 7KMWJV4U3J"JVGKÅW$$LGUTÁNO}jUTP BGFCEB K&FJ!NKMp°C ÐJVGKÅK&WGREKÅK $}&NOJ U3JVNCTLGpSUTFKÂWGKMp}MFNIKMWIq`JVGK¹ G Ó UT}¶´bzC$CWÂUTPREK&IGFVUÇpMÊqGK&p!K°K&WGREK©K $}MN J UÇJ!NCELp7UÇF!KHp!GC HLJVC hCEF!Á¡UrLG%K NOLGWC +p¬J UÇJ!>K HGNO} Np"}MUTPPOKMW`J!GKÅ} GNF!UTP/Í[DJJVNOLGREKMF°PNiDGNOW pKMKUTPOp!Ce¯H%K bÊ ' HEÄ (¶Ê9GNp"J!GKMCEF¬' HNPOPIKÅWGNOp!}MDp!p!K&WNOL JVGKLGK J½pKM}¶JVNCTL+Ê $ ! " ,» PKM}¶JVFCELGp°NOL`CELGK ´µWGNOÁK&LGp!NOCEL4UTP G ¨ ½p¬pJVK&Áp"hCEF!Á U iD4UTLqJ!DGÁ PONiDGNWAHGNO} :}jUTL`LCTJSIK>WGK ´ p!}&F!NIKMW HNOJ! J!GKÕË4K&F!ÁNPNiDGNOW JVGK&CEF2pNLG}&KlJ!GKMFK`UTFKlLCp!NOLGREPOKlK&PK&}&JVFCEL DGUTp!N8´µB4UTF¬JVNO}MPK&pMÊ ± JrNpSWGDGKJVCeJVKBF!KMpKMLG}&K C 7JVGK ³ CEDPCEÁÂIK&PKM}¶JVFCEL$´µK&PK&}&JVFCELNLqJVK&FVUT}¶JVNCTLGpMÊ+ ClWGKMp}MF!NOI KJVK WL4UTÁN}ÂC;K&PKM}¶JVFCELGpNL G ¨ pp¬JVKMÁp 7KÁÂDGpJrDGpKÅJVGKÍ[DJ!J!NLGRTKMF"PONiDGNW`J!GKMCTF ' GG1ÄF GGG( NL HGNO} zJVKrK $}MNOJVUÇJVNOCELGpH}MCTLGp!NOpJ©C /}&CEPPOKM}¶JVNOÆTKSKMPOKM}&J!F!CEL´µGCEPOKSK$}MN J UÇJ!NCELp©CJ!GK(HCEPKrË4KMFÁN p!KMUÊ 4 # ! F*,1 .!,- ..23 /*# K}&CELGp!NOWGKMFrUG ¨ ÁK¶J UTPOPN}(HNFK;HN JVB4UTF!UTICEPNO}ÂWGNpB K&F!pNCELI4UTLWUTpp!CHLNLËNOREDGF!K $A Ê IÊ GK iNLGK¶JVN}SBGUTFJHC JVK Ô UTÁNPOJ!CELGNUTL FKjUTWGp&~ H0 = X E(k)a†k ak , k A$Ê ¿ HNOJ! E(k) = k2/2m UTLGW a†k ak NpJVGKv}&F!KMUÇJVNOCEL UTLGLGNOGNPUÇJVNOCEL½CEBKMF!UÇJVCTFxCHCTLGK K&PKM}¶JVFCEL NL:p¬J UÇJ!K;HN JVÁCEÁK&LqJVDGÁ k Ê[²½CÇJVN}&KÂJV4U3J(7KWGClLGCTJNL}MPDWGKÅp!BGNOLWKMREFKMKC=hF!KMK&WGCEÁ KMF!KÇÊ ±µLzJ!GKx CEÁCEL4UTREUÁC$WKMP ' GG1Ä)(JVKxWGNOp!BKMFp!NCTLÕF!K&PUÇJ!NCEL E(k) NOp©PONLGKMUTF!NMK&WzNOLlJ!GKxÆN}&NLGN JµÉC Ë4KMF!ÁN KMLGK&F!RT EF ~ E+(k) = EF + vF (k − kF ) hCTF k ≈ kF UTLGW E− (k) = EF − vF (k + kF ) hCEF k ≈ −kF Ê,±µL JVGKlÍ[DJ!J!NLGRTKMFÅÁC$WGK&P ' GGG(9/JVGKlPNOLGKjUÇF!N MUÇJVNOCELNOpKJVK&LGWGKMW5JVCUTPP7ÆTUÇPDGK&p ¿ C k Ê ®½L`NLGLGNOJ!KxLiDGÁÂIKMF½C 4}¶JVNOÆTKÂpJVUÇJVK&p"UTFKÂUTWWGKMW'HGNO} `4UjÆEK>LGCvBGqp!NO}[email protected]}MUÇJVNOCEL+Ê Ô C7K&ÆEK&FÍD$J!JVNOLGREK&FrÁCWGKMP,4UÇp>UTLNÁBCEFJVUTLqJrUTWÆÇUTLqJ UTRTKC1ÆEK&FrJ!GKCEÁCEL4UÇRqUlÁCWGKMP|~+NOJrNp KUT}¶JVPO p!CEP ÆÇUTIGPKSDp!NLRJ!GKI CTp!CELGN jUÇJ!NCELvJ!KM} GLNiDGKTÊ KF!K HF!N JVK"JVKrK&PK&}&JVFCELl}MF!KMUÇJVNOCELeCEBKMFVU3JVCEFU #JVKMFHPONLGKMUTF!N jUÇJ!NCEL UTp AÊ I a†k = a†+,k Θ(k) + a†−,k Θ(−k) , HGKMFK Θ(k) Np°J!GK Ô KjUjÆNpNWGK hDLG}&J!NCELUTLW a†r,k HN JV r = ± °NOp°JVKÅ}MF!KMUÇJVNOCELCEBKMFVU3JVCEF hCTF F!NOREqJCEFxPO%K #JrÁC ÆNOLGReKMPK&}&J!F!CELGpr}MCTF!F!K&p!BCELGWGNOLGRzJ!ClJ!GK WNp!BKMFp!NOCELFKMPUÇJVNOCEL E+ (k) CEF E−(k) HGN} lCEIK&pJVCÅJVKxUÇLiJ!N ´ }MCEÁÁÂDJVUÇJVNOCEL FKMPU3JVNCTLGpMÊ ²½KMRTPKM}¶JVNOLGR©J!GK}MCELGp¬J UTLqJJVKMFÁp NLxKMLGK&F!RT3J!GK=NOLGK&J!N} Ô UTÁNP JVCELNUTL"}MUTL>IK;pNÁBGPO"HF!N J!JVK&L UTp X A$Ê D H0 = v F rka†r,k ar,k . SÌ L J!GK:CTJVKMFÉ4UTLW);JVNpÉLGCEL´µNLqJ!KMFVUÇ}&JVNOLGR Ô UTÁNOPOJ!CELGNUÇL5}MUTL I K:K BF!KMpp!K&W¼NLJ!KMF!Á C KMPOKM}&J!F!CELeWGK&LGp!N JµÉCEBKMF!UÇJVCEFHUÇp r,k H0 = πvF HNOJ! ar,k ~ UTLGW ρr (x) = ψr† (x)ψr (x) HGK&F!K XZ r ψr† (x) UÇLGW L/2 −L/2 dxρ2r (x) , ψr (x) UTFKJ!GKÉË4CEDGFNKMFSJ!FVUTLGphCEF!Á«C 1 X −ikx † e ar,k , ψr† (x) = √ L k 1 X ikx ψr (x) = √ e ar,k , L k A$Ê@G1Ä a†r,k UÇLGW AÊ GG AÊ G à ON pQJVGKPK&LGRTJ!lC HNOF!KTÊ KWGK4LGKJVGKLGCEL´µ} GNOFVUTP I CEpCELGNO}E4KMPOW+~ L i X 2π φr (x) = √ [ρr (k)e−ikx − ρr (−k)eikx ]e−α|k|/2 , L k>0 k AÊ GH HGKMFK α → 0 NpHUÅWGNOpJVUTLG}MK}&DJVC,HGN} lNpNOLiJ!F!CWGDG}&KMWlNLlÍ[DJ!J!NLGREK&FQPNOiDGNWvJ!GKMCEF¬vJVCNOLGp!DGFK JVGK}&CELqÆEKMFREKML}MK>CJ!GKNLqJVK&REFVUTPOpMÊ G<K hKMF!ÁNCTLGN}"CTB K&FVUÇJ!CEF ψr NO4p HF!NOJJVK&LzNOLeJ!GKI CTp!CELGNMK&WAhCTF!Á UÇp Mr irkF x+irφr (x) ψr (x) = √ e . 2πα AÊ GB GK UT}&J!CEF Mr /√2πα UTPPOCHp½JVClCEIJ UÇNL`JVKUÇWGKMiD4UÇJ!K UÇLiJ!N}&CEÁÁ>DJ UÇJ!NCELÕF!KMPUÇJVNOCELGp%+UTLGW kF NLvJ!GKK$B CTLGKMLqJ½NpLGK&KMWGK&WzJVCWGK&p!}MFNIK}MCEFF!KM}¶JVP JVKrK&LGKMFRTeI4UTLW+Ê U iNOLGR JVKÂWGKMFNOÆÇUÇJ!NOÆEKxC», Ê A$Ê@GH 6;KÂGU1ÆTK ∂xφr (x) = 2πρr (x) J!GKML'7KÂ}MUTL:F!KHF!NOJ!K JVGKLGCTL$´µNOLiJ!KMF!UT}&J!NLGR Ô UTÁNPOJ!CELGNUTL REN ÆEK&LzIqe», Ê AÊ G1Ä QUTp Z vF X L/2 dx(∂x φr (x))2 . H0 = 4π r −L/2 ¿ ? AÊ G ? ± 7KrWGK%4LKJ!GKSJVCTJVUTP4K&PW A$Ê@[email protected] φ(x) = φ+ (x) + φ− (x) , ϕ(x) = φ+ (x) − φ− (x) , JVGK&L vF H0 = 8π A$Ê@G ¿ AÊ GI L/2 Z dx[(∂x φ(x))2 + (∂x ϕ(x))2 ] . −L/2 KxNLG}&PDGWGKxLGC JVGK ³ CEDGPOCEÁÂIÕNOLqJVKMF!UT}&J!NCELp°IK&J*7KMKML:J!GK>K&PKM}¶JVFCELGpMÊGK>J!CTJ UÇP Ô UTÁNOPOJ!CÇ´ LGNUÇLÉNp H = H0 + Hint HNOJ! Hint = Z L/2 dx −L/2 Z AÊ GD L/2 −L/2 dx0 ρ(x)U (x − x0 )ρ(x0 ) , HGKMFK½J!GK°K&PKM}¶JVFCELvWGK&LGp!N JµCTB K&FVUÇJ!CEF ρ(x) }&CELqJ UTNOLGpQUrp!POC ÆÇUTFNUÇJ!NCEL ρ+(x) + ρ− (x) UTLGW UTp¬J p HN JVeUÂLGCEL´µPNOLGKjUÇF;WGK&B K&LGWGKML}MKrCEL φ ʱ 7KLGKMRTPKM}¶JJVGK&p!KE UTp¬JHCEp!}&NPOPUÇJ!NCELGp% 2kF CEp}MNPOPUÇJ!NCEL= JVGKÉNOLqJVKMF!UT}&J!NCEL Ô UTÁNP JVCELGNUTLI K&}MCEÁKMpÅiD4UTWFVUÇJ!N}vNOL±JVGKÉICEp!CTLGN} GKMPW9Ê U NOp>JVGK ³ CTDGPCEÁ>I NLqJVK&FVUT}¶JVNOCELzBCTJ!KMLqJVNUTP Ê4± ,J!GK ³ CEDPCEÁÂIlBCTJVK&LiJ!NUTPNOp°p!CTJ"F!UTLGREK U (x − x0 ) = U0 δ(x − x0) NOJ PKMUTWGpQJVC Z L/2 U0 AÊ ÃTÄ dx(∂x φ(x))2 , Hint = 2 4π −L/2 UTLGWÉJVKJ!CTJ UÇP Ô UTÁNP JVCELNUTLF!KjUÇWGp vF H= 8πg Z L/2 1 dx (∂x φ(x))2 + g(∂x ϕ(x))2 g −L/2 AÊ Ã7G , HGKMFK g = 1/p(1 + 2U0/πvF ) NOprJ!GK ³ CEDGPCTÁÂINOLiJ!KMF!UT}&J!NCELB4UTF!UTÁK¶JVK&FMÊ K;4LGWJV4UÇJ>», Ê AÊ Ã7G NpHiD4UTWGF!UÇJVNO}J!GKMLlJVGKSJVCÇJ UTP Ô UTÁNPOJ!CELGNUTL NOpKUT}&J!POÉpCEPOÆÇUTIPKUTLGWlNOJ!pKMNOREKMLGp¬J UÇJ!KMp©UTFK p!NOÁNOPUTFÐJVCÅJVKrK&NREK&LGpJVUÇJVK&pHC GUTÁCTLGN}"CTp!}MNOPPUÇJVCEF&Ê 4 *,1 .!- . .23 .! *,%01 .# !, 3 23 !" GK Ô UTÁNP JVCELNUTL HGN} lWGKMp}MFNIKMpJVGKK&WGREKÁCWGK&p©NOpp!NOÁBGP vUÇLlK&PK&}&JVFCEpJVUÇJVNO}"JVK&F!Á' HH)(|~ 1 H= 2 Z L dxV (x)eρ(x) , AÊ ÃEà 0 HGKMFK x NOp°U}MDGF¬ÆNPNOLGKjUÇF}MCCEF!WGNOL4UÇJ!KxUÇPCELGRÅJ!GK>K&WGREK L NpHJVKxPOKMLGRTJ!zC/JVGKrKMWREKÂUTLW V (x) NpQJ!GKS}MCEL74LGNLRBCTJVK&LqJVNUÇP9F!K&PUÇJ!KMWÉJVCÅJ!GKrUTBGBGPONKMWeK&PKM}¶JVFN} 4K&PW w ' HEÄF6Iq¿ (|~ V (x) = Eh(x) = Ô UTÁNPOJ!CELGNUTL}jUÇL IK vF Bρ(x)/ns HNOJ! ns NpÅJVGKl*J ;CWGNÁKMLp!NCTL4UTPQK&PKM}¶JVFCEL WGK&LGp!N JµEÊGK F!K HF!NOJJVK&LzUÇp Z πvF L A$Ê Ã H H= dxρ2 (x)dx , ν 0 UTLGWeNLÉJ!GKrË CTDGF!NOKMFQpB4UT}MKrNOJQNpQJVF!UTLGp hCEF!ÁKMWlUTp H= πvF X ρk ρ−k , ν k AÊ Ã HNOJ! ρk NOp½J!GKÅË4CEDGFNKMFHJVF!UTLGphCEF!Á C ρ(x) ~ ρ(x) = 1/√L Pk e−ikxρk Ê9GK>}MCELqJVNOLiDGNOJµlKMiD4U ´ JVNOCEL h CEF;J!GNp} NFVUÇP9WGK&LGp!N JµvF!KMUTWGp ρ˙k = ivF kρk Ê4Î7 }MCEÁB4UTFNLGRÂ»Ð Ê AÊ Ã =HN JVÉJVGKS}&PUTpp!NO}jUTP ¿GA Ô UTÁNP JVCELNUTLÂK&iD4UÇJVNOCEL C[ÁCÇJVNCTL);K>4LGW !J 4UÇJ=;K½}jUTL NWGK&[email protected]# ρ UTp7J!GK©}jUTLGCTLGN}MUTP b}MCCEFWGN ´ L4UÇJ!K qk UTLGWÂJ!GK;}&CEF!FKMp!BCELGWNLGR"}MUTLGCELN}jUÇP µ Á CEÁKMLqJVU Ð}jUÇLÂI K7NWGK&[email protected]ÅUTp pk = −i2πρk /νk Ê iDGp U #J!KMFiD4UTLqJVN jUÇJ!NCEL ;K4UjÆTK [ρk , ρk0 ] = − νk δk0 −k . 2π AÊ Ã ? GNprNpSJ!GK Ó UT} ´¬zCC$W$:}MCEÁÁÂDJVUÇJVNOCELF!K&PUÇJ!NCELUTLGWNOJrNprCEIJVUTNLKMWWGDGKJ!CzJ!GK} NFVUÇPNOJµ`C KMWGRTK ;UjÆEKrWGK&J!KMFÁNOLGKMWeIqeÁ UÇRELGK&J!N} GKMPW9ÊGGKrKMPK&}&J!F!CELe}MFKjUÇJ!NCELeCEBKMF!UÇJVCTFpVUÇJ!Np/GKMp A$Ê Gà A [ρ(x0 ), ψ † (x)] = δ(x − x0 )ψ † (x) , HGN} ÕNÁBGPNOKMpJV4U3J°JVKÂÁKMUTp!DF!KMÁKMLqJ°C ,JVGK>KMPK&}&J!F!CELGNO}xWKMLGpNOJµ`CTL:U pJVUÇJVKxCE' L HGNO} ψ† (x) NpHUT}¶JVNOLGRÂJVK&PPpDpHJ!4UÇJ©UTLlKMPOKM}¶JVF!CTLlGUTpHI K&KMLzUTWGWGK&W+Ê KNLqJ!F!CWGDG}MK"J!GK} GNF!UTP[Í[DJJVNLREKMFICEp!CTLGNE} 4K&PW i π X e−α|k|/2 −ikx √ √ e ρk . φ(x) = k L ν k AÊ ÃE¿ Ô K&F!K;J!GKp!ÁUTPPB UT}¶JVCEF α Np/U"pB4UT}&NUTP$}MDJ!CHGNO} ÅNp/NOLiJ!F!CWGDG}&KMWÅNLÍ[DJJVNOLGREKMFPNODNW>J!GKMCEF¬xJ!C NLGpDGF!KHJVK°}MCTLiÆTKMFREKMLG}&KC+JVK°NLqJ!KMREF!UTPp&ÊqGK"WGK&F!NOÆÇUÇJ!NOÆTK½C φ NpBGFCEB CTFJVNOCEL4UTPGJ!CxJ!GK°WGK&LGp!N Jµ~ π A$Ê Ã I ∂x φ(x) = √ ρ(x) . ν GNpHUÇPPCHpQJ!CFK¶´ K$BGF!K&p!pHJVGK Ô UTÁNOPOJ!CELGNUÇL NLÉJ!KMF!Á C φ ' HEÄ)( vF H= π Z L (∂x φ(x))2 dx . A$Ê ÃD 0 GK hCEF!Á·C©JVGKeKMPOKM}¶JVF!CTL CTB K&FVUÇJ!CEF ψ(x) Np hCTDGLGW Ii UTL UTL4UTPOCERT0HNOJV J!GKeBGF!CEBKMF¬JVNOKMpC }jUTLCELGN}MUTP+}&CEL3ÈDGRqU3JVK"ÆÇUTF!NUTIGPOKMpM~ p(x) UÇLGW q(x) HGNO} ÕUTFKSNWGK&Lq[email protected] ρ(x) UTLW φ(x)/√ν ~ ψ(x) = √ √ 1 e−ikx eiφ(x)/ ν . 2πa AÊ HEÄ GNpCEBKMF!UÇJVCTF[CEIqÆ$NOCEDGpPO"WKMBKMLGWGpCELJVGK 4POPNLR UT}&J!CEFMÊ Ë4K&F!ÁNCELSCEBKMF!UÇJVCTF!p UTF!K3iLGCHLxJ!C©UTLqJVN ´ }MCEÁÁÂDJ!K p! C H4U3JSNp©JVGKÂ}&CELGWGN JVNOCE, L hCE>F 4PPONLGR; UT}&J!CEF°NOL`CEFWGKMF°J!CÉNLGpDGF!KÂUTLqJVN ´ }MCEÁÁÂDJVUÇJVNOCEL F!K&PUÇJ!NCELGp {ψ(x), ψ(x0)} = 0%lGNpUTLqJ!N ´µ}&CEÁÁÂDJ U3JVCEFx}jUTL I Ke}&CEÁBDJVK&W5DGpNLGRJVKzÎ;U TK&F¬´ ³ UTÁBGI K&PP8´ Ô UÇDGp!WGCTF/ hCEF!ÁÂDPU eA eB = eA+B−[A,B]/2 [email protected] [A, B] NpU:}¶´µLiDGÁ>I K&FMÊÐGKlI CEpCELGNO} 4KMPOWeLGKMK&WGp©J!CpVUÇJ!N/p # JVGK}MCTÁÁÂD$J UÇJ!NCELvFKMPUÇJVNOCEL AÊ HG [φ(x), φ(x0 )] = −iπsgn(x − x0 ) , HGN} lNLWGDG}MK&pJVC A$Ê Hqà ψ(x)ψ(x0 ) = e±iπ/ν ψ(x0 )ψ(x) . Ë4F!CEÁ JVGK»,Ê AÊ HEà 7K;4LGWJ!4UÇJJVCeNOLGp!DGFKJ!GKUTLqJVN8´µ}MCTÁÁÂD$J UÇJ!NCEL`FKMPUÇJVNOCELGEp hCEFSK&PKM}¶JVFCEL CEBKMFVU3JVCEFp 7K"LGKMK&W JVCxp!K¶J ν = 1/m HNOJ! m NOp;UÇL CWGW NOLiJ!KMREK&FMÊiGNOp7}&CELG}MPODGp!NOCELNp}MCELGpNp¬JVKMLqJ HNOJ!eJ!GKrUTp!pDGÁBJVNOCELvJV4U3J½NOLÉJV[email protected];KrWGKMUTP)HNOJ!zU hF!UT}&J!NCEL4UÇP9iD4UTLqJVDÁ Ô UTPOP 4DGNOW+Ê ±µLCEF!WKMFxJ!C`CEIJVUTNLNL hCTF!ÁUÇJVNOCEL:CELJ!GKvWL4UTÁN}&pxC HKMPK&}&J!F!CELGp CEFxC 4hF!UT}&J!NCEL4UÇPiD4UÇp!N ´ B4UTF¬JVN}&PK&p B7KSLGKMK&WeJ!CÂp!BKM}&@N #JVGK"ICEpCELGN}"¸FKMK&)L 1 4p hDLG}&J!NCEL+Ê$GK»,DG}MPONWGKMUTLÉUT}¶JVNOCEL hCEFJVGNOp I CTp!CELGNOE} 4KMPOWeNp ' Iq¿ ( 1 S=− π Z dτ Z dx[∂x φ(x, τ )(vF ∂x + i∂τ )φ(x, τ )] . ¿E¿ A$Ê HH ,$<?<* ?<12' b [email protected]$<-* ;b;nG$(h b/-.0S'(! oG(%0ZDEb6m- @(-* ;b; 7 3 OO?A 4 q$'; 3j113 ( 3 ' ; A/@ $%+2$(`? ; %,?<; G%n1p-Bn' 13%G +H-H0 '(!,O?' G$$%0##b6 [email protected]$<?<* <;=;m79 3 B12; B+#' ' 3 '; A ; 1%? ')-$; G% 14-,n' 79 3 GKÐCEBKMFVU3JVCEFHN} rNp[NÁBGPON}MN JNLSJVGNOp[iD4UTWGF!UÇJVNO}ÐUT}&J!NCELUTPOPCHp9J!C©WK%4LGKÐJVK¸FKMKML1 p&hDGLG}¶JVNCTL L hDGLG}¶JVNOCEL:C ,JVKÂI CTp!CELGNO } 4K&PW+ÊGNOp¸FKMK&)L 1 p G(x, τ ) = hTτ φ(x, τ )φ(0, 0)i *lJ!GKÂ}MCTF!F!K&PUÇJ!NCEA hDGLG}¶JVNCTLlNOpWG%K GLGKMWzIi JVGKWGN 9K&F!KMLqJ!NUTP+K&iD4UÇJVNOCEL AÊ HG (i∂τ + vF ∂x )∂x G(x, τ ) = 2πδ(x)δ(τ ) . GKSJVGK&F!ÁUTP¸SF!K&KM)L 1 p hDGLG}&J!NCELeNOpCEIJ UÇNLGK&WzUÇp HNOJ! $ β = 1/kB θ Ê x/vF + iτ G(x, τ ) = − ln sinh π β AÊ H ? , K}&CELGpNWGK&FLCUSB CENOLqJ,}MCELqJVUT}&JNLÂJ!GK hFVUT}¶JVNOCEL4UTP$DGUTLqJVDGÁ Ô UTPPF!KMRTNÁKHN} NpJµBGNO}jUTPOPO UT} GNOK&ÆEK&WIq:BPUT}&NLGRÉÁK&JVUTPPON}rRqUÇJVK&pCEL:JVCEBCJVGKÅJ*7CzWNÁKMLGpNCEL4UÇPKMPOKM}¶JVF!CTLREUTpSNLUÉGNRT B K&F!BKMLGWN}MDPUTF/ÁUTRELK&JVNO} 4K&PW+Ê K©UTBGBGP ÅU"ÆECEP J UTRTKI K¶*J ;K&KML J!GK*J ;CrRqUÇJ!KMp HN} UTF!KBGPUT}MK&W CELzJ*7CÉKMWREKMp"C hFVUT}¶JVNOCEL4UTPiD4UTLqJVDÁ Ô UTPP 4DGNOW ËNREDF!K AÊ D ¶ÊÎ7lÆTUÇFNLGRJVGKxRqUÇJ!KxÆECTPOJ UÇREK ;K}MUTLzp HNOJ!} $ hF!CTÁ U ;KMU ÉI4UT} ip!}MUÇJ!J!KMFNLGRp!N JVD4UÇJ!NCELvJ!C UÅp¬JVF!CTLGRIGUT} ip!}jU3J!JVK&F!NOLGR pNOJVDGUÇJVNOCEL+Ê ±µL JVGK hCEFÁK&FÂ}jUTpK ,JVK Ô UTPP7PNODNW F!K&Á UÇNLGpÂNOL2CELGKÉBGNOKM}MK /JVGKeK&LiJ!NOJ!NKMp HN} 2JVDGLGLKMPHUTFK KMWGRTK½iD4UTpNB4UÇFJVNO}MPOKK$}MN J UÇJ!NCELGp&Êq±µL JVK½PUÇJ!J!KMF,}jUÇp!KiJVK Ô UTPP4DNWNpp!BGPONOJÐNLqJ!CrJ*7C>BGNOKM}&KMpQUÇLGW I K¶*J ;K&KMLlJVGK&p!K*J ;C;4DGNOWGp 4CELPOvK&PK&}&JVFCELGpH}jUÇLlJ!DGLGLGK&P Ê Ô KMFKjU #J!KMF 7K>HNOPPWNp!}&DGp!3p hCEF/J!GK ;KMU ÅI4UT} ip!}MUÇJ!J!KMFNLGRx}jUTpK HGK&F!KHJVGKHBGqp!NO}Mp;C +Ë = Ô » iD4UTp!NOB4UTF¬JVN}&PKNprÁCEpJrCEIqÆNCEDpMÊ GK WGK&p!}&F!NB$JVNCTLC QJ!GK p¬JVFCELGR`I4UÇ} p}jUÇJJVK&F!NLR`}jUTpK }MUTLIK F!KMUTWGNP rCEIJ UÇNLGK&WÅDGp!NOLGRrU"WGD4UTPONOJµrJVFVUÇLG/p hCEFÁ U3JVNCTL+ÊTÎ7KM}jUÇDGp!KJVKQJVFVUÇLGp!BCEFJÐIK&*J 7KMK&L*J 7CKMWGREK&p NpQCTDJ©CK&iDGNPONIGFNDGÁCiNOJNOpLGKM}&KMp!p!UTFeJ!CFKMpDGÁKSJ!GK Ó KMPW$$pChCTF!ÁÂDGPUÇJVNOCELvNOLvJVGKLGK $J©B4UÇFJjÊ 4 * .- +% # +!,!,..!,- [Í K&J DGp }MCELGpNWGK&FU©ppJ!KMÁ F!K&BGF!K&p!K&LiJ!KMWÅIqJVGKÐJVNOÁK ´µNOLGWGKMBKMLWGKMLqJ Ô UÇÁNOPOJVCTLGNUTL H = H0 +Hint HGKMFK H0 F!K&BGF!K&p!K&LiJ!p hF!KMK B4UTFJ!N}&PKMpxUTLGW Hint WGK&p!}MFNIKMpÅUÇLNLqJVK&FVUT}¶JVNOCEL HGNO} Np bWGN}&DGPOJ JVC UTWGWF!KMppMÊH±µL Á UTLq I CWBipN}Mp ' GG Ã7EGGH)(9QNOJeNOpÉ}MCELqÆEK&LGNK&LqJzJ!C 7CEF/ HNOJV U N}61 p JVGK&CEF!K&Á CTF,CELGKC NOJ!p,REKMLGK&FVUTPON MUÇJVNOCELGpNLCEFWGKMF/J!Cr}&CEÁBGDJVKHBGFC$WDG}&J!pC hK&F!ÁNCELUTLWICEpCEL CEBKMFVU3JVCEF=p HGK&F!K©JVGK½CEB K&FVUÇJ!CEF!p7UTFK½JVNOÁK ´µWGK&B K&LGWGK&LiJM~ AH (t) = eiHt Ae−iHt NLJVGK Ô K&Np!K&LiI K&F!R BGN}¶JVDGFKTÊ[GKBGFCEIGPK&Á HN JVJVGK Ô K&NpKMLiI K&F!RzF!KMBF!KMpKMLqJ U3JVNCTLNp"J!4UÇJJVKCTB K&FVUÇJ!CEF!p}MCTLiJVUTNLp JVGK µWGN }&DGPOJ B4UTFJ #JVGK NLqJVK&FVUT}¶JVNCTL B4UTFJ xC HJVK Ô UTÁNP JVCELGNUTL+Ê[± JÅNOLGWGDG}&KvJVGKÉBF!CEIGPOKMÁ·C JVF!UTLGp!PUÇJVNOLGRvJVKMp!KBGF!CWGDG}¶JVprNLqJVClNLqJ!KMFVUÇ}&JVNOCELF!K&BGF!K&p!KMLqJVUÇJVNOCELBGFC$WDG}&J! p HGK&F!KJVGKCEBKMFVU3JVCEFp ¿I t = −∞ µ1 |Gi µ2 OFF t∼0 µ1 µ2 ON t = +∞ µ1 |G0 i PSfrag replacements µ2 OFF |G0 i 6= eiη |Gi jq '3OD#%)'(! ?qDAOD* q 8 c' '(! [email protected]+; +G$E+* q'3OOD t = +∞ $'/; ' ; #13"D' - '3OD# G&A">2<$0,!E$%#'<h t = −∞ WGK&ÆTKMPOCEBDGLGWKMFxJ!GK NOL4DGK&LG}MKCQJ!GK µKjUTp¬ LCEL$´µNOLqJVKMF!UT}&J!NLGR Ô UÇÁNOPOJVCTLGNUTL H0 CTLGPO~ A(t) = eiH t Ae−iH t Ê ³ CELGp!NOWGKMFJVGK°REF!CEDLGWvpJVUÇJVK"U1ÆTKMF!UTREK"C[UxJVNOÁK ´µCEFWGKMFKMW BF!CWGDG}&JQC Ô K&NpKMLiI K&F!RÅCEBKMF!UÇJVCTF!pM~ 0 0 HN JV t0 > t1 > t2 > t3 > . . . $A Ê H A GKMLeJVF!UTLGpPUÇJ!NLGRÂJ!CJ!GKNLqJVK&FVUT}¶JVNOCELÉF!K&BGF!K&p!KMLqJVUÇJVNOCEL)GJVKrK¶ÆECEPODJVNOCELeCEBKMFVU3JVCEFFKjUTWGp&~ hAH (t0 )BH (t1 )CH (t2 )DH (t3 ) . . .i 0 S(t, t ) = T exp −i Z t dt1 Hint (t1 ) t0 GKBGF!CWGDG}¶J©CCTF!WGK&F!KMWlCEBKMF!UÇJVCEFpQJVGK&LzIKM}&CEÁKMpM~ . hS(−∞, +∞)T (AI (t0 )BI (t1 )CI (t2 )DI (t3 ) . . . S(+∞, −∞))i , AÊ Hq¿ AÊ HI HGKMFK T NpQJ!GKSJVNÁK"CEFWGKMFNLGRÅCTB K&FVUÇJ!CEFMÊ GK½}&KMLqJVF!UTPDGUTLqJVNOJµÅJ!C>}&CEÁBDJVKJVK½BGFC$WDG}&J7NL»,Ê A$Ê HI ,IqDGpNLGR( N}61 pJ!GKMCEFKMÁ Np JVGK½Ài´ Á U3JVF!N S(+∞, −∞) Ê GK&LJ!GKHppJ!KMÁ NpUÇJ MK&F!CJVK&ÁBKMF!UÇJVDGFKHCEFÐNLKMiDGNOPNIF!NDÁ$3JVGK REF!CTDGLGWÉpJVUÇJVK hCEF;J!GKMFÁ UTP ÐK $B K&}&J U3JVNCTLlC J!GNpÀ *ÁUÇJVFN NpÈDGpJ©UÂBGGUTp!EK UT}¶JVCE%F IKM}MUTDGp!<K 7K UTp!pDGÁKÐJ!4UÇJJVK7BKMF¬JVDGFI4UÇJVNOCELNp+J!DGF!LGK&W>CELxUTWGNUTI4UÇJ!N}MUTPP EʶGNpÁKjUTLGp[J!4UÇJ hS(+∞, −∞)i = N} 61 pJVGK&CEF!K&ÁzÊ eiη ÊGGKMF%K hCEFKrJ!GK *BGFC$WGD}&J©NpKMUTp!NOPOv}&CEÁBGDJVK&W HNOJVÉJ!GKGKMPOBzC ¿D Ô C;K¶ÆEK&F/N QJVGK ppJ!KMÁNOpxCTDJÂCHK&iDGNPONIGFNDGÁC9JVKMF!K NprLGC`RED4UÇFVUTLqJVK&K JVGUÇJ>J!GK p¬$p¬JVK&Á F!K¶JVDGFLJVCzNOJ!prNOLGNOJ!NUTPpJVUÇJVK hCEFrUTpÁBJVCTJ!N}MUTPP `PUÇF!REKÅJVNOÁKÇÊË4CEFKUTÁBGPK[NLËNREDGFK AÊ@GjÄA7K NPOPDGp¬JVFVU3JVKJVGNOp[BGGKMLCEÁK&LGCEL<HNOJ!UiWGFC$W$$LGUTÁNO}jUTPÇBGN}¶JVDGFKTÊj&7C½FKMp!K&FÆTCENFpj4UjÆNLR©[email protected] KMF!K&LqJ PK¶ÆEKMPOp TUTF!K7LGCTJ,}&CELGLGK&}&JVK&W U3J t = −∞ #JVGK UTDG}MK¶J,Np/p HN JV} GK&WC ¶Ê KQp HNOJ!} UTWGNUTI4UÇJ!N}jUÇPPO JVGK UTDG}&K&J½CELÕIK&*J 7KMK&L`JVGKr*J ;CF!K&p!KMF¬ÆECENOF!%p 9UTLW`Up¬J UÇJ!NCELGUTF GC NOpHKMpJVUTIGPONp!KMW+Ê ± ÐUÇJ½PU3JVKMF JVNOÁK t = +∞ .;KÅp HNOJV} C JVK UTDG}MK¶JUTRqUÇN)L 6;KÅLCTJVNO}MK>J!4UÇJ"J!GK>*J 7ClFKMpKMFÆTCENFpUTF!K>LGC UÇJJVGKpVUÇÁKPK¶ÆEKMPÐIGD$J>WGN 9K&F!K&Li( J hF!CTÁ JVGKPK¶ÆEKMPOpxNOLJ!GKpJ U3JVKUÇJ t = −∞ Ê KDGLGWGK&F!p¬J UTLGW NLJVGNOp>BGNO}&J!DGF!KJV4U3J>JV K ;UÇJVK& F 4C ¡JVF!CEDGRTJVGK UTDG}&K&JÂÁKjUÇLGprJ!GKJ!DGLGLGK&PNOLGRÕC QB4UTFJ!N}&PKMp hF!CEÁ«CELGKÉFKMpKMFÆTCENFÂJ!C`JVGKÉCÇJVGK&F HN} NpÂWGK&p!}&F!NIKMW2NOLJVGKvJ!DGLGLGK&PNLR Ô UTÁNP JVCELNUTL Hint Ê ²½C [JVGKREFCEDGLGWpJVUÇJVKUÇJ t = +∞ NOprLCÕPCTLGREKMFSFKMPUÇJVK&WJ!ClJ!GKRTF!CEDGLWp¬J UÇJ!K UÇJ t = −∞ IqzJ!GKÂBG4UÇp!: K UT}¶JVCEF&~ S(+∞, −∞)|Gi = |G0i 6= eiη |Gi Ê CvF!K&ÁK&WlJVGNOp°BGFCEIGPK&$ Á Ó KMPOWp! BGF!CTB CEpKMWeU>L%K }MCELqJ!CEDGF 7HGN} ÉNOp;p!C HLeNOLvËNOREDGF!,K AÊ GGEÊGNpQ}&CELqJVCEDF }&CEF!FKMpB CELWGNLGR>LG%K JVNOÁKÅCEFWGKMFNLGReCEBKMF!UÇJVCTF TK [RECKMp hF!CTÁ t = −∞ JVC t = +∞ UTLGWIGUT} :JVC t = −∞ ' GGB ( Ê Î;KM}MUTDGpKeJ!NÁKMpÂCTL2J!GKePC 7KMFÅ}MCTLiJ!CEDGFUTFK bPUTF!REK&F JVGUTL JVNÁKMp>CEL JVGKeDGBGBKMFÅ}&CELqJVCEDF /JVK BGF!CWGDG}¶JHC CEB K&FVUÇJ!CEF!pH}MUTLeI <K HF!N J!J!KMLzUTpM~ AÊ HD hTK (AI (t0 )BI (t1 )CI (t2 )DI (t3 ) . . . SK (−∞, −∞))i , HGKMFK>JVK>NLqJ!KMREF!UTPC ÆTKMF°J!GK Ó K&PWp!Õ}MCELqJVCTDGF K RECKM>p hF!CEÁ −∞ JVC +∞ UTLGWzJVGK&L:I4UT} ÉJVC −∞ ÊG±µLlJ!GNp}MUTp!K ;K4UjÆEK SK (−∞, −∞) = TK exp −i ( Z dt1 Hint (t1 ) K X Z = TK exp −i η η=± +∞ dt1 Hint (t1 ) −∞ ) . AÊ qÄ ½² CTJ!KJVGUÇJNL2REK&LGKMF!UTP8MJ!GK/JVNOÁK&p[UTBGBKjUÇF!NLRQNL"J!GK,CEBKMFVU3JVCEF+BF!CWGDG}&J AI (t0 )BI (t1)CI (t2 )DI (t3 ) }jUTLeIKPC}jUÇJ!KMWeKMN JVGK&FCELlJ!GKSDGBGBKMF©CEFCELÉJ!GKPC;K&FH}MCELqJVCTDGFMÊ GKŸSF!K&KM)L 1 "p hDGL}&JVNOCELlNp½UTLÕK GUÇÁBGPOKC ÐUÅJVNÁKCEF!WKMF!K&W`BGFC$WGD}&JjÊ GK¸FKMK&L)1 p>hDGLG}¶JVNCTL UTp!pC$}&NUÇJ!KMW HNOJ!JVGKÉJ*7CÇ´µIFVUTLG} KMp Ó K&PWp5}&CELqJVCEDGFÅNOpÂJVGK&F!KhCEF!KlU 2 × 2 ÁUÇJ!F!N IKM}jUÇDGp!K JVGK&F!KrUTFEK hCEDGFHB CTp!p!NOIGPK°CEF!WGK&F!NOLGREpM~ 0 G(t − t ) = G++ (t − t0 ) G+− (t − t0 ) G−+ (t − t0 ) G−− (t − t0 ) G0 (|t − t0 |) G0 (t0 − t) = , G0 (t − t0 ) G0 (−|t − t0 |) AÊ G HGKMFK G0 (t) }jUTLlIKr}&CEÁBGDJVK&WChFCEÁ J!GKSJVGK&F!ÁUTP¸SF!K&KML)1 p hDGLG}&J!NCELeDGpNLGRU N}vFCTJ U3JVNCTL+Ê ²½CTJ!KQJV4UÇJ,J!GKDGpK©C Ó K&PWp!¸SF!K&KML)1 p hDGLG}&J!NCELUÇPPCHp/DpÐJ!CEHF!N JVKWGNOF!KM}¶JVP xJ!GKCEBKMFVU3JVCEFp [email protected] TK°J!GK}MDGFF!KMLqJ IqÉ} C$CEpNLGRUTBGBF!CEBGFNUÇJ!KMP J!GKSJVNÁKMp;CELlJ!GK}MCELqJ!CEDGFMÊ Ë4CEF;KUTÁBGPK n o AÊ ià hψ2† (t)ψ1 (t)i = hTK ψ1 (t+ )ψ2† (t− )SK (−∞, −∞) i . ±µL:JVGNOpSKMiD4UÇJ!NCE)L JVGKCEBKMF!UÇJVCEFpSNL:PO%K #JS4UÇLGWpNWGKUÇF!KÅNL:J!GK Ô K&Np!K&LiI K&F!RlBGF!K&p!KMLqJVUÇJVNOCELUÇLGW CELGK&p;NLJVGK°F!NRTiJ74UTLWvp!NOWGK"UTFK°NOLJVK°NLqJ!KMFVUÇ}&JVNOCEL BF!KMpKMLqJ U3JVNCTL HNOJ! JVGK©JVNOÁKHCEFWGKM=F hCEPPOC Hp JVGK Ó K&PWp:}MCELqJVCTDGFMÊ GKUjÆEKMF!UTREKÂCÐCEB K&FVUÇJ!CEF!p°BGF!CWGDG}¶JSCELÕJ!GKÂF!NOREqJ"4UÇLGW:p!NOWGKÂNp°UJVNOÁK CEF!WKMF!K&WziD4UÇLiJ!NOJµ HGN} l}jUTLlIKS}jUTPO}MDGPUÇJVK&WlIqÉDp!NLR NO} .1 pJVGK&CEF!K&ÁzÊ 4 ) 1.!,- 1! ±µLvJVGNOpQp!K&}&J!NCEL)F;KS}&CELGpNWGK&F;JVGKSp¬J UÇJ!NCELGUTF}MDF!F!K&LqJ 4CHNLGRÅIK&J*7KMK&LlJ*7CKMWGREKSp¬J UÇJ!KMp4HGNO} NpÐ}jUÇPPK&WI4UT} ip!}MUÇJ!J!KMF!NOLGR>}&DGF!FKMLqJjÊGK½KMWGREK©pJVUÇJVK&p7NOp7NOLÅJVGK>hFVUT}¶JVNCTL4UTPGiD4UTLqJ!DGÁ Ô UTPOPGF!K&RENÁKTÊ IEÄ + K PSfrag replacements t0 × t × −∞ +∞ − ,n' 0G($% '(! - ; 3A1 1'#'Z GK Ô UTÁNOPOJ!CELGNUÇLEHGN} ÅWGK&p!}&F!NIKMp/J!GKQp¬$p¬JVK&ÁNp H = H1 +H2 +HB HNOJ! H1/2 WGKMp}MFNIKMp/JVK PK#J FNREqJHKMWREKxp¬J UÇJ!KMp H1/2 = (vF /π) R L(∂x φ1/2 (x, t))2dx 6HN JV φ1/2 UTFKrJ!GKx} NFVUÇP ICEpCELGN} 4KMPOWGp hCEFxK $}MNOJVUÇJVNOCELGp<HNOJ!} 4UÇF!REK νe 0UTLGWJ!GKJ!DGLGLGK&PNOLGR Ô UTÁNPOJ!CELGNUTL HB WGKMp}MF!NOIGNLRzJVK }MCEDBGPNOLGRÅI K¶*J ;K&KMLlJVGKS*J 7CK&WGREKMp 1 UTLGW 2 ~ A$<Ê H HB = Γ(t)ψ2† (t)ψ1 (t) + Γ∗ (t)ψ1† (t)ψ2 (t) , HGKMFK M1/2 i√νφ (t) AÊ ψ1/2 (t) = √ e , 2πα HNOJ!GNL M1/2 NpHU Ó POKMNL UÇ}&JVCTFHGNO} HNPP I KSCTÁN J!JVK&WÉGKMFKjU#J!KMFIqÉLGCÇJVN}&NLGRÂJ!4UÇJ M1/2 2 = 1Ê ¹½BCELÂJVKRqUTDGREK7JVFVUÇLG/p hCEFÁ U3JVNCTLxJVKQJVDGLGLKMPNOLGRSUTÁBGPONOJVDWGK7IK&*J 7KMK&LJ!GKQKMWREKpJVUÇJVK&pNOp Γ(t) = K hFVUÇ}&JVNOCEL4UTP} 4UTF!RTKTÊGK½REUTDGREK hDGLG}¶JVNOCEL χ WGKMBKMLGWp;CELGP ÅCEL Γ0 e−ie χ(t)/c HGKMFK e∗ = νe Np,JVG? JVNOÁK hCEFSUv}&CELGpJVUTLqJWG}ÅIGNUTp V0 NÁB CEpKMW`IK&*J 7KMK&L*J 7CeKMWGRTKMpM~ χ(t) = cV0t 9p!CvJVGUÇJSNLzJVGNOp }jUTpK Γ(t) = Γ0 eiω t HNOJ! ω0 = e∗ V0 Ê GKÂI4UT}ip}jUÇJJVKMFNLGRÉ}&DGF!FKMLqJCEBKMF!UÇJVCEF½}jUTLÕI KÂWKMF!N ÆEK&W'hF!CEÁ JVK Ô K&Np!K&LiI K&F!RÉKMiD4U3JVNCTL:C ÁCÇJVNCTL hCEFQJVKrWKMLGpNOJµÉCEBKMF!UÇJVCTF GCEF©UTP JVK&F!L4UÇJ!NOÆTKMPOIqÉ}jUÇP}MDPUÇJ!NLGR IB = −c∂HB /∂χ GJVGK&L 1/2 ∗ 0 IB (t) = ie∗ Γ(t)ψ2† (t)ψ1 (t) − Γ∗ (t)ψ1† (t)ψ2 (t) . AÊ ? GKU1ÆTKMF!UTREK IGUT}ip!}jU3J!JVK&F!NOLGRÕ}&DGF!FKMLqJÂNOprK $BGF!K&p!p!K&W DGp!NOLGR Ó K&PWp!}MCELqJ!CEDGF HGN} UTPPOCHp"J!C JVFKjUÇJHLGCTLGKMiDGNOPNIF!NDÁ pNOJVDGUÇJVNOCEL+~ hIB (t)i = n o R 1X hTK IB (tη )e−i K dt1 HB (t1 ) i . 2 η CPOC7KMp¬J©CEF!WGK&FHNLÉJVGKSJ!DGLGLGK&PNLRUTÁBPNOJ!DGWGK hIB (t)i = Γ0 AÊ A 7K4UjÆEK e∗ Γ20 X η1 eiω0 t+i1 ω0 t1 2 ηη 1 1 h i() h i(1 ) † η † η1 η1 η × TK ψ2 (t )ψ1 (t ) ψ2 (t1 )ψ1 (t1 ) . AÊ i¿ GK©}MCEFF!KMPUÇJVCTF/NpÐWGN 9K&F!KMLqJ3hF!CTÁ MK&F!C>CTLGPOHGKML 1 = − iJ!4UÇJÁKjUTLGp,J!GK½iD4UÇp!NBGUTFJ!N}MPOKMpÐUTFK }MCELp!KMF¬ÆEK&W:NLeJVGKrJVDGLGLKMPNOLGRBGF!C}MK&p!pMÊ ±µLzJ!GK>}MUTP}&DGPUÇJ!NCEL;K>UTF!KrPK&WzJ!C NLqJ!F!CWGDG}MKJVKx} GNOFVUTP ¸FKMK&)L 1 p hDGLG}&J!NCELC QJVKvI CTp!CELGNO } 4K&PW U3JÂBCEp!N JVNCTL x = 0 HN} WGCKMpÂLCTJÂWGK&B K&LGW CELJ!GK } GNF!UTPN Jµ 1/2 ~ G ηη 0 0 D n η 0η 0 (t − t ) = TK φ1/2 (t )φ1/2 (t ) oE 1 TK φ21/2 (tη ) − 2 IG 1 D n 2 0η0 oE TK φ1/2 (t ) . − 2 Ê A I GNp¸FKMK&L)1 phDGLG}&J!NCELNpÂUÇp!p!C}MNUÇJVK&W HNOJ!JVKJ*7CÇ´µIGF!UTLG} GK&p Ó KMPW$$p }MCELqJ!CEDGFÅUTp(;KÉGU1ÆTK WGNp}MDGpp!K&WI%K hCEFKTÊ[GKUjÆEKMF!UTREK}MDGFF!K&LiJx}jUÇLLGC¡IKK$BGFKMp!pKMW2UTprUTLNLqJ!KMREF!UTP/C1ÆEK&FrJ!NÁKÅC Ó KMPW$$p`¸SF!K&KML)1 p hDGLG}¶JVNCTL HGKMFK AÊ D Z +∞ ie∗ Γ20 X η−η dτ sin(ω0 τ )e2νG (τ ) , η hIB (t)i = − 2 2 4π α η −∞ =4LGN JVKJVK&ÁBKMF!UÇJVDGFKMp%iJ!GK°¸FKMKML1 p=hDGLG}&J!NCELNpÐREN ÆEK&LIq pKMK½»Ð Ê AÊ H ? ¶~ τ = t − t1 Êi®QJ Gη−η (τ ) = − ln sinh π (ητ β sinh AÊ ?TÄ + iτ0 ) , iπτ0 β HGKMFK τ0 = α/vF Êi®½BGBGPONOLGRSJVGNOp;¸SF!K&KML)1 p hDLG}&J!NCEL)7K©}jUTLCEIJ UÇNLÅJVGK½UTL4UÇPOiJVNO}jUTPGK $BGF!K&p!p!NOCEL hCEFQJVKxUjÆTKMFVUÇREK}MDGFF!KMLqJ°UTp e ∗ Γ2 hIB (t)i = 2 2 0 2π α Γ(2ν) HGKMFK 4 Γ α vF 2ν 2π β 2ν−1 sinh ω0 β 2 ω0 β Γ ν +i 2π AÊ ?7G 2 , NpQJVKrREUTÁÁU:hDGLG}¶JVNOCEL+ÊG®QJ &KMF!CJVK&ÁBKMF!UÇJVDF!KGJ!GKrUjÆEKMF!UTREK}MDF!F!K&LqJ°NOp a vF 2ν pREL AÊ ?Eà (ω0 )|ω0 |2ν−1 . ) 1.!,- 1! ! #. ±µL JVGNOpÅp!KM}¶JVNOCEL ;Ke}MCTLGp!NOWGKMFÂJ!GKepÁÁK&J!F!NMKMW2LGCTNp!KÉIKM}MUTDGpKlPUÇJ!J!KMFÂCTL LCENpKeUÇJ ω = 0 NOp }MCELp!NWKMF!K&W)4p!CJVGKNOp!p!DKxCpÁÁK¶JVFN M K&WlÆTKMFp!DGp°UTpÁÁK¶JVF!NM K&WlWC$K&p½LGCÇJ½ÁUÇJJVKMF&Ê4¹½pNLGRJVK e∗ Γ20 hIB (t)i = 2πα2 Γ(2ν) pÁÁK¶JVF!NO}"}MCTÁÂIGNLGUÇJVNOCELÉC}MDGFF!KMLqJ v * }MDGFF!KMLqJ©}MCTF!F!K&PUÇJ!CEF!p S(t, t0 ) = hIB (t)IB (t0 )i + hIB (t0 )IB (t)i − 2hIB (t)ihIB (t0 )i oE XD n R η 0−η −i K dt1 HB (t1 ) − 2hIB i2 , = TK IB (t )IB (t )e AÊ ?H η JVCPC7KMp¬J°CEFWGKMFHNOLeJ!GKSJVDGLLGKMPONLGRUTÁBGPN JVDGWK Γ0 4NOJHNOp©LCTJHLGKM}&KMppVUTF¬zJ!CK $B4UTLGWlJVGK Ó KMPW$$p K&ÆTCEPDJ!NCELÉCTB K&FVUÇJ!CEFI K&}jUTDp!KJVGK}MDF!F!K&LqJ½N [email protected]}MCTLiJVUTNL Γ0 Ê S(t, t0 ) = −(e∗ )2 Γ20 × TK = X 1 eiω0 t+i1 ω0 t 0 η1 h ψ2† (tη )ψ1 (tη ) i() h ψ2† (t0−η )ψ1 (t0−η ) i(1 ) AÊ ? (e∗ )2 Γ20 X 0 2νGη−η (t−t0 ) cos(ω (t − t ))e = S(t − t0 ) . 0 2π 2 α2 η Ë4F!CEÁ J!GNp©K $BGFKMp!pNCEL JVK>p!BKM}¶JVF!UTPWKMLGpNOJµzC,I4UT}ip}jUÇJJVKMFNLGR }MDGFF!KMLqJSLCENpKxNOp½CEI$J UTNOLGKMWzIi }jUTPO}MDGPUÇJVNOLGR>J!GKrË CTDGF!NOKMF;J!FVUTLGphCEF!ÁlÊ®QJ?4LNOJVKSJ!KMÁB K&FVUÇJ!DGF!KJVGKLGCTNp!KU3J &KMFCÇ´ hFKMiDGKML}&eNp (e∗ )2 Γ2 S(ω = 0) = 2 2 0 π α Γ(2ν) α vF 2ν 2π β 2ν−1 Iqà cosh ω0 β 2 ω0 β Γ ν +i 2π 2 , AÊ ? ? m<; +.< 'I 08 5$`1:G$$-0##E+>h ;=; 13 ν = 1/3 νL = 2/3 IB (+ ; % !# '* 9-!O Z 1; '2% !E' @(%+2$(`? ; % 3<113p;=b <3B; 1%- ' 3'##13 ; b$ ( e/3 * c W0%%3<D +q- &<*,< #16; 1%- ' -$;<+q- %+* 1' q<jq#* 2C- θ D*82- <I* νL = 4 θ UTLGWeNLÉJ!GKPNÁNOJMK&F!CJVK&ÁBKMF!UÇJVDGFK (e∗ )2 Γ20 S(ω = 0) = πα2 Γ(2ν) α vF 2ν |ω0 |2ν−1 . AÊ ?GA ±µL }&CELG}&PDGpNCEL);UÇJ MK&F!C VJ K&ÁBKMF!UÇJVDF!K ;K:FKM}MC1ÆTKMFeJVGKÀ$} CTJ!J iq´ PN ÇK:F!K&PUÇJ!NCEL HN JV VJ K e∗ = νe ~ A$EÊ ?E¿ S(ω = 0) = 2e∗ hIB (t)i . ®QJ 4LGN JVKSJVK&ÁBKMF!UÇJVDGFK J!GKp!GCÇJ J!GKMFÁ UTP9LGCENpK}MF!CTp!p!C1ÆTKMF©NpF!K&}MC1ÆEK&F!K&W A$EÊ ?I S(ω = 0) = 2e∗ hIB (t)i coth(ω0 β/2) . GKMpK7FKMpDGPOJ!pUTFKÐ}MCELGpNp¬JVKMLqJ HNOJ!rJ!GK7FKMpDGPOJ!p pGC HLÂNLr¯H%K bÊ ' Hqà (iNOLJ!GKREKMLGK&FVUTP hF!UTÁ%K 7CEF EhCEF JVFKjUÇJ!NLGRLGCELKMiDGNPONIGFNDGÁ JVF!UTLGp!BCEF¬J;BGKMLGCEÁKMLGUxNL Í[DJ!J!NLGREK&FÐPNiDGNOWGpMÊqK"À$} CTJ!J iF!K&PUÇJ!NCEL NLlJVGK(hFVUT}¶JVNCTL4UTP[iD4UTLqJVDÁ Ô UTPPK% KM}¶JUTPOPCHpJVGKxWGNFKM}&J°CEIGpKMFÆÇUÇJ!NCELzCU hF!UT}&J!NCELGUTP[} 4UTFREK HGN} A 7KrWNp!}&DGp!p©NLÉJ!GKLGK J©p!K&}&JVNOCEL+Ê hFVUT}¶JVNOCEL4UTP } 4UTF!RTK 4 L * , # .# ! # % %0 .#! *, 1- ³ TU L'hFVUÇ}&JVNOCEL4UTP} 4UTFREKÅ}jUTFFzJVGKÅ}&DGF!FKMLqJ %GK 4F!p¬JBGF!CEBCEp!UTPC3hF!UT}&J!NCEL4UÇPPOe} GUTF!REK&WK$}MN8´ J UÇJ!NCELp"UTIC1ÆEK>J!GKÂRqUTB';UTpSpGCHLIq`¯rÊ ÍUTDGREGPONL ' G1Äqà ( Ê+À$CCEL:U#J!KMF ¨ Ê ³ Ê+ p!DGN/pDGREREK&pJ!KMW JV4U3JQp!CTJLGCENOp!K"}&CEDGPOWÉF!K&ÆTKjUTP JVGK"} GUTF!REK°C JVGK&p!KSDGLiDGpD4UTP+K }&NOJVUÇJVNOCELGp ' G1ÄG (8UTLGWvÑÂÊG¸ÊBKML K JVK&LGWGK&WzJVNpH}MCEL}MKMB$J°C hFVUÇ}&JVNOCEL4UTP K $}MN J UÇJ!NCELpQJVCJVGKRqUTBPKMppHÁCWGKMp}MUTF!F¬NLGRJVGK}&DGF!FKMLqJ UÇJ©JVGKxKMWGREKxCÐ}MPOKjUTLzpVUTÁBGPOKMp©DGp!NOLGRÉU Í[DJJVNOLGREKMF©PNOiDGNWlÁCWGK&P ' HEÄ)(|Ê ±µL`JVGNOp>hF!UTÁK ³ Ê Ó UTLK UTLGWÊ[ËNpGKM F ' HEà (/BGFCEBCEp!K&W`JVCÉWGK¶JVKM}¶J"JVKÅDGUTp!NOB4UTFJ!N}&PKMp½DGp!NOLGR J!GKÂp!CTJLGCENOp!KÂUÇp!p!C}MNUÇJVK&W HNOJ!<U 7KjU ÅJVDLGLGKMP }&DGF!FKMLqJ;IK&*J 7KMK& L hFVUT}¶JVNOCEL4UTP$KMWGREK&p7J!GF!CEDREJ!GK½Ë = Ô » 4DGNOW+ÊqGKMFKjU #J!KM%F IH NLe¯HK%hp&Ê' HH)GG ? (84N J4;UTpHp!GCHLeJV4UÇJ©U CENpp!CELGNUTLÉiD4UTpNB4UTF¬JVNO}MPK°p!GCTJHLCENpKSÁÂDGpJHIKSF!K&}MC1Æq´ KMFKMW hCTF 7KjU }&CEDGBGPONLGRlUTLWJVGUÇJ>N JVpÁKjUTpDGF!K&ÁK&LiJxp!CEDGPWREN ÆEKUlWGNOF!K&}&JxWGK&J!KMF!ÁNLGUÇJVNOCELC JVG<K hF!UT}&J!NCEL4UÇP9} 4UTFREKTÊ GKMpKÕJVGK&CEF!K¶JVNO}jUTPBGFKMWGNO}&J!NCELGpC S}&DGF!FKMLqJ *iLGCENpKz} GUTFVUT}¶JVK&F!Np¬JVNO}Mp NOL hF!UT}&J!NCELGUTPiD4UTLqJVDGÁ Ô UTPOP;K9K&}&J4UjÆEKlI K&KML2Æ[email protected]&W NOL FKMÁUTF 3UTIPKvBCENLqJ}&CELqJ UT}¶JK $B K&F!NÁKMLqJ!pU3J 4PPONLGR' UT}¶JVCEF Ä G(|Ê,¯©K bÊ ' q)Ä ( QUTp ν = 1/3 NL KMN MÁUTLGL NOLGpJ!NOJVD$JVKvUÇLGW2NL2ÀUÇ}MPUjUÇJÂJ!GKÉpVUTÁK J!NÁK ' q B K&/F hCEFÁK&WrUÇJ PC J!KMÁB K&FVUÇJ!DGF!KÐNLJVK7pGCTJ LGCENOp!KÐWGCEÁNL4U3JVKMWrF!K&RENÁK %HGNPOKЯ©K bÊ ' G(iDGp!K&WÅ?U J JVCHJVGKJVGK&F!ÁUTP ´ p!GCÇJLGCENpK7}&F!CEpp!C1ÆEK&F/}MDFÆEKJVC½NWGK&[email protected] #"J!GK hF!UT}&J!NCELGUTPE} 4UTFREKTÊ3GK7W4UÇJ U©C 4¯H%K bÊ ' EÄ (4NOpWGNOp!BGPU1TKMWNLËNREDGF3K AÊ G Ã$ÊÇGNpK $B K&F!NOÁK&LiJ/p!GC 7KMWJ!4UÇJ hCTFÆEK&FÂPOC J!DGLGLGK&PG}MCTDGBGPNOLGR JVDGLLGKMPONLGRNpÐ}&CEGKMFKMLq!J $UTLGWJVGK 1/3 LCENpK©FKMWGD}&JVNOCEL NOpUrWGNOF!K&}&JK&ÆNWKMLG}&K°C hFVUT}¶JVNCTL4UTPG} 4UÇF!REK L KMN MÁUTLGLvUÇPp!CxÁKMUTp!DF!KMWvJ!G"K hF!UT}&J!NCELGUTP4} 4UTFREK e/5 NLJVGK e/3 Ê$À$DIGp!K&DKMLqJVP J!GK"REFCEDGB NO ν = 2/5 hF!UT}&J!NCELGUTP9p¬J UÇJ!K ' ià ( Ê IG ! $ * ? 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'>5$'; O <22; 13 V G% n1 *8 ## $3 G$0 ncqH1';b; 1%D'5$'; O * 1< 13 G%n1 ?$ EBG$0 $3 V 1';b; 1%D' 5$'; O '8- 2;b <D o$<%-CB ,'I12$& ' '(! ?$,@$<?<* 3' = # VP GKzJ!FVUTLGpB CTFJBGF!CEBKMF¬JVNOKMp C SKMPOKM}&J!F!CELN}eWGK&ÆNO}MKMpvUTF!KzDGp!D4UÇPPO} 4UTFVUÇ}&JVK&F!NMKMW CEL JVKÕI4U3´ !p NO}ÅCQ}&CELGWGDG}¶J UTL}MKÁKMUTp!DGFKMÁKMLqJVp&ÊÀ$DG} ÁKjUÇp!DGFKMÁKMLqJVpxUTF!KUTWKMiD4UÇJ!K hCEFWGK¶ÆN}MKNOL0HGNO} JVF!UTLGp!BCEF¬J7C}M}&DGF!p7NLG}&CEGK&F!KMLqJ!POiIGD$J hCEFÐÆEK&Fp!ÁUTPOP4WGK&ÆN}&KMp%$pDG} ÉUTpiD4UTLqJVDGÁ WGCÇJVpiJVK>;UjÆEK L4UÇJ!DGF!K:C xKMPOKM}¶JVF!CTLGpeBGPUjpeUTL NÁB CTFJ UÇLiJ F!CEPOKTÊÎ7KM}jUÇDGp!KJVGK:B4UTp!KC xUTL KMPOKM}¶JVF!CTL)1 p ;UjÆEK hDGLG}¶JVNCTLÉ} 4UTLGREK&p°UÇp©N JB4UTpp!K&p©J!GF!CEDREep!DG} ÕUWK&ÆN}&K BG4UTpKÁKMUTp!DGFKMÁKMLqJVp©UTFKF!KMiDGNOF!K&WeJ!C } 4UTF!UT}&J!KMF!N MKSJVKJ!FVUTLp!BCEFJHBGFCEBKMFJ!NK&4p hDGPP EÊ ±µLxUÇLrNOLqJVKMF hKMFKMLG}&K,K $B K&F!NOÁK&LiJ HN JVUiD4UTLqJVDÁ¼WGCTJ[NOÁÂIKMWGWGK&WNLSCELGKÐUÇF!Á C iJVGKЮ°GUTF!CELC Æq´ Î;CEGÁ F!NOLG R ' Gà ?FGà AF ÃE7¿ )[email protected] (9JVKr}&CELGWGD}&J UÇLG}MK QUÇp©pGC HLlJ!CWGKMBKMLGWlLGCTJHCTLGPOCELÉJVGKÁUTRÇ´ LGNOJ!DGWGKxC ,JVGKrJVF!UTLGp!ÁNpp!NOCELzJ!GF!CEDREÕJVKÂiD4UTLqJVDGÁ WGCTJ 9IGD$JSUTPOp!CvCELzJVGKxBG4UTpKUT}MiDGNOF!KMWÕIi I? MK POKM}&J!F!CELp7JVF!UjÆEKMFp!NOLGR>JVK"iD4UTLqJVDGÁ¤WGCTJMÊË4CEFNLpJ UÇLG}MK77K"}MCELp!NWKMF;KMF!K"UhCEDGFb´ JVK&F!ÁNL4UÇPG}MCEL´ 4REDGF!UÇJVNOCELJ!CÁKjUÇp!DGFKÉWGNFKM}&J!POJVGKeÁUTRELGN JVDGWKvUTLGW J!GKeBG4UTpKÉC½J!FVUTLp!ÁNp!pNCEL}MCK }MNOKMLqJ JVGFCEDGREeUÅiD4UTLqJ!DGÁ WGCTJQNLÉJVGK ³ CTDGPCEÁ>IvIGPC} 3UTWGKSFKMRENOÁK HN} eNpQK $B K&F!NÁKMLqJVUTPP NOLqÆEKMp¬JVN8´ RqUÇJ!KMW`NOL`¯H%K bÊ ' ÃE¿ (hp!KMKËNREDGFK¿@Ê G Ê KÂ}MCE7L 4REDGF!UÇJVNOCELÕ}&CELGp!NOpJ!p"C ÐKMÁNOJJVK&F°» UTLGWÕ}MCEPOPKM}¶JVCEF ³ }MCTLGpJ!F!N}¶JVNOCELGpUTLWUIGUTp!KzF!KMRTNCEL5Î NOL IK&J*7KMKML9Ê;GKzI4UTp!Kz}MCELqJVUT}&J!p p!K&FÆTK:UTp WFVUTNOLGNLGR F!K&p!KMF¬ÆECENOF!Ep HN JVUv} GKMÁN}MUTPBCTJ!KMLqJVNUTP µB = 0 Ê9GKÅÆECTPOJ UÇREKÂWGN 9K&F!KML}MKÅIK&*J 7KMKML» UTLWÎ NOp ³ UTLGWlÎNOp V ÊGKx»5UTLGW ³ }&CELGp¬JVF!NO}&J!NCELGpQUTF!KSpKMB4UTF!UÇJVK&WzIqvUIGUTF!FNKMF VEB UTLGWeI K¶*J ;K&KML HNOJ!l*J 7CvCEBKMLGNOLGREp&~4CELGK>CTB K&LGNLGRCB}MCTLGp!NOpJVp©C /J!GKrDGUTLqJVDGÁ WGCT>J HGCEpK>IKM4UjÆNCT>F 7K;UTLqJ½J!C ÁKMUTp!DF!K TUTLGWÅJVKQCTJVKMF,Np/U"F!%K hK&F!K&LG}MKHCEBKMLGNOLGR"NOLU hCEF!Á C UTLGCTJ!GKMFDGUTLqJVDGÁ¦BCENOLiJ}MCELqJVUT}&JMÊ ®QJ/POC J!KMÁB K&FVUÇJ!DGF!K&p/ICTJV>J!GK;B4UTp!K}&CEGK&F!KMLqJ,POKMLGRÇJVÅUTLW>JVGK;KMPUÇpJVNO}7ÁKjUÇ: L hF!K&KQB4UÇJ!ÂK $}MK&KMW JVGKKMLqJVNOF!KpVUTÁBGPKp!N MKÇʹ©p!NLRzJVKvÁÂDGP JVNBF!CEIK}MCELGWDG}&JVUTLG}M K hCEF!ÁÂDPU ' G1ÄE¿ (9 7K hCEDGLWJV4U3J JVGK}&DGF!FKMLqJ½UÇJJVGK}&CEPPOKM}&J!CEFQNOpRENOÆTKMLlIi IC = ¿Ê@G e2 (τEC VEB ± τC VCB ) , π HGKMFK τEC UTLGW τC UTF!KQJ!GKQJVF!UTLGp!ÁNpp!NOCELÂBGFCEI4UTIGNOPN JVNK&p hFCEÁ KMÁNOJ!J!KMFJ!CS}MCEPOPK&}&JVCTF/UTLGWÅJ!GF!CEDRE JVGK}&CEPPOKM}¶JVCEF/iD4UÇLiJ!DGÁ¡BCENLqJ,}&CELqJ UT}¶J7FKMpB K&}&JVN ÆEK&POEÊ KMLJVGKH}MCTPPK&}&J!CEF/}MNOF!}&DGNOJ,NOp,CEBKML IC = 0 67KÂ}MUTL`ÁKjUTpDGF!KxWGNFKM}&J!POÉJ!GK>JVF!UTLGpÁNOp!p!NOCELlBGF!CEIGUTIGNPONOJµ τEC IqzÁKjUÇp!DGFNLGRJVGKx}MCEPOPKM}¶JVCEF ÆECEP J UTRTK VCB = (VEB /τC )τEC Ê ±µL UT}¶JxJ!GKJ!FVUTLp!ÁNp!pNCELBGFCEI4UTINPN Jµ τEC NprUe}MCEKMF!K&LqJÅp!DGÁ C1ÆTKMFÂUÇPP,B4UÇJ!UTÁBGPONOJVDWGKMp hF!CEÁ KMÁNOJ!J!KMFJVC}MCTPPK&}&J!CEFMÊ4±µLlJVGK*J ;C3´µB4UÇJ!Õ}MUTp!K τEC = |tEC |2 HGK&F!K tEC = tQD + tsl tQD NpÂJ!GKÉJVFVUÇLGp!ÁNpp!NCTL2UTÁBPNOJ!DGWGK UTp!pC$}&NUÇJ!KMW HNOJ! J!GKeB4UÇJV J!FVUTLGp¬ÆEK&F!p!NOLGRJVGKeiD4UTLqJVDÁ WGCTJ! UTLGW tsl F%K hK&F!pJ!CJ!GKeJVFVUÇLGp!ÁNpp!NCTL2J!GF!CTDGRE2JVGKlCTJVKMFB4UÇJV9Ê GKML UÁUTRELGK¶JVN } GKMPW ;UTp UTBGBGPONK&)W /U:ÁUTRELGK¶JVN } G7D Φ /JVF!KjUÇWGNLGR:J!GKlUTF!KMU ® ,KMLG}&PCEpKMW IiJVGK&p!Ke*J ;CB4UÇJVpFKMpDGPOJ!p NLUTL®½4UTFCELGC1Æq´bÎ;CEÁ«B4UTp!K[email protected] KMFKMLG}&K ∆φ = 2πΦ/Φ0 Φ0 = h/e NOprJ!G K G7D DGUTLqJVDG Á I K¶*J ;K&KMLlJVGKS*J 7CNOLiJ!KMF hKMF!NOLGRB4UÇJ!GpMÊ Ë4CEFpNLGREPOK¶´ } 4UTLGLGK&P9JVF!UTLGpÁNOp!p!NOCEL+~ ¿Ê à τEC = |tQD + ei∆φ tsl |2 . ®°pp!DGÁNLGR hDGPP 2}MCEGK&F!K&LiJÉJ!FVUTLp!BCEFJJVGFCEDGREJ!GK`iD4UTLqJVDÁ WGCTJ 7J!GK`NLqJ!KM/F hK&F!K&LG}MKÕJ!KMFÁ Np BGF!CTB CEF¬JVNOCEL4UTPJVC |tsl||tQD | cos[∆φ + θ(tsl ) − θ(tQD )] HGK&F!K θ(tsl ) UTLGW θ(tQD ) UT}&}MCEDLi=J hCEF,JVK BG4UTpK"UT}&}MDGÁ>DGPUÇJ!KMWNLJVGK©*J ;CÂ}&CEF!FKMpB CELWGNLGRxB4UÇJ!GpMÊ®°pJVGK θ(tsl ) NOp;UrREC$CWUTBGBGFC $NOÁ UÇJ!NCEL }MCELpJ UÇLiJ QU} GUTLGREK`NOL5J!GK`BG4UÇp!K`C tQD POKjUTWpJVCUp!NOÁNOPUTFÅ} 4UÇLGREK`NOL JVGKÕBG4UTpK`C SJVK }MCEPOPK&}&JVCTFQp!NOREL4UTP|Ê ±µLJVGK K$B K&F!NOÁK&LiJ>CQº7UT}&CEIq ' à ? 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HQP C Ô UTLGW HQD UTFK UTÁNPOJ!CELGNUTL>C 9iD4UTLqJVDGÁ¡BCENOLiJ}&CELqJ UT}¶J UTLGWÉiD4UTLqJ!DGÁ WGCTJ F!K&p!BKM}¶JVNOÆTKMP EÊ4GK UÇÁNOPOJVCTLGNUTL Hint WGK&p!}&F!NINLGRÂJ!GKSNLqJVK&FVUT}¶JVNOCELÉI K¶*J ;K&KML iD4UTLqJVDGÁ¤WGCTJUTLGWÉiD4UTLqJVDÁ BCENLqJQ}&CELqJ UT}¶JNp;UTp!pDGÁKMWvJVCÅIK 7KjU ÊGK"ÁCÇJVNCTL KMiD4U3JVNCTLÉC JVGKCEBKMF!UÇJVCTFQRENOÆTKMp dc(t) ¿$Ê ? = i[H, c(t)] = −i[0 + W (t)]c(t) , dt HNOJ! W (t) = eiHt W e−iHt NpWGK&ÁCTLGpJ!FVUÇJ!KMWlNLeJVGNOp©}MUTp!KrUTpHJVKJ!NÁKSWGKMBKMLWGNLGRÁC$WGDPUÇJ!NCEL C [JVK"KMLGK&F!RT PK¶ÆEKMP 0 ÊG», Ê |¿$EÊ ? REN ÆEKMp7JVGK"K $BGF!K&p!p!NOCELeC c(t) 7HGCEpKUjÆEKMF!UTREKUTÁBGPONOJVDWGK½}MUTL I EK HFNOJJVKMLzUTp R ¿Ê A hc(t)i = hc(0)e−i t Tt e−i dtW (t) i , HGKMFK Tt ÁKjUTLGpvJ!NÁKÕCEFWGKMFNLGRÊQÎ7UTp!pDGÁNOLGRJVGUÇJÉJVGK:POK&ÆTKMP"ÁC$WDGPUÇJ!NCEL5NpÉU ¸UTDGpp!NUÇL BGF!C}MK&p!pWGK&p!}MFNIKMW Iq U:iD4UTLqJVDGÁ }&CEF!FKMPUÇJVCEF K(t) = [hW (t)W (0)i + hW (0)W (t)i]/2 &;K }jUTLeWGK&}MCEDBGPKSJVKxUjÆTKMFVUÇREKxNOLl»Ð Ê |¿$Ê A QUTp ¿Ê ¿ hc(t)i = hc(0)ie−i t e−Φ(t) , HNOJ! Z t Z t ¿Ê I Φ(t) = dt0 dt00 K(t0 − t00 ) . 0 0 ®°pp!DGÁNLGR"J!4UÇJ,J!GKiD4UTLqJVDGÁ¡WCTJ,WGCKMpÐLGCTJÐBKMF¬JVDGFIÅJVGKHiD4UTLqJVDÁ¦B CTNLqJ,}MCTLiJVUT}&J EJVGKHUjÆTKMFVUÇREK NL hW (t)W (t0)i F!K&WGDG}&KMp;J!C>JVGK°UjÆEKMF!UTREK HN JVvFKMpB K&}&JQJ!CxJVK"pJVUÇJVK½C +JVK"B CTNLqJ;}&CELqJ UT}¶JjÊGK }MCEFF!K&PUÇJ!CEF WGKM}MUj$prNL:J!NÁ K HNOJ!p!CEÁK>JVNOÁK>p!}jUÇPK HGNO} NOp°JVK}&CEF!FKMPUÇJVNOCELÕJVNOÁK>C JVGKiD4UTLqJ!DGK(t) Á WGCTJrpJVUÇJVKKMLGK&F!RTÁC$WDGPUÇJ!NCEL9Ê9Ë4CEF t ττCC +JVGKp!K&}MCELGWNLqJVK&REFVUTP,C1ÆTKMF t00 HNOPP pVUÇJ!DGFVU3JVK"JVCUÅ}MCELpJ UÇLiJ ;Kr4UjÆEK ¿Ê D hc(t)i = hc(0)ie−i t e−t/τ , HNOJ!UTp!pDGÁNOLGRSJV4U3!J hCEF,POCELGR"J!NÁKMp ;KH}MUTLFKMBGPUT}MKQJ!GKHp!K&}MCELGWNLqJ!KMREF!UTPGC t00 C1ÆEK&FJ!G4K HGCEPOK JVNOÁK"WCEÁ UÇNL+~ Z 1 ∞ |¿[email protected]Ê GjÄ −1 (τϕ ) = dtK(t) . t 0 0 0 0 2 II −∞ ϕ GK;WKM}jUj>C hc(t)i HN JVxJ!GK;J!NÁK}&CELGp¬J UTLqJ τϕ NOpWGK&BG4UTpNLGRÇLGCÇJ/KMLKMF!RÇ>FKMPUGU3JVNCTLxCEFKMp}jUTBK hF!CEÁ J!GKDGUTLqJVDGÁ«WGCTJ$' ÃI)( ÊGKvUTDJVCEFxpGC;K&WJV4U3JxJ!GK }&CELqJVF!NOIGDJ!NCELJVClJVGKWGKMBGGUTp!NOLGR FVUÇJ!K½WDGK½J!CxJVK°IGNUTp7WKMBKMLGWGp;CELJ!KMÁB K&FVUÇJ!DGF!K θ UTLW IGNUTp eV NLJVGK½pVUTÁK?QUj UTp7pGCTJ;LGCENpK NLlJVGKÂBCENOLiJ½}MCTLiJVUT}&JSUÇJ &KMF!C3´ hF!K&iDGKMLG}¶ +IDJ"WGCÉLCTJ hCEPOPC J!GK T (1 − T ) pDGBGBGFKMp!pNCEL9Ê9GK LGCELGK&iDGNPONIGFNDGÁ }MCTLiJ!F!NOIGDJVNOCELvJVCWGK&BG4UTpNLGRFVU3JVKNpK&pJVNOÁ U3JVKMWlUTp |¿[email protected]Ê GG (τϕ−1 )V ' λeV hCTFGNREÉINUTp eV θ , |¿[email protected]Ê G1à (τϕ−1 )V ' λ(eV )2 /θ hCEFPOC IGNUTp eV θ , HGKMFK λ NpJVK }MCTDGBGPNOLGRl}MCELGp¬J UTLq J HGN} WKMp!}&F!NOI K&p>JVGKNLqJ!KMFVUÇ}&JVNOCELIK&*J 7KMK&LJVGKiD4UTLqJVDGÁ WGCTJ©UTLWlJ!GKDGUTLqJVDGÁ BCENOLiJH}&CELqJ UT}¶JjÊ GKeBGF!CEIPKMÁ«C ©WGK&BG4UTpNLGRFVU3JVKÉC ½UTL2KMPOKM}¶JVF!CTL pJ U3JVKÉNL2U:BGNOLG} GK&W iD4UÇLiJ!DGÁ WGCÇJUTPpC pJ!DGWGNK&: W HNOJ!ÂU©LGKjUTFIqxÆTCEPOJVUTREK¶´ IGNUÇp!KMWxI4UTPOPNp¬JVNO}/L4UTLGCTpJVFDG}&J!DGF!K ' GG!D (|ÊDZµL>J!GNp ;CTF 63J!GKQUÇDJVGCTF BGF!K&p!K&LiJ!KMW UREK&LGKMF!UTPN jUÇJ!NCELC ½J!GKÉJVGK&CEFRENOÆTKML NL ¯©K bÊ ' à I (JVGUÇJÅJ U ÇKMpNLqJVCUT}&}MCEDLiJJVK p!BKM}&@N 4}"K 9K&}&J!pUTBGBKjUTFNLGRxWGDGK°J!C>JVK"}MCEÁBGPON}jU3JVKMWREKMCTÁK¶JVF UÇLGW JVK"} GNF!UTPN JµC +J!GK"pJVUÇJVK&p NLvJ!GKL4UTLGCEp¬JVFDG}&J!DGF!KÇÊ ID $ ®°p 7KS4UjÆEKSNOLqJVF!CWGDG}&KMWÉNOL JVGK°BGF!K¶Æ$NOCEDGpQ} GUTBJVK&FJ!GK"WGK&}MCEKMF!K&LG}MKC[J!GKSKMPOKM}¶JVF!CTL JVF!UTLGp!BCEF¬J JVGFCEDGREvJ!GKiD4UTLqJVDGÁ WGCTJHNp;WGDGKSJVCÂJ!GK} 4UTF!RTKE4DG}&J!D4UÇJ!NCELGpNOLvJVGKSiD4UÇLiJ!DGÁ BCENLqJ}&CELqJ UT}¶J 'IGà NI7UÇpà C D J!GGGKD7i D4G ÃÇUTÄLq(|JVÊqDG±µÁ LJ!B GCEKMNOpLqK½JHF}MKMCEp!LqDJ POJVUÇp%}&qJ©JVNGpKHNOLGWG}MK&FBGKj4UTUTppKMNW+LGÊ RrFVUÇJ!KJµ$BN}jUÇPPO>NLG}&F!KMUTp!K&p3HKMLJ!GKHÆECEP J UTREK GKÂBGDGFB CTp!K>C /JVGK>BGF!K&p!KMLqEJ 7CE/F zNOp©J!CvWGNp}MDGpp°JVK>}jUÇp!K>C ÐWGKMBGGUTp!NOLGR hF!CTÁ U iD4UTLqJVDGÁ B CTNLqJH}MCELqJVUT}&J½NLÉJVK(hF!UT}&J!NCEL4UÇP+iD4UTLqJVDGÁ Ô UTPP9K% KM}&J°F!K&RENÁK#' GjÄqà (|Ê ="D4UTLqJVDGÁ BCENOLiJ©}&CELqJ UT}¶J JVF!UTLGp!ÁNpp!NOCELe}jUTLlJVGK&LÕIKxWGK&p!}&F!NIKMWzIqlJ!DGLGLGK&PNOLGRI K¶*J ;K&KML:KMWREKxp¬J UÇJ!KM p ' HEÄ (84J!GKxiD4UÇLiJ!N &KMW UTL4UTPOCERSC [}&PUTpp!NO}jUTP4/p NOBGBGNOLGRxCTF!IGN JVpÐC +K&PK&}&JVFCELGp&ʱµLJVGNOpÐpJVFCELGREP Å}MCTF!F!K&PUÇJ!KMWKMPK&}&J!F!CELF!K&RENÁK KMWGRTKp¬J UÇJ!KMpF!K&BGF!K&p!K&LiJx}MCTPPK&}&J!NOÆEK>K$}MN J UÇJ!NCELGpSC7J!GKDGUTLqJVDGÁ Ô UTPP 4DGNW9~+WGKMBKMLWGNLGReCELJVK BGNL} GNLGRrC +J!GK°iD4UÇLiJ!DGÁ¤B CENOLqJ;}&CELqJ UT}¶!J NOJNpKMN JVGK&F hFVUÇ}&JVNOCEL4UTP4iD4UÇLiJ!DGÁ Ô UTPOP DGUTp!NOB4UTFJ!N}&PKMp CEFSKMPOKM}¶JVF!CTLG<p HGN} JVDGLLGKMP|Ê+± JrNpSB4UTF¬JVNO}MDGPUTF!P zNOLqJVKMFKMp¬JVNLRlIKM}MUTDGpKJ!GK}MDGFF!KMLqJ *iÆECEP J UTRTKUÇLGW JVGK°LGCENpK"} 4UTF!UT}&J!KMFNpJ!N}&pQWGK&ÆNUÇJVK"p¬JVFCELGREP hFCEÁ¤J!GK"}MUTp!KSC LGCTF!ÁUTP }&CELGWGD}&JVCTF!p ' HGHEFà HH)( ~ hCEFJVG, K 7KjU 5I4UÇ} p}jUÇJJVK&F!NLR }jUTpK 7J!GKÕ}&DGF!FKMLqJlUÇJ MK&F!CJVK&ÁBKMF!UÇJVDF!KÕÁUj2NLG}&F!KMUTp!, K HKML JVGKÉÆTCEPOJVUTREKÉIGNUTpÅNpPC 7KMFKM)W HGNOPKÉNOL2J!GKepJ!F!CELGRI4UT} ip}jUÇJJVKMFNLGR}jUTpKlJ!GK I(V ) NOpÅGNRTGPO LGCELGPONLGKMUTFMʱ J©Np;J!iDGp½NOÁBCEF¬J UTLqJ;J!CUTWGWGFKMp!pJ!GKNp!pDGKSC WGKMB4UTp!NOLG; R hF!CEÁ UÍD$J!JVNOLGREK&FPNiDGNOW+Ê Ô K&F!K)7K}MCELp!NWKMFSJVGK}MUTp!KC7p!NOÁBGPOKÅÍUTDGREPNL$hFVUT}¶JVNCTLGpHN JV 4POPNLR UT}&J!CEF ν = 1/m m C$WW>NLqJ!KMREK&F ¶Ê3®°pNL>¯©K bÊ ' à I)(93J!GK;WKMBG4UÇp!NLR"C U°p¬J UÇJ!K;NOLxJVK;WGCTJNp NLGWDG}MK&WÂIq>N JVp }jUTB4UÇ}MNOJ!NOÆTK }MCEDBGPNOLGRxJ!C>JVGK°IGNUTpKMWviD4UÇLiJ!DGÁ BCENOLiJ;}MCELqJVUT}&J GUTpp!DGÁNLGRxJV4UÇJ7JVGK°PK&ÆTKMP ÁCWGDGPUÇJVNOCELNOL JVK WGCTJHNOpHU¸UTDGpp!NUTLeBGF!C}MK&p!p½UTLGWlLGKMREPOKM}¶JVNLRÅI4UT} E´bUT}¶JVNOCELe%K KM}&J!pMÊ ±µL J!GNpe} 4UTB$JVKM%F ;K HNPOP 4FpJl}MCEÁBGDJ!K:JVGKWGKMB4UTp!NOLGR FVUÇJ!K:NL JVGK 7KjU UTLWp¬JVF!CTLGR I4UT}ip!}MUÇJ!J!KMFNLGRPNÁNOJHNOJ! J!GKlUTp!pDGÁB$JVNCTL J!4UÇJJVGK ³ CEDGPOCEÁÂI2NLqJ!KMFVUÇ}&JVNOCEL Npp!}&F!KMK&LGKMW Ii LGKjUÇF!IqSÁK&J UÇPPNO}REUÇJVKÇ~MJVGK ³ CTDGPCEÁ>ISNLqJVK&FVUT}¶JVNCTLSNp+JVKMLrF!KMWDG}MK&WrJ!C½UWGK&POJVU hDGLG}&J!NCELBCTJVK&LqJVNUÇP Ê ²½K $J 7K HNPOPK $J!KMLGWCEDF"F!K&p!DGP JVp°J!C JVGKÅ}MUTp!KÅC UTF!IGN JVF!UTFlI4UT} ip}jUÇJJVKMFNLGR 9DGpNLGRvJ!GKÂK UT}¶J p!CEPODJVNOCELÕC J!GKÅICEDGLGW4UÇFÀNLGK¸SCEFWGKML:ÁC$WKMP WGK&ÆTKMPCTB K&WIq:Ë4K&LGWGPK¶ 9ÍDW HNR 9UTLGWÀUTPOKMDGF ' GG ? (|ÊËNL4UTPOPO );K HNOPPÐp!C J!4UÇJ>J!GKMpKÉF!K&p!DGP JVp>}MUTL IK K J!KMLGWGK&W J!CÕJVK }jUÇp!KvC ©UÇF!IGN JVFVUÇF p!}&F!KMK&LGNLRGÊ DEÄ QD φ1 a) PSfrag replacements b) IB φ2 QD φ1 IB φ2 * & 3%0E+2? 'H'(! ? $p0%?<2< - @ (-* 3' D'2 B1'<2; 13 2(( & 5$; AHD' /@(-* 2'Ib 1 ' D$% bS?$H %+* 0 .'(! HG(%0ZDEb6cG 0>'(! 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HQD = 0 c† c , * +J-0/ b @ )](D) c† <=(*)Z2c6>)]?d27e)]:%)=<=6*(*93f-7[[email protected]"%?#Q9'6g"0?h<]93&.U:%)=Qe<Z2'UL27<]"06*"08')]:0iS6*946>@)A2'93?j6>(*&.<=6>&.(*) 2SUW93"0K6 <]9356>2'<=6k"%C6>@)O MOlX-J[[email protected])mMm2'BC"%:n6>93."`2' b @."%<;@oQ)=?*<=(*"%pW)]?q6>@)a)]Q.$3)mBC9JQ.)]?q"%!6>@)m2'p?*)=<])m9'G 6>&.)]:0"%$e"0? = vF H0 = π Z +J-sr dx[(∂x φ1 )2 + (∂x φ2 )2 ] , b 0" 6*@ φ (x) i = 1, 2 6*@)utv&J6*6>"0$3)=(epw93?D93"0<oxL)]:0Qf1 b @"0<;@y(D)]:`2c6>)]?F6*9z6*@){)]:0)]<=6*(*93|Q)=?*"n6i 93UW)](>2c6>93i ( p5i ρi (x) ∂x φi (x) = √πν ρi (x) }kiR872'(DiT"%.$!6*@)~$5276>)SUW9'6>)=56>"`27:9'GhIK&L2'56*&BUw93"056<]9356;27<=6Z1\93.)<]2'? b "06*<;@GH(*93B 2 b )Z2' pL2'<;K?*<]276*6*)](D"%$?*"06*&L276*"%93f1 b @.)](*)C6>@)_Mm27:%:g:%"0IK&"%Qz(*)=B2'"0?S"%z93)CU"%)=<]) #"0$3&(D)_+J-0/2 16>92 ?D6*(*93$!pL27<;T?D<Z276D6>)=(*"%.$?*"n6>&L2c6>"%9' b @)](D)~6*@)EMO2':%::%"%IK&"0Qo"%??DU:%"n6X"%{6 b 9 "%$3&(D)S+.-n/Zp -.6>@.) 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V e? = νe ν ω0 = eV 6>@)?D6*(*93$pL2'<;K?*<]276*6*)](*"0$R<Z27?*)'-w[[email protected])FIK&L2'?D"%UL27(D6>"0<]:0)~9'Uw)=(>276*93(m"06>@.)E<Z27?*)F9'G b )Z27pL27<;T?D<Z276j √ √ 6>)=(*"%.$"%? 6>@)o?DUL276*"`2':k<]&.6*9'"0? 1 "06>@ 6>@)o6*)]BCUw9'(>2': ψi (x) = ei νφi (x) / 2πα α = v F τ0 b τ0 <]&.6*9' 1w27Q"%u6>@)?D6*(*93$_pL2'<;K?*<]276*6*)](D"%$R<]2'?*)~6>@))]:%)=<=6*(*939'Uw)=(>276*93("%?a93p.6;27"%)=Q b "06*@6>@.) ?*&p.?D6>"n6>&.6*"%93 [[email protected])aMm2'BCν"%:n6>→ 93"%1/ν 2'~Q)]?D<](*"0p"%.$S6*@)"%56>)=(>2'<Y6>"093f1327?*?*&.B_)=QC6*9~pW) b )Z2'\1Kpw)Y6 b )=)]C6>@)Q9'6q27Q IK&L2'56>&BUw93"056<]9356;27<=6a(*)]2'Q? † † Hint = c cW ≡ c c ./ Z dxf (x)ρ1 (x) , +.- b 0" 6*@ f (x) "%?a2,93&.:%93BFpu"%56*)](>27<=6>"093u')](D)]: 1 b @"%<;@"0?a2'?*?D&B_)=Q6*9"%.<]:%&.Q)S?*<](D)])="%$pKi{6*@) )Z27(*p5iR$5276*)]? ? +Je−|x|/λs f (x) ' e2 √ , x2 + d 2 b @)](D) d "0?O6>@.)EQ"0?D6;27<])~GH(*93BV6>@.)EQ9'6O6>96*@)E)=Q$3)e27Q λ %" ?6>@.)E?*<=(*)=)]"0$R:%)=$'6>@- )F<]2' ?*)=)O6>@276 "0?k6>@.)m<;@L2'.$3)m9'G6*@)mIK&L27K6*&B Q9'6k?D6;2c6>)4)=)](D$'si Q&)a6>9E6*@)m"0K6*)](*2'<=6*"%93C9'G6>@.) W 0 )]:0)]<=6*(*93{"0{6*@)SIK&L2'56>&BVQ.9'6 b "06*@{6*@)S)]:0)]<=6*(*93{Q)=?*"n6i{"%R6>@.)~IK&L27K6*&B $ UW93"%56<=9356;2'<Y6Z- ! [[email protected])RQ)[email protected]'?D"%$9'GO2')=:%)=<=6>(D93?D6>276>)R"02Q9'6!<=93&U:0)]Q6*92PL&<=6*&L276*"%$<=&(*(D)]56!"%?F<]2'&?D)]Q p5i 6>@.)R)]:%)=<=6*(*93Q)=?*"n6iPL&<Y6>&L2c6>"%9'?]1 b @"%<;@|$3)=)](*276>){2PL&<Y6>&L2c6>"%.$Uw9'6*)]56>"%2':q"%6>@.)RQ9'6Z1 (*)=?*&:n6>"%.$e"%u2ep:%&(D(*"0$e9'G^6>@)S)].)](*$7i{:%)=8')]: [[email protected])SQ)][email protected]'?*"0$_(*276>)71L)YJU(D)]?*?D)]Q"0R6>)](DB_?X097G#"0(*(*)=Q&<="%p:0) <;@L2'(*$')SP&<=6*&L276>"093?X"0{6*@)~2'Q *2c <])=K6 b "0(*) "%R6*@) GH(>2'<Y6>"%9'L2':IK&L27K6*&BMm27:%:f)Y)=<=6Z1L"0? b (*"n6*6>)=27? r'+J1/3/ ' τϕ−1 1 = 4 Z ∞ dt −∞ Z Z dxf (x) ( A +.- dx0 f (x0 )hhρ1 (x, t)ρ1 (x0 , 0) + ρ1 (x0 , 0)ρ1 (x, t)ii . 93(DB27:27Q?*&UW)](D<]93.Q&<=6*"%$R?jiT?D6>)=B_?=1w6*@)FQ)[email protected]'?D"%$o(*276>)E<Z2'pW)E<]2':%<=&:`2c6>)]Q&?*"0$6>@.) ?*<]276*6*)](*"0$2'UU.(*952'<;@-d\9'(et&.6*6*"%$3)=(e:%"0IT&."%Q?E2'Q "%UL2'(j6>"0<]&:%2'(GH93(F6*@)u MOlX1hpw)=<Z2'&.?*)u9'G 6>@))]:%)=<=6*(*93"0<"056>)](*2'<=6*"%931k"n6"0?o<]9358')]"0)]56R6>9&.?*)6>@. ) )]:%[email protected] 2'UU(D952'<;@ 3 1k+ -, 6"%J Q&<=)]?,6*9E$3)=)](*276>)a6>@)O?DiTB_BC)=6*(*" )]Q<;@L2'(*$')mQ)=?*"n6i Q)]?D"06i_<]93(D(*)=:`276*93(]156;27T"0$~"%56>9E2'<]<=93&56 pL2'<;K?*<]276*6*)](D"%$ * hhρ1 (x, t)ρ1 (x0 , t0 )iisym = " = X η=± hTK ρ1 (x, tη )ρ1 (x0 , t0−η )e−i R 1 X − hTK ρ1 (x, tη )e−i K Ht dt1 i 2 η=± #" X η=± R K Ht dt1 ' ( i hTK ρ1 (x0 , t0η )e−i R K Ht dt1 i # . +J- M )](D)e2_6*&)=:%"%.$o)=8')]56 276 O <=(*)Z2c6>)]?42')YJ<]"n6;276*"%93 b @"%<;@.)])]Q.?46>9oU(D93UL2'$3276>)~6>9 x = 0 6>@)E:%9T<Z276*"%93u9'Gg6>@.)FQ9'6]-[[email protected])F)=IK&"%:0"%p(D"%&B )](D9RUw9'"%56 <]9356*(*"%p.&.6>"0936>9_6>@)FQ.)][email protected]?*"%.$R(>276*) <]93(D(*)=?*UW93Q?A6*9E6>@) )](D9e93(*Q)=(A"%o6*@)m6*&)=:%"%.$e2'BCU:%"n6>&Q) "n6A"0?A:`2'pW)]:0)]Q −1 (0) -J[[email protected])](D) "%?d9~<]9356*(*"%p.&.6>"093e6>9SxL(*?j6,93(DQ)](q"%e6*@)a6*&)=:%"0$EMO2'BC"%:06*93Γ "%2'0E?D"%<=) b )42'?D?*&(τBCϕ) ) 1 hφ(x)i = 0 ) J"%?D6*?]1 b @"%:0)m6>@.)~.93)]IK&"0:%"0p(*"0&B <=9356>(D"%p&.6*"%93R<=93(*(D)]?DUw93.Q"%$e6*9C6*@)S?*)]<=93Qu93(*Q.)]("% ' ( Γ0 τϕ−1 = (τϕ−1 )(0) + (τϕ−1 )(2) + · · · . +.- + [[email protected])XQ)[email protected]'?D"%$S(>2c6>)X<]9'K6*(*"0p&.6>"093?g"0F6>@) b )Z2'EpL2'<;K?*<]276*6*)](*"0$~<Z2'?D)O27(*)X<Z27:%<]&.:`276*)]Q!2'?hGH93:0:%9 b \9'(A6>@) )=(*97 93(*Q.)](X<]9'K6*(*"0p&.6>"093f1 b )[email protected]) ν hTK ∂x φ1 (x, tη )∂x0 φ1 (x0 , t0−η )i π2 ν 1 2 γφ1 (x,tη ) γ 0 φ1 (x0 ,t−η ) ∂ = e i|γ,γ 0 =0 . 0 hTK e xx π 2 γγ 0 hTK ρ1 (x, tη )ρ1 (x0 , t0−η )i = +J- 6>@"0?mGH93(*BE&:`2 b )FxLQ 0 GH93:%:09 b 0" $_6*@)eIK&L2'?D"%UL2'(j6>"0<]:%)<]93.?*)](j87276>"093f16*@T&.?46>@)e)=IT&."n γ = −γ :%"0p(*"0&B Q)][email protected]'?*"0$_(*276>) "0?X 39 p.6>2'"%)=Qu2'? / rJ/ ' ( (τϕ−1 )(0) ν = 2 4π Z ∞ dt −∞ Z dxf (x) Z 5r dx0 f (x0 ) X η=± η−η 2 ∂xx (x − x0 , t) . 0 G1 +.-n/ [[email protected])gpW93?*93."%< (*)=)]?GH&<=6*"%93 97GJ)=Q$3) "%? i i = 1, 2 Gηi 1 η2 (x−x0 , t1 −t2 ) = hφi (x, tη11 )φi (x0 , tη22 )− -5[[email protected])O<=9J) <]"%)=56>? 1 "0Q)]56>"nG iE6*@)O&.UUw)=(q2'Q:09 b )=(qp.(>2'<;@.)]?,97Gw6*@) )=:%Q.iT?*@C<]9'J φ2i i η η1,2 = ± 6>93&.(]\9'(m?*)=<]93.Q 93(DQ)]( <=9356>(D"%p&.6*"%936>9o6>@.)eQ)][email protected]'?*"0$R(>276*)'1w?*"%.<]) 2'(*)E"%Q)=Uw)=Q)=K6~"0 ψ1 , ψ2 6>@)~2'p.?*)].<])~9'G6*&)=:%"0$1 b )~93p.6;27"% X η 0 TK ρ1 (x, t )ρ1 (x , t 0−η η=± Γ2 =− 0 2 = X (−i)2 ) 2 η 0 Z hTK ρ1 (x, t )ρ1 (x , t dt1 K 0−η ) η=±;1 ,2 =± Z Z dt2 Ht (t1 )Ht (t2 ) K dt1 K Z dt2 ei(1 ω0 t1 +2 ω0 t2 ) K [ψ2† (t1 )ψ1 (t1 )]1 [ψ2† (t2 )ψ1 (t2 )]2 i X Z ∞ Z ∞ Γ20 ν dt1 dt2 ei(1 ω0 t1 +2 ω0 t2 ) η1 η2 = − 2 2π (2πα)2η,η ,η , , −∞ −∞ 1 2 1 2 D E √ √ η η η 0 0−η i ν1 φ1 (0,t1 1 ) i ν2 φ1 (0,t2 2 ) × TK ∂x φ1(x, t )∂x0 φ1 (x , t )e e D E √ √ η1 η2 × TK e−i ν1 φ2 (0,t1 ) e−i ν2 φ2 (0,t2 ) . &L2'?*"0UL2'(j6>"%<=:%)O<]93?D)](j8'2c6>"%9'u"0B_UW93?D)]? 1L?*9 1 = −2 ≡ Z Z ∞ Z X dt dxf (x) dx0 f (x0 ) η1 η2 Γ20 ν 4π 2 2(2πα)2 −∞ η,η1 ,η2 , Z ∞ Z ∞ 2 η−η η1 η2 η1 η2 (x − x0 , t) dt2 eiω0 (t1 −t2 ) eνG2 (0,t1 −t2 ) eνG1 (0,t1 −t2 ) ∂xx dt1 × 0 G1 (τϕ−1 )(2) = − + +.-n/3/ −∞ −∞ ηη1 ν[∂x G1 (x, t − t1 ) −ηη1 2 2 − ∂x Gηη (x0 , −t1 ) − ∂x0 G−ηη (x0 , −t2 )] 1 (x, t − t2 )][∂x0 G1 1 +.-n/ r . [[email protected])SQ)][email protected]'?*"0$_(*276>)SQ)=Uw)=Q?a93{6>@)S$3)=93B_)Y6>(jio9'G6>@)S?*)Y6>&U{8T"`2F6>@.):0)]$'6*@{?*<Z27:%)]? 1 1 d λs 2'Q -\[[email protected])E)=IK&"0872':0)]56m(D)]?D&:06aGH93(a?D6>(D93$_pL2'<;K?*<]276*6*)](*"0$o"0?O93pJ6;2'"0)]QpKiu(D)]U:%2'<]"0$ ν → 1/ν )YT6α6>9e6*@ ) S(D)])] ?GH&<Y6>"%9' Q&L2':0"06i [[email protected] ) S(D)])] ?GH&<Y6>"%9'2c6XxL"06*)46>)=B_UW)](*276>&(D)S"%?X$3"n83)]u2'? 0 G1ηη (x, t) = − ln sinh h π β (x/vF − t) η+η 0 sgn(t) 2 sinh − η−η 0 2 + iτ0 i iπτ0 β η + η0 η − η0 sgn(t) − 2 2 +.-n/ . 6*@) )](D9_6>)=B_UW)](*276>&.(*)~:%"%BC"06]1 b )E) JUL2'Q6*@) HG &<=6*"%93"%56>9_2![#2ZiT:%93(a?*)=(*"0)]?O2'Q 6;2'7)]?O"%56>92'<]<=93&56493.:0io6*@)E:09 b ]) ?j6493(DQ)](a<]9'K6*(*"0sinh(· p&.6>"093f· ·1 )b F ) 93p.6>2'"%{6>@.) (*)])=?OGH&<Y6>"0932c6 )=(*976>)]BCUW)](>2c6>&(D)S2'?"% )=G - 3 ' ( 0 Gηη 1 (x, t) = − ln τ0 + i(t − x/vF ) , *, '!- % # * : #% .!,- ! #!- 1!'! ! 1# !,- +.-n/ . ) [[email protected])R2'?D?*&BCU.6*"%9397G?D6>(D93$?D<](*)=)]"0$ "0?FB27Q)'1v6>@L2c6FBC)Z2'?6>@) ,9'&:%93BEp 0 "%56>)=(>2'<Y6>"093 2'? b )[email protected]')4<;@93?D)]{2'pWλ9s83)4∼27<=α6>?k= "%Cv83F)](jτi [email protected](j6A(>27$3)'- )]$52'(DQ"%.$~6*@)OGH93(DBF&:%2 f (x) ? '9 G "%ldI\- +.- b "n6>@ 1 )oxLQ6>@L2c6 −|x|/α Q)]<=(*)]2'?*"0$G2'?D6E6>9 )=(*9 b "n6>@6>@.) f (x) λs ∼ α b 872':%&)E9'G "%?m93J "%.xL."06>)=?*"0B2': -\6*@"%? <]2'?*)71"0646*&e(*.? 93&.6 2'Q6>@K&? √ 2 x xd x + d2 ' d [[email protected]) ,93&:093BFp{"%56>)=(>2'<Y6>"093R"%?X(D)]Q&<=)]Qu6>9e6>@)SQ)=:06>2eGH&.<=6*"%93f1 α f (x) ' 2e2 δ(x) . d ? +J-0/ M 9 b )=83)=(]1 b )!xLQ6>@276S6>@"0?27?*?*&.B_U.6*"%93"0? 9'6~)]<=)]?D?>2'(ji312'Qz"06 b "%:0:hpW)C(D)]:%2cJ)]Qz:`276*)]( 39 fO ?*)=(D6*"%$6>@.) S(D)])]?eGH&<Y6>"093|"%|lgI\- +.-n/ "06>@){Q)[email protected]'?D"%$(>2c6>) ldI\- +.-0/ 2'Q lg\ I +.-n/ r $3"n83)=? ( (τϕ−1 )(0) ' A +.-n/ 4e4 τ02 ν , = πβd2 2'Q b "06*@R6>@)S<;@L2'.$3)97G872'(D"`2'p:0)]? 1 1\27Q 1 )~9'p.6;2'"0f τ = t 1 − t 2 τ1 = t − t 1 τ2 = t 2 b (τϕ−1 )(2) sinh2ν πβ iτ0 sinh2ν βπ iτ0 h h i+ i dτ cos[ω0 τ ] = 2ν π 2ν π −∞ sinh β (ητ + iτ0 ) sinh β (−ητ + iτ0 ) η Z ∞ π π × dτ1 sgn(τ1 ) coth [−ηsgn(τ1 )τ1 + iτ0 ] + coth [ητ1 + iτ0 ] β β −∞ Z ∞ +.-n/ π π [ηsgn(τ2 )τ2 + iτ0 ] + coth [ητ2 + iτ0 ] . × dτ2 −sgn(τ2 ) coth β β −∞ e 4 ν 2 Γ2 − 2 2 20 2 4β π vF d XZ ∞ 6*@)k"056>)]$'(>2':5983)=( 1 b )k<;@2'$3)q8'27(*"`27p:%)=?v6*9 GH93(6>@)dxL(*?j6 ?*)]<=93Q 6>)=(*Bu1 τ t = −τ ∓iτ0 ±iβ/2 2'Q6*@)m"056>)]$'(>2':\9 b (*&.?A"%C6>@)O<]93BCU:%) _U:`2'.)aGH93(DB 6*9 −∞ ∓ iτ0 ± β/2 +∞ ∓ iτ0 ± β/2 )Ep(*"0$R"06mpL2'<;6*9 pKiQ)=GH9'(*BC"%$6*@)e<]9'K6*93&(4pW)]<]2'&?D)!6*@)](D)!2'(D)F9RUW93:%)=?4"% 6>@)S"0K6*)]$3(*2'Qf-\9'( (−∞,−1+∞)1 b )S93pJ6;2'"0 τ0 ω 0 , β (τϕ−1 )(2) R6>@) e4 Γ2 ν 2 τ02ν = 2 202 π vF d Γ(2ν) )=(*9!6>)=B_UW)](*276>&(D)S:%"%BC"06]1 b )[email protected]) (τϕ−1 )(2) 2π β 2ν−1 ω0 β ω0 β cosh Γ ν +i 2 2π (τϕ−1 )(0) = 0 2 . +.-n/+ +.-n/ 2'Q e4 Γ20 ν 2 τ02ν = 2 2 |ω0 |2ν−1 . πvF d Γ(2ν) ,93BCUL2'(*"0$46>@.)aGH93(DBF&:%249'Gf9')]IK&"0:%"%p.(*"%&.B Q)[email protected]'?D"%$(>276*)"%_ldI\- +J-0/+ b "06>@!6>@.)OpL27<;T?D<Z276j 6>)=(*"%.$e<]&(D(*)]56O93"0?*) "%ulgI\- - 2':0?*9!<]93BCUL2'(D"%$!lgI\- +.-n/ b "n6>@ldI\- - 1 b )SxQ ' A ?? (τϕ−1 )(2) = A ?A eτ 2 0 d S(0) , ( +J-sr lgIW- +.- r [email protected] b ?X6>@L276,6>@) 9')]IK&"0:%"%p.(*"%&.B Q.)][email protected]?*"%.$e(>276*) "%?AU(D93UW93(D6*"%93L27:W6*9F6>@) pL27<;T?D<Z276j 6>)=(*"%.$R<]&(D(*)]5693"0?*)'1 b @."%<;@z<Z2'zpw)!&.Q)](D?D6>9T9TQ p5i976>"%<="%$o6*@)!<]9356*"%K&"06i)]IK&L276*"%93 b "06*@ 2'?*?D&BCU.6>"093C6*@L276q6>@)O)]Q$')m<]&.(*(*)=56 b "06*@93&.6kpL2'<;K?*<]276*6*)](D"%$FQ.9J)=?,9'6qPL&<Y6>&L2c6>)m"0_6*"%BC)'-5[[email protected]"0? "%56>)=(*)=?D6>"0$S(*)=?*&:n6k?D&$3$3)=?D6k&?g6>9~)YT6>)=Q9'&(dU(*93p.:%)]B 6>9S6>@)<]2'?*)a9'G2'(*p"n6>(*2'(DiEpL2'<;K?*<]276*6*)](*"0$ b @"%<;@u"%?X<=93?D"%Q)=(*)]Qu:`2c6*6>)=(A93fO?*"0$~Q&L2':0"06i31 b )O(*)Z27Q"%:niF93p.6>2'"%!(D)]?*&.:06>?dGH93(g6>@.)O?j6>(*9'$EpL27<;T?D<Z276D6>)=(*"%.$E<Z27?*)'56*@)O)=IK&"%:0"%pJ (*"0&B<]9356>(D"%p&J6>"%9'"%6*@)FQ)[email protected]'?D"%${(>2c6>)F"0?O6>@.)e?>2'BC)e2'? "06>@) b )Z27pL2'<;K?D<Z276D6>)](D"%${<]2'?*)7[[email protected])e93.)]IK&"%:0"%p(D"%&BV<=9356>(D"%p&.6*"%93<Z2'pW)e93p.6>2'"%)=QGH(*9'B 6*@) b )]2'p2'<;K?*<Z2c6*6>)=(*"0${<Z2'?D)epKi (*)=U:`2'<="%$ (D)]<]2':%:6*@L276 "0?XQ)=xL.)]Q"0u2!Q."0w)](*)=56B2'.)]("%o6*@) 6 b 9_:%"0B_"n6>? ν → 1/ν ω 0 PSfrag replacements β=5 (τϕ−1 )(2) 1 β = 10 0.5 β = 50 0 0 0.2 0.4 ν 0.6 0.8 1 n23<<1:'(!s? $'[email protected]+; bG(E+* 1'-EbG<& ':'(!m-,32(%+p# ':-;=; + !E$%#'< [email protected] $?* !E' G'- 0 n !D;=;Q; b$ $3.O- ' 3<1$13o;=b G$%0ZDE+. β = 5, 10, 50 2'Ibm1'O%)G $ OD(63 * '38<38(+ ; 2'IbZ 1'%12'$3:D'- <; + !D%? ' eV = 0.1 6 * 'I33 bD%h ν = 1/m [[email protected])g93.)]IK&"%:0"%p(D"%&B <=9356>(*"0p&.6*"%93497GJ6*@)gQ)][email protected]'?*"0$(>276*)h"%?U.(*93UW93(D6*"%932':6>9X6>@) )](D97 GH(D)]IK&)].<=i 93"0?*)!"%6>@)CIK&L2'56>&.B MO2':%:g:0"%IK&"0Qf1 b @"0<;@ "0?<=93B_U.&.6>)=Q "0 )=GH?=- rJ1 3 1^+ J1v3+ -v[[email protected])!6>@) 93(*)Y6>"0<Z2':vU(D)]Q"0<=6>"093?a9'Gg9'"%?*)"%u6>@) b )]2'2'.Q?D6*(*93$_pL2'<;K?*<]276*6*)](D"%$R:0"%BC"06>[email protected]8')FpW)])]8')]( "0xL)=Q "0Uw93"056S<]9356;27<=6)YJUw)=(*"0B_)=K6*?E2c6~x:%:%"0$oG2'<Y6>93( .1 /31 r -[[email protected]"%?S"%?S&J ν = 1/3, 1/5 Q)](D?D6*9J9TQ GH(D93B 6>@.)<=9356>"%K&"n6i|)]IK&L276*"%931 b @"%<;@(D)]:%276>)=?!6*@)u<]&(D(*)]5693Uw)=(>276*93(e6>9z6>@)uQ)]T ?*"n6iR93UW)](>2c6>93( / r - A6 )](*9C6>)=B_UW)](*276>&(D)'16*@).93)]IK&"0:%"0p(*"0&B Q.)][email protected]?*"%.$_(*276>)~9'GglgI\- +.-n/ GH93( b )]2'pL2'<;K?D<Z276D6>)](D"%$Q)]UW)]Q.?E936>@)CIK&L2'56>&.B UW93"0K6~<=9356;2'<Y6Ep"%2'? b "06*@z6*@)_) JUw93.)]56 -v[[email protected]"%?~"%?~"% ?*@L27(*U<=9356>(>27?D6 b "n6>@ a)YG - r'+ 1 b @)](D)_6>@.)IK&L2'56>&.B UW93"%56<]9356>2'<=6 2ν − 1 < 0 p"`27?4Q)]UW)].Q)]<=)"0?4:%"0)Z2'(=- )!2':%?D9{<Z2':0<]&:%276>)FK&BC)](D"%<]2':%:niu6*@"%?m<]9356>(D"%p&J6>"%9'276 xL"n6>)F6>)=B! 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' ( ( ' ( ' ( "! ! #! βeV ' 1 ? $! PSfrag replacements 1.5 5 β = 100 (α/d)2 F 4 (τϕ−1 )(2) 1 β = 10 3 2 1 β = 50 β = 50 0 β = 10 0 1 2 0.5 d/λs 3 4 5 β = 100 0 0 1 eV 0.5 1.5 2 m'[email protected];=bG(Eb<* 1'?E+G? '8bq-32$%bS# !D%? ' '(! @$<-* 2'Ibn1'O% G -8- ;=;=b"!$%#'< 0'* 5(<;=$%'(!n#* 2- ! '<m- n 4<3 ν = 1/3 β = 10, 50, 100 O- 'oG$%0ZDE+S0% 1'< %12'3/D'B- 0'; 3/<3/?$83<113 ; +% q>b0% q-B? '>'(! '[email protected];=bG(Eb<* 1'?E+G? 'b - 3<2$%b4#G% 1 - (G -(A 0 1b,$3 O? 'q0 1+ * ; ?b2;= 13.G&A 2 q8!#$%& '8'(! (α/d) d/λs 6>)=B_UW)](*276>&(D) 16>@.)_Q)[email protected]'?D"%${(*276>)!"%.<](*)]2'?*)=? b @.)] 6*@)exL:%:0"%$_G2'<=6*93( "%<=(*)]2'?*)=?] )xL.Q 6*@L276R1/β 6>@.)z> Q.)][email protected]?*"%.$y(*276>)=?u)Y872':%&L2c6>)]Q 276{Q."0w)](*)=56u6*)]BCUw)=(>276*&(*)=?{<]93"0<]"0Q)276{6>@.) &[email protected]?D"%<]2': 872':0&) 1,pW)]<Z27&?*)6>@.)@KiTUW)](Dpw93:0"%<<=93?*"0)BF&:n6>"0U:%"0)]Q pKi6>@)?DIT&2'(*)=Q ν = 1/2 B_9TQ&:0&?,97G6*@) ~27B_B_2GH&<Y6>"093R"%{lgI\- +.-n/+ Q9J)=?9'6Q)=Uw)=Q9'o6>)]BCUW)](>2c6>&(D)'1 b @"0:%) 2c6 6>@)?>2'BC)6*"%BC)~6*@)F) .UW93)=56 "%? )](D9w6*@"%?O"%?aK9 b GH9'(mUW)](D6*&(*p276>"n83)E<]2':%<=&:`2c6>"%9'? (2ν − 1) 9'G^6>@)SpL2'<;K?D<Z276D6>)](D"%$C<]&.(*(*)=5642'.Qu.93"%?D)'!#"%$3&.(*)k+.- 176>@.)AQ)]UW)].Q)]<=)9'GL6>@),93)=IT&."%:%"0p(*"0&B <]9'K6*(*"0p&.6>"093E97GL6>@)AQ.)][email protected]?*"%.$ (>276*) 93!6>@)aIK&L2'56>&B UW93"0K6d<=9356;2'<Y6,p."`2'?g839':06;27$3)"%?dU:%976*6>)=QeGH93(d?*)=8')](*2':W6*)]BCUw)=(>276*&(*)=?]-K!6>@)a<Z2'?D) 9'G?D6>(D93$!pL2'<;K?D<Z276D6>)](D"%$.16>@)SQ)[email protected]'?D"%$C(>276*) "%<=(*)]2'?*)=? b @)=6*@) p"`27? "0<](D)Z2'?D)]?=- @)] 6>@)S6>)=B_UW)](*276>&.(*)~"%?a:%9 b )].93&[email protected] 1 L6>@)Q)][email protected]'?*"0$(*276>)?>276*&(>eV 2c6>)]?=-w6>@)~<Z2'?D)F9'G @"%$'@o6>)]BCUW)](>2c6>&(D)]? 1.6*1/β @)SQ )][email protected] 2'?*"0$C(>276*)27:%?*9e"0<](D)Z2'?D)]? b @)] "%<=(*)]2'?*)=?]1\p&.6 > eV "06E"%<=(*)Z27?*)]?FGH(D93B 21/β xL"06*) 872':%&.) 976e?*@9 b 1 b @"%<;@|"0?FU(*9'Uw93(j6>"093L2':g6>9eV 6*@)o6>)=B_UW)](*276>&.(*)'[[email protected]"%.$3?S2'(*)!IK&"n6>)eQ"n)=(*)]56276 b )Z2'p2'<;K?*<Z2c6*6>)=(*"0$- [email protected]"%$'@6*)]BCUw)=(>276*&(*)=?]1f6*@)!Q)][email protected]'?*"0$ (>276*)!Q)]<=(*)Z27?*)]? b @.)] b )"0<](D)Z2'?D) v6*@"%?~pW)]@L2Z8T"%93(S"0??jiJBCU.6*93B2c6>"%<e97GX<]&(D(*)=K6F2'Q 93"%?D) <;@L2'(*2'<=6*)](*"0?D6*"%<a"%R2tv&J6*6>"0$3)=(,:%"0IK&"%QfeV -5_6>@)4:09 b 6>)=B_UW)](*276>&.(*)a<Z2'?D) 1TGH9'(k?DB2':0: 1 1/β eV eV 6>@)O:%9 b )](,6*@)O6*)]BCUw)=(>276*&(*)71J6>@.)m:`27(*$3)=(q6*@)4Q)[email protected]'?D"%$F(*276>)42'Q6*@)mG2'?j6>)](k"06kQ)]<=(*)]2'?*)=? b @.)] b )a"%.<](*)]2'?*) eV - A6 θ = 0 156>@.)OQ)[email protected]'?D"%$(>276*)"%? "%JxL"06*) 276 eV = 0 -K[[email protected]"%?kt&.6*6*"%$')](d:%"0IT&."%Q pw)[email protected]"%93(X"0?X"%{?*@L27(*Uu<]9356>(*2'?D6 b "n6>@R6>@)S(D)]?*&.:069'G a)YG - r7+ - ! ! ' ( !, " 1.# ! # % *, .!,- 1) % # 8 , ) .!,)[email protected])?D)])]!pW)=GH93(D)'1'6*@)AQ)[email protected]'?D"%$S(*276>)A"0?#U(D93UW93(D6*"%93L27:T6*9m6>@) )](D97 GH(D)]IK&)].<=iF6*&)=:%"%.$ !m? b ]< &(D(*)=K6.93"%?D) ?D)])SlgI\- +J-sr -T[[email protected]"0?X"%?,pW)]<]2'&?D)46>@) <;@L27(*$3)mP&<=6*&L276>"093?X27(*)4Q"0(*)=<=6>:niC(*)]:%276>)=Q 6>9C6>@)<]&(D(*)]564P&<=6*&L276>"093?m2':%93.$C6*@)E)=Q$3)]?OGH93:%:09 b "0$e6>@)<]9356*"%K&"06iu)=IT&276>"093f-\[[email protected])F<=9356>"n K&"06iu)]IK&L2c6>"%9'(*)=:`276*)]?O6*@)E<=&(*(D)]56 93UW)](*276>9'(O6>9_6>@)Q)]?D"06i93UW)](>2c6>93(O2'?4"[email protected]':%Q?a"%?D)]<=93Q A IK&L2'56>" )]Q{GH93(DB ρ(x, ω) = +J-srJ/ 1 ∇.J(x, ω) . iω L(*93BVlgI\- +.- rJ/ 1 b )E<]2' b (*"n6>)2_<=93)=<=6>"093GH93(DBF&:%2!pW)=6 b )])=6>@))]Q$3)<]&(D(*)=K6m93"0?*)E<=93(j (*)=:`276*93(X2'QR6*@)SQ)]?D"06i5Q.)]?D"06i{<]9'(*(*)=:`276*93(276XxL."06>)4GH(D)]IK&)=<=i2'? hhρ(x1 , ω)ρ(x2 , −ω)ii = Z +∞ −∞ dt iωt e ∂x1 ∂x2 hhI(0)I(t)ii . ω2 +.- r3r {93&(<Z2'?D)'16*@)SQ)]?D"06i5Q.)]?D"06i{<]9'(*(*)=:`276*"%93.?A"%?X<=93?*"0Q)](D)]Q b "06*@ -[#27T"0$e6>@)SQ)=(*"n8'2 ω=0 6>"n83) b "06>@(*)]?DUw)=<=6k6>9E6*@)mUW93?*"n6>"093?]136*@) 2 6>)=(*B "%!6>@)OQ)]9'B_"0L276>9'(q"0?,<]2'<])=:%)=Qf1TGH9'(,2':0:\p"`2'? ω (*)=$3"%BC)]?=1W$'"08T"%$2xL"n6>)E<=9356>(D"%p&.6*"%93u6>9o 6*@)FQ)=?*"n6iPL&.<=6>&276>"093?]-w"%2_Uw)=(D6>&.(*pL276*"08')F<Z2':0<]&J :`276*"%93 9'Ga6>@)o6*&)=:%"%.$Mm2'BC"%:n6>93"%2'f - O976>)o6>@276C276 )=(*96>)=B_UW)](*276>&.(*)'16>@)R<=&(*(D)]56o27:%93$ 6>@)S)=Q$3) b "06*@93&.6pL27<;T?D<Z276D6>)=(*"%.$_Q9T)]?976B2'7)[email protected]'6a93"0?*)716>@)=u6*@) PL&<=6*&L276*"%93?9'G^6>@.) <]&(D(*)=K6*?e2':093$6*@))=Q$3)=?!2'(*)2':%?D9"%Q.)]56>"0<Z2':g6>96>@)P&<=6*&L276>"093?9'GA6>@.)6>&.)]:0"%$<]&(D(*)]56][[email protected])6>&.)]:%"0$~<]&(D(*)]56qPL&<Y6>&L276*"%93.? b )](D)m<]9'B_U&J6>)]Q_93_Uw)=(D6>&.(*pL276*"08')]:0iF&?*"0$E}k)=6>@.)Y 27?>276 6>)=<;@"%IK&)=? /3/ 1f/r3rJ1f/ r MO)](D)'1^pw)YGH93(*)C?*@.9 b "0$6*@)C(*)]:%276>"093pw)Y6 b )=)]6*@)Q)[email protected]'?D"%$(>276*)27Q 6*@)_<=&(*(D)]56F.93"%?D) GH93(2'(Dp"06*(>2'(jipL27<;T?D<Z276D6>)=(*"%.$1\ b "0:%:(D)]Q)=(*"n83)S6>@)S$3)=)](*2':)YJU(D)]?*?D"%939'G^6>@)S<]&.(*(*)=56O.93"%?D)27? ?*@9 b _"% a)YG - /3/ -KC6>@"0? b 93(DW1K6>@.)m2'&J6>@93(D?k<=93?D"%Q)=(q6*@)aGH(*2'<=6*"%932':IK&L2'56>&.B MO2':%:LBC9TQ)]: √ -L[[email protected]) "%R6>)=(*BC?9'G#)Y83)=2'Qu9JQ.Q:0)=G 6D B_98T"%.$Fpw9'?*93? e,o φ (x + t) = [φ1 (x, t) ± φ2 (−x, t)]/ 2 )=8')]o27Q_9TQQ!<;@L2'(D$3)]?k2'(D)6*@K&?k(D)]:%276>)=Q!6*9 6>@)a<;@L2'(D$3)]?d9'Gw6>@)93(D"%$3"0L2':T:%)=G 6j2'QC(*"[email protected]BC98T"%$ √ √ )]Q$')]?4p5i 2'Q "0?a6*@)E6>976;2':^<;@L2'(D$3)E93pw9'6*@ ∆Q = Q1 − Q2 = 2Qo Q1 + Q2 = 2Qe Qe )]Q$')]?k2'QC"%?d<]93.?*)](j83)=Qo)=8')]o"0C6*@)U(*)=?*)].<])m97G6>@.)O"0K6*)](*2'<=6*"%93f-'[[email protected])Op2'<;K?*<Z2c6*6>)=(*"0$E<=&(*(D)]56 6>@K&?eQ)=Uw)=Q?!93:ni 9'|6*@)R9TQQpw9'?*936>@.)]93(ji3-h[[email protected]){Q)=?*<=(*"%UJ6>"%9'9'Ga93&(FBC9JQ.)]:,"06>)=(*BC?e9'G IK&L2'?*"0UL2'(j6>"%<=:%)=?O27:%:%9 b ?&?a6>9<]2':%<=&:`2c6>)S)Y.2'<Y6m6>(*2'?DUw93(j6mU.(*93UW)](j6>"%)=?]-W56>)]$'(>2'p"0:%"n6io)].?*&(D)]?46>@.) )YJ"%?j6>)=<])R9'GO2IK&L2'?D"%UL27(D6>"0<]:0)_pL27?*"%? b @)](D){6*@)o?D<Z276D6>)](D"%$"0?F93)pKi 93)7-h[[email protected])]?D){(*)]?D&:06*?e2'(D) K9 b "% )=GH?=- / r 1L/ r 1L/r -KL93(d2'5i 156>@.)a?DUw)=<=6*(*&B <]9'K6>2'"%."%$S2SK"%e27Qo2'_2'56>"0K"% ν b "06*@6>@)E<;@L2'(*$')]? Qo = 1/√2 2'Q −1/√2 (*)=?*UW)]<Y6>"08')]:ni31"0?m<;@L27(>2'<Y6>)](D" )]Qp5i6*@)F(>27U"%Q"n6i θ Q)=x)]Qup5i 1 @)=(*) "0?2'u2'(*p"n6>(*2'(Di?*<Z27:%)'E = −pvF = M eθ /2 b M @)]2~UW93?*"n6>"n83)X8393:n6;2'$')a"0?g6>&(D)]Q_93f156>@)aUW93?*"n6>"08')]:niF<;@L2'(D$3)]Q_IK&L2'?*"0UL2'(j6>"%<=:%)=? 6>@)K"%.T? xL:%:h6*@)C?*)Z2J- GA6*@)=izQ99'6~"%56*)](>27<=6Z16*@)K"%.T? b "%:0:hxL:%:d27:%:gBC93B_)=56>&B ?j6;276*)]? b "06>@ p< 276 )](D96>)=B_UW)](*276>&.(*)'-[[email protected])_UW93?*"n6>"0939'G,6*@)L)=(*BC"g:%)=8')]:d"%[email protected]"0G 6*)]Q 2'Q 6*@)CQ)]v?D"0F6iz 9'G eV /2 IK&L2'?*"0UL2'(j6>"%<=:%)=?_"0?<;@L2'.$3)]Q Q&)6>96>@.)"%56>)=(>2'<Y6>"093f-d G b )Q)YxL) "0?_6*@)Q)=?*"n6iy6*@)] ρ(θ) GH93( "%?A6*@)S?*@"nG 69'G^6>@)~L)](DB_"f:0)=8')]:b "06>@ ρ(θ) = 0 "06>@.93&.6SθpL2'><;KA?D<Z276D6>)](D"%A$"%56>)=(>2'<Y6>"%9'f16>@.)_<=&(*(D)]56F2c6 )](D96>)=B_UW)](*276>&.(*)C2'(*"0?*)=?~GH(D93B 6>@.) K"%K?B_98T"%.$F6>9e6>@)S:0)=G 6a276X6*@)~\)=(*BC"83)=:%9T<]"n6i ' )( ? ' ?( ' ? A( I0 (V ) = evF Z A dθρ(θ) = ν −∞ e2 V . 2π +.- r [[email protected])~pL2'<;K?*<]276*6*)](*"0$<=&(*(D)]564"0?X6>@)~(>276*)~276 b @."%<;@6*@)~<;@L2'(*$')9'G6>@)~:%)YG 6D B_98T"%$e)=Q$3)~"%?Q) √ U:%)Y6>)=Q -L{6>@.)IK&L2'?D"%UL27(D6>"0<]:0) pL2'?*"0?]16*&)=:%"%.$C<=93(*(D)]?DUw93.Q? IB = ∂t [e∆Q/2] = ∂t [(e/ 2)Qo ] 6>9e6>@.)~U.(*9T<])]?D?9'Gg2eK"%?*<Z2c6*6>)=(*"0$!9'6>@.)~<=9356;2'<Y6a"%56>9C2'u2'56>"0K"%\ )4<=93?D"%Q)=(A6>@)4"0B_U.&(*"n6i S B_276>(D"n_)]:0)]BC)]56 Sjk(p/TB ) b @"%<;@{Q)=?*<=(*"%pW)]?2F?D"%$':%)mIK&L27?*"n UL2'(j6>"%<=:%)97G~6iTUW) 2'Q B_9'B_)=K6*&B ?D<Z276D6>)=(*"%.$ )=:`2'?j6>"0<Z2':0:0i 9'G~6>@)UW93"%56{<=9356;2'<Y6"056>9|2 IK&L2'?*"0UL2'(j6>"%<=:%)F97Gk6jiTUW) -f[[email protected])C)])=(*$'i p <;@L2'(*2'<=6*)](D" )=?~6*@)!<]9356>2'<=6 UW)](j6>&( k TB TB ∝ λ1/(1−ν) pL276*"083)S27(*$3&BC)]56X97G6*@) (*)=93(*B_2':0" 2c6>"%9'$3(D93&Uf1?D)]) )=G - / r 2'.Q{6*@)S(>2'U."%Q"n6i <Z2'{pW) θB Q)=x)]Qep5iS6>@),(*)]:%276>"093 ?*9m6*@L276^6>@)k"%BCU&(*"n6i B2c6>(*" S)]:0)]BC)]56>?#2'(*)qGH&<Y6>"%9'? θB TB = M e /2 ' ( S 9'G θ − θB ' ) ( -.[[email protected])E}k)=6>@.)Y 27?>276 6>)]<;@."%IK&)~$3"08')]?X6>@.)S6*&)=:%"0$eU(*9'pL2'p"0:%"06i2'? / r |S+− (θ − θB )|2 = 2'Q |S++ |2 = 1 − |S+− |2 1 1 + exp (θ 2 1−ν ν )=(*9!6>)=B_UW)](*276>&(D)'1 b -J9e6*@L276O276 IB (V, TB ) = −evF Z − θB ) )[email protected]) A −∞ +.- r +.- r ? dθρ(θ)|S+− (θ − θB )|2 [ @)~6*9'6;27:<]&(D(*)]56 X Q9J)=?9'6aQ)]UW)].Q93{6>"%BC) ?*"0<])S6>@.) I I(V, TB ) = I0 (V ) + IB (V, TB ) ?DiT?D6*)]B"%?a"%2!?D6>)]2'Q.iu?D6>276>)7-wuG2'<=6]1L6>@"0?a<=&(*(D)]564"0? b @)=(*)E6*@)<=&(*(D)]5649'Uw)=(>276*93( "0<]:0&Q)]?A6*@)S<]&(D(*)=K6 b "n6>@{"06*?APL&<=6*&L276*"%93?=-L[[email protected])SI<]&=(D(*hj(t)i )=K6aPL&<Y6>&L2c6>"%9'?2'(*) <;@2'(>2'<Y6>)=(*" )]Q j(t) p5io6>@)S<=93(*(D)]:`2c6>93( 1 C(ω) = 2 Z +.- r dteiωt h[j(t), j(0)]i . A !6*@"%?gIK&L27?*"%U2'(D6*"%<]:0)AU"0<=6>&.(*)'176>@)IK&L2'?D"%UL2'(j6>"0<]:%)=?g2'(D)X<]93(D(*)]:%276>)=Q!p&.6k276 )=(*9 6>)=B_UW)](*276>&.(*)'1 2':%:q6>@){K"%. ?D6>276>)=? b "n6>@|(>27U"%Q"n6i :0)]?D?!6*@L2' 2'(*)RxL:0:%)=Qf1h2'.Q 6*@)R(*)=B2'"0"%.$K"%?j6;2c6>)]?=1 2'? b )]:%:A2'?C2':%:,2'56>"%K"0 ?j6;276*)]?=1d2'(*)R)=B_U.6i'[email protected]'6!.93"%?D)R9J<=<]&(D? b @.)]|6>@){pL2'<;K?D<Z276D6>)](D"%$z"%? "%<=:%&Q.)]Qf- ) <Z27Q)=?*<=(*"%pW) 6>@)SQ<[email protected]'693"0?*)4GH(D93B 6*@)SIT&2'?*"0UL2'(D6*"%<=:%) 2'U.U(*9527<;@pW)]<]2'&?*)S6>@.) ?*<]276*6*)](*"0$97z6*@)FUW93"%56a<]9'K6>2'<=64"%?a)]:`27?D6>"0<E27Q93.)Ep5i9')':0)=G 6OB_983)=(mpL27<;T?D<Z276D6>)=(*? "%56*9 2C(D"%[email protected]BC983)](a<]9'(*(*)=?*UW93Q?a6>9_2'u9JQQTpW93?*93uK"%?*<Z2c6*6>)=(*"0$_"056>9_2'27K6*"%K"%.W-L G b )Q)=x) @)] 2K"0 97GB_9'B_)=K6*&B ?*<]276*6*)](*?e"056>92'2'56>"%K"0\1#2'.Q "0GX"n6F?*<]276*6*)](D? f = 1 b "%56>92K"0e6>@)=6>@)m2Z83)](*2'$3)O9 8')](AB_2'p5i!)=8')]56>?X"0? -2 Tf=6>@0)aIK&L2'?D"%UL2'(j6>"0<]:%) 2'UU(D952'<;@f1v6>@.)o93"0?*)"0?~6*@)]U.(*93UW93(D6*"%932':h6>96*@)_hf P&i<== 6*&L27|S6>"0+− 93(p/T 9'G B )| f C(0) ∝ (hf 2 i − hf i2 ) J"%.<]) "%?)]"n6>@)=( 93( 1 2 1 b )[email protected]) ' 1 ! f 0 1 hf i = hf i Z 2 C(0) = e vF L93:%:09 b "0$6*@) GH93(*B '<? ?( A −∞ dθρ(θ)|S+− (θ − θB )|2 [1 − |S+− (θ − θB )|2 ] . 9'G^6>@) 6>(*2'?DB_"0?*?*"093{2'BCU:%"n6>&Q.)4"%ulgI\- +.- r |S+− |2 [1 − |S+− |2 ] = 1 b +.- r +.- r'+ )~<Z27 b (D"06*) ∂|S+− |2 ν . 2(1 − ν) ∂θB J"%.<])E.)]"06*@)]( 93( Q )]UW)].Q? 93 1 F ) <Z27U&:%:6*@) 93&.6497Gg6>@)F"056>)]$'(>2': - a?*"0$6>@.) ρ A θB b ∂ θB )YJU(D)]?*?D"%93.? +.-sr 2'Q J+ -sr GH93( '2 .Q976>"%<="%$6>@L276 1 b )[email protected]) I(V, T ) T ∂ =∂ ? B C(0) = − B TB eν TB ∂TB I(V, TB ) . 2(1 − ν) )_9'6*"%<])!6>@276 "%?~6>@)CGH&<Y6>"%9' 9'GX9':0i ?D)])ldIK?=- I(V, TB )/V V /TB /'/ 16*@L276BC)Z2'? ' ?( V ∂ I(V, TB ) V 2 ∂ I(V, TB ) =− , ∂TB V TB ∂V V 6>@)= b ) xLQ2'9'6*@)](XGH93(DBF&:%2EGH93( )=(*97GH(*)]IK&)=<=i{.93"%?D) C(0) = ' /3/ ? ( eν (V Gdif f − I) , 2(1 − ν) "0?A6>@)SQ"n)=(*)=K6*"`2':<]93Q.&<=6>2'<])7b @)](D) G dif f = ∂V I 3+ θB +.- r' 2'Q / 9'G )=G - +.- +J- / ) <Z27u6*@)](D)=GH93(D)~"0583937)~<=&(*(D)]56O<]93?D)](j8'2c6>"%9'2c6X6>@)SUW93"%56<]9'K6>2'<=66*9CQ.)](*"n83)~2!$3)])=(>2': S GH93(*BE&:`2GH93(!6>@)Q)]<[email protected])](D)]<=)(>276*)'1 b @"%<;@ Q.)]?*<=(*"0pw)=?6>@)<](D93?*?D9 8')](oGH(D93B 6>@) b )]2' 6*9 6>@.) ?D6*(*93$!pL27<;T?D<Z276D6>)=(*"%.$_(D)]$3"0B_)71 (τϕ−1 )(2) = +J- r e3 τ02 ν (V Gdif f − I) , d2 1 − ν lgIT&276>"093 +.- r 27:%:%9 b ?a&?m6*9Q)=?*<](D"%pW)E6>@.)E<](D93?*?D983)]( "06*@)FQ)[email protected]'?D"%$o(*276>)~GH(*93B 6*@) b )Z2' 6>9R6>@.)e?D6>(D93$upL2'<;K?*<]276*6*)](D"%$u(*)=$3"%BC)e276 )](*9R6*)]BCUw)=(>276*&(*)7-f 6~"%? BC93(*)FQ." <]&:n646>9{Q)=(*"n83)C2 $3)].)](>27:^(D)]?D&:06m276xL"n6>)S6>)]BCUW)](>2c6>&(D)pW)]<]2'&?*)~6>@)<]&(D(*)=K6m"%{6>@)2'p?D)]<=)F9'G#pL2'<;K?*<]276*6*)](*"0$ PL&<Y6>&L276*)]?a276xL"n6>)46*)]BCUw)=(>276*&(*)7- ,*, '!-1) .!/*( # % , :1!.!,- )9 b <=93?*"0Q)](F2'$52'"0 6>@.)RUw)=(D6>&.pL276>"n83)o<]2':%<=&:`2c6>"%9' 9'G6*@)oQ)[email protected]'?D"%$(*276>)7- )]B_2'(*c2'p:ni31 GH93(6*@) b )Z2' 2'Q|?D6*(*93$p2'<;K?*<Z2c6*6>)=(*"0$(*)]$'"%BC)]?]1^"06E"%?FUW93?D?*"%p.:%)6*9$39pW)=i39'Q6>@)R?j6>(*9'$ ?*<=(*)])="%.$_:0"%BC"06Z1T2'.Q b )~<Z2'u<]93BCU&.6*)SlgIW- +.-0/r GH93(2'2'(Dp"06*(>2'(ji ,93&.:%93BFp{7)](*.)]: f (x) ν 2 Γ20 dx dx0 f (x)f (x0 ) 4β 2 π 2 vF2 α2 2ν π 2ν π Z ∞ sinh sinh iτ iτ X 0 0 β β h i+ h i dτ cos[ω0 τ ] × 2ν 2ν π π −∞ sinh (ητ + iτ0 ) sinh (−ητ + iτ0 ) η β β Z ∞ π π [−ηsgn(τ1 )(x/vF − τ1 ) + iτ0 ] + coth (−η(x/vF − τ1 ) + iτ0 ) × dτ1 sgn(τ1 ) coth β β −∞ Z ∞ π π 0 0 × dτ2 −sgn(τ2 ) coth [ηsgn(τ2 )(x + τ2 ) + iτ0 ] + coth [η(x /vF + τ2 ) + iτ0 ] . β β −∞ (τϕ−1 )(2)=− Z Z +.- 3 [[email protected])d6>(*"0U:%) 6*"%BC)"0K6*)]$3(*2':5"% 6*@)d?*)]<=93QF9'(*Q)=(^<=9356>(D"%p&.6*"%93 6*9a6*@)dQ)][email protected]'?*"0$O(*276>)d"0?^<=93BCU&.6>)=Q 2'L2':niK6>"%<]2':%:niC2'? ω0 β Γ20 ω0 β ν 2 (−i)2ν 22ν−2 2ν π Γ ν +i =− sinh iτ0 cosh ω0 τ0 − 4βπ 3 vF2 α2 Γ(2ν) β 2 2π h i 2 sinh βπ (ηx/vF + iτ0 ) X Z +.- β h i × dxf (x) i(2τ0 + ηβ) + . ln ηπ sinh π (−ηx/v + iτ ) 2 (τϕ−1 )(2) G η F β G 0 ) 2':0?*9a<]93?D"%Q)=(^6*@)qQ)][email protected]'?*"0$m(*276>)q"%lgIW- +J- GH9'( k "06*@(D)]$527(*Q"0$a6*@),U.(*93UJ τ0 ω0−1 , β b i h i h )](j6>"%)=?g9'GGH&.<=6*"%93 2'QeGH&(j6>@)=(d?*"%BCU:0"n ln sinh βπ (ηx/vF + iτ0 ) / sinh πβ (−ηx/vF + iτ0 ) G iT"%$F6*@L276 )=83)=f1 b )~93p.6>2'"% f (x) (τϕ−1 )(2) 4Γ2 ν 2 τ02ν−2 = 2 04 π vF Γ(2ν) 2π β 2ν−1 2 Z ∞ 2 ω0 β ω0 β cosh Γ ν +i dxf (x) . 2 2π 0 +.- ? [[email protected])q(*)]?D&:06^<]2'pW)dQ"%?DU:`2Zi')]QE"0 6>)](DB_?v9'GJ6>@)d(*276>"09apW)=6 b )])=E6>@.)k27(*p"n6>(>27(Di4?D<](D)])]."%$OQ)][email protected]'?*"0$ (>276*)S2'Q{6>@)S?j6>(*9'$!?*<](D)])="%$CQ)[email protected]'?D"%$C(>276*) pW9'6>@u93.)]IK&"%:0"%p(D"%&B <=9356>(*"0p&.6*"%93? F ≡ (τϕ−1 )(2) (2) (τϕ−1 )λs →α d2 = (eα)2 3 Z ∞ dxf (x) 0 2 , +.- A b @ )](D)6>@)"%56>)=$3(>2':O"%?2 GH&<=6*"%93 97G d/λ 2'Q b )(*)]<]2':%:a6>@L2c6 α "0?6>@)?*UL276*"`2':O<]&.6*9'-, G 6>@) ,93&:093BFp "%56>)=(>2'<Y6>"093y7)](D)]: "%?!s<;@9'?*)] 27?_?D&$3$3)=?D6>)=Q pw)YHG 93(*) ?*)])lgIW- +J- 1g6*@) f (x) Q)][email protected]'?*"0$!(>276*)~276a2'(*p."06>(*2'(Di @L2'?a2'u2'L2':niK6>"%<]2':) JU(*)=?*?*"093 ' ? ( λs F = πd 2α 2 E0 d λs + N0 d λs , +.- ' ( b @ )](D) E (d/λ ) 27Q N (d/λ ) 2'(D)F6>@) )=pw)=(S2'Q O)=&B27GH&<Y6>"%9'? / r'+ 1fpW9'6*@9'G )](D9 93(*Q.)](]- 0"%?#U:%9's6D6>)=Qe"%F6>@.0)X"%?D)=s6g9'G#"0$3&(*),+.- 132'.Q "%?#6>2'')=C6*9 pw)2 ?*B_2':0:J<=93?D6>2'56ZF (α/d)2 "%?k"%.xL."06>)O"%6*@)S2'p?D)]<=) 9'G^?*<=(*)])="%.$1.p&J6"%oU(*2'<=6*"%<]2':w?D"06*&L276>"093?=1T6*@) U(*)=?*)=<])S9'GvBC)=6>2':%:0F"%< $5276*)]? 2': b 2ZiT? "%BCUW93?*)=? 2xL"n6>)F?D<](*)=)]"0$u:%)].$'6>@fQ)=<](D)Z2'?D)]? b "06*@ 2'Qz2'UU(D952'<;@)=? 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