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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.
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HAL Id: tel-00175563
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Submitted on 28 Sep 2007
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HÊ [email protected]Ê G »,POKM}&J!F!CELeJVDLGLGKMPONLGRÂJ!€GF!CTDGRE€zUÅJ!DGLGLGK&P$ÈDGLG}¶JVNCTLÕÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ
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HÊ Ã[email protected]Ê G ƒ„€KxNOLGKMPUTpJ!N}"}&DGF!FKMLqJ ÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ
HÊ Ã$Ê Ã ="D4UÇLiJ!DGÁ BCENLqJH}&CELqJ UT}¶J°UÇp½UÅpCEDGF!}&KxC LGCTNp!K ÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ
HÊ H ²½CENOp!KSWGK¶JVK&}&JVNOCELlNLeK $BKMF!NOÁ•K&LqJVprÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ
HÊ [email protected]Ê G ²©CENpKF!KMWDG}&J!NCELeÁKjUTpDGF!K&Á•K&LiJ!p`ÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ
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?$Ê [email protected]Ê G ¨ K&JVK&}&J!CEF½}&CELGpNpJ!NLGR–C UÅp!NOLGREPK"²"ÀxÈDGLG}&J!NCEL ÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ HI
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?$Ê Ã[email protected]Ê G ­lC$WGK&P Ô UTÁNP JVCELNUTL¡ÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ EÄ
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?$Ê Ã$Ê H ÀNLGREPOK"KMPOKM}¶JVF!CTLlJ!DGLGLGK&PNLRŸÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ qÃ
?$Ê Ã$<Ê &ƒ 7C KMPOKM}¶JVF!CTLGpQJVDGLLGKMPONLGRUTp„*J 7C•‚iD4UTpNB4UÇFJVNO}MPOKMp ÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ G
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?$Ê [email protected]Ê G ®½BGBKMLGWGN e® ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ ATÄ
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?$Ê ?$Ê A ®½BGBKMLGWGN lË ÊxÊrÊrÊxÊÊrÊxÊrÊxÊrÊrÊxÊrÊxÊÊrÊxÊrÊrÊxÊrÊxÊrÊrÊrÊrÊrÊxÊrÊxÊ A ,*, '!- # % 23!, #). .! * 1!, # % ,)'!- 1!
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®QJ„LGCEL´ MK&F!CÅJVK&Á•BKMF!UÇJVDF!KJV€GK&F!Á•UTP GDG}&J!D4UÇJVNOCELGpQ}&CELqJVFNIGDJ!K"}MCTLGp!NOWGKMF!UTIGPOJVCÂJ!€GKSLGCENOp!KK&ÆTKML
NLJ!€GKeUTIGpKMLG}&KzC HJ!€GKeIGNUTp>}MDGFF!KMLqJ N|Ê KÇÊ/NL K&‚iDGNPONIGFNDGÁ¶Êƒ„€GK&p!KeJV€GK&F!Á•UTP 4D}&JVDGUÇJVNOCELGp–UTFK
}jUTPOPK&WJV€GK&F!Á•UTP;LGCTNp!KeUÇLGW UÇPp!C iLGC HL UTpÒTCE€GLGpCEL$´µ²©‚iDGNp¬J•LCENpKÉI K&}jUTDGpKlJ!€GK& ;K&F!A
K 4F!p¬J
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F!K&B CEF¬JVK&W`K$BKMF!NOÁ•K&LqJ UTPOPOÉIq`ÒÊ+ΰÊ9ÒTCE€GLGpCEL ' H A (UTLGW‹UTLGUTPO MK&WÕJV€KMCEFK&JVNO}jUTPOPOÉIq Ê ²©‚iDGNp¬J
MONQPSR%T$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¿ (|ʱµLJV€GK•PNÁNOJ k θ eV, ω [J!€GKMFÁ
TU PÐLGCTNp!KWGCEÁNLGUÇJVK&prC1ÆTKMF>CÇJV€GK&FxJµB K&p>CQLGCENpKTʃ„€GK
Á UÇRELGNOJ!DGWGK"CJ!€GKLGCENpKSB C7KMF©Np„BGFCEBCEFJ!NCEL4UÇPJ!CJ!€GK}MCELGWDG}&JVUTLG}MK G CJ!€GKppJ!KMÁC6H€N} €
NpHUTLeNOPPDpJVF!UÇJVNOCEL CJ!€GK<4DG}&J!D4UÇJ!NCEL$´ WGNpp!NOB4UÇJVNOCEL J!€GKMCEFKMÁl~
|Ã$<Ê SI = 4kB θG ,
H€GKMFK G = 1/R HNOJ!€ R Np„F!K&p!NOpJ UÇLG}MKÇ6Ê KrLGCTJVKSJ!€4UÇJHJV€GKrK $BGFKMpp!NCTLÕNLl»,‚ Ê Ã$Ê QNp„ÆÇUTPNOWÉNL
I CÇJV€ÉJV€GKr}MPUTp!pN}MUTPUTLGWeJV€GK‚iD4UTLqJ!DGÁ F!KMRTNÁKTÊG±µLeJV€GKrPU3J!JVK&F„}jUTpK 7Kx€4UjÆTK>p!NOÁ•BPOJVCF!KMBPUT}&K
G Iq JV€K}&CELGWGD}&J UÇLG}MK‚iD4UTLqJVDGÁ NOL Í UÇLGW4UTDGK&F 1 p hCEF!ÁÂDPU G = 2e2 T /h H€GK&F!K T NpÉJ!€GK
JVF!UTLGp!ÁNpp!NOCELÉBGF!CTI4UTIGNOPNOJµ hCEF„p!NOLGREPKS} €GUTLGLGK&P[}jUTpK ¶Ê
ƒ„€GKMFÁ UTP9LGCENpKNpHUTPOp!CÅ}jUÇPPK&W H€GNOJ!KSLGCENpK * J!€GKp!BKM}¶JVFVUÇP[WGKMLGpNOJµeNOp„NLGWGK&B K&LGWGK&LiJ©C f Ê
B
' .: ! /
23!,
*,#)+!,# .
$À €GCTJ7LGCENOp!K½NL•UTLvKMPOKM}¶JVF!NO}jUTP }&CELGWGDG}¶JVCEF7Np7UxLGCTL$´µK&‚DNPNOIGF!NODGÁ hIGNUTpÐÆECTPOJ UÇREK V 6= 0›LCENpK½CEFNRÇ´
NL4U3JVKMWhF!CEÁ JV€GKSWNp!}&F!K¶JVKMLKMp!p©CJV€K"} €4UTFREKMpHC KMPOKM}¶JVF!NO}jUTP}MDF!F!K&LqJjÊÀ$€GCÇJ„LGCENOp!K"NOp„UÂWGCEÁNL4UÇLiJ
}MCELqJ!F!NIDJVNOCELzNOLzJ!€GK>LGCTNp!(
K H€GKML eV kB θ, ω Ê À$€GCTJ°LGCENpK;UTp"GF!pJ°WGKMp}MFNIKMW:Iq:À} €GCTJ!J q
' HI ( H€GCpJ!DGWGNK&WlJ!€GK} €4UTFRE<K 4DG}&J!D4UÇJ!NCELeBG€GK&LGCEÁKML4UÅNOLzUÂÆÇUT}&DGDGÁ JVDI KWGNOC$WGKÇÊ
± 7KÕUTpp!DGÁKÉJV€4U3JJ!€GKeKMPOKM}&J!F!CELpBGUTp!p}MCEÁBGPOK&JVK&PONLGWGK&B K&LGWGK&LiJJV€F!CEDGRT€ U:}MCTLGWGDG}¶JVCE%F JV€GK&LJ!€GK•LiDGÁ>I K&FxC ;‚iD4UTLqJ U N NOLUÉJVNOÁ•KÅNOLiJ!KMF¬ÆTUÇP T0 4D}&JVDGUÇJVK&pÂUTLGW}jUTLI KWGK&p!}MFNIKMWIi
CENOp!pCELGNUÇLp¬J UÇJ!Np¬JVN}&pMʃ„€GK•UjÆEKMF!UTREK•LDÁÂI K&FNprRENOÆTKMLIi‹J!€GK•ÁKjUÇL}&DGF!FKMLqJ hN i = hIiT /e
UTLGWvJ!€GK"ÁKjUTLÉp‚DGUTF!KSWGK&F!N ÆTU3JVNCTL Np h(∆N )2 i = hN i 7H€N} €ÉNp;DGp!K&WeJ!CÅ}jUTPO}MDGPUÇJVKHJV€KS}MDGFF!0KMLqJ
4DG}¶JVD4UÇJ!NCELpHUÇJ MK&F!;C hF!KM‚iDGK&LG}&
e2 h(∆N )2 i
T02
ehIi
=
.
T0
h(∆I)2 i =
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PNOÁ•N J
|Ã$<Ê A SIP oisson = 2ehIi .
ƒ„€GKlp!€GCTJ LGCENOp!KlB C 7KMF NOpÅ*J HN}MKlJV€GKlBGFC$WGD}&J•C "JV€Kz} €4UÇF!REKz‚DGUTLqJVDGÁ UTLGW JV€GKzÁ•KMUTL }MDGFb´
F!K&Li"J 4C HNLGR•JV€GFCEDGRE€:UWGK&ÆNO}MKTÊ ƒ„€GNp½%K KM}¶JS}jUTLÕI KÂCTIGp!K&FÆEK&W‹NLzÆÇUT}&DGDGÁ JVDGIKMp½CEF°NOLzJ!DGLGLGK&P
ÈDGLG}¶JVNCTLGp H€GK&F!KÅJV€GK–} €4UÇF!REKłiD4UTLqJVUzUTFKÂJVF!UTLG/p hK&F!FKMW‹NLGWKMBKMLGWGK&LqJxC ;KMUT} €CÇJV€GK&FMÊ+ƒ„€GNOpSp!€GCÇJ
LGCENOp!KÂNOp°}jUÇPPK&W‹CENOp!pCELGNUÇL`LGCENOp!KÅUTLW»,‚ Ê |Ã$Ê A ½N"p LC HLUTpÀ} €GCTJ!J q'
 hCEF!Á>DGPUʃ„€GNOp"F!K&p!DGP J
NprÁ•CWGN 4KMWNL‚iD4UÇLiJ!DGÁ·J!FVUTLp!BCEFJ>WGDGK•J!CÕJV€K UT}&JxJV€4U3J>K&PKM}¶JVFCELGpÅUTF
K hK&F!ÁNCELGNO}–B4UTFJ!N}&PKMp
H€GN} €eCEIK&•JV€GKrUTDGPN9BGF!NOLG}MNOBGPKÇÊ$±µLÉJV€Np„}jUÇp!K 4N &;K}MCTLGp!NOWGKMF„NOLG}MNOWGKMLqJHKMPOKM}¶JVF!CTLGp„CELlUÂB CÇJVKML´
JVNUTP/I4UTFF!NOKMF H€GN} €UTF!KFVUTLWGCEÁPO:K&NOJV€KMFJVFVUÇLGp!ÁNOJJVKM
W HNOJ!€BF!CEI4UÇIGNPONOJµ T CTFxF%K 4K&}&JVK&W HNOJ!€
BGF!CTI4UTIGNOPNOJµ
#JV€GNOp;Np›JV€GK°F!KjUÇp!CELvJ!€4UÇJQp€GCTJ„LCENpK"Np7UTPpCÂ}jUÇPPK&W B4UÇFJVN JVNOCEL LGCTNp!K p!C
JV€G<K hFKM‚iDGKML}&Re=NLWG1KM−BKMTLGWKMLqJ°p€GCTJ©LGCENOp!KSpB K&}&JVF!UTP[WGK&LGp!N JµeNp ' HD (
Ã$Ê ¿ SI = 2ehIi(1 − T ) .
± JQNOp;DGp%K hDGP9J!CÂWGK 4LGK"JV€KSË UTLG
C UÇ}&JVCTFQUTp;J!€GK"F!UÇJVNOC>IK&*J 7KMKMLeJV€K"NLGWKMBKMLGWGK&Lq4J hF!K&‚DKMLG}¶Ép!€GCÇJ
LGCENOp!KSWGN Æ$NOWGKMWeIqvJ!€GKrCENOp!p!CTLzLGCTNp!K
F ≡
SI
,
2ehIi
D
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±µLiJ!KMFKMpJ!NLGRTPO [}&CEF!FKMPUÇJVNOCEL‹BG€GKMLCEÁ•K&L4UePN ÇKÅJV€GK•UTDPN,BGFNLG}&NBGPOKÅCEF ³ CEDGPCTÁÂI‹NLqJVK&FVUT}¶JVNCTL
}jUTLp!DGIpJ UÇLiJ!NUTPOPO:pDGBGBGFKMppÂp!€GCÇJ>LGCTNp!KNLÁKMp!CTp!}MCTBGN}ppJ!KMÁpMÊË CTFrp¬pJVK&Á•pxN0
L H€GN} €}MDGFb´
F!K&LiJ>NpLGCTJ>}jUTFF!NOKMWNLDLGNOJ!pxC QKMPK&}&J!F!CEL} €4UTF!RTK JV€GK•REKMLGK&FVUT3P hCEFÁÂDGP
U hCEFJV€K p!€CTJ>LCENpK Np
SI = F 2qhIi H€GK&F!KJV€GKKMPOKM}¶JVF!CTLz} €4UÇF!REK e NOp„F!K&BGPUT}&KMWlÔ IqlUTLzK 9K&}&J!NOÆEK} €4UÇF!REK q Ê4À$€GCTJ©LGCENOp!K
Á•KMUTp!DF!KMÁKMLqJ!p"B K&F/hCEFÁ•K&W`NLlJV€GKhFVUT}¶JVNCTL4UTP‚iD4UTLqJVDGÁ UTPOPF!KMRTNÁK>UTPOPC7KMWÕJ!€GKÂCEIGpKMF¬ÆTU3JVNCTL
C +J!€GK hFVUÇ}&JVNOCEL4UTP4} €GUTF!REK°}MCEFF!KMpB CTLGWGNLR>JVCrJV€GK"‚iD4UÇp!N ´ B4UTF¬JVN}&PK&p ' qÄ Già (8$UTLGWvp€GCTJ„LCENpK°Np
KMLG€GUTLG}MK&WIq‹
U UT}¶JVCEF [NL‹Ul²"Ài´ ÈDGLG}&J!NCEL:IKM}MUTDGpK•C ›JV€K®½LGWGFKMK&Æq´ F!%K GKM}&J!NCEL ' H)( Ê9ƒ„€GKMFK¶´
hCEF!K qUSp!€GCÇJ7LCENpK„Á•KMUTp!2DGFKMÁKMLqJ›RENOÆTKMpÐUTWWGNOJ!NCEL4UÇPN7L hCEF!Á•UÇJVNOCELÂUÇI CED$J,JV€GKHKMPOKM}¶JVF!NO}jUTPJVF!UTLGp!BCEF¬J
H€GN} €lNp„LCTJ½UT}M}&KMp!pNIGPOKrÆNU–}&CELGWGD}&J UÇLG}MKrÁ•KMUTp!DF!KMÁKMLqJ!pMÊ À$€GCTJ©LGCTNp!KÁKjUTpDGF!K&Á•K&LqJ°UÇPp!C–}MUTL
I KDGpKMWÕJ!C•WNpJ!NLGRTDGNp€lIK&*J 7KMK&L:}MPUTp!pN}jUÇPUTLGWՂiD4UÇLiJ!DGÁ p!}MUÇJ!J!KMFNLGR•NLzU} €4UÇCTJVNO}r}MUjÆ$N Jµ ' G (8
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p ' A)(9 H€GNPOKr;
U hK&F!ÁNCELGNO}"JV€GK&F!Á•UTP[p!CTDGF!}&K>KMÁNOJ!pHB4UTFJ!N}&PKMp
p!K&B4UTFVU3JVKMP  |UÇLiJ!N ´ IGDGLG} €GNOLGR QPKMUTWGNOLGRÅJVCp!DGI$´µCENOp!pCELGNUÇLÕpJVUÇJVNOpJ!N}M
p ' i¿ (|Ê ƒ„€GN?p UT}¶J½}&CELGp¬JVNOJ!DJVK&p
UÂÆEKMF¬eNÁB CTFJ UÇLiJ7JVCCEP hCEFQJ!KMp¬JVNLRK&LqJ UTLGRTPKMÁKMLqJ©NLÉJ!€GK}MCELqJVK J½C ‚iD4UTLqJ!DGÁ }MCEÁBGDJVUÇJVNOCEL+Ê
±µL Á UT}&F!CEp}MCEBN}Õp¬pJVK&Á•%p „p!€GCÇJÉLGCENOp!KÕNOpvLGCTJvBGF!K&p!KMLqJeIKM}MUTDGp!K:}&DGF!FKMLqJ 4DG}¶JVD4U3JVNCTLGpÉUTFK
UjÆEKMF!UTREK&WzCED$J½IqÉK&PK&}&JVFCELGp„JVF!UTLGp hKMF!FKMWeJV€GFCEDGRE€lÁÂDGP JVNOBGPK©JVFVUÇLGp!BCEFJH} €GUTLGLGK&PpMÊ
, !" !,# .()$*.-*/%012 ,!,:.
±µLJ!€GKH€GNRT€;hFKM‚iDGK&LG}&–PONÁNOJ ω k θ, eV & KMFCÇ´µBCENLqJ 4D}&JVDGUÇJVNOCELGpÐNLÅJV€K©WK&ÆN}&K©NOLqJVF!CWGDG}&K
B
UTLUÇpÁ•ÁK&J!F NL5JV€GKzp!BKM}&J!F!DGÁ S(ω)
6= S(−ω) Ê7±µL5JV€GKzWGK%4LNOJVNOCEL CLGCENpK »Ð‚ Ê Ã$Ê H Ô
ˆ =
I(t) NOp„F!KMBPUT}&KMWlIi JV€GKSJVNOÁ•K°WGKMBKMLGWKMLqJ"}&DGF!FKMLqJ©CEB K&FVUÇJ!CEF„NLÉJ!€GK KMNOp!K&LIKMFR•BN}&J!DGF!K I(t)
Ô
K $B
HNOJ!€ IKMNOLGRSJV€GK„JVNOÁ•K„NOLGWGK&B K&LGWGKMLqJ UTÁNP JVCELNUTLxC J!€GKHppJ!KMÁlÊiƒ„€GK
ˆK $B
p!BKM}¶(iJVĤt)
FVUÇP[WGIKMLGpNO(−i
JµÉNOĤt)
p„LGC WGK 4ĤLGKMWzUTp
SI (ω) = 2
± 7K>LC5JV€GKHNOLGNOJ!NUTP REFCEDGLGW pJ U3JVKMp
SI (ω) = 4π
X
i,f
Z
∞
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ˆ
ˆ
dth∆I(t)∆
I(0)ie
.
−∞
|ii
UTLGW–JV€GK 4L4UÇP NOLiJ!KMFÁ•K&WGNUÇJ!K pJ U3JVKMp
ˆ 2 P (i)δ(Ef − Ei − ω) ,
|hf |I|ii|
|f i
7K©CTIJ UTNOL
ÃÊ@1G Ä hCEFQNOLGNOJ!NUTP p¬J UÇJ!KMpMÊ4±µLeCEFWGKMFQJ!CÅNLqJVK&F!BGFK&JHBG€qp!NO}jUTPOPO
NpH}&CEDGBGPOKMWeJVCvU–WGK¶JVKM}¶JVCEF&Ê K 4LGWlJV€4UÇJ S (ω) NOp
BGF!CTB CEF¬JVNOCEL4UTP4JVCxJV€GKSK&LGKMFRT J!FVUTLGphKMFQF!UÇJVK°I K¶J*;K&KMLeJV€GKSp¬$p¬JVK&Á UTLGWvJ!€GK"WGK¶JVKM}¶JVCEF&ÊG± E > E
ω = E − E > 0 9KMLGK&F!RT‹NOp"JVFVUÇLGp/hK&F!F!K&W0h FCEÁ J!€GK–WGK&J!KM}&J!CEFSJVCÉJV€Kp¬pJVK&Ázʃ„€4U3JxÁKjUÇLGp
B CTp!NOJ!NOÆTKhF!K&‚iDGKMLG}&NK&pÅ}MCEFF!KMpB CTLGWJVC:UÇL2UTIGp!CTF!BJ!NCELK&LGKMFRTBGF!C}MK&p!p h F!CTÁ«J!€GKvKMLqÆNFCELGÁKMLqJ!
H€GNPOK/LGK&RqUÇJVN ÆEK hFKM‚iDGKML}MNK&p}&CEF!FKMpB CELWrJ!C½UÇLrK&Á•NOp!pNCELSBGFC$}&KMppMÊ ƒ„€GKÐF!KMUTp!CELrCi JV€K7UÇpÁ•ÁK&J!F
NLvJ!€GKp!BKM}&J!F!DGÁ NOpQJV€GKBGFKMpKMLG}&K>C MK&F!CÇ´ B CENOLqJ44 DG}¶JVD4UÇJ!NCELpMÊ4± JV€GKpp¬JVKMÁ NOp„NLeKM‚iDGNOPNOIGF!NODGÁ
UÇJ MK&F!CJ!KMÁB K&FVUÇJ!DGF!KÐLGCK&LGKMFRT NpUjÆÇUTNPUTIGPKh CEF–KMÁNp!pNCEL p!CJV€4U3J S (−ω) = 0 › IDJ•J!€GK
ppJ!KMÁ }jUTL`[email protected];Uj$p°UTIGpCEF!I`K&LGKMFRTzUTLGWÕJ!€GKMFK%h CEFK S (ω) 6= 0 Ê ƒ„€GKÂUÇpÁ•ÁK&J!FlNp©NÁBCEFJVUTLqJ
UTPpC–UÇJ 4LGN JVKSÆECTPOJ UÇREK V UTLGWÉJVK&Á•BKMF!UÇJVDGFK θ [email protected] JV€GK}&CELGWGN JVNCTL ω eV, k θ pJ!NPOP ÆÇUTPNOW+Ê
±µLÕKM‚iDGNOPNOIGF!NODGÁ$ UÇJ>4 LGNOJ!KrJ!KMÁB K&FVUÇJ!DGF!K θ 4 J!€GK>BC;K&F°WGK&LGp!N JµzCTI K¶$p©JV€GKrWGK¶J UTNOPKMWlI4UTPUTLG}&K
F!K&PUÇJ!NCEL ' )I (
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S (ω) = e
S (−ω) .
G1Ä
HNOJ!€
P (i) NOp;JV€KSBGF!CEIGUTIGNPONOJµWGNOpJVFNIGD$JVNCTL
SI (ω) 7KÂUÇp!p!DÁ•KJV€GKrLGCENOp!Kp!CTDGF!}&K>pp¬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
±µLÉJV€GKPNOÁ•N J;CPOC hF!K&‚DKMLG}¶ ω kB θ ;KFKM}MC1ÆTKMFHJV€GK}MPUTp!pN}MUTP+}jUÇp!K SI (ω) = SI (−ω) Ê
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Z
NL (E)
dEe
−iEt
X χLα (y, z) p
aLα (E)eikLα x + bLα (E)e−ikLα x .
υLα (E)
α=1
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0
ei(E−E )t a†nα (E)Aαα
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0
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0
kLβ (E)kLβ (E )
β
h
i
0
0
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+[kLβ (E) − kLβ (E 0 )]
h
io
0
0
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CEL H€GKMFKxN J„Np„ÁKjUÇp!DGFKMW+Ê
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0
0
0
0
Aαα
nn0 (L; E, E ) = δαα0 δnL δn0 L −
X
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β
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L hDGLG}&J!NCEL UTp!pC$}&NUÇJ!KMW HN JV€vPOKjUTW
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0 0
e
hIi =
2π
Z
0
0
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0
Z
e X
hIi =
dETα (E)[fL (E) − fR (E)] .
2π α
0
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2
S (ω) = lim
T →+∞ T
+
Z
T /2
dt
Z
∞
−∞
−T /2
|Ã$Ê Ã 0
dt0 eiωt [hI(t)I(t + t0 )i − hIihIi] .
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JV€GK&F!KhCEF!KSNOLiÆTCEPOÆTKMp„REF!UTLGWÉ}MUTLGCELGNO}jUTP9UjÆEKMF!UTREK&p©C hCTDGF hKMFÁ•NOCELvCEBKMF!UÇJVCTF!p7H€GN} €e}jUÇLÉI KS}&CEÁ–´
BGDJ!KMA
W HNOJV€ NO} .1 p„JV€GK&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 .
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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)]
H€GKMFK Aαα
0 NOp„K$BGF!K&p!pKMW:NLe»,‚ Ê Ã$Ê ÃTÄ ÊG®°pp!DGÁNLR–J!€4UÇJ
C KMLGK&F!RTnn;K(L;
r€4E,
UjÆEK E )
0
0
2e2
S + (ω) =
π
+
Z
X
X
α
Ã$Ê ÃGA .
Np„NOLGWGKMBKMLWGKMLqJ
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)])
.
$Ã Ê ÃE¿ ƒ„€GKxLCENpKxU3J ω = 0 NOpHCEIJ UÇNLGK&WCHN JV€GCED$J°UÇp!p!DÁ•NOLGR snn ;αβ (E) NpHNLWGKMBKMLGWKMLqJ°C/KMLKMF!Rǁ
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$Ê ÃI ƒ„€GNp hCTF!ÁÂDGPU`Np–UTPOp!C:CEI$J UTNOLGKMW hCEF hK&F!ÁNCELGp>IiDGp!NOLGRJ!€GK;U1ÆTKÕB4UÇ}TK&J UTBGBF!CqUT} € '<?H (|Ê,±µL
JV€GKvUTIGp!K&LG}MKeCHINUTpÂCTFÅUÇJ€NRE€J!KMÁB K&FVUÇJ!DGF!K θ |µL − µR| << kB θ JV€GKvJ*7C 4F!p¬JÂJVK&F!Áp
WGCEÁNL4U3JVKTÊG¹½p!NOLGR•J!€GKrF!KMPUÇJVNOCEL fi(1 − fi ) = −kB θ∂fi /∂E .7K>FKM}&C ÆTKMF°J!€GKÅÒTCT€GLGp!CTL`²©‚iDGNOpJ
hCEF!Á>DGPU ' H AHq¿ ()hCEFQJ!€GKMFÁ UÇP[KM‚iDGNPONIGFNDGÁ 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$Ê ÃD H€GKMFK G = e2 Pα Tα /π NOpÂJV€GKlÍUTLGWGUTDGKMF–}&CELGWGDG}¶J UTL}MKlC½J!€GKeÁ•K&p!CEp}MCEBN}e}MNOF!}MDNOJjʱµL J!€GK
CEBGBCEp!N JVKxPNÁNOJHJ!€GKÅIGNUÇp°PUÇF!REK&F½JV€GUTL:JV€GK>J!KMÁB K&FVUÇJ!DGF!K L − µR| >> kB θ .;K–REK¶JUvp!€GCÇJ
LGCENOp!K H€N} €lNpHUTPOp!C–}jUTPOPK&WÉF!KMWDG}MK&WÕp!€CTJ©LGCENpKSCEF„‚iD4UTLqJV|µ
DÁ p!€GCTJ©LGCTNp!K
Ã$Ê HEÄ S + (ω = 0) = 2eF hIi ,
HNOJ!€ F NpÐJV€GK°ËUÇLG
C UT}¶JVCE=F HNOJ!€•JVF!UTLGpÁ•NOp!p!NOCEL NOp7K&LGKMFRT•NLGWGK&B K&LGWGK&LiJQCEF;NOL•JV€K°PNOLGKjUÇF›FKMRENOÁ•K
F =
P
α
Ã$Ê HG Tα (1 − Tα )
P
.
α Tα
ƒ„€GK:pVUÇÁ•K`FKMpDGPOJ!p;K&F!K‹WGNp}MDGpp!K&WˆIq ÍUTLW4UTDGK&FeUTLGW ­ÕUTF¬JVNOL ' ?HF ? ? (UTBGBKjUTPONLGRJVC ;UjÆEK
B4UT} ÇK&JVp&Ê
±µL`JV€GKÂBFVUT}¶JVN}MUTPP ÉNÁBCEFJVUTLqJ°}MUTp!K H€GK&L‹J!€GKÂp}jUTPOKÂC,JV€KÂKMLGK&F!RTzWGK&B K&LGWGKML}MK–C,JVF!UTLGpb´
Á•NOp!pNCEL}MCK }MNOKMLqJVp
Np›ÁÂDG} €ÉPUTF!REK&F,JV€4UTL I CTJ!€ JV€K½JVK&Á•BKMF!UÇJVDF!K"UÇLGWÉUTBGBPNK&W•ÆTCEPOJVUTREK
JV€GKłiD4UÇLiJ!NOJ!NKMpSNOL»Ð‚ TÊ α|Ã(E)
$Ê Ã I °Á UjÕI K–F!K&BGPUÇ}MKMW‹Iq`J!€GKMNOF"ÆTUÇPDGK&p"J U ÇKMLU3JSJV€GKË4KMFÁ•NKMLGK&F!RTTÊ
ƒ„€GKMA
L 7KxCEI$J UTNOL
"
#
X
X
2e2
eV
2
2kB θ
Tα + eV coth
Tα (1 − Tα ) ,
S (ω = 0) =
π
2kB θ
α
α
+
Ã$Ê Hqà µ± LÕJV€K>}jUÇp!KÅUTPOP[JV€GKrJVFVUÇLGp!ÁNpp!NCTLÕ}MCK }&NKMLqJ!pSUTF!Kxp!Á•UTPP}&CEÁ•BGUTF!K&WzJVC,GJ!KMFÁ•p½BGF!CEBCEF¬JVNCTL4UTP
JVC Tα2 UTF!KLGK&REPK&}&JVK&W)GJV€KML
+
S (ω = 0) = 2ehIi coth
eV
2kB θ
ƒ„€GNp4hCTF!ÁÂDGPUÂWGKMp}MFNIKMp„JV€GKp€GCTJ©LGCENOp!K¶´ JV€GK&F!Á•UTP+LGCTNp!KS}&F!CEpp!C1ÆEK&FMÊ
" , !" # .!"+# !" Ã$Ê HH .
® B CENOLqJ½}&CELqJ UT}¶J"NOp©Dp!D4UTPOPOÉWK%4LGK&W:UTp½U•}&CELGpJ!F!NO}&JVNOCELeI K¶J*;K&KML`J*7CvÁK&J UÇPPNO}°FKMp!K&FÆTCENFpMʃ„€GK
}MCELWGDG}&JVUTLG}&K•C7‚iD4UTLqJVDÁ B CENOLqJ}MCELqJ UÇ}&JrWGNpBGPUjprUÉpJVK&BHNp!K•NLG}&F!KMUTp!KUTprUhDGLG}¶JVNCTLC;JV€K
RqUÇJ!K"ÆECEP J UTRE#
K ' GH ( Ê
ƒ„€GKMFKQUTFK7WGN 9K&F!K&LiJ QUjp/C 7 UTIGFN}jU3JVNLR½U©B CTNLqJ}&CELqJ UT}¶JjÊ ± J/}MUTL>IK›F!KjUÇPN &KM(
W hCEFNLGp¬J UTLG}&K;NOL
U"IGF!KMU E´OÈDGL}&JVNOCELÅIqÂBGDPPNOLGR"UÇB4UTFJ,U"BGNK&}MK„C }MCTLGWGDG}¶JVCEF,DGLqJ!NPN J/IGFKjU ip&Êq±µL–UÁCEFK;}MCTLiJ!F!CEPOPK&W
QUj ;BCENLqJv}&CELqJ UT}¶JVpÉUÇF!,
K hCEFÁ•K&W NOL à ´µWNÁKMLGpNCEL4UÇP„KMPK&}&J!F!CEL RqUÇp!KM%p ;KÇÊ RGÊ7NL¸rUT®½p ®½P#¸UT®°p
€GK&J!KMFCÇ´µp¬JVF!D}&JVDF!KMp&Ê9ˁÕUÇBGBGPONOLGR U–ÆECEP J UTRTKJ!C pDGNOJVUTIGPOq´ p!€4UTBKMWlRqUÇJ!KxK&PK&}&JVFC$WKMp JV€KxK&PKM}¶JVFCEL
RqUTp„}MUTLeI KPC}jUÇPPOWGK&BGPK¶JVK&WzUÇLGWÕUÅBCENLqJ„}&CELqJ UT}¶J©}jUTLlI KSW%K 4LGK&WÕPOC$}MUTPP EÊi®°LGCÇJV€GK&F©ÁKjUÇLGpHC }MFKjUÇJ!NLGR>U>B CTNLqJ;}&CELqJ UT}¶J„Np7Iq BCEp!N JVNOCELGNLR>UTLÉÀ$ƒ©­e´ J!NBv}&PCEpK½J!C>JV€GK°p!DGF UT}MK"C U>}MCTLGWGDG}¶JVCEF&Ê
®°PP/J!€GK pN &KMprC QJV€K•}&CELGpJ!F!NO}&JVNOCELUTF!K•UTp!pDGÁKMWJ!CÕIK•p€GCEFJ!KMFrJV€4UTLJV€GKÁKjUT0
L hF!KMKB4UÇJ!€
WGDGKÂJ!CeUTLqÕJµB K–C ›p}jUÇJJVKMFNLGR [UTLGW`J!€DpJ!FVUTLp!BCEFJ"J!€GF!CTDGRE€:JV€GKÅBCENOLiJ"}MCELqJ UÇ}&JNp°I4UTPPONp¬JVN}ÇÊ
±µL U‹‚iD4UÇLiJ!DGÁ BCENLqJÅ}&CELqJ UT}¶!J /J!€G
K HNWJ!€2C ½J!€GKÉ}MCTLGpJ!F!N}¶JVNOCEL NpÅ}&CEÁ•BGUTFVUTIPK J!C`JV€GKlË4KMFÁ•N
QUjÆTKMPK&LGRTJ!€+7Ê ="D4UTLqJVDÁ BCENLqJQ}&CELqJ UT}¶JHNp„UÂpNÁBGPK©}MCTLGWGDG}¶JVCE4F H€GNO} €eNp;Dp!KMWeJVCxJVKMp¬JHCEDGFQLCENpK
Á•KMUTp!DF!KMÁKMLqJHpK&JVDB+Ê
G?
.!, %0123!, !,# .
³ CELG}&KMF!LNLGRrJV€GK GLGNOJ!K>hFKM‚iDGKML}&vLCENpKB K&F/hCEFÁ•NOLGRSJV€GK½NLqJVK&REFVUTPC E NL »Ð‚ Ê Ã$Ê ÃE¿ HNOJ!€ JV€K
LGCTJ!N}MK–JV€GUÇJ R ∞ dEf (E)[1 − f (E + x)] = x/(1 − e−βx) 7K CTIJ UTNOLJ!€GK LCELGpÁÁ•K¶JVFN MK&W
LGCENOp!KSp!BKM}¶JVFDGÁ−∞hCEF„KMPOKM}¶JVF!CTLlp¬pJVK&Á UTp ' ÃÇÄ)(
NL
2e2 X
2ω
S (ω) = −
Tα2
π α
1 − eβω
+
Ã$Ê HG NL
2e2 X
ω − eV
eV + ω
−
+
,
Tα (1 − Tα )
π α
1 − eβ(eV +ω) 1 − eβ(ω−eV )
H€GKMFK NL Np›JV€GK"LiDGÁ>I K&FQC} €4UTLLGKMP9 V NOp7JV€K"UTBGBPNK&W ÆECTPOJ UÇREKTÊ$»,‚Ê Ã$Ê HG ÐNp7LGCTJQp¬Á•ÁK&J!F!N}
hCEF„BCEp!N JVNOÆTKSUTLGWlLGK&RqUÇJVN ÆEK<hF!K&‚iDGKMLG}&NK&pMÊ
±µLKM‚iDGNOPNOIGF!NODGÁ V = 0)7K FKM}&C ÆTKMFrJV€GK 4DG}&J!D4UÇJ!NCEL$´ WGNpp!NOB4UÇJVNOCELÕJ!€GKMCEFKMÁ«UÇJ 4LGN JVK hF!K ´
‚iDGKMLG}&NK&
p '<?GA)(
2(−ω)
Ã$Ê H ? S + (ω) = 2G
,
1 − eβω
±µLxJV€K MK&F!C©JVKMÁBKMFVU3JVDGFK7PONÁNOJVNOLGR;}MUTp!K 7K;FKM}&C ÆTKMFJ!€GK›‚DGUTLqJVDGÁ LGCENOp!K H€N} €ÂNOp WNp!}&DGp!pKMW
NL¯H%K bÊ ' ÃÇÄ (7UTLGWp!€GC HLNOL»,‚ Ê 9HÊ Ã I "NOL‹J!€GKLGK Jx} €4UTBJ!KMF>C ;JV€NpSJV€GK&p!NOpMʱ ;J!€GK•FKMpKMFÆTCENFp
€4UjÆEKCELGK} €4UÇLGLGKM9P GJV€GKFKMp!DPOJVp©UTFKNPPODGpJ!FVUÇJ!KMWÉUÇ
p '<?EB¿ ?I)(
2
(2e /π)T (1 − T )(eV − ω)Θ(eV − ω) ,
S (ω) =
(2e2 /π)[−2T 2 ω − T (1 − T )(eV + ω)Θ(−eV − ω) + T (1 − T )(eV − ω)] ,
+
N
N
ω ≥ 0,
ω < 0,
Ã$Ê H A H€GKMFK Θ(x) NpSJV€GK Ô KMUjÆ$NOp!NOWGK hDGLG}&J!NCELUTLGW T NOpJ!€GKÅJVF!UTLGp!ÁNpp!NOCEL‹BGF!CEIGUTIGNPONOJµTÊ9±µL UÇ}&J!)7K
CEIJVUTNLz»,‚Ê Ã$Ê H A „IqÕDp!NLR JV€GKxF!K&PUÇJ!NCEL S +(ω) = SI (−ω) ʃ„€GKÂBPCTJ½C S +(ω) NOp°p!€CHL:NOL
JV€GKDGBB K&F©BGUTLGKMP[NOLeËNRTDGF!KrÃ$Ê HÊ
!#.
­zKjUÇp!DGFNLGRJV€GKrLGCEL$´ pÁÁ•K¶JVF!NMK&WzLGCTNp!KrÁKjUTLGp©IKMNLRvUTIGPOKSJVC WNpJ!NLGRTDGNp€lIK&J*7KMK&L:KMÁNpp!NCTL
ω > 0 7K&LGKMFRT 4C HpvJ!CJV€GKÕWGK&J!KM}&J!CEF •UTLGW UTIGpCEF!B$JVNCTL ω < 0 ;KMLKMF!Rǁ#4CHphFCEÁ J!€GK
WGK&J!KM}¶JVCEF C;JV€GK•WGK&ÆN}&K DGLWGKMFrJVKMp¬JjÊ Ô C7K&ÆTKMFK$B K&F!NOÁ•K&LiJVUTPP :NOJrNOprWN}MDGP JxJ!CzWNpJ!NLGRTDGNp€
DGL4UTÁ>IGNREDCEDGp!P ÅIK&*J 7KMKMLep¬$ÁÁK&JVFN &KMWvUÇLGWvLGCEL´µpÁÁ•K¶JVFN MK&W LGCTNp!K $BGUTFJ!PO–IKM}jUÇDGp!K H€GUÇJ„Np
C#JVK&LeÁ•KMUTp!DGFKMWlNpQJ!€GKK $}MK&p!p°LCENpKTÊ
EÌ #J!KM)L *J ;C–pDG} €zÁKjUÇp!DGFKMÁKMLqJVpHUTFK"BKM/F hCTF!ÁKMWÉCELvJ!€GKSpVUTÁK"p¬$p¬JVK&Áz~J!€GEK 4FpJ H€GNOPK°N J„Np
WGF!N ÆEK&L:CEDJ"C 7K&‚iDGNPONIGFNDGÁ hKTÊ [email protected]Ê Iq`UTBBGPONLRÉU WG}ÂÆTCEPOJVUTREK ©UTLGW`J!€GKÂp!K&}MCELWNOL`KM‚iDGNOPNOIGF!NODGÁ
hJ!€GKrÆTCEPOJVUTREKrp!CEDF!}MKxNpHJVDGFLÕC ¶Ê ƒ„€GK>K $}MKMppSLGCENOp!KrNp©WG%K 4LKMW:UTp©JV€GKrWGN 9K&F!KML}MK>NOLzJ!€GKxLCENpK
I K¶*J ;K&KMLlJV€G<K 4FpJ½UTLGWeJV€GKpKM}MCTLGWzÁKjUÇp!DGFKMÁKMLqJj~
ÃÊ Hq¿ SM,excess(ω) = SM,noneq (ω) − SM,eq (ω).
±µLvÁCEpJ;}jUTpKMpQÁKMpCEp!}&CEBGN}½pVUTÁBGPOKMpQUÇF!K°WF!NOÆTKMLÉCED$JQC K&‚DNPNOIGF!NODGÁ¦Iq UTLvK J!KMF!LGUTP9WG}½ÆECEP J UTREK
V 4pC–J!€4UÇJ
|Ã$Ê HI SM,excess(ω) = SM (ω, V 6= 0) − SM (ω, V = 0).
ƒ„€GK©K $}MK&p!p„LGCENOp!KHNp›DGp%K hDG6P H€GKM
L ;K°UTF!K©NLqJVK&F!K&pJVK&WvNLPCC iNLRSNLqJVCrJV€GK©} €4UTLGRTK°NL–JV€K½p¬$p¬JVK&Á
H€GN} €UTF!KlWGDGKlJVCWF!NOÆNOLGRCED$J C "KM‚iDGNPONIGFNDGÁlÊ/± J•Np•UTPpCDGp%K hDG?P H€GK&L UB4UTF¬JVNO}MDGPUTF–p!K&J!DGB
[email protected]
PSfrag replacements
2
S+
1.5
1
0.5
0
-2
-1
0
1
2
1
2
h̄ω/eV
0.5
+
Sexcess
0.4
0.3
0.2
0.1
0
-2
-1
0
h̄ω/eV
[email protected]$<-* 'I 0 121%-;?3% A '([email protected](-* 2'Ib91'O% + n!D%? ')'(!j!D [email protected]$-A
S
b>< #'(! 3
-
,2<; 'D13pb8?B22 2(;+ $:(1%O:'I 0 +
~ω/eV 4e V /h T = 0.5
Sexcess(ω)
+B Z '(! 3
6
2<; 'D13/b:- ; ' nm'
4e V /h T = 0.5 U KM}&J!p°JV€KÅÁ•KMUTp!DGFKMÁKMLqJIq`NOLiJ!F!CWGDG}&NLGRÉUÇL‹UTWGWGN JVNCTL4UTPLGCENOp!K:H€GNO} €‹Np"NOLGWGKMBKMLWGKMLqJCÐJV€K
pVUTÁBGPOKSpJ U3JVK pC IqeJ UiNLR–J!€[email protected] KMFKMLG}&K>IK&J*7KMKMLzJV€GKJ*;C•LGCENOp!KrB C7KMF!p>;Kr}jUÇLÕREK¶J°FNWlC
JV€GKNOLGpJ!F!DGÁKMLqJ U3JVNCTL$´µWGK&B K&LGWGK&LiJHLCENpKSB C7KMFMÊ
²½C ;K•UTBGBGP :»,‚ Ê Ã$Ê HI ½J!Cz}MUTP}&DGPUÇJ!KÂJV€GK–K }&KMpp>LGCENOp!K–C ;‚DGUTLqJVDGÁ BCENLqJ}&CELqJ UT}¶J>U3J
MK&F!CÅJVK&Á•BKMF!UÇJVDGFK}MCEFF!KMpB CTLGWGNLR–J!C–NOJVp„pB K&}&J!F!DGÁ C LGCENOp!K S + NLl»,‚Ê |Ã$Ê H A ¶~
|Ã$Ê HD +
Sexcess
(ω) = (2e2 /π)T (1 − T )(eV − |ω|)Θ(eV − |ω|) .
ƒ„€GK„p!BKM}¶JVF!UTP4WGKMLp!NOJµÂC K $}MK&p!p›LGCENpK„I KMUTF!p,ÁCEp¬J,C NOJVp ;K&NRE€qJÐLGKMUTF MK&F!<C hF!K&‚DKMLG}&NKM%p iIGDJ,JV€K
K hFK¶´
LGCENOp!K>WKM}MFKjUTpKMpPNOLGKjUÇF!PO JVC MK&F!CÉC1ÆEK&FSUvFVUÇLGREK [0, ±eV ] hCEF½I CTJ!€`BCEp!N JVNOÆTKÂUTLW:LGKMRqU3JVNOÆT
‚iDGKMLG}&NK&p $UTLGW•ÆÇUTLNp!€KMp;IK&TCELGW J!€GK©B CENOLqJVp ω = eV hp!KMK©JV€GK½PC 7KMF›B4UTLGK&PNOLvËNOREDGFK©Ã$Ê H ¶Êiƒ„€GK
K $}MK&p!pHLGCENOp!K°J!€GKMF%K hCEFKS}MCELqJVUTNLGpQUÂp!NLREDGPUÇF!NOJµ~NOJ!p;WGK&F!NOÆÇUÇJ!NOÆTK"WGNOÆTKMFREKMpHUÇJ;JV€GNOpQB CTNLqJjBÊ KEGLGW
JV€4U3J J!€GKÐp!BKM}¶JVFVUÇPWKMLGpNOJµC $JV€GKÐK $}&KMp!pLGCEL$´ pÁ•ÁK&J!F!N MKMWLGCTNp!K Sexcess
L hDGLG}¶JVNOCEL
+
(ω) Np UTLrK&ÆEK&:
³
C,JV€K:hF!K&‚DKMLG}¶ Sexcess
+
+
(ω) = Sexcess
(−ω) Ê CELp!KM‚iDGK&LqJVPO+JV€GKÅK $}MKMpppÁ•ÁK&J!F!NMKMWÕLGCENpK
[email protected] KMFp°CELGP eIq`
U UÇ}&JVCTF½*J 7C ÁKjUTpDGF!K&W`NLÕ¯©K bÊ ' 7à G(
sym
+
+
Sexcess
Sexcess
hF!CEÁ JV(ω)
€GKK =$}MSK&p!excess
pÂLCE(ω)
L$´µp¬+Á•Á
K&J!F!N &(−ω)
KMWLGCENOp!KTʃ„€iDGpxK $}MK&p!pÅLGCENOp!K–K $BKMFNÁKMLqJVpxNL‹JV€GK‚iD4UTLqJVDGÁ
F!K&RENÁKx}MUTL`DGpD4UTPOPO[email protected] KMF!K&LqJVPOeIKÂK BPUTNOLGKMW`IqzDGp!NOLGRvLGCTL$´µp¬$ÁÁK&JVFN &KMWÕCTF°pÁÁ•K¶JVFN MK&W
LGCENOp!KK $BGF!K&p!pNCEL+Ê,À$C JV'
C iLGC ¤BGF!K&}MNOp!KMP  H€4UÇJłiD4UTLqJ!NOJµNpxÁ•KMUTp!DGFKMW NOLp!D} € K $B K&F!NÁKMLqJ!p
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NprUzpKMFNKMpxC „K¶ÆEKMLqJ!pÂN0
L H€GN} €K UT}&J!PO‹CELGK•KMPOKM}&J!F!CELB4UTpp!KMp hJVDLGLGKMPOp "JV€GFCEDGRE€JV€GK•I4UTFF!NK&FMÊ
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∆E =
Q2
(Q − e)2
−
>0.
2C
2C
HÊ@G
= e/2C Ê
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GI
Insulator
Normal metal
Normal metal
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CEL:JV€GK½ÈDLG}&J!NCEL‹IKMPC JV€GKÂJ!€GF!K&p!€GCTPW e/2 Ê9ƒ„€GNpBGFC$}&KMp!pC}M}&DGF!p<HNOJ!€U hF!KM‚iDGK&LG}& f = I/e
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p!CEDF!}MKQUTLGW hKMK&W–U°}MDF!F!K&LqJ,JVC°J!€GK ÈDGLG}¶JVNOCEL>JV€F!CEDGRT€–U°PUTFREK›F!KMpNp¬JVCEF&ÊT± JVpF!KMpNp¬J UTLG}&KQNpUTp!pDGÁKMW
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}jUTBGUT}MN J UTLG}&KMpHIKMPC 10−15 Ë JV€GKSJ!KMÁB K&FVUÇJ!DGF!KÁÂDGp¬JHI KIKMPOC UTICEDJ G Ó Ê
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KMÁÂIKMWWGKMWNLÅJV€K©K&PK&}&JVFN}MUTPG}MNOF!}&DGNOJ' AG AEÃF AH)(|Ê ¨ L4UÇÁ•NO}jUTP ³ CTDGPCEÁ>IIPC}ÇUÇWGK„NpÐU‚iD4UTLqJVDGÁ
%K KM}¶J H€GNO} €‹UTBGBKjUTFp H€GKML‹U•‚DGUTLqJVDGÁ }MCE€KMF!K&LqJS}MCELWGDG}&J!CEF"NOp°}MCTLGLGKM}¶JVK&W:NL`pKMF!NOKM>p HNOJV€`UTL
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ÈDGLG}¶JVNCTL+Ê ƒ„€GKQ‚iD4UTpN ´µBGUTFJ!N}MPOKMp NL>J!€GK7J*7CSÁK&J UÇPK&PK&}&JVFC$WKMp,UTFK;WGK&p!}&F!NIKMWÅIqrJV€GK Ô UTÁNP JVCELGNUTL
X †
X
9H$Ê Ã Hqp =
k ckσ ckσ +
q c†qσ cqσ ,
qσ
kσ
H€GKMFK k UÇLGW q UÇF!KÕJ!€GK`KMLKMF!RTNKMp C‚iD4UTp!N8´µB4UÇFJVNO}MPOKMp;HNOJV€#QUjÆEK`ÆTKM}¶JVCEF k UÇLGW q H€GNPOK σ
WGKMLCTJVK&pJV€GK&NFp!BGNOL+Êǃ„€K 4F!p¬JÐUTLGWxJV€GKQpKM}&CELGW–p!DGÁ }MCTF!F!K&p!BCELGWÂJ!C°JV€GK;PK%#JUTLGWÅFNRE€qJKMPOKM}&J!F!CWGK
F!K&p!BKM}&J!NOÆTKMPOTÊ
GD
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HT =
X
A(
HÊ H Tkq c†qσ ckσ e−iφ + h.c. ,
kqσ
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ÈDGLG}¶JVNCTL+Ê NOJ!€ J!€GK‹BGDGFB CEpKCÂ−∞
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} €4UTFREKCELlJ!€GK;ÈDGLG}¶JVNCTLlK&PK&}&JVFC$WKMp©IiÉJ!€GKCEBKMFVU3JVCEF e−iφ 7Kx}&CELGpNWGK&F©CTLGPOvJ!€GK<4DG}¶JVD4UÇJ!NCELp
UTF!CTDGLGWÅJV€GK„ÁKjUÇL–ÆÇUTPODGKQWGK¶JVK&F!ÁNLGK&WIqÂJ!€GK„K J!KMF!LGUTPGÆECEP J UTRTK V ÊEƒ„€4U3J7NOLGWGDG}&KMp,DGp,J!Cr}&CELGpNWGK&F
φ̃(t) = φ(t) − eV t UTLGW Q̃ = Q − CV Êq±µLiJ!F!CWGDG}&NLGR φ̃ NLqJ!C HT 7K½BKMF hCEF!Á¦UJ!NÁK¶´µWKMBKMLGWGK&LqJ
DGLGN J UTF¬ JVF!UTLGp hCEF!Á•UÇJVNOCEL H̃ = U † HU − iU † ∂U/∂t HN JV€
U=
Y
h
i
exp ieV tc†kσ ckσ .
X
Tkq c†qσ ckσ e−iφ̃ + h.c.
kσ
9H$Ê< ƒ„€GKLGK%ˆJVDGLLGKMPONLGR Ô ÇU Á•NOPOJVCTLGNUTLJV€GK&LzFKjUTWp
H̃T =
HÊE? kqσ
UTLGWÉJV€KrLK% Ô UTÁ•NOPOJ!CELGNUÇL•C JV€KrK&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!RǁePK&ÆTKMPOp©IK&J*7KMK&LzJ!€GKPKjUÇWGpMÊ
±µL•JV€K>hCEPOPC HNLR 7K"HNPOPGDGpK½JV€K©J!DGLGLGK&PNLR Ô UTÁ•NOPOJ!CELGNUÇLÂNLJV€GK?hCEF!Á HÊ ? iUTLW LGC VJ €K
JVCTJVUTP Ô UTÁ•NOPOJ!CELGNUÇL•C JV€Krp¬pJVK&Á Np
9H$Ê ¿ H = H̃qp + Henv + H̃T ,
H€GKMFK Henv WGK&p!}MFNIGNOLGReJV€GKKMLqÆNFCELGÁKMLqJjʱµLCTDGFx}MUTp!K [JV€GK•KMLqÆNF!CTLGÁ•K&LqJ>NOprFKMBGFKMpKMLqJVK&W Ii
JV€GKWGK¶ÆN}M<K H€GK&F!K ;K<7CEDGPOWzPON TK½JVCÁ•KMUTp!DGFKSLGCENOp!KrUTLGWÉJV€KQÈDLG}&J!NCELeNp„Dp!KMWÕUTp©UÅWGK&J!KM}&J!CEFMÊ
HNOJ!€
eV
.: .).# ! # % !,!,..!,- $.! * ,!,! ,! .#!
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;KzÁ UT KeJ*;CNOÁ•BCEF¬J UTLqJUTpp!DGÁBJVNOCELGp&Ê7ËNF!p¬J!/ J!€GKzJ!DGLGLGK&PNLRFKMp!NOpJVUTLG}MK R NOp•PUTF!REKe}&CEÁ–´
B4UTFKMWJ!CÕJV€GK•F!K&p!Np¬J UTL}MKv‚iD4UTLqJ!DGÁ R = 2π/e ʃ„€GNpxNÁBGPNOKMp"JV€GUÇJ>J!€GKvpJVUÇJVK&pÂCELJ!€GK•J*7C
KMPOKM}&J!F!CWGKMpÐCELPOÂÁN xÆEK&F;7KjUi POÂp!CJV€4UÇJÐJ!€GK Ô UTÁNP JVCELGNUTL H Ê<A NpÐURECC$WÅWGK&p!}&F!NB$JVNCTLC9JV€K
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T
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ƒ„€GKSJVDGLGLKMPNOLGR–FVUÇJ!KSNp„REN ÆEK&LzIqvJ!€GKrË K&F!ÁN+RECEPOWGKMLeFDGPK
9H$Ê I Γi→f = 2π|hf |H̃T |ii|2 δ(Ei − Ef ) .
ƒ„€GNpHNOpHJ!€GKrFVUÇJ!KrC JVF!UTLGpNOJVNOCELGp„IK&*J 7KMK&L`JV€GKNLNOJVNUTP9p¬J UÇJ!K |ii UTLGWeJV€(
K 4LGUTPpJ U3JVK ÊÀ$B K&}MN ´
N}MUTPP  7K>pK&J |ii = |Ei|Ri UTLGW |f i = |E 0i|R0i H€GK&F!K |Ei |E 0i UTF!Kr‚iD4UTpN ´µBGUTFJ!N}M|fPOKipJ U3JVKMp°C ÃTÄ
F!K&p!BKM}&J!NOÆTK•K&LGKMFRT E E 0 UTLGW |Ri |R0i UTF!K–F!K&p!K&FÆECTNFrpJ U3JVKMp } €GUTF!REKpJVUÇJVK&pEHNOJ!€K&LGKMFRENK&p
ER ER0 Ê4ƒ„€GKÁ•UÇJVFN KMPK&Á•K&LqJHNL 9HÊ I 7J!€GKMLlI K&}MCEÁKMp
H D
Ê
hf |H̃T |ii = hE 0 |HTe |EihR0 |e−iφ̃ |Ri + hE 0 |HTe † |EihR0 |eiφ̃ |Ri ,
P
UT}&J!pvNL J!€GK`‚iD4UTpNB4UÇFJVNO}MPOKzpB4UT}MKÇÊQƒ„€GKÕJ!KMFÁ hE 0|Tkq c†qσ ckσ |Ei
†
=
kqσ Tkq cqσ ckσ
HNOJ!€ HTe
RENOÆTKMpJV€GK`LCEL$´ &KMFC }MCELqJVFNIGD$JVNCTL CELGP  H€GKMLJ!€GK`NLNOJVNUTPHUTLGW#4L4UTP½pJ U3JVKMpvUTF!K`C"JV€GK$hCEF!Á
|Ei = |..., 1kσ , ..., 0qσ , ...i UTLGW |E 0 i = |..., 0kσ , ..., 1qσ , ...i F!K&p!BKM}¶JVNOÆTKMP EÊ ƒ„€GNprÁ•KMUTLGpJV€4U3J>NOL
JV€GK½NLGN JVNUTPGpJVUÇJVK°UTLvK&PKM}¶JVFCELvNp›C}M}MDBiNOLGRÂJV€GK½pJVUÇJVK (k, σ) NOL•JV€K°PK#J;K&PK&}&JVFC$WKBH€GKMFKjUTpQJ!€GK
pJVUÇJVK (q, σ) Np>DGLGC}M}MDBGNK&W NOLJV€GKÉFNRE€qJ>KMPK&}&J!F!CWGK /POKjUTWNLGRÕJ!C Pβ (E) UTp–U`}MCEÁ>IGNL4U3JVNCTLC f (k )[1 − f (q )] Ê
± SJV€GK:UÇBGBGPNOKMW2ÆECEP J UTRTK eV NOp ÁÂD} €p!Á•UTPPOKMFÅJV€GUTLJV€GKÕË4KMF!ÁNHKMLGK&F!RT 7K:Á Uj5UTpp!DGÁK
JV€4U3JÉUTPOP©‚iD4UÇp!N ´ B4UTF¬JVN}&PKepJVUÇJVK&pÉNLqÆECTPOÆEK&W€4UjÆEK:K&LGKMFRENK&pv}MPOCEp!KlJVCJ!€GK:Ë4KMFÁ•NHK&LGKMFRTEÊ7ƒU iNOLGR
JV€GKSJ!DGLGLGK&PNLR•Á•UÇJVFN vKMPK&Á•K&LqJ©J!C•IKxUTBGBGFC $NOÁ UÇJ!KMP  NOLGWGKMBKMLWGKMLqJ"C k q 7K>Á•U1vF!KMBPUT}&K
P
p hCEF›JV€GK°WGKMLGpNOJµ•C pJVUÇJVK&p
|T |2 IqvUÇLeU1ÆTKMF!UTREKMWÉÁ•UÇJ!F!N K&PK&Á•K&LiJ |T |2 7H€GNO} €eUT}M}&CEDGLqJV4
UÇJ;k,q,σ
J!€GKË4KMkqFÁ•NKMLGK&F!RTTÊG®°POP }&CELGp¬J UTLqJ„JVK&F!Áp„UTFK"}MCEPOPK&}&JVK&WvNLvJ!€GK°J!DGLGLGK&PNLRÂF!K&p!Np¬J UTL}MK RT Ê$ƒ„€GK
JVCTJVUTP+F!UÇJVEK hCEF„K&PK&}&JVFCELÉJVDGLLGKMPONLGR hF!CTÁ PK #J„JVC–F!NORE€qJHNp
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$Ê@GG
R,R0
½² C 7K,JVFVUÇ}MKÐCEDJ[J!€GKÐKMLqÆNF!CTLGÁ•K&LqJp¬J UÇJ!KMp&Ê1ƒ„€GK7BF!CEI4UÇIGNPONOJµ©CF4LGWGNOLGR„JV€GKÐNLNOJVNUTP3F!KMpKMF¬ÆECENOF
pJVUÇJVK |Ri Np Pβ (R) = hR|ρβ |Ri HNOJV€SJ!€GK,KM‚iDGNOPNIF!NDÁ WGKMLGpNOJµSÁ•UÇJ!F!N ρβ = Zβ−1 exp(−βHenv ) Ê
Ô K&F!K Z = ƒ F {exp(−βH )} NOpQJV€GKB4UÇFJVN JVNOCEL hDGLG}¶JVNOCELeCJ!€GKKMLqÆNFCELGÁKMLqJjÊ
¯©K HFβ!NOJ!NLGR–JV€KÂWGKMP J ;
U henv
DGLG}&J!NCELzNL HÊ G1Ä „NLzJ!KMFÁ C ÐNOJVp½Ë4CEDGFNKMFHJVF!UTLGp hCEF!Á UTLGWÕDGp!NOLGR JV€K
Ô K&Np!K&LiI K&F!RBGF!K&p!KMLqJVUÇJVNOCEL)7KrCTIJ UTNOL
R,R0
*, @ *, :# 1.)'# ! ! , . , .# ! % ,! .#!
KSWK%4LGKJV€KrK&‚iDGNPONIGFNDGÁ¤}MCEFF!K&PUÇJ!NCEL h DLG}&J!NCEL
heiφ̃(t) e−iφ̃(0) i =
=
p!CÅJV€GUÇJ?;KREK¶J
X
R
Pβ (R)hR|eiφ̃(t) e−iφ̃(0) |Ri
1 X
hR|eiφ̃(t) e−iφ̃(0) e−βHenv |Ri ,
Zβ R
Z ∞
1
Γ→ (V ) = 2
dEdE 0 f (E)[1 − f (E 0 )]
e RT −∞
Z ∞
dt
exp (i(E − E 0 + eV )t) heiφ̃(t) e−iφ̃(0) i .
×
−∞ 2π
Ã7G
9H$Ê@G à 9HÊ GH ± QJV€KvLGCENOp!KNpŸUTDGpp!NUTL)[JV€GK•}MCEFF!K&PUÇJ!NCELhDGLG}¶JVNCTLWGK%GLGKMW NOL 9H$Ê@G à }MUTLIK !p NOÁ•[email protected]
UTBGBGP NLGR>J!€GKSREK&LGKMF!UTPN MKM
W N} •J!€GKMCTF!KMÁ UTp |UÇPp!CÅpKMKS®½BGB K&LGWGN É® ' »Ð‚ip&Ê ?$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}MK„JV€K°UTIIGF!K¶Æ$NUÇJVNOCEL–}jUTPOPK&WB€4UTp!&K *BG€4UÇp!K°}&CEF!FKMPUÇJVNOCEL hDGLG} ´
JVNOCEL+~
[email protected]Ê G ? J(t) = h[φ̃(t) − φ̃(0)]φ̃(0)i ,
UTLGWÉJV€KxË4CEDF!NK&F;JVF!UTLGp hCEF!Á C J!€GK}MCEFF!KMPUÇJVNOCE
L hDGLG}¶JVNOCEL 9HÊ GB ~
1
P (E) =
2π
Z
∞
−∞
H€GN} €lNp„}MUTPPOKMWvJ!€GKWGNp¬JVF!NOIGDJ!NCEL hDLG}&J!NCEL+Ê
ƒ„€GKNLqJVK&REFVUTP9C ÆTKMF©KMLGK&F!RTvC P (E) Np„LCEF!Á•UTPNMK&W JVCAGH€N} €l}MCELGF!Áp
UTIGNOPNOJµWGK&LGp!N Jµ
Z
∞
®°LGCÇJV€GK&F„BGF!CEBKMF¬JµeC
9HÊ [email protected] dt exp [J(t) + iEt] ,
P (E)
UTpHU–BGF!CTI$´
P (E)dE = eJ(0) = 1 .
P (E) NOpQJV€GKp!C3´µ}jUÇPPK&WÉWGK&JVUTNPOKMWÉI4UÇPUTLG}&KSpÁÁ•K¶JVF
9H$Ê@G ¿ −∞
P (−E) = e−βE P (E) ,
9H$Ê@GI H€GN} € Á•KMUTLGpÐJV€4UÇJ7JV€GK½BGF!CTI4UTIGNOPNOJµxJVCxK$}MN JVK½J!€GK°K&LqÆ$NOF!CELÁ•K&LiJ;Np›PUTFREKMFÐJV€GUTL•J!€GK°BGFCEI4UTINPN Jµ
JVC•UTIGpCEF!IzKMLGK&F!RTAhF!CTÁ JV€GKrKMLqÆNFCELGÁKMLqJ"IqlU Î7CEPOJ MÁ•UTLGLA UT}&J!CEFMÊ ³ CELp!KM‚iDGK&LqJVPO9LGC•KMLGK&F!RT
}jUTLlIK>UTIp!CEFI K&$
W hF!CTÁ JV€GKrKMLqÆNFCELGÁKMLqJ°UÇJ MKMFC•J!KMÁB K&FVUÇJ!DGF!K UTLGW P (E) JV€GK&LzÆÇUTLGNOp!€GK&p>hCTF
LGKMREUÇJVN ÆEKSKMLKMF!RTNKMp&Ê
L
,!,!.'!- 1)
¹½pNLGRÅJV€KrWK%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 „ƒ €GKÂK BF!KMpp!NOCEL 9HÊ GD QJ UÇKMpSNOLiJ!CÉUT}&}MCEDGLqJ"J!€GKÂBCEp!pNIGNOPN JµÉCÐKMLKMF!Rǁ`K$} €4UÇLGREKÅIK&J*7KMKML‹JV€K
JVDGLLGKMPONLGRKMPOKM}&J!F!CELUTLGW J!€GKÕK&LiÆNOF!CELGÁKMLqJMÊ=K`Á•U12NOLqJVKMFBGF!K¶J P (E) UÇp•JV€K`B CTp!p!NOIGNPONOJµJVC
KMÁNOJJV€KvKMLGK&F!RT E JVCÕJ!€GK K $J!KMFL4UTP;}&NF}MDGN JjÊ ³ CEF!FKMpB CELWGNLGRTPO P (E) hCTFxLGK&RqUÇJ!NOÆEK KMLGK&F!RENOKMp
WGKMp}MFNIKMpÐJV€GK½UTIGpCEF!BJ!NCELC 9KMLGK&F!RT–Iq–JV€GKHJVDLGLGKMPONLGRK&PKM}¶JVFCEL+Êi±µLqJVKMRTFVUÇJ!NLGRC1ÆEK&F7ÆÇUTF!NUTIGPOK E 0 ;KCEI$J UTNOLeJ!€G<K 4L4UTP hCEF!ÁÂDP:
U hCE4F hCEF QUÇF!WÉJVDGLLGKMPONLGR–FVU3JVKSIKMNLR
H Ê ÃÇÄ À$NÁNPUTF!P  7K }jUÇP}MDPUÇJ!KÂJV€GKI4UT};UTF!WJ!DGLGLGK&PNOLGRzF!UÇJVKÇÊ Ô C7K&ÆEK&FNOJrNpFVU3JV€GK&FxCEIqÆNCTDGp<hF!CEÁ
JV€GKp¬$ÁÁK&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/ UpNLGREPOKSJVDGLGLKMPÈDGLG} ´
JVNOCELzFKMWGD}MKMp½JVC–JV€GKxWGK&J!KMF!ÁNLGUÇJVNOCELeC P (E) CTF©J!€GK>BG€GUTp!Kx}MCEFF!KMPUÇJVNOCEL h DGLG}¶JVNCTL J(t) Ê ±µLÕJ!€GK
LGKJHpKM}&J!NCEL;K<HNOPPBGF!K&p!KMLqJ°UTLeKUTÁBGPK"h CEFQp!BKM}&NUTP9NÁB K&W4UTLG}&K H€GN} €eFKMPU3JVKMpQJ!€GKE4 DG}¶JVD4U ´
JVNOCELGpC J!€GKÐÆECEP J UTREK7UT}&F!CEpp J!€GK[ÈDLG}&J!NCEL>UTLGWxJ!€GK›}MDGFF!KMLqJ 4 DG}&J!D4UÇJ!NCELGpNOLÂUHLGKjUTFIqxÁKMpCEp!}&CEBGNO}
1
Γ→ (V ) = 2
e RT
Z
∞
−∞
dE
E
P (eV − E) .
1 − exp(−βE)
←
→
WGK&ÆNO}MK ' ÇÃ Ä |( Ê
ÃEÃ
$ +! ! " " ±µL ¯HK%bÊ ' ÃTÄ)9( EJV€K°UTD$JV€GCEFpÐBGF!CEBCEpK°UÁ•KMUTp!DF!KMÁKMLqJ›p!K¶JVDGBhCEF,WGK&J!KM}¶JVNLR>‚iD4UTLqJVDÁ¡LGCENpKHC ÆTKMF7U
HNWGK=hF!K&‚DKMLG}¶ÅFVUTLREKQDGpNLGRSU°J!DGL4UTIPK7J*7CÇ´ PK&ÆTKMPGp¬pJVK&Á¡UTp,U½WGK&J!KM}&J!CEFMÊTƒ„€GK„WGK&J!KM}¶JVCEF/}&CELGpNpJ!p
C;UeWCEDGIGPOKłDGUTLqJVDGÁ WGCÇJ ¨ = ¨ EH€GN} €NpS}MUTB4UT}&NOJ!NOÆEK&POz}MCEDBGPK&WJ!CeJ!€GK–PKMUTWGpC;UeLKjUTFIi
Á•K&p!CEp}MCEBN}S}MCTLGWGDG}¶JVCEF&Ê4ƒ„€GKxp} €GKMÁKrCJ!€GKppJ!KMÁ NOp„p!€GCHLÕNOLlËNREDF!K<HÊ Ã$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
*(.!,. .(:!"
®WGCEDIGPK7‚DGUTLqJVDGÁ¦WCTJ,NpU hDGPP }MCELqJ!F!CEPOPUTIGPOK›J*7CPK¶ÆEKMPppJ!KMÁ HN JV€>J!€GK„p!K&B4UTFVU3JVNCTLÂI K¶J*;K&KML
PK¶ÆEKMPOp = EL − ER }MCTLiJ!F!CEPOPK&WlIqlRqUÇJ!KrÆTCEPOJVUTREK U•WG}rNLGK&PUTp¬JVNO}S}MDGFF!KMLqJ°}jUTL`}&NF}MDGPUÇJVKSNOLzJV€K
WGK&J!KM}¶JVNCTL}&NF}MDGN JSCELGP `@N ›JV€;
K hFKM‚iDGK&LG}&
NpSBGFC ÆNOWGKMWIq`JV€GK–ÁKMpCEp!}&CEBGN}–WGK¶ÆN}MKÇÊ[ƒ„€GK
JVDGLLGKMP,F!UÇJVK–IK&*J 7KMKMLJV€GKWGCTJ!pSNprUTpp!DGÁKMωW=ÁÂDG} €p!Á•UTPPOKMF°J!€4UTL‹JV€GKÅJ!DGLGLGK&P,FVUÇJ!KMprUT}MFCEp!pJV€K
PK #JHUTLGWlF!NORE€qJHI4UTF!FNK&F!p„p!CÅJ!€4UÇJHJV€GKNOLGKMPUTpJ!N}"}&DGF!FKMLqJ©Np„REN ÆEKMLlIq
9H$Ê ÃEÃ Iinel () = eTc2 P () ,
H€GKMFK Tc NOprJ!€GKJ!DGLGLGK&P7}&CEDGBGPONLGReIK&*J 7KMK&LJV€GK•WGCTJVp&ʃ„€G
K hCEFÁÂDGPU HÊ G ? "}jUTLI KF!K HF!N J!JVK&L
UTp
H€GK&F!KeLGC J!€G
K 4DG}¶JVD4UÇJ!NLGR:B€4UTp!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!pJV€Gφ̃(0)i
K ¨ = ¨ ÈDGLG}¶JVNCTL δVDQD (t) = V (t) − hV δ(t)i
Iq‹JV€GK•F!K&PUÇJ!NCEL
Rt
Ê KÅ}MUTP}&DGPU3JVKÂJV€;
K GF!pJBGUTFJrNL`J!€GK–BG€4UTpK•}&CEF!FKMPUÇJVCEF"UTLGW:J U ÇK
δ φ̃(t) = e −∞ dt0 δVDQD (t0 ) )
NLqJVC>UT}M}&CEDGLqJQJ!€GK°WGK 4LGN JVNCTL hCEFÐJV€GK"LCEL$´µp¬Á•ÁK&J!F!N &KMW•B C 7KMFQpB K&}&JVF!UTP9WKMLGpNOJµ•C [JV€GK½ÆECEP J UTREK
4DG}¶JVD4UÇJ!NCELpHUT}MFCEp!p„JV€KQÈDLG}&J!NCEL
SV (ω) = 2
Z
∞
−∞
dteiωt hδVDQD (t)δVDQD (0)i ,
ÃH
9H$Ê ÃH JV€GK&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 hCEFÁ•NOLGR•J!€GKÂNLqJ!KMREF!UTPp©C1ÆEKMF"J!€GK>J!NÁK t0 UÇLGW t00 UTLGW:LGCÇJVNLR•JV€GUÇJ"JV€K:4DG}¶JVD4UÇJ!NCELp°CÐJV€K
ÆECEP J UTRTKQUT}&F!CEppJV€GK ¨ = ¨ ÈDGLG}&J!NCEL>F!KMPUÇJVK›JVC°J!€GKQ}&DGF!FKMLqJ 4DG}¶JVD4UÇJ!NCELpJV€GFCEDGRE€xJV€GKQÁKMpCEp!}&CEBGNO}
WGK&ÆNO}MKvUÇp SV (ω) = |Z(ω)|2SI (ω) &HN JV€ Z(ω) NpJV€GK•JVFVUÇLGp!NOÁ•BKMW4UÇLG}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©NLlULCEL$´µp¬Á•ÁK&J!F!N &KMW
−∞
hCEF!C
Á 77K€4UjÆEK
Z ∞
|Z(ω)|2
π
9H$Ê Ã ? dω
J(t) =
SI (ω)(e−iωt − 1) ,
RK
ω2
N pQJ!€GK‚iD4UTLqJVDGÁ CFKMp!NOpJVUTLG}MKÇÊ
±µLÕJV€GKxPNOÁ•N JHC,p!Á•UTPOP)4DG}&J!D4UÇJ!NCELGp½C,ÆECTPOJ UÇREK>UÇ}MF!CTp!p½J!€GK©ÈDGL}&JVNOCEL)
hCEF„PCTLGRÂJVNOÁ•K&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 „ƒ €GK"GF!pJ;B4UTF¬J;FKMLGCEFÁ UÇPN &KMp,JV€K½K&PUTp¬JVN}©}MDF!F!K&LqJ H€GK&L = 0 ʱµL UT}¶J!iIq }&CELqJVF!CTPPNOLGR"JV€K°RqUÇJ!K
ÆECEP J UTRTKQNLÂJ!€GK„WGK&J!KM}¶JVCEFÐ}MNOF!}&DGNOJ/J!CS}MCELqJ!F!CEP 6= 0 ;KHWGCLGCÇJ,}MCELp!NWKMF hDGFJ!€GKMF/J!€GNpJVKMFÁzÊT± J!€GK
NÁB K&W4UTLG}&K Z NOp/p!Á•UTPOP$K&LGCEDGRE€ Z = 0.1R ÇJ!€GK„}MCTDGBGPNOLGR"C J!€GKQLGCENOp!KQNOLqJVC"JV€K„WGK&J!KM}&J!CEF,Np
p!D }&NKMLqJ!POÂK 9SK&}&J!NOÆEKHIGD$J,JV€GK„JVF!UTLGpSNÁB K&W4UTLG}&KKQNpÐUTBBGF!!C $NÁ•UÇJVN ÆEK&POrNLGWKMBKMLGWGK&LqJ7C hFKM‚iDGKML}&~
J hCEF!ÁÂDPU 7KÅCEIJVUTNC
L hCEF½J!€GKÂNOLGKMPUTpJ!N}r}MDGFF!KMLqJSJ!€GF!CEDRE€
2 Ê ƒ„€GKÂPUTp"
|Z(ω)|2 ' |Z(0)|2 ≡ κ2 RK
¨
¨
JV€GK = Np
T 2 SI ()
9HÊ ÃE¿ Iinel () ' 2π 2 κ2 c 2 .
e K 4LW5JV€GUÇJ–JV€GKl}MDGFF!K&LiJ 4DG}¶JVD4U3JVNCTLGp U3J hFKM‚iDGK&LG}& ω FKMpDGPOJNL2JV€KlNOLGKMPUTpJ!N}É}&DGF!FKMLqJvU3J
PK¶ÆEKMP›WGN 9K&F!KML}MK = ω Ê ƒ„€GUÇJÂÁKjUTL(
p 7Kv}jUÇLpJ!DGWJV€GK•BGF!CEBKMF¬JVNOKMprC H}MDGFF!K&LiJ>LGCENOp!K NOLJ!€GK
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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)
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NLvJ!KMF!ÁpHC JV€GK&NFHË4CEDGFNKMF;}MCEÁB CTLGKMLqJVp©UTp
Ψσ (x) =
X
eikx ckσ ,
Ê Ã k
Ψ†σ (x)
=
X
e−ikx c†kσ ,
k
HNOJ!€ c†kσ ckσ NpJV€K;}MFKjUÇJ!NCEL |UTLGLN€GNOPUÇJ!NCEL CEBKMFVU3JVCEF&hCEF/UÇL>KMPOKM}¶JVF!CTL) H€N} €Â€4UÇp/Á•CTÁ•K&LiJ!DGÁ
³
k UTLGW2pBGNL σ NLJ!€GKeÎ ÀJV€GK&CEF ' IG ( ʃ„€GK 4F!p¬JÅJVKMFÁ—NOL GÊ G xWGK&pJ!F!C1p UTLW }&F!KMUÇJVK&pCTLGK
KMPOKM}&J!F!CELeUÇLGWlJ!€GKMFK%hCEFKS}MCELp!KMF¬ÆEK&p½JV€KSLDÁÂI K&F„CB4UTFJ!N}&PKMp&ÊGÎ;D$J„JV€GKSJ*7CPUTpJ„J!KMFÁ•p„NOLG}MFKjUTpK
CEFWGKM}&F!KjUÇp!K`*J 7CB4UTF¬JVN}&PK&pMÊЃ„€GK`ÁKjUTL C "JV€KÕBGFC$WGD}&J Ψ† Ψ† ΨΨ UTFKÕLGCTL$´ ÆÇUTLGNOp!€GNOLGRUTLGW
JV€GK&p!K–JVK&F!Áp HNOPP/BPUj‹UTLNÁB CTFJ UÇLiJ"F!CEPOKTÊ ∆(x) NOpr}MUTPPOKMW:
JV€KBGUTNFSBCTJVK&LqJVNUÇP Ê+ƒ„€GK%K KM}¶JVN ÆEK
Ô UTÁNP JVCELNUTL GÊ G ;NOp„WGNUTRTCEL4UTPON MK&WvIqÉJV€GKrÎ;CTRECEPDI C1ƍJVFVUÇLGp/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 H€GKMFK γ γ † UTF!KLGK% CTB K&FVUÇJ!CEF!prpJ!NPP,p!UÇJVNOp/#NLRzJV€K hK&F!ÁNCEL}MCEÁÁÂDJVUÇJVNOCEL‹F!K&PUÇJ!NCELGp&Ê[ƒ„€GK&
UTF!K }jUTPOPKMW hK&F!ÁNCEL‚DGUTp!N8´µB4UTF¬JVNO}MPK•CEBKMFVU3JVCEFpMÊ/ƒ„€GKep¬J UÇJ!K u (x) v (x)>}MCEFF!K&p!BCELGWGpÅJVC:J!€GK
QUjÆT"K hDGLG}¶JVNOCEL•C+UrKMPK&}&J!F!CEL$´ [email protected] €CEPK ´µ[email protected]‚iD4UTp!N8´µB4UÇFJVNO}MPOKHUÇkJ›B CEpNOJ!kNCEL x Êiƒ„€GK©}MCEFF!K&p!BCELGWGNOLGR
HqÃ
Ô UTÁNP JVCELNUTL€4UTp©UÅWGNUÇRECEL4UTP hCEF!Á
Hef f = Eg +
X
Ê< †
Ek γkσ
γkσ ,
H€GKMFK Eg Np/JV€K©RTF!CEDGLWp¬J UÇJ!KHKMLGK&F!RTÅC Hef f UTLGW Ek NOp/JV€GKHKMLKMF!RǁÅCJV€GKHK$}&NOJ U3JVNCTL n Êqƒ„€GNOp
Ô UTÁNP JVCELNUTLBGF!C1ÆNWKMpÅJV€4U3JÅJV€GKeKMPOKM}¶JVF!CTL5UÇLGW5€CEPK ;UjÆEKChDGLG}&J!NCELpÅpVUÇJ!Np/#JV€GKlÎ;CTRECEPDI C1Æ
KM‚iD4UÇJ!NCELp
kσ
∇2
− µS + V (x)]u(x) + ∆(x)v(x) ,
2m
−∇2
Ev(x) = −[
− µS + V (x)]v(x) + ∆∗ (x)u(x) .
2m
&JV€KMp!K›KM‚iD4U3JVNCTLGp[LGKMK&WxJ!C©IK,p!CTPOÆEK&WxpKMP ´µ}MCTLGp!NOpJVK&LqJVPOTÊjƒ„€GK uk UÇF!K,KMNOREKML
Eu(x) = [−
±µLBGF!NOLG}MNOBGPK
CU–PONLGKMUTFQp¬$p¬JVK&Á HNOJV€e}&CEF!FKMp!BCELGWNLGRKMNOREKMLqÆÇUTPDKMp
Ek ~
vk
u
u
E
= Ω̂
.
v
v
GÊE? 7hDGLG}&J!NCELp
Ê A ƒ„€GKCEBKMFVU3JVCEF Ω̂ Np Ô K&F!ÁNOJ!NUTLvpCJ!€4UÇJ„J!€[email protected] KMFKMLqJ©KMNRTKMLhDGL}&JVNOCELGp u UTFKSCEFJ!€GCERECTL4UTP Ê
± u Np;J!€GKSp!CEPODJVNOCEL hCEF7JV€GKSK&NREK&LiÆÇUTPODGK E J!€GKML −v Np7JV€GKSpCEPDvJ!NCELhCEF;JV€KSKMNRTKMLqÆÇUTPDGK
v
u
−E Ê
±µ0
L H€4U3<J hCEPPOC Hp";K;HNPOP/UTpp!DGÁK V (x) = 0 Ê)K GLGWJ!€GK–KMNOREKMLhDLG}&J!NCELGpSNOL:JV€GK–REK&LGKMF!UTP
hCEF!Á u(x) = u0 eikx Ê9± 37KCTLGPOl}MCTLGp!NOWGKMFSK&LGKMFRENK&p E > ∆ J 9JV€KMF!K HNOPP I KÂUvB4UTNF°C Á UÇRELGNOJ!DGv(x)
WGK&pQC k UÇp!vp!0C}MNUÇJVK&W HNOJV€eKMUT} €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!KC ;J!€GKÎ ³ ÀBGUTNFNLGRÉC k UTLGW
2m
−k 7K–ÁÂDGpJ"}MCELGpNWGK&F"ICTJV€:pNRELGp°C k 9pCÉJV€4UÇJ°JV€GK&F!KÂNOpSU hCEDGF hCEPW`WKMREK&LGKMF!UT}&ÕC ›F!KMPOK&ÆÇUTLqJ
pJVUÇJVK&"p hCEF½KjUT} € E pKMKÅËNOREDGFK GÊ Ã HNOJV€lJV€GKxLGCTJ!N}MKrJV€4U3J°JV€Kx*J 7ChCEPWzp!BGNOL`WGK&REKMLGK&FVUT}¶`CELPO
U KM}&J!p LGCTF!Á•UTPNjUÇJ!NCEL pNLG}&KÕJV€GK&F!KzNp LCp!BGNOL$´ GNB BGF!C}MK&p!p!K&pMÊ ³ CTLGp!NOWGKMFNLGRJ!€GKÕFKMPUÇJVNOCELGpC
CEBKMFVU3JVCEFp7NOLvÎ;CERTCEPDGIC1ÆÅJVF!UTLG/p hCTF!Á•UÇJVNOCE)L 7K 4LGW J!€4UÇJ7JV€GK½K $}MN J UÇJ!NCELGp;UÇJ ±k+ UÇLGW ±k− UTFK
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F!K&‚DNF!K&WJVC`Á•U ÇKÉUTL K $}&NOJ U3JVNCTL HNOJ!€ } €4UTF!RTK e Np Eek = µ + Ek H€GNOPKJV€4UÇJ>JVC`Á•U TKvUTL
K $}MN J UÇJ!NCE'
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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 GrUTLGWl€GCEPOK"NLz
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C2
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12 1'$3%D' ?$%& ' ? 3"G UTIC ÆTKrJ!€GKp!DGBKMF}MCELWGDG}&J!CEF°} €KMÁN}jUÇP[B CTJ!KMLqJVNUTP8GUTLGWlBGPUT}&KMWC;K&PP)HNOJ!€GNLÉJ!€GKxREUTBzNOLzCTF!WGK&FHJ!C
UjÆECENOW‚iD4UTp!NOB4UTF¬JVN}&PK•BGF!C}MK&p!p!K&pMÊ,Î7KM}jUÇDGp!KeWGCEDIGPK•C$}&}MDGB4UÇLG}&NpÂBGFCE€GNOIGNOJ!KMWIqJV€GK ³ CTDGPCEÁ>I
IGPC} 3UTWGK /®°LGWF!KMK¶Æ JVF!UTLGp!BCEF¬J•C}M}&DGF!pÅÆNU:pKM‚iDGKMLqJ!NUTP„JVDLGLGKMPONLGR‹C ©JV€Ke*J 7CK&PK&}&JVFCELGp&ʺ›K&J!
I K&}jUTDp!KÂC ›K&LGKMFRTÕ}&CELGpKMFÆÇUÇJ!NCEL JV€KÂpVUTÁK>K&LGKMFRT`FKM‚iDGNOF!KMÁKMLqJ!pSUTp"NOL:ËNRTDGF!K ?$Ê ÃÇI`€4UjÆEKÂJ!C
I Kp!UÇJVNO/p 4K&,
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}jUTL–NK&PW–CEF,REN ÆEKHp!CEÁK„C 9NOJVp,K&LGKMFRTÅJVCSJ!€GK©² ¨ À–WGK&J!KM}&J!CEF qK&PK&}&JVFCELGNO}QJVFVUÇLGp!N JVNCTLGp/ÆNU"JV€GKHWGCÇJ
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JVClJV€4UÇJ>C„JV€KvWGCTJ>PK&ÆTKMP|Ê/À$D} €JVFVUÇLGp!N JVNCTLGpx}MUTLJV€iDGp>C$}&}MDGFxK&ÆEK&[email protected]!€GKv} €GK&Á•NO}jUTP›B CÇJVKMLqJ!NUTP
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p!N JVD4UÇJ!NCEL4p H€GN} ,
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hJ!€GK bK&LiÆNOF!CELGÁKMLqJ ¶Ê
a)
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∆
µS
µL ∆
c)
d)
D h̄ω ∆
µS
∆
D h̄ω ∆
D h̄ω ∆
µ
L
µS µS
µL
PSfrag replacements
b)
∆
∆
3 1-5 -71%& ':+q-
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3 1-5 -71%& '8+q-; O? %+*G $3
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ÁKMpCEp!}&CEBGN}S}&NF}MDGN J7p¬$p¬JVK&Á FKjUTWp
[email protected]Ê G H0 = H0 + H0 + Henv ,
H€GKMFK
X †
?Ê Ã H0 =
k ck,σ ck,σ ,
L
S
L
k,σ
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B K&F!}&CELGWGDG}¶JVCEF Ô UTÁ•NOPOJ!CELGNUÇL €4UTp„J!€GKSWGNUÇRECEL4UTP hCEF!Á
X
?Ê H †
H0 − µ S N S =
Eq γq,σ
γq,σ ,
S
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q,σ
H€GKMFK γq,σ , γq,σ
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†
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©JV€GK>‚DGUTp!NOB4UTFJ!N}&PKrKMLGK&F!RT ζq = q − µS Np©JV€GKÂLCEF!Á•UTP pJ U3JVKÂpNLGREPOK¶´
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HNPOP,I KÉUÇp!p!DÁ•K&WJVC`IK•J!€GKvPUÇF!REK&pJxKMLGK&F!RTp!}MUTµPKv
NOLJ!€GKMpKÉ}jUTPO}MDGPUÇJVNOCELGpMÊ Ô KMFKjU#J!KMF ;KvUTPpC
WG%K GLGK eV = µL − µS UTLGWlUTp!pDGÁK µS = 0 Ê
Ô KMFKF;KSWGC>LGCTJQpB K&}MN #•J!€GK Ô UÇÁ•NOPOJVCTLGNUTL–C[J!€GK°K&LqÆ$NOF!CELÁ•K&LiJQI K&}jUTDGpK"JV€GK°KMLqÆNF!CTLGÁ•K&LqJ
F!K&BGF!K&p!KMLqJ!p"UTLzCEB K&Lzp¬pJVK&Áz~ JV€KrÁKMpCEp!}&CEBGNO}x}&NF!}&DGNOJ H€N} €ÕFKMBGFKMpKMLqJVp°J!€GKrKMLqÆNF!CTLGÁ•K&Lq>J HNOPP
CELGP ‹Á [email protected] hKMp¬J>N [email protected] ;Æ$NUlJ!€GKvBG€4UÇp!K 4D}&JVDGUÇJVNOCELGp hφ(t)φ(0)i )H€GNO} €2UTF!K•NLGWGD}MKMW2U3JxJ!€GKɲ"À
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C JV€GKPC}jU3JVNCTLÉC J!€GKr}jUTB4UÇ}MNOJ!CEFQBGPUÇJVK&pMÊ4±µA
L H€4UÇ>J hCEPPOC Hp77Krp!€4UTPOPUTp!pDGÁKJ!€4UÇJHJV€KrDLGpÁÅ´
Á•K¶JVFN MK&W LGCTNp!K½p!BKM}¶JVFVUÇP9WGK&LGp!N Jµ + UTpQWGK 4LGKMWÉNOLv»,‚ Ê |Ã[email protected]Ê G1à ÐNOLv} €4UTB$JVKMF„Ãr}MCTF!F!K&p!BCELGWGNOLGR
JVCBG€GCTJ!CELÕK&Á•NOp!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!BCELGWNLGReJVCzBG€GCTJ!CELUÇIGp!CEFBJVNOCE)L [NOpxpB K&}[email protected] GKMWIq‹JV€KJ!FVUTLGpB CTFJxBGF!CEBKMF¬JVNOKMprC JV€GK°Á•K&p!CEp}MCEBGNO}"}MNOF!}&DGNOJ ' G!IGDFIq¿ ( Ê Ô KMFK hh· · · ii p¬J UTLWGp4hCEFQUÇLÉNF!FKMWGD}MNIPK°LCENpK°}MCTF!F!K&PUÇJ!CEF
H€GKMFKSJV€GKBGFC$WGD}&J©C U1ÆTKMF!UTREKr}MDGFF!KMLqJ!p½€4UÇp©IKMK&LzpDGIJ!FVUT}¶JVKMWzCEDJMÊ
ƒ„€GK©J!DGLGLGK&PNOLGR Ô UTÁNP JVCELNUTLÂWKMp!}&F!NOIGNLGRSJ!€GKHKMPK&}&J!F!CEL–JVF!UTLG/p hK&F!FNLGRIK&*J 7KMKML•JV€GKHp!DB K&F!}MCTL$´
WGDG}¶JVCEFHUTLWlJ!€GKLGCEF!Á•UTP Á•K¶J UTP+POKjUTWeNOLeJ!€GKr²"ÀxÈDGLG}¶JVNCTLlNOp
HT =
X
Tk,q c†k,σ cq,σ e−iφ ,
?ÊE? k,q,σ
H€GKMFK„JV€GK„NOLGWGN}&KMp k UTLW q F!%K hK&F/JVCSJV€K„LGCEF!Á•UTP$Á•K¶J UTP$PKjUÇW•UÇLGW–p!DGBKMF}MCELGWDG}&J!CEFMÊ KH}MCELGpNWGK&F
hCEF„p!NOÁ•BPN}&NOJµÅJ!€4UÇJ Tk,q = T0 Ê ²½CTJ!K"JV€4UÇJHJ!€GK"JVDGLLGKMPONLGR Ô UÇÁ•NOPOJVCTLGNUTL•}MCELqJVUTNLGpHU 4DG}¶JVD4U3JVNLR
BG€4UTp<K UT}&J!CEF H€GNO} €zFKMBGFKMpKMLqJVp°J!€GK}MCEDBGPNOLGRÅJVCÅJV€GKrÁKMp!CTp!}MCTBGN}}MNOF!}&DGNOJMÊG±µLGWGK&KMW) IKM}jUÇDGp!KxC
JV€GKH}MUTB4UT}&NOJVN ÆEK„}&CEDGBGPONLGRSIK&*J 7KMK&L•JV€K©pNWGK&p7C J!€GK©²"À©ÈDGLG}¶JVNCTL UTLW–J!€GKHÁ•K&p!CEp}MCEBN}H}MNOF!}MDNOJ!TU
}MDGFF!K&Li?J 4D}&JVDGUÇJVNOCELÉJVFVUÇLGp!PUÇJVK&p„NLqJVC–UÂÆECTPOJ UÇREEK 4DG}¶JVD4UÇJ!NCELlUT}&F!CEpp„JV€GKr²"ÀrÈDGL}&JVNOCEL+Ê4Î7CTJV€lUTFK
F!K&PUÇJ!KMW:ÆNUvJV€GKÅJ!FVUTLGpb´µNOÁ•BKMW4UÇLG}MKÅC ›JV€K}&NF}MDGN #
J ' ÃÇ)Ä ( V (ω) = Z(ω)I(ω) Ê+²½K J![JV€GKÅÆTCEPOJVUTREK
4DG}¶JVD4UÇJ!NCELpvJVFVUÇLGp!PUÇJVK:NOLiJ!C2BG€4UTp0
K 4D}&JVDGUÇJVNOCELGplUT}MFCEp!pÉJ!€GK ÈDGLG}&J!NCEL HUTpÉJV€GK‹BG€4UÇp!K‹NpÉJV€K
}jUTLCELGN}MUTP+}&CEL3ÈDGRqU3JVKC J!€GK} €4UTFREK>U3J©J!€GK;ÈDGLG}¶JVNCTLG
p '<AG)( ~GJ!€GKxB€4UTp!KNOpHJ!€iDGp½}&CELGpNWGK&F!KMWÕUTp©U
‚iD4UTLqJVDGÁ Á•K&} €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 °JV€K
BG€4UTpKS}MCEFF!K&PUÇJ!CEF7NOp;JV€KMF!K hCEF!KSK $BGF!K&p!p!K&WzNOLvJVK&F!ÁpQC [JV€K"JVFVUÇLGp¬´ NÁB K&W4UTLG}&K°C JV€K"}MNOF!}MDNOJ„UÇLGW
JV€GKpB K&}&JVF!UTP[WGK&LGp!N JµÉC LGCENpK ' ÃTÄ (8H€N} €lNp„p!€C HLzNLl»,‚ Ê 9HÊ Ã ? Ê
ƒ„€GKvBGFKMp!K&LqJ•p¬pJVK&Á—IKjUTFpÂp!NOÁ•NOPUTFNOJ!NKM"p HNOJV€JV€GKvp¬JVDGW$C HNOLGKMPUTpJ!N}®°LGWGFKMK¶ÆF!K 4KM}¶JVNOCEL
NL JV€Kl}MUTp!C
K H€GKMFKlJ!€GKep!DGBKMF}MCELWGDG}&J!CEF }&CELqJ UTNOLGp–BG€4UTpC
K GDG}&J!D4UÇJVNOCELGp ' II3ID)(|ʛÀ$DG} €5BG€4UTpK
4DG}¶JVD4UÇJ!NCELpÅWGKMp¬JVF!C1 J!€GKepÁ•ÁK&J!F IK&*J 7KMK&LKMPOKM}¶JVF!CTLGpUTLGW €GCEPK&p ,UÇLGW U 9K&}&J–JV€GKe}&DGF!FKMLqJ
ÆECEP J UTRTK"} €4UTF!UT}&J!KMF!NOpJ!N}MpHC JV€Kx²°À>ÈDLG}&J!NCEL+Ê
L
,!,!.'!- 1!"
ƒ„€GKÉJVDGLLGKMPONLGR:}&DGF!FKMLqJ•UTp!pC$}&NUÇJ!KMW HNOJ!€J*;CKMPOKM}&J!F!CELpNOpÂREN ÆEKML2IqJ!€GKlË K&F!ÁN;RECTPWGK&L FDGPK
I = 2eΓi→f 7HNOJV€ÉJ!€GKSJVDGLGLKMPNOLGR–FVUÇJ!K
Γi→f = 2π
X
f
|hf |T |ii|2 δ(i − f ) ,
G
?Ê A H€GKMFK i UTLW f UÇF!KvJV€K JVDGLLGKMPONLGR`K&LGKMFRENK&pÅC„JV€KÉNLGN JVNUÇPÐUTLGW 4L4UÇP;pJVUÇJVK&p/NOLG}MPODGWGNOLGRÕJV€K
KMLqÆNFCELGÁKMLqJ!4UÇLGW T NpQJ!€GKSJVFVUÇLGp!N JVNCTLÉCEB K&FVUÇJ!CEF7H€GN} €lNp„K BF!KMpp!K&W:UTp
T = HT + HT
∞ X
n=1
1
HT
iη − H0 + i
n
,
?Ê ¿ H€GKMFK η Np„BCEp!N JVNOÆTK"NL74LGNOJ!KMpNÁ•UTP Ê
ƒ„€GF!CEDRE€GCEDJÅJ!€GNp>} €4UTBJ!KMF,CTLGKÉ}MCELp!NWKMF!pÂJ!€GKÉBG€GCÇJVCÇ´µUTp!pNpJ!KMW JVDLGLGKMPONLGR  ®,ƒ >}&DGF!FKMLqJ
WGDGK"J!CÂJV€GKS€GNORE
€ hF!K&‚iDGKMLG}¶É}MDF!F!K&Lq>J GDG}&J!D4UÇJVNOCELGp„C J!€GKSÁ•K&p!CEp}MCEBN}"WGK¶ÆN}MK UTp;J!€[email protected] KMFKMLG}&K
'<A )(|~
?$Ê I IP AT = I(K&LqÆ$NOF!CELÁ•K&LiJ ) − I(LCK&LqÆ$NOF!CELÁ•K&LiJ ) ,
H€GKMFK 7NLREK&LGKMF!UT8P ÐJV€KÕJVCTJVUTP©}MDGFF!KMLqJ hCEFJVDGLLGKMPONLGRC KMPK&}&J!F!CELGp•JV€GFCEDGRE€ J!€GKÈDGL}&JVNOCEL Np
I = I → − I← Ê
Ô C;K¶ÆEK&F+K BKMFNÁKMLqJ UÇPPO4NOJ©Np©WGN•}&DGPOJ©JVC•}MCEDBGPKx}jUTB4UÇ}MNOJ!NOÆTKMPOÉUTLGWÕJ!€GKMLÕJ!C FKMÁC ÆTK>JV€K
Á•K&p!CEp}MCEBN}WGK&ÆNO}MKr}MNF}MDGN J hF!CEÁ JV€GKrWGK&J!KM}¶JVCEF©}MNOF!}MDNO
J ' ÃEà (|Ê €4UÇJ©N%p 4NL UT}&J 4C #JVK&LÕÁKjUTpDGF!K&W
NpJV€GK•K $}MK&p!p–LGCENOp!K N|Ê KÇ@Ê JV€K WGN 9K&F!KML}MKvIK&*J 7KMK&L }&DGF!FKMLqJ 4D}&JVDGUÇJVNOCELGpxUÇJÅUÕRTNOÆEK&LIGNUTp>UÇLGW
MK&F!C–IGNUÇp„NLvJV€KrÁKMpCEp!}&CEBGNO}S}MNF}MDGN JjÊÍ U3JVKMF„CT)L 7K HNOPP }jUTPO}MDGPUÇJVK°J!€GKSK $}&KMp!p½LGCENOp!K Sexcess
+
C S +(ω) ÊG±µLeJV€GNO4p ;CE/F 6;KJV€iDGpHÁKjUTpDGF!KSJV€[email protected] KMF!K&LG}MKIK&*J 7KMK&LzJV€Kr}&DGF!FKMLqJVpHJV€F!CEDGRT€l(ω)
J!€GK
WGK&J!KM}¶JVCEF H€KMLU°IGNUTpÆTCEPOJVUTREKQUÇLGW–U MK&F!CSIGNUTp/UÇF!K„UTBBGPNOKMWxJ!C°JV€K„WGK&ÆN}&KQ}MNOF!}&DGNOJ ÇUÇp,U hDGLG}¶JVNCTL
C WGK&J!KM}&J!CEFHIGNUÇpQÆECEP J UTREK H€GNO} €zNOp„WG%K 4LKMW`UÇp
?$Ê D ∆IP AT (eV ) = I(eVd 6= 0, eV ) − I(eVd = 0, eV ) .
ƒ„€GNprUTPOp!Cl}MCEFF!K&p!BCELGWGprJVCeJV€GK[email protected] KMFKMLG}&K•IK&*J 7KMK&LJV€GK• ®,ƒ™}MDGFF!KMLqJ!pxJV€F!CEDGRT€J!€GK•WK&JVK&}&J!CEF
H€GKML U IGNUTp ÆECTPOJ UÇREK:UTLGW U &KMF!C2IGNUTpÉUTFK‹UTBGBGPONK&WJVCJV€K:WGK&ÆN}&K:}MNOF!}MDNO!J ∆IP AT (eV ) =
p HNOJV€‹LGClKMLqÆNFCELGÁKMLqJ
IP AT (eVd 6= 0, eV ) − IP AT (eVd = 0, eV ) IKM}MUTDGpK•JV€K}&CELqJVFNIGDJ!NCELE
}jUTL}MKMPHCED$JjÊЃ„€GKÕ[email protected] KMF!K&LG}MKzI K¶*J ;K&KML ®,ƒ }MDGFF!K&LiJ!p•J!€Dp BGFC ÆNOWGKMp•}MF!D}MNUÇP©NOL hCEFÁ UÇJ!NCELCEL
JV€GKpB K&}&JVF!UTPWGKMLp!NOJµlC K $}&KMp!p°LGCENpKC J!€GKxÁKMpCEp!}&CEBGNO}rWK&ÆN}&KTʲ½CTJ!N}&KJ!€4UÇJ©CEDGF©}jUÇP}MDPUÇJ!NCEL
UTBGBGPONK&p°JVCvJ!€GK MKMFCÉJVKMÁBKMFVU3JVDGFKÅ}jUTpK hCEFS}MCELqÆTKMLGNOKMLG}&K IGDJN JS}jUTL‹IKÅREKMLKMFVUÇPN &KMW:JV
C GLGNOJ!K
JVK&Á•BKMF!UÇJVDGFKMp&Ê
.!,- . . 1# ! !,!'.!-
®°P JV€GCEDRE€ CEDF hC}MDGp•CNLqJ!KMF!K&pJ }MCELG}&KMFLGpvBG€GCÇJVCÇ´µUTp!pNpJ!KMW®½LGWGFKMK&Æ F!K4KM}¶JVNCTL)=7K`LGK&KMW J!C
}MCEÁBGDJ!KSUTPOP9BCEp!pNIGPOK°}MCTLiJ!F!NOIGDJVNOCELGp&ʃ„€GK}MDF!F!K&LqJ©UTp!pC$}&NUÇJ!KMW HNOJ!€ÉCELGKSKMPOKM}¶JVF!CTLvJVDGLGLKMPNOLGRÅNp
RENOÆTKMLeIqÉJV€GKrË4KMFÁ•N RECEPOWGKMLeF!DPK I = 2πe
X
f
|hf |HT |ii|2 δ(i − f ) .
?$Ê G1Ä ƒ„€GK;}MUTP}&DGPU3JVNCTLrCGJV€GK7}MDGFF!K&LiJ/BGF!C}MK&KMWGpNLxJ!€GK;p!UTÁK=;UjÂUTp JV€4U3JC4U½LGCEFÁ UÇPÁK&JVUTPMÈDGLG}¶JVNCTL
'<A )(9 K $}MKMB$J>JV€GUÇJrCELGK€4UTpJVCeJ U ÇK NLqJ!CzUÇ}M}MCTDGLqJ>JV€K•pDGB K&F!}&CELGWGDG}¶JVNOLGRÕWGK&LGp!N JµCQp¬J UÇJ!KMprCEL
JV€GK©F!NORE€qJ;pNWGK©C +J!€GK/ÈDGLG}&J!NCEL BH€N} €vNOp›WCELGK°IqK $BGPOCENOJ!NLGRSJ!€GK"Î;CTRECEPNODGIC ÆxJVF!UTLG/p hCTF!Á•UÇJVNOCEL+Ê
Ë4CEF>JV€Kz}MUTp!KeC °KMPK&}&J!F!CELGpÅJ!DGLGLGK&PNL'
R hF!CEÁ—J!€GKeLGCEFÁ UTP;Á•K¶J UTPQPOKjUTW J!C‹J!€GKlp!DGBKMF}MCELGWDG}&J!CEF
iÃ
>
%
(eV > ∆) GJV€Kr}&DGF!FKMLqJ hF!CEÁ PK #J„J!C–F!NRT€iJHFKjUTWp
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 JµÕC/pJ U3JVKMp°C,JV€GKxJ*;CvPOKjUTWp NOLÕJV€K>LGCEFÁ UÇP p¬J UÇJ!K Ê K>LGCTJ!N}&K
JV€4U3J•J!€GNNp•}jUTPO}MDGPUÇSJVNOCEL2Np pNÁNPUTFÂJVCJV€GK`}MUTP}&DGPU3JVNCTL CSJV€GKlJVDGLGLKMPNOLGRF!UÇJVKzCSK&PKM}¶JVFCELGpvNOL
JV€GK:BF!KMpKMLG}&KC ÂUTL K&LiÆNOF!CELGÁKMLqJAH€GNO} € NOpÉp!€GCHL NL p!K&}&J!NCELGpAH$Ê@GEÊ H2UTLGW HÊ GEÊ hCEPPOCHNLGR
JV€GKxpVUTÁK>p¬JVK&BGp Ê+ƒ„€GNp½}MDGFF!K&LiJ"NLG}&PDGWGK&p°ICTJV€`UTL`KMPUTpJ!N}>UTLGW:UTLÕNLGK&PUTp¬JVN}r}MCTLiJ!F!NOIGDJVNOCE)L 4JV€K
hCEF!ÁKMF„IKMNOLGR–F!K&LGCEF!Á•UTPON MK&WÉIqÉJV€GKBGFKMp!K&LG}MKxCJ!€GKKMLqÆNFCELGÁKMLqJ' ÃTÄ ( Ê Ô KMF!K NOpHJ!€GKKMLGK&F!RT
C °UTL K&PKM}¶JVFCEL5NOL J!€GKeLGCEFÁ UTP„ÁK&JVUTPQPOKjUTW5UTLGW E NOpÅJV€GKlKMLGK&F!RT2C SU‹‚DGUTp!NOB4UTFJ!N}&PKÉNL2JV€K
p!DGBKMF}MCELWGDG}&J!CEF;POKjUTW9Ê ³ €4UTLGRENOLGRÆTUÇF!NUÇIGPK&p,NL–JV€GK©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„}MCEÁBGDJ!NLGRxJV€GKS}&DGF!FKMLqJ hF!CEÁ
F!NORE€qJQJVC–PK #JHNLlUÅp!NOÁ•NOPUT3F 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 H€GKMFKvJV€GK•JVFVUÇLGp!ÁNpp!NCTL}MCK }&NKMLqJ–C„J!€GKɲ°ÀzÈDLG}&J!NCELNOLJV€KvLGCEFÁ UTP›pJVUÇJVK NpÂWGK4LGK&W5UÇp
K 7KMNORE€q>J hDLG}&J!NCELGpHUÇF!KWG%K GLGKMWÕUTp
T = 4π 2 NN NS T02 C1e = eT /RK Ê4ƒ„€G<
p
?$Ê GH el
K1e
(eV ) = (eV )2 − ∆2 ,
p
?$Ê GB inel
K1e
(Ω, eV ) = (Ω − eV )2 − ∆2 .
À$NÁNPUTF!P  7KxCEI$J UTNOLvJV€G<K hCEFÁÂDGPUÂC ∆IP AT hCTFQJV€GK}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 ≤ ∆ JV€GK&F!K„UTFK;LGC°KMPUÇpJVNO}›J!FVUTLp!NOJ!NCELpIKM}MUTDGp!K;KMPK&}&J!F!CELGp}jUTLGLCTJ/K&LiJ!KMF
JV€GK°p!DGBKMF}MCELGWDG}&J!NLGRÅRqUÇB+ʲ©K&ÆTKMFJ!€GKMPOKMpp4WGDGK"J!C>JV€K"BGF!K&p!K&LG}MKC[J!€GK"KMLqÆNFCELGÁKMLqJ!$KMPOKM}&J!F!CELp
}jUTL‹UÇIGp!CEFI:CEF"K&Á•N J"KMLKMF!RÇ,
 hF!CTÁ CEF©JVCvJ!€GKÅKMLqÆNFCELGÁKMLqJSp!CvJ!€4UÇJUÇL:NLGK&PUTp¬JVN}r‚iD4UTpNB4UTF¬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 ƒ„€GK:p!NOLGREPOK`KMPOKM}&J!F!CEL h‚DGUTp!NOB4UTFJ!N}&PKÅJVDGLLGKMPONLGR }&DGF!FKMLqJlNpvC(4FpJeCEFWGKMFÉNOLJV€GK`J!DGLGLGK&PNLR
UTÁBGPN JVDGWGKÇÊ K½LC 5JVDGFL•J!CxBGFC$}&KMpp!KMp=H€GN} €•NLqÆECTKHJV€GKHJVDGLLGKMPONLGRC J*;CxKMPK&}&J!F!CELGp,J!€GF!CEDRE€
JV€GK"²"À NLqJ!KM/F UÇ}MKTÊ$±µLGWGK&KM)W 4IKM}MUTDGp!KSCTDGF„UTNOÁ¤NOp7J!CÅp!€GC JV€4UÇJQ®°LGWGFKMK¶ÆvF!K4KM}¶JVNOCELÉ}jUTLeIK"DGpKMW
JVC"ÁKjUÇp!DGFK„LGCENpK ;K„LGK&KMWÅJVCSK UTÁ•NOLGK„UTPOP*J 7CKMPOKM}&J!F!CELÅBF!C}MKMpp!K&p 7K„pJ UÇFJ HN JV€ÂJV€KQJVFVUÇLG/p hK&F
CQJ*7C`KMPOKM}¶JVF!CTLGpÅUTpx‚DGUTp!NOB4UTFJ!N}&PKMpxUTIC1ÆEK•J!€GKvRqUÇB+Ê ³ UTP}&DGPUÇJ!NCELprC„JV€GK•Á UÇJ!F!N ‹KMPOKMÁKMLqJÂNOL
»,‚Ê ?Ê A HUTFKrJ!€GKML`}MUTF!FNK&WÕCEDJ©JVC•p!K&}MCELGW:CTF!WGK&F½NOLzJV€KxJVDLGLGKMPONLGR Ô UÇÁ•NOPOJVCTLGNUTLÉDp!NLR•JV€K T
Á U3JVF!N 2
X
1
?$Ê G ¿ HT |ii| δ(i − f ) .
I = 4πe
|hf |HT
ƒ„€GKNLGN JVNUTP p¬J UÇJ!KNpHUÅBGF!CWGDG}¶J!
iη − H0 + i
f
?$Ê@G!I |ii = |GL i ⊗ |GS i ⊗ |Ri ,
H€GKMFK |GLi WGKMLCTJVK&p;UREF!CTDGLGWpJ U3JVKH€N} € }&CEF!FKMpB CELWGp›J!CrU 4POPK&W•Ë4K&F!ÁNGp!KMU hCEF,J!€GKHLGCEF!Á•UTP
KMPOKM}&J!F!CWGKTÊ |GS i NpJV€GK„Î ³ À>REF!CEDLGWÅpJVUÇJVKQNOL>JV€KQp!DGBKMF}MCELWGDG}&J!CEF,PKMUTW+Ê |Ri WKMLGCTJ!KMp/J!€GKQNLNOJVNUTP
pJVUÇJVKSC JV€GKK&LiÆNOF!CELGÁKMLqJMÊ Ì"LlJ!€GKCTJV€GK&FH€4UTLGW)CEDGFHREDGK&p!p?hCEFQJV€GK<4LGUTP+pJVUÇJVKp€GCEDGPOWzFKjUTW
?$Ê GD †
|f i = c†k,σ c†k ,σ γq,σ
γq† ,σ |GL i ⊗ |GS i ⊗ |R0 i ,
H€GKMLe*J 7C KMPOKM}¶JVF!CTLGp©UTF!KSK&Á•N J!JVK&W hF!CEÁ¤JV€GKrp!DGBKMF}MCELWGDG}&J!CEFMÊ ƒ„€GK µREDGKMpp C ,»,‚ Ê ?$Ê GD ;NOpHUTL
NL hCTF!ÁKMWÉCELGKÇ~ UTLeKMPOKM}&J!F!CELlB4UTNOFQNpHIGFC TK&LzNOLÉJV€GKp!DB K&F!}MCTLGWGDG}¶JVCE%F UÇLGWzCTLGKKMPK&}&J!F!CELeJVDGLLGKMPOp
JVC>J!€GK"LGCTF!Á•UTP ÁK&JVUTPPKjUÇ)W FH€NPK©JV€GK°CTJV€GK&FQIKM}MCTÁ•K&p„U‚iD4UTp!NOB4UTF¬JVN}&PK©NL J!€GKSp!DB K&F!}MCTLGWGDG}¶JVCEF JV€GKp!UTÁ•KrBGFC$}&KMp!p©Np„J!F!DGK hCEF„JV€GKrp!K&}MCELWÕKMPOKM}¶JVF!CT$
L H€GNO} €zJVDLGLGKMPOp„JVC•J!€GKLGCEFÁ UTP[ÁK&JVUTP[PKMUTW+Ê
€GKMLeJV€KrpDGBKMF!}&CELGWGD}&JVCTF½POKjUTWlUTIGpCEF!IpH*J 7CKMPK&}&J!F!CELG%p ?$Ê ÃTÄ †
|f i = ck,σ ck ,σ γq,σ
γq† ,σ |GL i ⊗ |GS i ⊗ |R0 i .
ƒ„€GK bREDGK&p!p ©C [»,‚ Ê ?$Ê ÃTÄ /Np,J!€4UÇJÐ*J ;CxKMPOKM}&J!F!CELp7}MUTL•J!DGLGLGK&6P hFCEÁ™J!€GK½LCEF!Á•UTPGÁK&JVUTP4PKMUTW UÇLGW
I K&}MCEÁKr‚DGUTp!NOB4UTFJ!N}&PKMpHNOLzJV€KxpDGB K&F!}&CELGWGDG}¶JVCEF&Ê Ô KMF!K |R0i NpHJV€G(
K 4L4UTPpJVUÇJVKxC /J!€GKxK&LqÆ$NOF!CEL´
Á•K&LqJjÊ
KlUTFA
K GF!pJ}&CELGp!NOWGKMFNLGR:J!€GKe}jUTpKzC ©*J 7CK&PKM}¶JVFCELGp–JVDGLGLKMPNOLG'
R hF!CTÁ—JV€GKepDGB K&F!}&CELGWGDG} ´
JVCEFÅPOKjUTW2JVC:J!€GKzLCEF!Á•UTPQÁK&JVUTPQPKMUTW+Ê,±µLqJVFC$WDG}MNOLGRJ!€GKe}MPOCEp!DGFKeF!KMPUÇJVNOCEL hCEFÅJV€KlK&NREK&LGpJVUÇJVK&p
C JVR€GK:LCELG}MCTLGLGKM}¶JVK&W pp¬JVKMÁ {|υii} ;UTLGW DGp!NOLGR JV€G'
K UÇ}&JÉJV€GUÇJ 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
LGKMFRTeWGK&LGCEÁNL4UÇJ!CEF!p&7Ê K€4UjÆEK
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 ƒ„€GKMLTIq>J!FVUTLGphCEF!ÁNLR©J!€GK;JVNOÁ•K7WGKMBKMLWGKMLqJÐBG€4UÇp!KMp,NOLqJVCSU½J!NÁK7WKMBKMLGWGK&LG}MKHC4J!€GK;J!DGLGLGK&PNLR
Ô UTÁNP JVCELNUT)L 7K}jUTL‹NLqJ!KMREF!UÇJVKÅCED$J>UÇPP GL4UTP,UTLWÆNF¬JVD4UÇP/pJVUÇJVK&pMÊ[ƒ„€GNOpxUÇPPC Hp"DpJ!CeF!%K HFNOJ!K
JV€GK½JVDGLGLKMPNOLGRÂ}&DGF!FKMLqJHNLvJ!KMF!ÁpQC [J!DGLGLGK&PNOLGRÂCEBKMF!UÇJVCEFpQNL•JV€GKSNOLiJ!KMF!UT}&J!NCELvFKMBGFKMpKMLqJ UÇJ!NCELvJ!C
PC 7KMpJ©CEFWGKMF O(T04) ÊGƒ„€GKr}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 H€GKMFKSJV€GKSJVNOÁ•K°WGKMBKMLGWKMLG}&K>C JV€GKCEBKMF!UÇJVCTF!pHNp„REC1ÆTKMF!LKMWÕIq
HT (t) = ei(KL +KS +Henv )t HT e−i(KL +KS +Henv )t ,
HNOJ!€ KL = H0 − µLNL UTLGW KS = H0 − µS NS ÊEƒ„€GKHNLqJVK&FVUT}¶JVNOCELÅBGF!K&p!K&LiJVUÇJVNOCEL–NL–»,‚ Ê ?$Ê ÃEà €4UTp;J!€GKSUTW$ÆTUÇLiJVUTREK©JV€4UÇJ ;KSCTLGPO€4UjÆEK"J!CÂ}jUÇP}MDPUÇJ!K°pJVUÇJVNOpJ!N}jUÇP UjÆTKMFVUÇREKMp„C #JVNOÁ•K ´µWGK&B K&LGWGKMLqJ
}MCEFF!K&PUÇJ!NCEL hDGLG}&J!NCELpMÊ KSK$BGFKMp!pHJV€KJ!DGLGLGK&PNOLGR Ô UÇÁ•NOPOJVCTLGNUTLNLvJ!€GK}MFKjUÇJ!NCEL UTLGLGNO€GNPUÇJVNOCEL
CEBKMFVU3JVCEFp,NLÅJV€GK©LGCEFÁ UÇP ´UÇLGW•pDGBKMF¬´ }MCELWGDG}&J!NLGRrPKMUTWGp&ÊiÎ;KM}MUTDGp!K©JV€GK&F!KHNpÐLGCÇJ;UÇLiÂNLqJVK&FVUT}¶JVNCTL
I K¶*J ;K&KMLzB4UTFJ!N}&PKMp„NOLÉ*J ;CPKMUTWGp 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*;CKMPOKM}¶JVF!CTLGp„JVDGLGLKMPNOLGR–UTp„‚iD4UTpNB4UÇFJVNO}MPOKMp„UTLWÕUTp„®½LGWGFKMK&ÆeFK%4K&}&J!NCEL+Ê
Ë4CEF©J!€GK>}&CEF!FKMPUÇJVNOCELzC,CEBKMFVU3JVCEFp°NLlJV€GKxLGCEFÁ UTPÁK&JVUTPPKjUÇW) Dp!NLR•JV€K: N}61 p½JV€KMCEFKMÁC.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!NWKMFH‚iD4UTp!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 ? ƒ„€GKrK$B CTLGKMLqJVNUTPp©CBG€4UTpKxCEBKMF!UÇJVCTF!p½UTF!K}jUÇP}MDPUÇJ!KMWzUTp½NOLlJ!€GKr®°BGBKMLGWN l® pKMK»,‚ Ê?$ÊE? I UTLGWÉJV€KrB€4UTp!K ´µBG€GUTp!Kr}MCEFF!K&PUÇJ!NCELhDGL}&JVNOCELeNp„WGK4LGK&WÕNLe»,‚ Ê 9HÊ G ? Gp!CÅJ!€4UÇJ' DEÃF6DH)(
$? Ê ÃGA he
e
e
e
i = e
.
Ë4CEPPOCHNOLGR»,‚ Ê 9HÊ Ã ? 7K hDGF¬JV€GK&FUÇp!p!DÁ•K JV€GUÇJ J(t) 1 EH€GN} € Á•KMUTLGpU 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 )
NÁB K&W4UTLG}&KHUTBGBGFC!NOÁ U3JVNCTL)ÇJ!CEREK&J!€GKMF3HNOJ!€ÅJV€GK? UT}&JÐJ!€4UÇJ J(t) NOp ;K&PP IKM€4UjÆEK&WÉUÇJ›PUÇF!REKQJ!NÁKMpMÊ
ƒ„€GNpQUTPPOCHp7DGpQJVC>K$B4UTLGWeJV€K"K$BCELGKMLqJ!NUTP+CBG€4UTpKr}&CEF!FKMPUÇJVCEFp eJ(t) ≈ 1 + J(t) ʃ„€GKSFKMp!DPOJ
hCEFxJ!€GKv}&DGF!FKMLqJÅ}MCTLiJVUTNLpÂI CÇJV€ UTL K&PUTp¬JVNO} UTLW2UTLNLKMPUÇpJVNO}•}&CELqJVFNIGDJ!NCEL I← = I←el + I←inel H€GKMFK
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`UTp„NOLz»,‚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 H€GKMFK K2e←
inel
(Ω, eV, η) NOp–WGK%4LKMW NOL »,‚ Ê ?$Ê<AG >NOL5®½BGBKMLGWGN ΰÊЃ„€GKlKMPUTpJ!N}É}MCTLiJ!F!NOIGDJVNOCEL
K $Np¬JVpHCELGP vN eV < −∆ ÊË4CEF −∆ < eV CELGP  J!€GKNLGK&PUTp¬JVN}½B4UTF¬J½}&CELqJVFNIGDJ!KMp„JVC I← Ê
?
$À NÁNPUTF!P ;K–}MUTP}&DGPUÇJ!K:hCEF I→ Ê[ƒ„€GKMFK–NpSUÉpÁÁ•K¶JVF¬:I K¶J*;K&KMLJV€GKÅÁ•UTRELGN JVDGWKÂI K¶J*;K&KML
JV€GK°F!NRT€iJ;UTLGWÉPOK%#J;Á•C1ÆNLGRx}MDGFF!K&LiJHDB CEL IGNUTp7F!K¶ÆEKMFpVUTP|~$J!€GK"K$BGFKMpp!NCTLAhCTF I← NOp7J!€GK"pVUÇÁ•KSUÇp
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!CLGKMK&WÕJVC•}jUTFFeCEDJ½}jUÇP}MDPUÇJ!NCELGp„C/JV€GKrÁ•UÇJVFN ÉK&PK&Á•K&LiJ½NLz»,‚ Ê $? Ê A QJ!C
p!K&}MCELGW CEF!WGK&F;NOLJ!€GK½J!DGLGLGK&PNLR Ô UTÁ•NOPOJ!CELGNUÇL+ÊTƒÐBGN}MUTPP EJV€K°NLNOJVNUTPp¬J UÇJ!K"HNOPP4IK"UTp›p!€GCHLÉNOL
»,‚Ê $? Ê GI ¶Ê4ÌSLÉJ!€GKCTJV€GK&FH€4UTLGW)4CTDGF„REDGK&p!p?hCEFQJV€GK<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
H€GKMLzU ³ C$CTB K&F©BGUTNFQNOp„KMÁNOJJVKMW 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 H€GKML`J!€GK>pDGBKMF!}&CELGWGD}&JVCTFSPKjUÇW`UTIGpCEF!IpSU ³ CCEB K&F°B4UÇNFMÊ Ô KMF!K |R0i NOp½JV€K4L4UÇPp¬J UÇJ!KÂC,JV€K
KMLqÆNFCELGÁKMLqJjÊ7ƒ„€GK bRTDGKMpp ÕCr»,‚p&Ê ?Ê HEÄ •UTLGW ?$Ê HG NOpvUTREUTNL UTL NLhCTF!ÁKMW CTLGKT~7NLGWGK&KMW)
JV€GK s8´ QUjÆEKÅp¬Á•ÁK&J!F`C ›J!€GKÅp!DB K&F!}MCTLGWGDG}¶JVCEFNÁBCEp!K&p°JV€4U3JSCELGP ÕpNLGREPOK&JSBGUTNFp"C›KMPOKM}&J!F!CELp
}jUTL:IKÅKMÁNOJJVK&W:CEFSUTIGpCEF!IKMW9Ê[ƒ„€GNp"B€GKMLGCTÁ•K&LGCEL€4UTpSIKMKML‹WGK&p!}&F!NIKMW‹NOL`JV€GKÅKMUTF!P ,
 7CEF zCEL
KMLqJ UÇLGREPK&Á•K&LqJÐNLÅÁKMp!CTp!}MCTBGN}„BG€qp!NO}Mp ' DT7Ä DG(8qUÇLGWÅJV€GKHF!K&p!DGP JVNOLGER 4L4UTPpJ U3JVKH}jUTL ENL–BGFNLG}&NBGPOK I KSWK&JVK&}&J!KMWzJ!€GF!CTDGRE€zUÅÆNOCEPUÇJ!NCEL C /Î7KMPP9NLGK&‚DGUTPN JVNK&p ' GHG (|Ê
Î;KM}MUTDGp!K ;K UTF!K•UTPOp!Cl}MCELGpNWGK&F!NOLGRlJ!€GK–JVDGLGLKMPNOLGRlBGF!C}MK&p!p>C ;*J 7C`KMPOKM}&J!F!CEL(
p hFCEÁ JV€GKp!D´
B K&F!}&CELGWGDG}¶JVCEFHJVCJV€GKxLGCEFÁ UTP[ÁK&JVUTP[PKMUT)W 4JV€Kx}&DGF!FKMLqJ"NOp½K BF!KMpp!K&WUÇp½NOL`»,‚ Ê ?$Ê ÃTà ¶Ê4±µLlJV€GNOp
}jUTpK JV€GKxBG€qp!N}MUTP NLqJVK&F!BGFK&J U3JVNCTLÕC л,‚ Ê ?$Ê ÃEà „Np½U €GCEBBGNLGRBGFC$}&KMp!p°C /*J 7CÉKMPK&}&J!F!CELG?p HNOJ!€
CEBGBCEp!N JVKp!BGNOLGp:hFCEÁ JV€GK•p!DGBKMF}MCELGWDG}&J!CEFJ!€GKMFKMIqFKMÁC1Æ$NOLGR`U ³ C$CEBKMFxB4UTNOF>NLJV€GK p!DGBKMFb´
}MCELWGDG}&J!CEF 9UTLGW‹I4UT} ÕUTRqUTNOL+ʃ„€GKÅWGKMPUjzJ!NÁKMp°IK&*J 7KMK&LJ!€GKÂ*J 7CeJ!DGLGLGK&PNOLGRvBGFC$}&KMp!pKMpC ÐJV€K
KMPOKM}&J!F!CEL(
p HN JV€GNOLUlB4UTNOFSNpRENOÆTKMLIi 0 UTLGW 00 FKMpB K&}&JVN ÆEK&PO &H€GK&F!KjUÇpxJV€KJ!NÁKÅI K¶*J ;K&KML WGK ´
pJ!F!C1NLGRlUTLGW}MFKjUÇJ!NLGRlU ³ CCEB K&FB4UTNF°t NpRENOÆTtKML‹Iq t ÊË4CEPPOCHNOLGRÉ»,‚ Ê ?$Ê ÃH "UÇLGW}&CELGpNWGK&F!NLR
CELGP `JV€GK®°LWGF!K&K&ÆBGF!C}MK&p!p &7K }jUÇLLGC HF!N JVK–JV€GK–JVDLGLGKMPONLGRl}MDF!F!K&LqJÅUTpxC
U hDGL}&JVNOCELC QJV€K
LGCEFÁ UTP UTLGW:UTLCEÁ UÇPCEDGp ½¸SF!K&KM)L 1 p hDGL}&JVNOCELGp"C ,JV€GKÂLCEF!Á•UTPÁ•K¶J UTP PKMUT)W GLσ JV€K‚iD4UTLqJVDGÁ
WGCTJ G UTLGW–JV€GK©p!DB K&F!}MCTLGWGDG}¶JVCEF% F p!K&K°®½BGB K&LGWGN ³ H€GNO} € Np,J!€GK½p!UTÁK½UÇp›NOL ¯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 )
.
ƒ„€GKF!K&p!DGP J hCEFQJV€GK}&DGF!FKMLqJ°}&CELqJ UTNOLGpHI CÇJV€lUTLeKMPUTpJ!N}UTLGWlUTLlNLKMPUÇpJVNO}"}MCTLiJ!F!NOIGDJVNOCEL+~
el
inel
I← = I ←
+ I←
,
A
?$Ê Hqà ?$Ê HH H€GKMFKSJV€GKKMPUTpJ!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 ƒ„€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 ? H€GKMFK,JV€GK,WGK&LGCEÁNL4U3JVCEFp[UTF!K,pB K&}MN 4KMWrNLS®½BGB K&LGWGN ¨ Ê I→ NOp[WGKMFNOÆTKMWrNLUQp!NOÁ•NOPUTFÁ•UTLGLGK&FjIGD$J
NOJ!p+K$BGFKMp!pNCELxNp[CEÁNOJJVK&WS€GKMFKTÊ ²½K¶ÆEK&FJV€KMPK&p!p%ÇN JVp+K9K&}&JVp&HNPOPTIKÐWGNpBGPUjEK&WNLSJV€GKÐÁKjUTpDGF!K&Á•K&LqJ
C JV€GKLGCENOp!KSC UÅB CTNLqJH}MCELqJVUT}&JMÊ
ƒ„€GK–UTIC ÆTKÂK $BGFKMpp!NCTLGp}MCELGp¬JVN JVDJ!K>JV€GK>}MKMLqJ!FVUTP/FKMpDGPOJSC ,JV€GNOp ;CTF ~ 7KÅDGLGWGK&F!p¬J UTLGW‹LGC
€GC ¦JV€K•}&DGF!FKMLq:
J 4D}&JVDGUÇJVNOCELGprNL‹JV€GK•LGKMNORE€iI CEFNLGReÁKMpCEp!}&CEBGN}}MNOF!}&DGNOJrREN ÆEKF!NpKÅJVCÕNOLGKMPUTpJ!N}
UTLGWKMPUTpJ!N}Å}MCTLiJ!F!NOIGDJVNOCELGpSNOL‹J!€GK}MDGFF!KMLqJxI K¶*J ;K&KML UeLGCEFÁ UÇP/Á•K¶J UTPÐUÇLGWUep!DB K&F!}MCTLGWGDG}¶JVCEF&Ê
K?4LGWJV€4U3J;ICTJV€•}MDGFF!K&LiJ;}MCELqJ!F!NIDJVNOCELGpЀ4UjÆEK©JV€GK©pVUTÁ?K hCEF!C
Á iUTLGWJV€GK©}MDGFF!K&LiJ 4DG}¶JVD4UÇJ!NCELp
C +J!€GK°ÁKMpCEp!}&CEBGN}½WGK&ÆNO}MKSU KM}¶J;J!€GK"}MDF!F!K&LqJ„NLJV€GK½WGK&J!KM}&J!CEF„UÇJ›JV€K"KMLGK&F!RT•}MCTF!F!K&p!BCELGWGNOLGRxJ!C
JV€GKSJ!CTJ UTP[K&LGKMFRTeC *J ;C•KMPK&}&J!F!CELGpHNOLlJ!€GKLGCEF!Á•UTP[POKjUTW+Ê®°LGWGFKMK¶ÆzF%K 4K&}&J!NCELeJV€GK&F!K hCEF!KxUT}&J!p°UÇp
UTLeKMLGK&F!RT GPOJVK&FMÊ
²½K J;K}MUTLÉ} €4UTLGRTKÆÇUTFNUTIPKMp;UTpQNOLvJV€GKSBGFK&ÆNCTDGp„p!K&}&JVNOCELGp&Ê NOJ!€eUTLeUTF!INOJVF!UTFIGNUTp 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 Ê9ƒ„€GK
ÇKMFLGKMPhDLG}&J!NCELGp
KNelS
UTLGW
KNinel
S
?$Ê HGA UTFKÂp!€CHL‹NOL`®°BGBKMLWGN
, !" # .!"+# !" # 1:(# % !,# .
µ± L:JV€GNOp"p!K&}&JVNOCEL)7K–NPOPDGp¬JVFVU3JVKxJ!€GKÅBGFKMp!K&LqJ>FKMp!DPOJVpEHN JV€Uvp!NOÁ•BPKÂK GUÇÁ•BGPOKTÊ.KÅ}&CELGp!NOWGKMF<hCEF
JV€GNOp–BGDGF!BCEpK`U‹‚DGUTLqJVDGÁ B CTNLqJ}MCELqJ UÇ}&J! H€GNO} €NpUWGK&ÆN}&K$H€GCTp!KÕLCENpKlpB K&}&J!FVUTP©WGKMLp!NOJµ
Np ;K&PP›} €4UTF!UT}&J!KMF!N MK&W IiDGpNLGRÕJ!€GK p}jUÇJJVK&F!NLRÕJV€GK&CEF '<?I)?E¿ (|Ê Ô KMFK €C 7K&ÆTKM%F ;Kv}&CELGpNWGK&F
DGLGp¬$ÁÁK&JVFN &KMWeLGCENOp!KS}MCTF!F!K&PUÇJ!CEF!%p H€GNO} €z€4UjÆTKxIKMK&Lzp€GC HLÕNLl»,‚ Ê Ã$Ê H A Ê
®°p–B CTNLqJVK&W CTDJUTIC ÆTK 7KlUTF!KÉ}&CEÁ•BDJVNOLGR`JV€GKeWGN 9K&F!K&LG}MKeC ©JV€GKe®,ƒ }&DGF!FKMLqJVpNL JV€K
BGF!K&p!K&LG}MK°UTLGW NOLJ!€GK½UTIGp!K&LG}MK½C 9J!€GK©WG}½INUTp&Êqƒ„€GNp›Á•KMUTLGp,JV€GUÇJ 7K©NLGpKMFJ›JV€K½pB K&}&JVF!UTP WGKMLp!NOJµ
C K $}MK&p!p›LGCENOp!KQC J!€GK„Á•K&p!CEp}MCEBN}QWGK¶ÆN}MK H€GN} € hCEF/U"B CENOLqJ,}MCELqJVUT}&JÐIKjUTFp,Á•CTpJ,C N JVp 7KMNORE€qJ
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'hFKM‚iDGKML}MNK&pMÊ,»$}MK&p!p–LGCENOp!K•WGKM}&F!KjUÇp!KMp–PONLGKMUTF!P `JVC&KMFC:C ÆTKMF–UÕF!UTLGREK [0, ±eVd ] hCEF
B CTp!NOJ!NOÆTK•UÇLGWLGK&RqUÇJVN ÆEK;hF!K&‚DKMLG}&NKMp €GK&F!K7KvFK%HF!N JVK•»,‚Ê ÃÊ HD <HN JV€J!€GKLGCTJVNO}MK€GKMFKjU#J!KMF%
;KDGpKrJ!€GKLGCTJVUÇJVNOCEL Vd NLGp¬JVKMUTWeC V J!Cp€GC¼JV€KÆTCEPOJVUTREKSCÁKMp!CTp!}MCTBGN}WGK¶Æ$NO}MK
+
Sexcess
(ω) =
2e2
T (1 − T )(eVd − |ω|)Θ(eVd − |ω|) .
π
?Ê Hq¿ KÅ} €GCCEpK–UvREKMLGK&F!NO} hCEF!Á hCEF°J!€GK>JVF!UTLGpNÁB K&W4UTLG}&K+pNÁNPUÇF©NOL`p!BNF!N J"JVCvJ!€4UÇJS} €CEp!K&LNOL
¯©KbÊ ' ÃÇÄ)( Ê ³ CELp!NWKMF!NOLGRJV€GK©}MNOF!}MDNOJÐNL•ËNOREDGF!K&p"?$Ê@G°UTLGW&?$Ê HUÇJ ω = 0 EJV€GK©WGK¶Æ$NO}MK°UTLGWWGK&J!KM}&J!CEF
UTF!KSLCTJ„}MCTDGBGPK&WÕUÇLGWÉJV€GK"J!FVUTLGpNÁB K&WGKML}MKSp!€CEDGPWÉJ!€GKMFK%hCEFKÆÇUTLNp!€9Ê Ì"LeJ!€GKSCTJ!€GKMF„€4UÇLGW)GJV€K
JVF!UTLGp!NOÁ•BKMWGUTLG}MK›Np BGFKMWGNO}&JVK&W>JVC°€4UjÆEK„U©}MCELGp¬J UTLqJ,IKM€GU1ÆNOCEFU3J/PUÇF!REK hF!K&‚iDGKMLG}&NK&pMÊK;J!€GKMFK%hCEFK
} €GCCEp!KSJV€K hCTPPC HNOLGRÂREK&LGKMFNE} hCEFÁ hCEFQJ!€GKSJVFVUÇLGp!NOÁ•BKMW4UÇLG}MK
|Z(ω)|2 =
(Rω)2
,
ω02 + ω 2
?$Ê HI H€GKMFK R Np©JV€GKrJµBGN}MUTP€GNORE€$hFKM‚iDGK&LG}&ÕNÁB K&W4UTLG}&KÂUTLWÕJV€GKx}MFCEp!pC ÆTKMF<hF!K&‚iDGKMLG}¶ ω0 Np½KMp¬JVN ´
Á U3JVKMWhF!CEÁ JV€GK„K $B K&F!NÁKMLqJVUTPW4U3J U"C¯HK%bÊ ' ÃEà ( E} €C$CEpNLGR"U 4LGNOJ!KQ}MD$JVChF!KM‚iDGK&LG}& ω0 ÁKjUTLp
JV€4U3J›UÇJ hF!K&‚iDGKMLG}&NK&p ω ω0 TJ!€GK„ÁKMp!CTp!}MCTBGN}H}MNOF!}&DGNOJ,€GUTp,LGCNLGDGKMLG}&KHCELJ!€GK„WGK¶JVKM}¶JVCEFÐ}&NF!}&DGNOJ
I K&}jUTDp!KPC hF!KM‚iDGK&LG}MNOKMp½WGC–LGCTJHBGFCEB4UTRqU3JVK"JV€F!CEDGRT€ÕUÅ}jUÇB4UT}MN JVCEF&Ê
K–}jUÇP}MDPUÇJ!KÅLDÁ•K&F!N}MUTPP zJ!€GK ®,ƒ¦}&DGF!FKMLqJVprNL:J!€GK UÇI C1ÆEK–JV€F!KMK–}MUTp!K&pM~ p!NLREPKÅUÇLGW*J 7C
‚iD4UTp!NOB4UTF¬JVN}&PKHJVDGLLGKMPONLGR>UTLGWv®½LGWGF!K&K&Æ F!%K GKM}&J!NCEL9Ê®°POP K&LGKMFRENK&pQUTFK"ÁKjUTpDGF!K&
W HNOJV€ F!KMpB K&}&JQJ!C
JV€GKHpDGB K&F!}&CELGWGDG}¶JVNOLGRxREUTB ∆ Êq±µL–JV€KMp!K©DGLGN JVp%;K©} €GCEpK ω0 = 0.3 UTLW η = 0.001 Ê ³ DGF!FKMLqJVp7UTFK
JµBGN}MUTPP –BGPCTJJVK&WeUTp„
U hDGL}&JVNOCELÉC [J!€GKSWG}"IGNUTp;ÆTCEPOJVUTREK eV C [JV€GK°WGK&J!KM}&J!CE4F hCEFQpK&ÆTKMFVUÇP9ÆÇUTPDGK&p
C /J!€GKÂÁKMp!CTp!}MCTBGN}xIGNUTp©ÆECEP J UTRTK eVd hJV€K>KMLqÆNFCELGÁKMLqJ Ê[Ì"DGF½Á•CTJ!NOÆÇUÇJ!NCELeNpHJVC }MCELGpNWGK&F°JV€K
 ®,ƒ2}MDF!F!K&LqJVp HNOJ!€ÂJV€GK„}&CELGWGN JVNCTL |eV | < 1 |eV | < ∆H€GK&F!KQJ!€GK„%K KM}¶JÐC JV€GK„K&LiÆNOF!CELGÁKMLqJ
CELÅJV€GKH®,ƒ }&DGF!FKMLqJ;NOp,Á•CTpJ,BGFCELGCEDLG}MK&)W UTLGW ;KHp€4UTPPBGF!K&WGN}¶J,JV€4UÇJÐ*J 7CKMPK&}&J!F!CELGp,J!DGLGLGK&PNLR
UTpxU ³ CCEBKMFrB4UTNF h®°LGWGFKMK¶ÆBF!C}MKMpp!K&px}&CELqJVF!NOIGDJ!KMpJV€GK•Á•CEp¬JJVCzJ!€GK• ®,ƒ¦}&DGF!FKMLqJ! K$}MK&BJ
}MPOCEp!KvJ!C eV = ∆ ÊÐÎ;KM}MUTDGpKzC ©JV€KÉpÁ•ÁK&J!F IK&*J 7KMK&L LGK&RqUÇJ!NOÆEK eV UTLW BCEpNOJVN ÆEK eV ;K
I
GW NpBGPUjrJV€GKHFKMp!DPOJVp hCTF eV > 0 ÊEƒ„€K½®,ƒ }&DGF!FKMLqJVp hCEF/J!€GK©UTI C1ÆTKHJ!€GF!K&K„BGF!C}MK&p!pKMpQUÇF!K„BGPOCTJ!J!KMW
LGKJHJVC–CELKxUÇLGCTJV€KMFHNLeËNOREDGFK ?$ÊE?:hCEF„}&CEÁB4UTF!NOp!CEL9Ê
K(4LGWÕNOL`ËNOREDGFK ?EÊ ?ÅJV€4UÇJ½[email protected]€GKx} €GKMÁN}MUTPB CÇJVKMLqJ!NUTP[CJV€GKxLGCEF!Á•UTP[ÁK&JVUTPPKjUÇWzNOp½}&PCEpK
JVC`J!€GKvBCTJVK&LiJ!NUTP7C HJV€GKÉpDGBKMF!}&CELGWGD}&JVCTFÅPKjUÇW eV ∆ JV€Ke ®,ƒ¤}MDGFF!KMLqJ!pNOLJV€GK•JV€GFKMK
}jUTpKMpSUÇF!KÂpDGBGBGFKMp!pKMW+ÊË4CEF©JV€GKx*J ;CÉ}MUTp!K&pSC ЂiD4UTp!NOB4UTF¬JVN}&PKJVDGLLGKMPONLGR JV€KÅ ®,ƒ }MDF!F!K&LqJVpUTFK
KM‚iD4UTPSJ!C &KMFC IKMPC U }&KMFJVUTNL JV€F!KMp€GCEPOW+ʄƒ„€GKp!NLREPK:‚iD4UTpNB4UÇFJVNO}MPOK:}[email protected] KMFp hF!CEÁ
MK&F!C UÇJ UJ!€GF!K&p!€GCEPO)W 3H€GN} €NOp•[email protected] 4K&W UTp ∆ − eV ÐJV€4UÇJ•N%p 7‚iD4UÇp!NBGUTFJ!N}MPOKMpUTFK`UTIGPOKlJ!C
JVDGLLGKMP9UTI C1ÆTK"JV€4UÇJ;p!DGBKMF}MCELWGDG}&J!CEF„RqUTBÉCTLGPO@N [J!€GK&•}jdUÇLlICEFF!C JV€GK°LGKM}&KMp!p!UTFeK&LGKMFRT hF!CEÁ
JV€GKÁKMp!CTp!}MCTBGN}WGK&ÆNO}MKTÊ ƒ„€GNOpxK BPUTNOLG<p H€iJV€K•}&DGFÆTKMpÅUTpp!C}MNUÇJVK&W HNOJ!€WGN 9K&F!K&LiJrÆÇUTPDKMprC JV€GK:ÁKMpCEp!}&CEBGNO}:WGK&ÆN}&K‹IGNUTpvÆTCEPOJVUTREK:UÇF!K‹p!€GN #JVK&W J!C JV€GK:FNRE€qJlUTp eVd WGK&}MFKjUTpKMpMÊ©Ë4CEF *J 7C
‚iD4UTp!NOB4UTF¬JVN}&PK°J!DGLGLGK&PNOLGR 7K>CEIp!KMF¬ÆEKrJV€4UÇJ„J!€GKx ®,ƒ¼}&DGF!FKMLqJ°€GUTp°UÅpNÁNPUÇF7JV€F!KMp€GCEPO)W H€GN} € }MCEÁB4UTFKMW J!CÅËNOREDGF!K ?$Ê ?TU Np;BDGp!€GK&WlJ!C QUTFWvJV€GK°F!NRT€iJ;NLÉËNREDGFK ?EÊ ?ÇI IKM}jUÇDGp!KSÁCEF!K½KMLGK&F!RT
Np–LGK&KMWGK&W JVCJVFVUÇLG/p hK&F*J 7CK&PKM}¶JVFCELGpvUÇI C1ÆEKeJV€KzRqUÇ)B Ð}MCEÁB4UTFKMW5J!CUp!NOLGREPKÉK&PK&}&JVFCEL+ʛ²½CÇJ
p!DGFBGF!NOp!NOLGREPO•JV€GKr}MCTF!F!K&p!BCELGWGNOLGR}MDGF¬ÆEKMp°UTF!KrCELG}&K>UÇRqUTNLep€[email protected] #J!KMWeJVC–JV€GKrF!NORE€q>J HN JV€lWGKM}&F!KMUTp!NOLGR
eVd Ê4ƒ„€KMp!Kr}MDFÆEK&p"UTPOP9€4UjÆTK>UÅp!€GUTF!BlÁ U NOÁÂDGÁ UÇJ eV = ∆ Ê
KSJVDF!LeLGC J!C–J!€GK®°LGWF!KMK¶Æl ®,ƒˆ[email protected] KMFKMLG}&Kr}&DGF!FKMLq!J H€GNO} €zWCEÁ•NOL4UÇJ!KMp;J!€GKrUTI C1ÆTK*J 7C
BGF!C}MK&p!pKMpvU3J–p!Á•UTPP„UÇLGW5ÁCWGKMF!UÇJVKeIGNUTp!K&pMÊв½CÇJVKÉJV€4U3JJ!€GKeJVCTJVUTPQ®°LWGF!K&K&Æ }MDGFF!K&LiJ•}MCTLiJVUTNLp
UTL K&PUTp¬JVNO}l}&CELqJVFNIGDJ!NCEL5UT;
p ;K&PP©UTp•UTL NOLGKMPUTpJ!N}e}MCTLiJ!F!NOIGDJVNOCEL I K&PC J!€GKÕRqUÇ)B Ð}MCELqJVF!UTF J!C
‚iD4UTp!NOB4UTF¬JVN}&PK JVDGLGLKMPNOLG4R H€GNO} €x€4UÇp[}MCELqJVFNIGD$JVNCTLGp[I K&PC JV€GK,RqUÇBrCTLGPO½I K&}jUTDp!KÐJV€GK&p!KÐBGF!C}MK&p!pKMp
UTF!KrBG€GCÇJVCÇ´µUTp!pNpJ!KMW+ÊÎ;K&}jUTDp!:
K 7KÂUTFK>}&CEÁ•BDJVNOLGRJ!€[email protected] KMFKMLG}&K>IK&*J 7KMKML®,ƒ }MDGFF!KMLqJ!"p HNOJ!€
UTLGA
W HNOJ!€GCEDJHJV€KxÁKMpCEp!}&CEBGN}IGNUTp„ÆECEP J UTRTK 7K>K $BKM}¶J°JV€GUÇJHJV€GKrKMPUTpJ!N}S}MCTLiJ!F!NOIGDJVNOCELl}jUTLG}&KMPOp
CEDJMÊ Ô C7K&ÆEK&FJV€GK 4FpJrJVK&F!Á C;»,‚ Ê ?$Ê H A ½JVK&PPOp"DGpJV€4UÇJSJ!€GK–BGF!K&p!K&LG}MK•C;J!€GK–KMLqÆNF!CTLGÁ•K&LqJ
UTPpC–RENOÆTKMpHF!NOp!KSJVC•UTLl%K KM}&J!NOÆTKxK&PUTp¬JVN}S}&CELqJVFNIGDJ!NCELÉJ!CJ!€GK>®,ƒ ®½LGWGF!K&K&ÆlWGN 9K&F!KML}MKr}MDGFF!KMLqJMÊ
¹½L hCTFJVDL4UÇJVK&PO JV€GNOp„KMPUTpJ!N}"}&CEF!FKM}&J!NCELeNOp„LGCTJHp!Á•UTPOP9}MCTÁ•B4UÇF!KMWÉJ!C–J!€GKSJVFDGK®°LGWGFKMK¶Æe}MDGFF!KMLqJMÊ
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FVUÇJ!KSC KMPK&}&J!F!CELGp hFCEÁ J!€GKLGCEF!Á•UTP+POKjUTWÉJ!C–J!€GKWGCTJ hp!KMKrIKMPC hCEFHUT}¶JVD4UTP[LiDGÁ>I K&F!p Ê
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}MCELWGDG}&J!CEF°NOp½J!FVUTLGpÁ•N J!J!KMWzJ!C•J!€GKÂLGCEFÁ UÇP POKjUTWzCEF½ÆNO}MKxÆTKMFpVUÊ ƒ„€G:
K 4FpJSBGFC$}&KMpp"NLqÆECTPOÆEK&p°JV€K
KMPOKM}&J!F!CEL hFCEÁ U ³ C$CTB K&F/B4UTNOFJVDLGLGKMPONLGR°CELqJVC"JV€K„WGCTJ ÇLGK JTJV€GNOp/KMPOKM}¶JVF!CTLÅKMp!}MUTBKMpÐNL>J!€GK„PKMUTW JV€GK„CÇJV€GK&F,KMPK&}&J!F!CEL hFCEÁŸJ!€GK„pVUTÁK ³ CCEB K&F,B4UTNOFJV€KMLDGLGWGK&F!RECKMp,JV€K„pVUTÁK;J*7CrJ!DGLGLGK&PNLRSBGF!CÇ´
}MK&p!p!K&pMÊ À$NOÁ•NOPUTF,JVF!UTLGpNOJVNOCELG%p NLvJ!€GKCEBGBCEp!N JVK"WNF!K&}&J!NCE)L UTF!KSLKM}MK&p!p!UTF hCEF7*J ;CKMPOKM}&J!F!CEL4p hF!CEÁ
JV€GK•LGCEFÁ UTP›PKMUTWJ!C`KMLGW DB UTpÅU ³ CCEBKMFxB4UTNFxNLJ!€GK pDGBKMF!}&CELGWGD}&JVCTFMÊ,²©CTJVK J!€4UÇJxJV€GNOp>WGK ´
p!}&F!NB$JVNCTL C K¶ÆEKMLqJ!p½UÇp!p!DÁ•K&pQNÁBGPNO}MN Jµ>JV€GUÇJQJV€K°p!DB K&F!}MCTLGWGDG}¶JVCEF„POKjUTWvFKMÁ•UTNLp;NLJV€GK°REF!CEDLGW
pJVUÇJVKÂNOLÕJV€KÂNLGN JVNUTPUTLG'
W 4LGUTPp¬J UÇJ!KMp ®½LGWGFKMK&ÆÕBGF!C}MK&p!p ¶ÊÌ"L`JV€KÂCTJV€KMF"€4UÇLG)W 9N ,JV€GK>LGCEF!Á•UTP
Á•K¶J UTP PKMUTW`NOp°NLNOJVNUTPP vNOLÕJV€GK>REF!CEDLGW:pJVUÇJV
K 4PPOKMW:Ë4K&F!ÁNpKjU 9NOJ°Np½P%K #J°NL:UÇL:K $}MN JVKMW‹p¬J UÇJ!K
HNOJ!€‹*J 7ClKMPK&}&J!F!CELGp€4UjÆNLRlK&LGKMFRENK&prUTI C1ÆTK•Ë4K&F!ÁN/KMLGK&F!RT EF NL:JV€K 4L4UTP,p¬J UÇJ!KTÊ[ƒ„€GKK JVF!U
KMLGK&F!RTv€4UTpHI K&KMLlBGF!C1ÆNWKMWlIiÉJ!€GKKMLqÆNFCELGÁKMLqJjÊ4ƒÐBGN}MUTPP 
?$<Ê qà |ii = |GL i ⊗ |GS i ⊗ |0QD i ⊗ |Ri ,
H€GKMFK |GL,S i WKMLGCTJ!KMpU©REFCEDGLGWxpJVUÇJVK H€N} €>}&CEF!FKMpB CELWGpJ!C°U 4PPOKMWrË4KMFÁ•NqpKj>U hCTFJ!€GK›LGCEF!Á•UTP
KMPOKM}&J!F!CWGKTÊ |0QD i NpJV€GK–ÆÇUT}MDGDÁ«C ;JV€K ‚iD4UTLqJVDÁ«WCTJÂUÇLGW |Ri WKMLGCTJ!KMp>J!€GKNLGN JVNUÇPpJVUÇJVK•C JV€GKK&LiÆNOF!CELGÁKMLqJMÊ Ì"LlJ!€GKCTJV€GK&FH€4UTLGWeCEDGFHREDKMp!p hCEFQJ!€G<K 4L4UTP[p¬J UÇJ!Kp!€GCEDPWeF!KMUTW
?$Ê H |f i = 2−1/2 [c†k,σ c†k ,−σ − c†k ,σ c†k,−σ ]|GL i ⊗ |GS i ⊗ |0QD i ⊗ |R0 i ,
H€GKMLzU ³ C$CTB K&F©BGUTNFQNOp„KMÁNOJJVKMW hFCEÁ pDGB K&F!}&CELGWGDG}¶JVCE%F CEF
?$Ê |f i = 2−1/2 [ck,σ ck ,−σ − ck ,σ ck,−σ ]|GL i ⊗ |GS i ⊗ |0QD i ⊗ |R0 i ,
H€GKML`J!€GK>pDGBKMF!}&CELGWGD}&JVCTFSPKjUÇW`UTIGpCEF!IpSU ³ CCEB K&F°B4UÇNFMÊ Ô KMF!K |R0i NOp½JV€
K 4L4UÇPp¬J UÇJ!KÂC ,JV€K
KMLqÆNFCELGÁKMLqJjÊ ƒ„€GK°È[email protected] 4}MUÇJVNOCE,
L hCEF°J!€GKÅ} €GCENO}MKÂC ;»,‚ipMÊ ?$Ê H °UTLW ?$Ê ½NOp°JV€KÅpVUTÁKÅUTpSNOL
p!K&}&JVNOCEL ?$Ê Ã$EÊ ?Ê
0
0
0
0
!, % # 1 . % # $*, ,*,#)# F . ! 1 1!"
„ƒ €GK>J!KM} €GLGNO‚iDGK ;KÅDp!K>J!Cv}jUTPO}MDGPUÇJVKrJV€GK>}MDGFF!KMLqJ"J!€GF!CTDGRE€`JV€KŲ ¨ À–ÈDLG}&J!NCEL`NOp½J!€GKÂpVUÇÁ•KÅUÇp
JV€GK>QUj ;K°CEIJ UÇNL–JV€GKS»,‚ Ê ?$Ê ÃEà ʮ½POJ!€GCEDGRE€ ;K"UTPpC>}MCTLGp!NOWGKMF›JV€GK©JVDGLGLKMPNOLGR>BF!C}MKMppQC+J*7C
KMPOKM}&J!F!CELp7IGD$J;JV€K°KMPOKM}¶JVF!CTLGp;ÁÂDpJ7€4UjÆEK°ÆNF¬JVD4UÇPp¬J UÇJ!KMp7CELv‚iD4UTLqJVDÁ¤WCTJ!iJV€GK&L JV€K°}jUÇP}MDPUÇJ!NCEL
?EÃ
C JV€GKÁ•UÇJVFN KMPK&Á•K&LqJHNLl»,‚Ê $? Ê<A 7JVC hCEDGF¬JV€lCEF!WKMF„NLÉJ!€GKSJVDGLGLKMPNOLGR Ô UÇÁ•NOPOJVCTLGNUTL•RENOÆTKMp
I← = 2e
XZ
f,υi
∞
dte
i(i −f )t
−∞
hi|HT† |υ1 i
∞
Z
Z
∞
0
0
dt01 e−i(i −υ1 −iη)t1 hυ1 |HT† |υ2 i
Z
∞
Z
∞
0
dt02 e−i(i −υ2 −iη)t2
0
dt3 ei(i −υ4 +iη)t3
dt03 e−i(i −υ3 −iη)t3 hυ3 |HT† |f ihf |HT |υ4 i
0
Z0 ∞
Z ∞
×hυ4 |HT |υ5 i
dt2 ei(i −υ5 +iη)t2 hυ5 |HT |υ6 i
dt1 ei(i −υ6 +iη)t1 hυ6 |HT |ii
×hυ2 |HT† |υ3 i
0
0
= 2e
XZ
f,υi
∞
dt
−∞
Z
0
∞
dt1
0
Z
∞
dt2
0
Z
∞
dt3
0
Z
∞
dt01
0
Z
∞
Z
dt02
0
∞
0
0
0
dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 )
0
hυ2 |HT† |υ3 ieiυ3 t3 hυ3 |HT† |f i
hυ1 |HT† |υ2 ie
e
×hi|HT† |υ1 ie
×e−if t hf |HT |υ4 ieii (t1 +t2 +t3 ) e−iυ4 t3 hυ4 |HT |υ5 ie−iυ5 t2 hυ5 |HT |υ6 ie−iυ6 t1 hυ6 |HT |ii .
ii (t−t01 −t02 −t03 )
0
iυ2 t02
iυ1 t01
?$Ê ? ,» ‚Ê ?$Ê< ? ;WGKMp}MFNIKMp°K¶ÆNWGK&LiJ!POÉJ!€GKrJ!DGLGLGK&PNOLGR–CJ*7C K&PKM}¶JVFCELGp?hF!CEÁ J!€GKrp!DGBKMF}MCELGWDG}&J!CEF©J!C
JV€GKSLCEF!Á•UTP+ÁK&JVUTP9POKjUTWÉÆNU–UłiD4UTLqJVDGÁ WGCTJ! UÇLGWeI4UT}ÉUTREUTNL)FHNOJ!€eJ!€GKWGKMPUjvJVNOÁ•K&pQI K¶J*;K&KML
J*;CŀCEBGBGNOLGRÅpJ!KMBGpHUTFK"REN ÆEKMLeIq t1 t2 t3 t03 t02 t01 F!K&p!BKM}¶JVNOÆTKMP 4UTLWvJV€GK"J!NÁK½IK&J*7KMK&LzJV€K
WGKMp¬JVFC NOLGR:UTLGW }&F!KMUÇJVNOLGR:U ³ CCEBKMF>BGUTNFxNp>RTNOÆEK&LIqJMʃ„€GNOp>JVDLGLGKMPONLGRzBGF!C}MK&p!pÅÁ•UTK&p>JV€K
}MCEFF!K&p!BCELGWGNOLGR–}MDGFF!KMLqJ½I K&NLGR–UÇp
I← = 2e
Z
∞
dt
Z
∞
dt1
Z
∞
dt2
Z
∞
dt3
Z
∞
dt01
Z
∞
dt02
∞
Z
−∞
0
0
0
0
0
0
iµS (2t−t01 −t02 −2t03 −t1 −t2 ) −iµL (2t−t02 −t03 −2t1 −t2 −t3 )
0
0
0
dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 )
e
×e
†
×hHT 1 (t − t01 − t02 − t03 )HT† 2 (t − t02 − t03 )HT† 1 (t − t03 )HT† 2 (t)
×HT 2 (t1 + t2 + t3 )HT 1 (t1 + t2 )HT 2 (t1 )HT 1 (0)i .
?Ê< A ƒ„€GK–BGF!CTIGPK&Á Np½JV€iDGprF!KMWDG}MK&WJ!CeJ!€GK–}jUTPO}MDGPUÇJVNOCEL`C7}MCTF!F!K&PUÇJ!CEF!p°C7J!€GKÂJVDLGLGKMPONLGR Ô UTÁ•NOPOJ!CÇ´
LGNUÇL NLJV€KlRTF!CEDGLW p¬J UÇJ!KTÊ,¹©p!NOLGR N}61 pÅJ!€GKMCEFKMÁCJV€KMp!Ke}MUTL2I KeK$BGFKMpp!KMW5NL JVK&F!ÁpÅC½U
p!NOLGREPKB4UTF¬JVNO}MPKv¸FKMK&L)1 phDGLG}&J!NCEL IKM}MUTDGpKvJV€GK Ô UTÁNPOJ!CELGNUTL‹CHJV€GKvNOp!CEPUÇJVK&W}&CEÁ•BCELGK&LqJVpÅNp
‚iD4UTWGF!UÇJVNO} hK$}MK&BJÅÁ•U1IK hCTFrJ!€GK K&LiÆNOF!CELGÁKMLqJ H€GNO} €NprWGKMUTPOJxp!K&B4UTFVU3JVKMP  ¶Êƒ„€GKvWGK¶J UTNOPÐC JV€GNOpx}MUTP}&DGPU3JVNCTLNOpxp€GC HL NL®°BGBKMLGWN »H&Ê K }jUÇLLGC HF!N JVK•J!€GK•J!DGLGLGK&PNOLGRÕ}&DGF!FKMLqJÅUTp>U
hDGLG}¶JVNCTL•C JV€GK½LGCEFÁ UTP UTLGW UTLGCEÁ•UTPCTDGp ,¸SF!K&KM)L 1 p hDGL}&JVNOCELGp›C 9J!€GK©LGCEF!Á•UTP4ÁK&JVUTP PKjUÇ)W GLσ JV€GK‚iD4UTLqJ!DGÁ WGCTJ G UTLWlJ!€GKp!DGBKMF}MCELGWDG}&J!CEF F H€GNO} €ÕUTFKp!€GCHLÕNOLÉ®°BGBKMLWGN ³ Dσ
I← =
2eT14 T24
Z
∞
dt
σ
∞
Z
dt1
∞
Z
dt2
Z
∞
dt3
Z
∞
dt01
Z
−∞
0
0
0
0
0
iµS (2t−t01 −t02 −2t03 −t1 −t2 ) −iµL (2t−t02 −t03 −2t1 −t2 −t3 )
∞
dt02
Z
∞
0
0
0
0
dt03 e−η(t1 +t2 +t3 +t1 +t2 +t3 )
e
×e
X ∗ 0
0
0
0
>
0
×
−Fσ (q , −t1 − t02 )F−σ (q, t1 + t2 )G>
Lσ (k, t − t2 − t3 − t1 − t2 − t3 )GL−σ (k , t − t1 )
k,k 0 ,q,q 0 ,σ
× Gt̃Dσ (−t01 )Gt̃D−σ (−t03 )GtDσ (t3 )GtD−σ (t1 ) + Fσ∗ (q 0 , −t01 − t02 )Fσ (q, t1 + t2 )
0
0
>
0
t̃
0
t̃
0
t
t
× G>
Lσ (k, t − t2 − t3 − t1 )GL−σ (k , t − t1 − t2 − t3 )GDσ (−t1 )GD−σ (−t3 )GD−σ (t3 )GDσ (t1 )
0
0
0
0
0
0
0
0
0
0
×eJ(t−t1 −t2 −t3 )+J(t−t3 )+J(t−t1 −t2 −t3 −t1 −t2 )+J(t−t3 −t1 −t2 )−J(−t1 −t2 )−J(t1 +t2 ) .
?H
i
?$Ê i¿ ƒ„€GKÕFKMpDGPOJ hCEFÅJV€GKz}MDGFF!K&LiJ }MCELqJVUTNLGpICTJV€ UTL K&PUTp¬JVN}lUTLWUTL5NLGK&PUTp¬JVN}É}&CELqJVFNIGDJ!NCEL9Ê,ƒ„€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←
?$Ê I H€GKMFK D←0 Np7J!€GKSCEFNRENOL4UTP4WGK&LGCEÁNL4UÇJ!CEF3H€GN} €eNp7LGCTJHU 9K&}&JVK&WÉIq JV€K"KMLqÆNFCELGÁKMLqJ!FH€GN} €eNp
WG%K GLGKMW‹Iq‹»,‚ Ê ?$Ê ¿ ? +UTLW D←el Np½JV€GKÅWGK&LGCEÁNL4U3JVCEF½BGF!CWGDG}¶JU KM}&J!KMW‹Iq`JV€KÅKMLqÆNF!CTLGÁ•K&LqJ
pKMKr»,‚Ê ?$Ê ¿ A 7C®°BGBKMLWGN lË ¶Ê4ƒ„€GKNLKMPUÇpJVNO}"}MCTLiJ!F!NOIGDJVNOCELÉF!KMUTWGp
inel
I←
eγ 2 γ 2
' 21 2
π RK
×√
Z
∞
d
eV
Z
∞
d
eV
0
Z
∞
dE
∆
Z
∞
dE 0
∆
|Z(−( + 0 ))|2 SI (−( + 0 ))
∆
p
,
inel
( + 0 )2
D←
E 2 − ∆2 E 0 2 − ∆2
2
?$Ê D H€GKMFK D←inel Np[J!€GKÐWGKMLGCTÁ•NOL4UÇJVCTF+BGF!CWGDG}¶J/U3J!JVFNIGD$JVKMWSJ!C©J!€GKÐNLGK&PUTp¬JVN}}MDGFF!KMLqJ H€GNO} €xNOpWGK%4LKMW
NLe»,‚ Ê ?$Ê ¿E¿ ;C®°BB K&LGWGN eË;Ê
Ô KMFK γ = 2πN T 2 UÇLGW γ = 2πN T 2 WGK4LGKrJV€GKrJVDGLGLKMPNOLGR F!UÇJVK&p½IK&J*7KMK&L:JV€GKxp!DGBKMFb´
}MCELWGDG}&J!CEF7UT1 LGW J!€GK½WSCTJQ1 UÇLGWvIK&2J*7KMKMLvJ!€GK°N W2CTJQUÇLGW•J!€GK°LGCTF!Á•UTP4Á•K¶J UTP POKjUTW)qFKMpB K&}&JVN ÆEK&POBHNOJ!€
NS UTLW NN UTpxJ!€GKvWGK&LGp!N JµC Hp¬J UÇJ!KMpÂBKMF>p!BGNOLC „JV€K *J ;C:ÁK&JVUTPprNLJV€GKvLCEF!Á•UTPÐpJVUÇJVKÉU3J
JV€GKr} €GK&Á•NO}jUTPBCTJVK&LiJ!NUTPOp µS UTLGW µL FKMp!BKM}¶JVN ÆEKMP EÊ®°POP[}MCELqJVFNIGD$JVNCTLGp©J!C•J!€GKx}&DGF!FKMLqJS}&CELqJ UTNOL
WGKMLCEÁ•NOL4UÇJ!CEF!p H€GK&F!KÕJ!€GKÕNOL 4LGN JVKMpNÁ•UTP η UTWGNUÇI4UÇJVNO}e/p HNOJ!} €GNLRB4UTF!UTÁK&JVK&F ÅNpNLG}&PDGWKMW NOL
CEF!WKMF;J!C–U1ÆTCENWÉWNOÆEK&F!REK&LG}MK&pMÊ ±µ
L UT}¶!J GN JQ€4UTp„IKMK&Lep!€GC HLlNLɯH%K hp&Ê ' DT)Ä ([UTLW ' D ? (+JV€4U3JHUÂBGF!CTB K&F
F!K&p!DGÁÁ U3JVNCTLC +JV€GK½B K&FJ!DGF!I4U3JVNCTL•pKMF!NOKM%p $NOLG}MPODGWGNLRxUÇPP F!CEDLGW•J!F!NB=p hFCEÁ¦JV€GK½WGCTJ›JVCrJV€GK½LGCEF¬´
Á UÇPGPKMUTWGp qPOKjUTWp›J!C>UIGF!CEUTWGKMLNLGRrC JV€GK©WGCTJ7PK¶ÆEKMP|Ê K„JVU TK½NLqJVCrUT}&}MCEDGLqJ›JV€GNOpÐIGF!CqUÇWGKMLGNOLGRrIi
F!K&BGPUT}&NLGR η Iq γ/2 6HNOJV€ γ = γ1 + γ2 NLqJ!CvCEDGF"}MUTP}&DGPU3JVNCTLGp hNLG}&PDGWNLGR•CELGPOzUvIGFCqUTWGK&LGNLR
WGDGKQJ!CJ!€GK„p!DB K&F!}MCTLGWGDG}¶JVCEF ¶Êq®°pÐÁKMLqJVNOCELGK&W–UTI C1ÆTK 7K©€GU1ÆTK©UTp!pDGÁ•K&WÅJV€4UÇJ γ1 γ2 ÊT±µLCTF!WGK&F
JVC–UjÆECTNWvJV€KSK $}MN J UÇJ!NCELÉC ‚iD4UTpNB4UÇFJVNO}MPOKMp„UTIC1ÆEK"J!€GKSRqUT)B $JV€GK&p!KFVUÇJ!KMpHUTPOp!CÅLGK&KMWÉJVC hDGP 4PP JV€K
L H€4UÇ=J hCEPOPC Hp7K ÇKMKMBvJ!€GK©LGCTJ U3JVNCTL η NOL CEDGFÐK BF!KMpp!NOCELGp $I KMUTF!NOLGR
}MCELWGNOJ!NCEL D + γ < ∆ ʱµ
NLxÁ•NOLGWxJ!€4UÇJ/N JFKMBGFKMpKMLqJVp,JV€KQPNOLG%K HNOWJV€>UTp!pC$}&NUÇJ!KM:
W HN JV€>J!€GK;POKjUTWGp&ÊEË4CEF LiDGÁ•K&F!NO}jUTPBGDGFB CEpKM%p NO4J HNOPPI KSpD }MNOKMLqJHJVCUTp!pDGÁK"JV€4UÇJ η N4p ÇKMBJHÆTKMFepÁ UÇPP9}&CEÁB4UTF!K&WvJVCÅJV€KSp!DGBKMF}MCELGWDG}&J!NLGR
RqUT)B GUTp ;K&PP[UTp©UTPOP9JV€KrFKMPOK&ÆÇUTLqJHPK¶ÆEKMP[K&LGKMFRENK&p WGCTJHPOK&ÆTKMPB CTp!NOJ!NCEL $IGNUTpQÆECTPOJ UÇREKM%p 4K&JV}ÇÊ Ê
ƒ„€GK>UÇI C1ÆEKrK $BGFKMpp!NCTLGp°}&CELGpJ!NOJ!DJVKSJV€KrpKM}&CELGW`Á•UTNLÉFKMpDGPOJ©C JV€GNO?p 7CE/F ~ 7KxDLGWGKMFpJVUTLGW
LGC €GC JV€GK›}MDGFF!K&Li&J 4D}&JVDGUÇJVNOCELGpNLSJV€K›LKMNRT€ICEFNLGRHÁKMp!CTp!}MCTBGN},}&NF!}&DGNOJRTNOÆEKÐFNp!K/J!C½NOLGKMPUTpJ!N}
UTLGWeKMPUTpJ!N}S}&CELqJVF!NOIGDJ!NCELGp„NOLvJV€GK}&DGF!FKMLqJ°NOLÉJV€GKr² ¨ ÀeWGK¶ÆN}MKÇÊ
K 4LGWrJV€4U3JICTJ!€rFNRE€qJ UTLW>PK #J}MDGFF!K&LiJ}MCTLiJ!F!NOIGDJVNOCELGp[€4UjÆTK›J!€GK7p!UTÁ3K hCTF!ÁlÊjƒ„€GKÐ}MDGFF!K&LiJ¬´
}MDGFF!K&Li;
J 4D}&JVDGUÇJVNOCELGp–C ©JV€GKeÁKMp!CTp!}MCTBGN}eWGK¶Æ$NO}MKlU KM}¶JJ!€GKeWGK&J!KM}¶JVCEF–}MDF!F!K&LqJvUÇJÅJ!€GKeKMLGK&F!RT
}MCEFF!K&p!BCELGWGNOLGR"JVC"J!€GKQJVCTJVUTPK&LGKMFRTÂC *J 7CrKMPK&}&J!F!CELGpK $NOJ!NLGRSNO
L CEF,K&LqJVKMFNLGER hFCE
Á JV€K„LGCEF!Á•UTP
PKMUTW+ʃ„€KMF!K hCEF!K ;KrBGF!C}MK&KMWeJVCÅJV€Krp!UTÁKS} €4UTLGREKC ÆÇUTF!NUTIGPOKMp„U hCEFQJV€KSp!NLREPK²°À>ÈDGL}&JVNOCEL+Ê
Ë CTF eV > 0 KMPUTpJ!N}}MDGFF!K&LiJ–}&CELqJVFNIGDJ!NCELpxNOL I→ UTF!K•BGF!K&p!K&LiJ–IDJÂJV€KvpVUTÁK }&CELqJVFNIGD$´
JVNOCELGp NOL I← ÆÇUTLNp!€ #JV€GKÕCEBGBCEp!N JVKÕNOp JVFDGK,hCEFJV€GKÕ}jUTpK:C eV < 0 Ê ³ €4UÇLGRENLRÆÇUTF!NUTIGPK&p
NL JV€KzNOLGKMPUTpJ!N}É}MCTLiJ!F!NOIGDJVNOCELGpUTLGW5WG%K GLGNLGR‹JV€$
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 ƒ„€G$
K 4FpJJVK&F!Á NL5»,‚ Ê ?$EÊ ?ÇÄ ÂWGKMp}MF!NOI K&p•J!€GKzK&PUTp¬JVNO}l}&CELqJVFNIGDJ!NCEL NL2JV€KÕ®,ƒ }MDGFF!KMLqJMÊ
®°P JV€GCEDRE€<;K7UTF!K,POKMppNOLiJ!KMFKMpJ!KMWxNLSJV€GNOp[}MCELqJ!F!NIDJVNOCEL)7K7}MUTLGLGCÇJNORELGCEFK,NOJNOLrBFVUT}¶JVN}&KÐI K&}jUTDGpK
NOJ;}MCELqJVFNIGD$JVKMpQJ!CÂJV€GK"J!CTJ UÇP ∆IP AT ÊGƒ„€GKSKMLqÆNFCELGÁKMLqJ©U KM}¶JVpQJV€Np„}MDF!F!K&LqJH}MCELqJVFNIGD$JVNCTL)IGD$J
UÇJÂJ!€GKÉKMLGW2C ©JV€KvJVDGLGLKMPNOLGR`BGFC$}&KMpp!KM%p ,JV€GK&F!KeNpÂLC:KMLGK&F!RT K $} €4UTLGREKeIK&*J 7KMK&L5JV€KÉWGK&ÆN}&K
UTLGW JV€KWGK¶JVK&}&JVCTFl}&NF}MDGN JjÊQƒ„€GKp!K&}MCELWˆJVK&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!Á•UTPqPKMUTWxJ!C°JV€GK7p!DGBKMF}MCELWGDG}&J!CEFÆNU„JV€K;‚iD4UTLqJVDÁŸWGCÇJ!HNOJV€xKMLGK&F!RT
K$} €4UTLREKTÊ4ƒ„€GKK&PKM}¶JVFCELGp„}jUÇLzUÇIGp!CEFIeKMLGK&F!RT hNLÉ}MUTp!KSJV€KMNF7JVCTJVUTP9K&LGKMFRTÉNp;p!Á•UTPPOKMF7J!€4UTLÉJV€K
p!DGBKMF}MCELWGDG}&J!CEF;} €KMÁN}jUÇP BCTJVK&LqJVNUÇP µS ÐCEF›KMÁNOJ›K&LGKMFRT [email protected] JV€GK&NFÐJVCTJVUTP KMLGK&F!RTNp7IGNRTREKMF,JV€GUTL
³ CCEB K&FB4UTNF
µS ¶Êƒ„€GKPUÇpJJVK&F!Á NL»,‚ Ê ?$Ê ?TÄ "WGKMp}MFNIKMpJV€GKNLqÆEK&F!pK•JVDLGLGKMPONLGRlBGFC$}&KMp!p&~ U
UTIGpCEF!IGp°KMLGK&F!RTC
 hF!CEÁ J!€GKÂLGK&NRE€iICEF!NOLGR•WK&ÆN}&K 9N JVp½}MCELGp¬JVN JVDGK&LiJ"KMPK&}&J!F!CELGp½JV€GK&L`JVDGLGLKMPJ!C JV€K
LGCEFÁ UTPPKjUÇW+Êq±µL•J!€GNp›K&ÆTKMLq!J iJV€GK©JVCTJVUTP4KMLKMF!Rǁ–C +JV€K½CED$JVRECENOLGRKMPOKM}&J!F!CELp7NOp›BCEpNOJVN ÆEKÇÊq± CTL•JV€K
}MCELqJ!FVUTF¬[JV€GNOpxJ!CTJ UTP,K&LGKMFRTNprLGKMRqU3JVNOÆTK[JV€KMLJ!€GK ³ CCEBKMF>BGUTNF€4UTpxKMÁNOJJVKMWKMLKMF!RǁJ!CzJV€K
WGK&ÆNO}MKTÊ
±µLÉCEF!WGK&F7J!CÂDGLGWGK&F!p¬J UTLGWe€GC JV€K"WGK&J!KM}¶JVCEFQ}&NF}MDGN JQU KM}¶JVp;J!€GK"IKM€4UjÆNCTF;C [J!€GK°}&DGF!FKMLqJ NOL
JV€GKSBF!KMpKMLG}&K>C JV€KrK&LqÆ$NOF!CELÁ•K&LiJ 7KNLqÆEK&pJVNORqUÇJ!K"JV€G<K 7KMNRT€i?J hDGL}&JVNOCELGp KNelDS (ω, eV, D , η)
UTLGW KNinelDS (Ω, eV, D , η) pKMB4UTF!UÇJVK&POTÊ[ƒ„€GK 7KMNORE€qJ hDGLG}&J!NCEL KNelDS (ω, eV, D , η) NOpBGPCTJJVK&WNOL
ËNRTDGF!K ?$Ê I–UTp½;
U hDGL}&JVNOCELeC hFKM‚iDGK&LG}& hCTFQ*J ;CÆTUÇPDGK&p©C JV€KxIGNUTpQÆECTPOJ UÇREKrUTLGWe*J ;CÆTUÇPDGK&p
C QJ!€GKvWGCTJxPK¶ÆEK&P;BCEp!N JVNCTL+Ê[ƒ„€GNp>KMPUÇpJVNO} ωÇKMFLGKMP7Np>p¬Á•ÁK&J!F!N}DGLGWKMFÅUÕINUTpÆECEP J UTRTK•FK&ÆEK&F!p!UTP
' KNelDS(−eV ) = −KNelDS (eV )(|ÊQË4F!CEÁ J!€GK`F!NORE€qJvB4UÇLGKMP½C SJV€GNOp 4RTDGF!K H€GK&F!'K 7K:}MCELGpNWGK&F
|eV | ≥ D 47K4LGW JV€4UÇJeJ!€GKMFKNOpeU pÁ UÇPP"p¬JVKMBˆUÇJ ω = ∆ − D UTLGWˆU p€4UTF!B BKjU U3J
ω = −∆ + D ʃ„€GKB KMU ‹NprUTp¬$ÁÁK&JVFN} UTLGWN JVp€GK&NRE€qJxNprÁÂDG} €PUTF!RTKMFSJV€4UÇLJV€GUÇJrC QJV€K
??
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!NORELUTLGW‹IKM}MCTÁ•K&prLKMRqUÇJ!NOÆTKTÊ[ƒ„€GK–ÆECEP J UTRTKÅIGNUTp
eV Á•UTNLPOeU KM}¶JVp½JV€GKÅUÇÁ•BGPONOJ!DGWGKC,JV€K>BKjUzUTLGW`C,JV€GKxpJVK&B:NL KNelDS Ê9ƒ„€KÂPK#J°B4UÇLGKMPC
ËNRTDGF!&
K ?$Ê IÕWGKMp}MFNIKMp KNelDS H€GKML |eV | < D ʃ„€GKvBKjU€GK&NRE€qJÂWKM}MFKjUTpKMpłiDGN JVK UTp¬JÅUTp>U
hDGLG}¶JVNCTLlC eV UTLGWÕN JVp„POC$}MUÇJVNOCELÉNp„p€[email protected] #J!KMWzUÇJ ω = −∆ + eV Ê4ƒ„€GKrB KMU vNpHp¬$ÁÁK&JVFN}EhCEFHU
PUTFREK"IGNUTpMÊ
K°J!DGF!LÉLC JVCxJV€GK"J!F!DGP •B€GCTJVC3´bUTpp!Np¬JVK&WÉBGF!C}MK&p!p!K&p H€GNO} €ÉNLqÆECTPOÆEKSK&NOJ!€GKMF;UTIGp!CTF!BJ!NCELvCTF
KMÁNpp!NCTL:C ;K&LGKMFRTEʃ„€G;
K ÇKMF!LKMP inel eV, D , η) NpSBPCTJJVKMW‹NOLËNREDGFK ?$Ê DlUÇpxU hDGLG}¶JVNOCEL
C hF!K&‚iDGKMLG}¶ Ω 7H€N} €É}MCTF!F!K&p!BCELGKWGpQNJ!DSC>(Ω,
JV€K°JVCTJVUTP9K&LGKMFRT•C *J ;CÅK&PKM}¶JVFCELGp FhCEF D > 0 ʱµLvJ!€GK
PK #J7BGUTLGKM9P eV NOpÐLGKMRqU3JVNOÆTK $UTLGW•NL–JV€GK©}MK&LiJ!KMFQBGUTLGKM9P eV NOp›BCEpNOJVN ÆEK©IGDJ eV < UTLGW 4L4UTPOPO
JV€GKxF!NORE€qJ"B4UTLKMPC ›ËNREDF!K ?$Ê D•WGKMp}MF!NOI K&p eV ≥ D Ê KGLGW`JV€4U3J H€GK&L eV < DD J!€GKMFKÂNp°U
pJ!KMBUÇJ Ω = D + eV Ê €GK&L 7KNOLG}MFKjUTpK eV }MPCTp!KÂJVC D 9J!€GK–pJ!KMBpJVNOPPWGCEÁNL4UÇJ!KMp KNinelDS
IGDJ©JV€GK&F!KxNp½U•p!Á•UTPOP BKjU eUÇJ Ω = −∆ + eV Ê €GKML eV ≥ JV€Np hNLqÆEK&FJVK&W ½BKjU eNOp©ÆTKMF
p!€4UÇF!B+Êiƒ„€GNOp;Np›K $BGPON}&NOJ›NL–JV€GK½F!NORE€qJ;B4UÇLGKMP|ʃ„€K°NLqÆTKMFJ!KMWvBKjU .DBH€GNO} €v€4UTp;UxPUÇF!REK©UTÁBGPN JVDGWK Á U TKMpÐN J,LGC WGN }MDGP J/JVCrCEIGpKMFÆTK©J!€GKHpJ!KMB+Êqƒ„€GK hNLqÆEK&FJVK&W ,B KMU >NOp,PC}jUÇJ!KMWUÇJ Ω = −∆ + D Ê
®°RqUÇN)L eV Á•CTpJ!POÕU 9K&}&J!p"JV€GK–UTÁBGPONOJVDWGKxC KNinelDS Ê[Ë4CEF D < 0 LGCÇJSp!€GC HL +JV€GKÅFKMpDGPOJNOp
p!NOÁ•NOPUTF"J!C`JV€4UÇJÂC D > 0 HN JV€ CEBB CEpNOJ!K eV IGDJÂJ!€GKÉUTÁBGPONOJVDWGK•C ©J!€GKvBKjU NpxWGCEDGIGPOKMW
}MCEÁB4UTFKMWÉJVCÅJ!€4UÇJHC D > 0 H€GKML |eV | ≥ |D | Ê
²½CTJ!K½JV€GUÇJ;DLGWGKMFpJVUTLGWGNOLGR>JV€K°IKM€4UjÆNCEF›C +J!€GK½*J 7C:7KMNORE€qJ hDGLG}¶JVNOCELGpQUÇp;
U hDLG}&J!NCEL•C +JV€K
[email protected] KMFKMLqJÅB4UÇFVUTÁK&J!KMF!p eV UTLW D xC HJV€GKÉWK&JVK&}&J!CEFÅNpÂ}&F!DG}&NUTP|ʱ JÅUTPPOC HpxDGpÂJVC:}&CELqJVFCEP;JV€K
%K KM}¶J°C JV€GKrWGK¶Æ$NO}MKÆECEP J UTRTKrINUTp CELeJV€GKrWG}r}MDGFF!KMLqJ°C J!€GKxWK&JVK&}&J!CEF UTLGWzNOJHNOp©J!€GKMF%K hCEFK
JV€GK ÇK& hCEF„K $J!FVUT}¶JVNOLGRÅJV€GKLGCENOp!EK heVFCEdÁ J!€GKÁ•KMUTp!DGFKMÁKMLqJ©C JV€GNOp„WG}x}&DGF!FKMLqJjÊ
, ' .: .# ! # 23!, #.!" :#! 7
KLGC¡}MUTP}&DGPUÇJ!K ∆I hF!CEÁ »,‚ Ê $? Ê T? Ä HNOJ!€J!€GK•pB K&}&J!FVUTP›WGKMLp!NOJµC;K$}MK&p!pÂLCENpK•CQU
‚iD4UTLqJVDGÁ BCENLqJ½}MCELqJ UÇ}&J! RENOÆTKML`IqÕ»,‚Ê ? Ê Hq¿ ¶Ê6Kx}MCELGpNWGK&F½J!€GKÅ ®,ƒ }&DGF!FKMLqJUTp°UhDGLG}¶JVNCTL
C9J!€GK©WGK&J!KM}&J!CEFÐÆECEP J UTREK eV hCEFÐp!K¶ÆEKMF!UTP ÆTUÇPDGK&p7C JV€GK©WGK¶Æ$NO}MK©ÆECEP J UTREK eV BH€N} €vUÇF!K©p!€GCHLvNOL
ËNRTDGF!K,$? Ê G1ÄÊBK GLGWvJV€4U3JQJV€GK&F!KSUÇF!K°J*7CÂÆÇUTPDGK&pQC eV UÇJ4H€GNO} € ∆I } €4UTLREKMp„WGF!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
ƒ„€GK hLGKMRqU3JVNOÆTK [WGKMFNOÆÇUÇJ!NOÆEK›UÇJ = −D pKMKMÁpJ!C°WGN ÆEK&F!REKÇÊ3ƒ„€K;€GK&NRE€qJ C 4ICTJV€JV€K;BKjU TU LGW:JV€Kp¬JVK&BNOLG}MFKjUTpKMpNLUÉÁ•CTLGCTeV
JVCTLGCEDGp"Á•UTLGLKMFUTprUhDGLG}¶JVNCTL:C7J!€GK–FVUÇJ!NCÉC›JV€KÅWGK&ÆN}&K
ÆECEP J UTRTK eVd WGNOÆNOWGKMWIq JV€GKzWGCTJ PK&ÆTKMP D Ê3 €GKML eVd Npp!Á•UTPP9/JV€GKzB KMU NpÁÂDG} € €GNORE€GKMF
JV€4UÇLJV€GK•pJ!KMB+ʱµLG}MFKjUTpNLGR eVd J!€GKvBKjU hDGF¬JV€GK&F>NOLG}MFKjUTpKMp IGDJ>J!€GK p¬JVK&B €GKMNORE€qJÂNL}MF!KMUTp!K&p
UTpJ!KM%F 9pJVUTFJ!NLGR hF!CEÁ JV€K>JV€GFKMp€GCEPW:WK&ÆN}&K>ÆECEP J UTRTK eVd = ∆ − D Ê9±µL‹ËNOREDGFK ?$Ê G1Ä 6;K GLGW
JV€4U3EJ hCEF D = 0.4 UTLGW eVd = 0.8 JV€GK–BKjU z€GKMNORE€qJNpSpJ!NPOP€GNORE€GKMF½JV€4UTL:J!€GK–pJVK&)B [IGD<J HN JV€
Ô
D = 0.6 UÇLGW eVd = 0.6 4JV€K>pJ!KMB`IKM}&CEÁKMp°€NRE€GK&F©J!€4UTLlJV€GKxB KMU Ê KMFK hCTF©pB K&}MN 4}MN Jµ6;K
CELGP É}MCELp!NWKMF½J!€GK>}MUTp!(
K H€GKMFK > 0 IGD$J°FKMp!DPOJV>p hCEF D < 0 }jUÇL`I KrCEIJVUTNLKMWÕNOLÕUp!NOÁ•NOPUTF
Á UÇLGLGKM%F GK $BGPOCENOJ!NLGR>KMPK&}&J!F!CELl€GCEDPOKSpÁ•ÁK&J!FTÊ
±µLCTF!WGK&FJVEC hDGF¬JV€GK&F,DGLGWGK&F!p¬J UTLGW–JV€K„I K&€4UjÆ$NOCEF/C ∆IP AT 7KH}MCELGpNWGK&F/JV€GK;[email protected] KMFKMLqJÐ}MCELqJ!F!N ´
IGDJ!NCELGp C GJ!€GNpBG€GCTJ!CÇ´bUÇp!p!NOpJ!KMW>}&DGF!FKMLq!J H€GN} €ÂUTF!K7p!€GC HLÅNL>ËNRTDGF!K ?$Ê GGEÊÇÀ$BKM}MN 4}jUÇPPO !;K;BGPCTJ
JV€GK©KMPUTpJ!N}©}MDGFF!K&LiJ;F!K&LGCEF!Á•UTPON MK&W•Iq–JV€K½K&LiÆNOF!CELGÁKMLqJ UT=p ;K&PPUTp›JV€GK©F!NORE€qJQUTLW PK #J;NOLGKMPUTpJ!N}
}MDGFF!K&LiJ!pMÊ K;4LGWJ!€4UÇJrJV€GK•KMPUTpJ!N}–B4UTF¬J>NOpxp¬$ÁÁK&JVFN}–IK&*J 7KMKML eV B CTp!NOJ!NOÆTK•UÇLGWLGK&RqUÇJVN ÆEKÇÊ
± JÂNOpÂUTPOÁ•CEp¬J>K&‚iD4UTP7J!C MK&F!'
C H€GK&L
ʱ JÅp!€C Hp–UzpJVK&B2UÇJ
Ê ƒ„€GNpÂ}MUTL IK
DGLGWGK&F!p¬JVCC$W hF!CTÁŸJV€4K UT}&J,J!€4UÇJÐUÇJ,J!|eV
€GKQJV| €<F!KMp€GDCEPOW eV = D EKMPOKM}&J!F!CELp/|eV
JVDGLG| L=KMP hF!DCTÁŸJV€K„LGCEF!Á•UTP
Á•K¶J UTPPKMUTWvJVCÂJ!€GK"pDGB K&F!}&CELGWGDG}¶JVCEFHBGFKMWGCTÁ•NOL4UTLqJVP •Iq Á U iNOLGRÂF!K&p!CELGUTLqJ„JVFVUÇLGp!N JVNCTLGp7J!€GF!CEDRE€
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 4POJ!KMFNLGRrWGK&ÆNO}MK BHNOJ!€
JV€GK–UÇNÁ C 7UT} €GNK¶ÆNLGReUÉp!K&PK&}&JVNOCEL:IK&*J 7KMKMLBG€GCTJ!CEL‹KMÁNpp!NCTL:UTLGWUTIGpCEF!B$JVNCTL:BGF!C}MK&p!p!K&pMÊ K
WGKM}&NWGK&W J!CFKMp¬JVF!NO}&JCEDGFp!K&POÆEK&p•J!CJ!€GKeBG€GCTJ!CÇ´bUTpp!NOpJVK&W ®½LGWGF!K&K&Æ hp!DGIGREUTB ÅF!KMRTNÁK ,UTpp!DGÁNLR
JV€4U3JxJ!€GK WGCÇJ>POK&ÆEK&P7NO(
p 7KMPOP HNOJV€NLJ!€GKRqUTB+ÊÎ7‹}MCTÁ•BGD$JVNLRzJV€KJ!CTJ UTPÐK }&KMppÅBG€GCTJ!CÇ´bUTpp!NOpJVK&W
}MDGFF!K&LiJ,UTp 7KMPOPUTpNOJVp WGN 9K&F!KMLqJ,}&CELqJVFNIGDJ!NCELp hCEFUTIGpCEF!BJ!NCEL>UTLGWÂK&Á•NOp!pNCE)L 3UTLWÂF!NORE€qJ,UTLGW>P%K #J
}MDGFF!K&LiJ!p ;K=hCEDGLGW>J!€4UÇJ hCEFW};IGNUTp ÆTCEPOJVUTREK&p}&CEÁ•BGUTFVUTIPK,JVC eV = −D N JNOpBCEpp!NIPKÐJVC°Á•U ÇK
p!DG} € U`WGNOpJ!NLG}¶JVNCTL+ʃ„€Ke² ¨ ÀWGK¶JVKM}¶JVNOCEL p!K¶JVDGB2}MCTDGPWJ!€GKMFK%hCEFKvBGF!C1ÆNWKÉÁ•CEFK NOLhCEF!Á•UÇJ!NCEL
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I K&KMLNLqÆEK&pJ!NRqUÇJ!KMW:JV€KMCEFK&JVNO}jUTPOPO ' DGA (›UTLGW‹K BKMFNÁKMLqJ UÇPPOzF!K&}MK&LiJ!PO ' Dq¿ (|Ê+±µLp!DG} 0
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NLJV€GK"}&DGF!FKMLqJQÆTCEPOJVUTREK©} €4UTFVUÇ}&JVK&F!NOpJVNO}Mp&ʱµLɯH%K bÊ ' Dq¿ (8JV€GK½‚iD4UTLqJVDGÁ WGCTJ;}MCELGpNp¬JVKMWÉC U>}jUÇF!ICEL
L4UTLGCÇJVDGIK"Á•U iNLGRrJV€GK7ÈDGL}&JVNOCELvIK&*J 7KMK&LÕUÂLGCTF!Á•UTP Á•K¶J UTP PKjUÇWÉUTLGWeUÂpDGB K&F!}&CELGWGDG}¶JVCEF&Ê Ô K&F!K ;KrWGNOWlLCTJ©}MCELGpNWGK&F„JV€GKr‚iD4UTLqJVDGÁ WCTJ©NLÉJV€GK Ó CTLGWGCF!KMRTNÁK4UTLWC7KxNOLG}MPODGWGK&WzNOLiJ!KMF!UT}&J!NCELGp
CELÉJV€GKWGCÇJHNLÉJV€GK ³ CEDPCEÁÂIeIGPOC$} 3UTWKSF!KMRTNÁKTÊ
® }&KMLqJVF!UTP7BCENOLiJrC ;JV€GNOprp¬JVDGW$NOpJ!€G
K UÇ}&JrJV€4UÇJxUTPOP,}MCELqJ!F!NIDJVNOCELGpJVCzJ!€GKBG€GCTJ!CÇ´bUTpp!NOpJVK&W
¨
}MDGFF!K&LiJhCEF„ICTJV€eJV€GKr²°ÀeUÇLGWÕ² ÀÉp!K&J!DGBGp%}MUTLlI K}jUÇpJ©NLÉJV€Krp!UTÁKEhCEF!ÁC
Z
|Z(Ω)|2 +
Sexcess(±Ω)Kprocess (Ω, eV, · · · ) .
∆IP AT (eV ) ∝ dΩ
Ω2
?ÊE?FG G€ KMFK Kprocess NpxU,TK&F!LGK&P H€GNO} €WGKMBKMLWGpÂCELJV€GK•L4UÇJVDF!K KMPUTpJ!N}CEFrNLGK&PUTp¬JVN}SUTp(7KMPOP;UÇp
JV€GKlÁKM} €4UTLNp!Á hp!NOLGREPKe‚iD4UTpN ´ B4UTFJ!N}&PKÉCEF–B4UTNOFÅJVDGLGLKMPNOLGR ÂC½JV€GKl} €4UTFREKzJ!FVUTLGphKMFBGF!C}MK&p!p&Ê
€GKMLzWGKjUÇPNL;
R HN JV€zUTLÕK&PUTp¬JVNO}SBGF!C}MK&p!p67K>DGLGWKMF!p¬J UTLWzJV€GUÇJ½J!€GKrKMLqÆNF!CTLGÁ•K&LqJ°FKMLGCEFÁ UÇPN &KMp
JV€GKÅW}}&DGF!FKMLqJrK&ÆTKML H€KMLLGCeBG€GCTJ!CEL‹NpSK $} €4UÇLGREKMWI K¶*J ;K&KMLJV€GKÅ*J 7Cz}&NF!}&DGNOJ!pMÊ ±µLJ!€GKÅ}jUTpK
C HNLKMPUÇpJVNO}J!DGLGLGK&PNOLGRÕCELPO JV€K hF!K&‚DKMLG}¶ Ω }MCEFF!KMpB CTLGWGp>J!C`JV€GK J!CTJ UÇP7K&LGKMFRTC „JV€GK *J 7C
KMPOKM}&J!F!CELp H€GN} €ÉK&LqJVKMF hK$NOJ ÐJ!€GK"p!DB K&F!}MCTLGWGDG}¶JVCEF hF!CTÁ #JVC ,JV€GKSLCEF!Á•UTP Á•K¶J UTP PKMUTW+Ê$ËNL4UÇPPO JV€GK½p!NORELvC [J!€GK hF!K&‚iDGKMLG}¶ |UÇLGWvJV€iDGp;J!€GK"ICEDGLGWÉC [JV€GK"NOLqJVKMRTFVUTP NLv»Ð‚ Ê ?$Ê ?7G H€GNO} €eUTF!K°P%K #J
bIGPUTL ŀKMF!K rWGK&}MNOWGKM(p H€GK&J!€GKMF>UzREN ÆEKML}MCELqJ!F!NIDJVNOCEL}&CEF!FKMpB CELWGpxJ!ClJ!€GKvUTIp!CEFBJVNOCELCTFJ!C
JV€GKK&Á•NOp!p!NOCELÉC UÅBG€GCTJ!CEC
L hFCEÁ J!€GKÁ•K&p!CEp}MCEBN}S}MNOF!}&DGNOJMÊ
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I K&}jUTDp!KeJV€GKep!BKM}¶JVF!UTP„WGKMLp!NOJµ2C °K }&KMpp LGCENOp!KÉNO;
p 7KMPOPQ} €4UTF!UT}&J!KMFN MK&WUTLW5IKM}MUTDGp!KlNOJ€4UTpU
p!NOÁ•BGPO:
K hCEFÁzÊ+± J 7CEDGPOWIK–DGp!K hDGP›J!CÉJVKMp¬JxJ!€GKBGF!K&p!KMLqJxÁ•CWGK&P/JVCep!N JVD4UÇJ!NCELEp H€GKMFK•JV€KLCENpK
p!BKM}¶JVF!DÁ K$€GNOIGNOJ!p„}MDGpBGpHCEFQpNLGREDPUTFNOJVNOKMp&Ê ³ Dp!BGp½UTF!K LCHLeJVCÅC}M}&DGF„NLvJ!€GKS€GNORE€hF!K&‚DKMLG}¶
}&PCEpK;JVC"J!€GK„RqUTB LGCENpKQC LGCEFÁ UÇPp!DB K&F!}MCTLGWGDG}¶JVNLRQÈDLG}&J!NCELGp&ÊiÀ$NOLGREDGPUTF!N JVNK&pNLÂJV€K„LGCENpK„UTFK
iLGC HLvJVC>C}M}&DGF;NOL } €GNOFVUTPÍ[DJJVNLREKMF›PNO‚DN)W EJ!KMpJ!KMWÉNOL•JV€K°}MCTLiJ!K J„C 9JV€G>K hFVUT}¶JVNCTL4UTP ‚iD4UTLqJVDGÁ
Ô UTPOPÐK% KM}&J ' DI ( Ê/À$DG} €2p!NOLGREDGPUTF!N JVNK&pCEF>}&DGp!BGp–p€GCEDGPOW I KvKMUTpJVC:FKM}&CERELGNMKvNOLCEDGFÂBF!CEBCEp!K&W
Á•KMUTp!DF!KMÁKMLqJHC JV€KrB€GCTJVC3´bUTpp!Np¬JVK&Wz}&DGF!FKMLqJjÊ
ÌSL‹REK&LGKMF!UTP/REFCEDGLGWp 7K–€4UjÆEKBGF!CEBCEpKMWUÉÁ•K&} €4UTLGNOp!Á H€N} €‹}MCEDBGPK&pUeLCEF!Á•UTPÁK&JVUTP *
p!DGBKMF}MCELWGDG}&J!CEF;}&NF}MDGN J›J!CÂUrÁ•K&p!CEp}MCEBGNO}½WK&ÆN}&"K HNOJ!€•J!€GK°RECEUTP4C [DGLWGKMFpJ UÇLGWGNLR>JV€GK©LGCENOp!K½C JV€GK›PU3J!JVK&FMÊ1ƒ„€GK7BGF!K&p!K&LiJp!K¶JVDGBÅpDGREREK&pJVp JV€4U3JN JNOpBGPUTDp!NIPK/JVC½K JVF!UT}&JNL hCTF!Á•UÇJVNOCELSUTICEDJ€GNRT€
hF!K&‚DKMLG}¶2LGCENpKTÊ Ô NRE€ hF!K&‚iDGKMLG}¶ LGCENpKeWGK&J!KM}&J!NCEL LGC }&CELGp¬JVNOJ!DJVK&p UÇL5NOÁ•BCEFJVUTLqJÂpDGI4K&PW
C HL4UTLCEp!}&CEBGN}ÉCTFÂÁ•K&p!CEp}MCEBN}vBG€qp!NO}Mp&ʛ­zKjUTpDGF!K&Á•K&LqJ–p!K&J!DGB p!} €GK&Á•K&p H€GN} €2}jUÇL IKvBGPUT}MK&W
bCEL} €GNB ©LK JQJVCxJV€GK½}MNF}MDGN J7J!CÂIK°ÁKjUTpDGF!K&WeUTF!K½DGp!K hDGP hCEF;(U hDGF¬JV€GK&FQDGLGWKMF!p¬J UTLWGNLGR>C €GNRT€
hF!K&‚DKMLG}¶e}MDGFF!KMLqJ½Á•CTÁ•K&LiJ!pMÊ
, ! ±µLÉJV€GNOp½UÇBGB K&LGWGN 7;Kr}MCELp!NWKMF„JV€GKrUjÆEK&FVUTRTKrCK BCELGK&LqJVp°CJ!€GKBG€[email protected] KMFKMLqJHJVNÁKTÊ
Ë CTPPC HNOLGRJ!€GKÕJ!€GKMCEFKMÁ NL ‚iD4UTLqJVDÁ ÁKM} €4UTLN},hCEF•CEBKMFVU3JVCEFp;N C ≡ [A, B] pVUÇJ!Np4KMp
[A, C] = [B, C] = 0 JV€KML
?$Ê ?EÃ eA eB = eA+B eC/2 ,
AEÄ
[email protected]
A
Np„PONLGKMUTFQNOLzÎ7CEp!K}MFKjUÇJ!NCELlUTLWÕUTLLGN€GNOPUÇJ!NCELCEBKMF!UÇJVCEFp7K€4UjÆEK
Ë4F!CEÁ J!€GKMpKxJ*7CvBGFCEB K&FJ!NKMp% [email protected]
CEIJVUTNL
heA i = ehA
2 i/2
NOp°U›´µLiDGÁÂIKMFHJV€GK&L
?$Ê ?H .
C = hCi = hABi − hBAi
A2 + B 2
A B
he e i = exp AB +
.
2
C
67K>KMUTp!NOPO
?$Ê ? À$NÁNPUTFhCEFHU 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, 4›NOp©UTPpCPONLGKMUTFQNOLzÎ7CEp!K}MFKjUÇJ!NCELlUTLWÕUTLLGN€GNOPUÇJ!NCEL•CEBKMFVU3JVCEFp
;K}MUTLzBKMFhCEF!Á¤JV€GKrUjÆEK&FVUTRTKxC JV€KrK $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 ÉJV€GK&p!K>FKMpDGPOJ!p½JVC•CEDGF©}jUTPO}MDGPUÇJVNOCELGp4HN JV€zJ!€GK>B€4UTp!K φ(t) = e R t dt0V (t0) V (t)
ÆECEP J UTRTK,CJ!€GKÐÁ•K&p!CEp}MCEBGNO}ÐWGK&ÆN}&K=H€KMF!K37K=7CEDGPOWxPON TK/J!C½ÁKjUÇp!DGFK›LCENpKTÊ1± GL−∞
CENpK›NOp¸rUTDp!p!NUTL)
Ê K"UÇPp!(
C 4LGW J!€4UÇJ [φ(t), φ(t0)] = 0 Êiƒ„€GKMpKSBGF!CTB K&FJVNOKMp›C φ(t) pVUÇJ!Np #
φ(t) NOp7PONLGKMUTFÐNLI CTp!CELGp&F
JV€GK}&CELGWGN JVNCTLGp;JVCUTBBGPOv»Ð‚ip&Ê ?$Ê ? QUTLGW ?EÊ ?GA ¶ÊG±µLGWKMKMW hφ2(t)i = hφ2(0)i 7;Kr€4UjÆEK
?$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 )] ,
H€GKMFK<;KrF!K&}jUTPOP+€GKMFK
, ! J(t) = h[φ(t) − φ(0)]φ(0)i Ê
±µLJ!€GNprUTBB K&LGWGN )7K•WK%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»Ð‚ip&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
, ! ±µLÉJV€GNOp½UÇBGB K&LGWGN 7;KrF!K&}jUTPOP9JV€GKWGK4LGN JVNCTLÉC Ó K&PWp!€`¸FKMK&L)1 p hDGL}&JVNOCELGp' DD ( Ê
ËNFpJ! 7KÉWGK%GLGK JV€KÉUTLGCEÁ•UTPOCEDGpx¸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 JV€GKÉPOC;K&F
IGFVUÇLG} €),UTLW t0 > t JV€GK&L Fσ (q, t+ − t0+ ) = Fσ∗(q, t− − t0−) = pREL (σ)uq vq e−iE (t−t ) ʃ„€GK&p!K
¸FKMK&)L 1 >p hDGLG}¶JVNCTLGp°K&LiJ!KMF©JV€GKÂWKMp!}&F!NOBJVNOCELÕCJV€GKx®°LGWGFKMK¶Æ`BGFC$}&KMp!p&Ê9± 7KÂ}&CELGp!NOWGKMF©JV€GKxp!NOLGREPK
‚iD4UTp!NOB4UTF¬JVN}&PK>J!DGLGLGK&PNOLGRÉNL:JV€KpDGBKMF!}&CELGWGD}&JVCTF ;KDGpK–J!€GK–}MCELqÆTKMLqJVNOCEL4UTPÐWGK4LGNOJ!NCEL:C;JV€K
¸FKMK&)L 1 p hDGL}&JVNOCELlUTp hCEF„LGCEFÁ UÇP9ÁK&J UÇPpMÊ
À$KM}&CELGWGP 7KrWK%4LGKJV€K>¸SF!K&KML)1 p?hDGLG}&J!NCELeC JV€KrCTLGKSPK¶ÆEKMP = ¨
0
q
?Ê Aqà GDσ (t − t0 ) ≡ hTK cDσ (t)c†Dσ (t0 )i .
À$NÁ[email protected] 4}MUÇJVNOCELGpC$}&}MDGFI K&}jUTDGpK„JV€GK„‚iD4UTLqJ!DGÁ¦WGCTJ/€4UTpÐUSp!NOLGREPOrC}M}&DGBGNK&W–PK&ÆTKMPHN JV€ÅK&LGKMFRT Ê
„ƒ €GK?4F!p¬JQKMPOKM}¶JVF!CTL Np,J!FVUTLGphKMFF!KMW•JVCxJ!€GK©PKjUÇW•IK%hCEFK©J!€GK°pKM}&CELGWv€GCTBGp;CEL–JV€K°‚iD4UTLqJVDÁ¤WCTJ;DpC
JV€4U3JQNL CEDGF 7CEF/.7;KSCELPO}MCELp!NWKMF;J!€GK = ¨ ¸FKMK&L)1 p4hDLG}&J!NCEL H€KMF!KSICTJ!€ JVNOÁ•K©‚iD4UTLqJVN JVNK&p t
UTLGW t0 UÇF!K"NOL JV€GKSDBGB K&FQCEFQPOC ;K&FQIGFVUÇLG} )€ 4UTLWvJV€GK¸SF!K&KML)1 p4hDGLG}¶JVNOCELvÆÇUTPDGK&pQCELGP ;H€GKML t > t0
@N t t0 NLÉJV€GKrDGBGBKMFHIGF!UTLG} € GtDσ (t − t0) = e−i (t−t ) UTLW t0 > t N t t0 NLeJV€KrPOC ;K&F©IFVUTLG} € JV€GK&L Gt̃Dσ (t − t0 ) = e−i (t−t ) Ê
ƒ„€GKx¸FKMKML 1 ?p hDGLG}¶JVNCTLlNOLÉJV€GKSLGCTF!Á•UTP+ÁK&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 H€GKMFK>*J ;CÉK&PK&}&JVFCELGp°J!DGLGLGK&PNOLG
R hFCEÁ CEF½J!C JV€GK>LGCEF!Á•UTP
Á•K¶J UTP POKjUT)W ipC>JV€4U3J 7K"CELGP –}MCELGpNWGK&FQLGCEFÁ UÇP4Á•K¶J UTP9¸FKMK&)L 1 =p hDGLG}¶JVNCTLGp H€GK&F!K t UÇLGW t0 UTFK°NOL
JV€GK©WGN 9K&F!KMLqJ7IGFVUÇLG} €GKMp&ÊGË4CEFÐJV€GK©}jUÇp!K°C [KMPOKM}&J!F!CELp7JVDLGLGKMPONLGR hF!CTÁ¦JV€GK©p!DGBKMF}MCELWGDG}&J!CEF;J!CxJV€K
LGCEFÁ UTPÁ•K¶J UT9P );K•DGp!K–JV€GK•REF!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 JV€GK„LCEF!Á•UTPÁK&JVUTP$J!CSJV€GKHp!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± LvJV€GNOpQB4UTF¬J!F;KSBGFKMpKMLqJHJV€GKSWGK&LGCEÁNL4U3JVCEF›BGF!CWGDG}¶JVp4H€GNO} €lUTBGBKjUTFQNOL JV€K"JVDGLGLKMPNOLGRÂ}&DGF!FKMLqJ
JV€GFCEDGRE€ JV€GK`²"À:ÈDGL}&JVNOCEL UTp U ³ CCEBKMF BGUTNF&Ê;ÀDG} € WGKMLCEÁ•NOL4UÇJ!CEF!p}MCEÁK$hFCEÁ JV€GKzKMLGK&F!RT
WGKMLCEÁ•NOL4UÇJ!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
AqÃ
?$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
∞
JV€GK&L)7KrWK%4LGKJV€K 7KMNORE€qJ?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 , ! ±µLJV€GNOpvUTBB K&LGWGN =;K:}&CEÁ•BDJVKÕJ!€GK`BGFC$WDG}&JÉCSJVDGLGLKMPNOLGR Ô UTÁ•NOPOJ!CELGNUÇL CTB K&FVUÇJ!CEF!pvNOL JV€K
NOLGNOJ!NUTP ,RTF!CEDGLWzp¬J UÇJ!KH€GNO} €zNOp„p!€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 NÁBGPN 4}jU3JVNCTLGpC}M}&DGFÐI K&}jUTDp!K„JV€GKH‚iD4UTLqJ!DGÁ¡WGCTJЀ4UÇp7USpNLGREP >C}M}&DGBGNOKMWPK&ÆTKMPHN JV€–KMLGK&F!RT
D Ê4®½p½NOLz¯HK%bÊ ' DEÄ (94JV€GK<4FpJ½KMPOKM}&J!F!CELlNp„J!FVUTLGphKMFF!KMWlJVC–JV€GKPOKjUTWlI KhCEF!KSJ!€GKxpKM}&CELGWՀCEBGp©CEL
JV€GK‚iD4UTLqJ!DGÁ WGCTJMÊ4ƒ„€GKMF%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!NOIGNLR–J!€GK®°LGWF!KMK¶ÆeBGF!C}MK&p!p%67KxUTp!pDGÁ•K
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°JV€GKÅ}&CEF!FKMPUÇJVNOCEL`C›CEBKMF!UÇJVCEFp"NL:LCEF!Á•UTPÁK&JVUTPPOKjUTW9DGpNLGRÉJV€K
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!LGNOLGRvJV€KÅBG€4UTpK 4DG}&J!D4UÇJ!NCELGp% J!€GK:hCEDGFb´µBCENLqJ°}MCEFF!KMPUÇJVCTF6H€GN} €:NOp"NÁBGPNO}MN JHNL`JV€K
K$BGFKMp!pNCELCÐJV€KÂJVDGLGLKMPNOLGRÉ}MDF!F!K&LqJxNOp HF!N J!J!KMLUTpSUvJVNOÁ•KxCEF!WKMF!K&WBF!CWGDG}&JMÊÌ"LG}MKÅCTF!WGK&F!KMW
JV€GKÅBF!CWGDG}&JrC ›JV€KK $B CELKMLqJVNUTP,RENOÆTKMpSJV€KK $B CELKMLqJVNUTP/C›JV€GK–pDGÁ C;UTPPB4UTNFNLGRTp"I K¶J*;K&KML
BG€4UTpKÅCEBKMFVU3JVCEFpSUTp"p€GC 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 JV€GNOpÅUTBGBKMLGWN 7KlBF!KMpKMLqJ–JV€GKÉWKMLGCEÁNLGUÇJVCEFxBGF!CWGDG}¶JVp H€GNO} € UÇBGB KMUTFÂNOLJV€GKvJ!DGLGLGK&PNLR
M} DGFF!K&LiJHJ!€GF!CEDRE€lJ!€GKr² ¨ ÀxÈDGLG}¶JVNOCEL+Ê
D 0 NOpQJV€GKCEFNRENOL4UTPWGKMLCEÁ•NOL4UÇJ!CEF H€GNO} €zNOp„LGCTJ©U KM}¶JVKMWlIqÉJV€GKK&LiÆNOF!CELGÁKMLqJ
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 ¿ ? NOpQJV€GKWGK&LGCEÁNL4U3JVCEFQBF!CWGDG}&J½U KM}¶JVK&WzIqÉJV€KrK&LqÆ$NOF!CELÁ•K&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 H€GKMFK Dinel NOprJ!€GK WKMLGCEÁNLGUÇJVCEFSBF!CWGDG}&JÂUÇJ!J!F!NOIGDJVK&WJ!CzJV€K NLKMPUÇpJVNO}–}MDGFF!KMLqJ |U KM}¶JVKMWIi
KMLqÆNFCELGÁKMLqJQUTLWzNOp„WGK%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 )
CEF„KMPOp!K
Π(x1 , x2 , η) =
ƒ„€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} €GUTLGREKrJV€GKxL4UTÁKrCÆÇUTF!NUTIGPOK
0 J!€GKMLÕ} €4UTLGREKÆÇUTF!NUTIGPOK
0
;K;HNPOPCEI$J UTNOL‹J!€GK–pVUTÁK hCEF!Á>DGPU hCEFICTJV€‹}MUTp!K&p eV >→0 UÇLGW eV < 0 Ê À$NOLG}MK UTLGW =0 −
UTFK
NLGWKMBKMLGWGK&LqJVPOÂKM‚iDGNOÆÇUTPOKMLqJ!TNOJÐNp,K¶ÆNWGK&LiJÐJ!€4UÇJ Ψinel
Ê Ô K&F!KjU #JVKMF%
0
inel
0
→ (, , D , η) = Ψ← (, , D , η)
;KLGK&REPK&}&JHJV€K ← CEF → NLWGK eNOLÉJV€GK&p!<K hDGLG}¶JVNCTLGpMÊ
± eV > 0 GJV€GKrKMPUTpJ!N}S}&DGF!FKMLqJ"}&CELqJVF!NOIGDJ!NCELGpHNOL I→ K $Np¬J½IDJ½J!€GKrKMPUÇpJVNO}S}MDGFF!K&LiJ°}MCELqJ!F!N ´
IGDJ!NCELGp„NOL I← ÆTUÇLGNp€ hNLe}MCTLiJ!FVUTp¬JHJVCÅJV€GK}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
„ƒ €GK Ô UTPOPGK% KM}¶J!WGNp}MC1ÆEK&F!KMW Ii»,WHNOL Ô UTPP4NOLCG!Iq¿Dq€4UÇBGB K&LGp3H€GKMLÉUÇL•K&PKM}¶JVFN}H}MDGFF!K&LiJ 4CHp
JV€GFCEDGRE€eUÂ}&CELGWGD}&JVNOLGR–BGPU3JVK°NOLeUÅÁ UTRTLGK&J!N}"GKMPWÉBKMFB K&LGWGNO}MDGPUTF;JVCÂJ!€GKSBGPUTLGKTÊ$ƒ„€GKSÁ•UTRELGK¶JVN}
4KMPOW‹K $KMF¬JVpÂUlÍ[CEF!K&LqJ ;hCEF!}&KCTLJ!€GK–Á•C1ÆNLGRe} €4UÇF!REK}jUTFF!NOKMF!p<H€GNO} €J!KMLGWGpJVCeBGDp!€J!€GKMÁ J!C
CELGK½p!NWK"C [JV€K°}MCTLGWGDG}¶JVCEF&Ê®¼IGDNPWDGB C } €GUTF!REK"UÇJ;J!€GK"p!NOWGKMp;C [JV€GK°}MCELGWDG}&J!CEF!p HNOPP I4UTPUTLG}&K
JV€GNOp Á•UTRELGK¶JVNO}zNOL 4DGK&LG}MK ;BGFC$WGD}MNLR UÁKjUTpDGFVUTIPKzÆTCEPOJVUTREK VH I K¶*J ;K&KML *J ;C2pNWGK&pvC SJV€K
}MCELWGDG}&J!CEF hp!K&KvËNOREDGF!K AÊ G Ê ƒ„€K Ô UTPOP,F!K&p!Np¬JVN Æ$N Jµ‹NprBGF!CTB CEF¬JVNOCEL4UTPJ!CÕJV€K UTÁBGPONOJVDWGK–CQJV€K
Á UÇRELGK&J!N} GKMPW9Ê
!"- 23 "! ' 1G ÄEÄ (&hCEDGLGWÕJ!€4UÇJ°UÇJHJVK&Á•BKMF!UÇJVDGFKMp½C,CELGP eU hK Ó KMPOÆNOL
±µL GDIEÄ Ó PUTDp©ÆTCEL Ó PN J M NOLGR
UTLGW€NRE€Á•UTRELK&JVNO}CGKMPW 9H´G1ă KMp!PU ›JV€GK Ô UTPOP½FKMpNpJVUTLG}&K`WGNW LGCTJ ÆÇUTF¬ PONLGKMUTF!P  HNOJ!€
JV€GK 4KMPOW+Ê ± J>ÆÇUTF!NOKMWNLUzpJVK&BHNp!K UTp!€GNOCEL hp!K&KÉËNRTDGF!K A Ê Ã ¶Ê± J:;UTpÅUÇPp!CChCEDGLGWJV€4UÇJH€GKMFK
4 JV€GK Ô UTPP,FKMp!NOpJVUTLG}MK;;UTp(4UÇJ![JV€KPOCELGREN JVDGWNL4UTPF!K&p!Np¬J UTL}MKWGNp!UTBGBKjUTFKMW+ʃ„€GK;4KMPOWU3JH€GNO} €
JV€GK°BGPUÇJ!KjUTDpQUTBGBKjUÇF!KMWCEF=H€GKMFK"JV€GK"POCELGREN JVDGWNL4UTP4FKMpNpJVUTLG}&K°ÆÇUTLGNp€GKMW7;UTp„NLGWKMBKMLGWGK&LqJHC
JV€GK•Á U3JVKMFNUT9P +JVK&Á•BKMF!UÇJVDGFK CEF>CÇJV€GK&F>ÆÇUTF!NUTIGPK&pxC „JV€GKvK $B K&F!NÁKMLqJ IGD$JÅCELGP WKMBKMLGWGK&W2CEL
UÕ}MCTÁÂIGNLGUÇJVNOCELC ?hDGLW4UTÁKMLqJ UTP7}MCELpJ UÇLiJ!p h/e2 Ê/ƒ„€GNOpÂBG€GK&LGCEÁKMLGCTL }jUTL IKvDGLGWKMF!p¬JVCC$W2NOL
JVK&F!Áp„C J!€GKrÍUTLGW4UTDlPK¶ÆEK&P4p hCEFÁ•K&WlNOLlU–Á•UTRELK&JVNO} 4K&PW+Ê
ƒ„€GK Ô UTÁNPOJ!CELGNUTL#hCEFeU2B4UTF¬JVNO}MPKÕp!DGIÈK&}&J!KMW J!C5U2Á•UTRELGK¶JVN}'GKMPW BKMFB K&LGWGN}&DGPUÇFvJVC NOJVp
WGNFKM}¶JVNCTLÉC ÁCTJ!NCELv}MUTLlI <K HF!N J!J!KMLeWGC HL`UTp
H=
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PUTFREK;p!UTÁBGPK ÇJV€KMF!KHUÇF!K„Á•UTLqÂKMWREK„pJVUÇJVK&p,LGKjUTF/J!€GKHË4KMFÁ•N$KMLGK&F!RT EF C KMPOKM}&J!F!CELp,K&ÆEK&L H€KML
JV€GK&F!KrUTFKSLGC•p¬J UÇJ!KMpHLGKMUTF EF NLÉJ!€GKIGDGP Ê
Ë CTPPCHNOLGRÅÀJ!CELGK ' G1Ä A)( 7K}MCELGpNWGK&F„JV€GK 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!€GKPNLKjUTF
RqUTDGRTK Ax = −By ʱ 7U PNOLGKjUTF©}MCTL 4LGNOLGRvBCTJVK&LiJ!NUTP V (y) = Ey NOp°BGD$JSN)L JV€K>WGK&REKMLGK&FVUT}¶:Np
PN #JVKMW 9JV€GKKMLKMF!RǁC „UepJ U3JV;
K HNOJV€ k = k IKM}&CEÁ•K&p (k) = Ek/B UTLGWKMUT} €p¬J UÇJ!K•}MUTF!FNK&p
U Ô UTPP;}MDGFF!KMLqJMÊ7Ë4DGF¬JV€GK&F,JV€KzBG€qp!NO}jxUTPHpDGF/ UT}&KzC°JV€KeJ*7CWNÁKMLGpNCEL4UÇPQKMPOKM}&J!F!CEL RqUTpNp–U3J
K hDGPP UTLW•JV€CEp!K°UÇJÐJV€GK½F!NRT€iJ7UTFK½K&Á•B$Jµ p!K&KSËNOREDGF!K A$Ê A Ê
y = 0 JV€KML•J!€GK°p¬J U3JVKMp›JVCJV€GK½PK #J;UÇF!?
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W HN JV€eJV€GK ¬Ë4KMFÁ•N9p!DGF UT}MK xUÇJ
2
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(1/2)ωc
kx
edge
bulk
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ν=1
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E− (k)
kvF
−kvF
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0
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kF
k
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ƒ„€GK iNLGK¶JVN}SBGUTFJHC JV€K Ô UTÁNPOJ!CELGNUTL FKjUTWGp&~
H0 =
X
E(k)a†k ak ,
k
A$Ê ¿ HNOJ!€ E(k) = k2/2m UTLGW a†k ak NpJV€GKv}&F!KMUÇJVNOCEL UTLGLGNO€GNPUÇJVNOCEL½CEBKMF!UÇJVCTFxCHCTLGK K&PKM}¶JVFCEL
NL:p¬J UÇJ!K;HN JV€‹ÁCEÁ•K&LqJVDGÁ k Ê[²½CÇJVN}&KÂJV€4U3J(7K–WGClLGCTJNL}MPDWGKÅp!BGNOLWKMREFKMK–C=hF!KMK&WGCEÁ €KMF!KÇÊ
±µLzJ!€GKxƒ CEÁCEL4UTREUÁC$WKMP ' GG1Ä)(JV€KxWGNOp!BKMFp!NCTLÕF!K&PUÇJ!NCEL E(k) NOp©PONLGKMUTF!NMK&WzNOLlJ!€GKxÆN}&NLGN JµÉC
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hCEF k ≈ −kF Ê,±µL JV€GKlÍ[DJ!J!NLGRTKMFÅÁC$WGK&P ' GGG(9/JV€GKlPNOLGKjUÇF!N MUÇJVNOCELNOp–KJVK&LGWGKMW5JVCUTPP7ÆTUÇPDGK&p
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AÊ I a†k = a†+,k Θ(k) + a†−,k Θ(−k) ,
H€GKMFK Θ(k) Np°J!€GK Ô KjUjÆNpNWGK hDLG}&J!NCEL‹UTLW a†r,k HN JV€ r = ± °NOp°JV€KÅ}MF!KMUÇJVNOCEL‹CEBKMFVU3JVCEF hCTF
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UTp
X
A$Ê D H0 = v F
rka†r,k ar,k .
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L k
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L k>0 k
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JV€GK}&CELqÆEKMFREKML}MK>CJ!€GKNLqJVK&REFVUTPOpMÊ
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Mr irkF x+irφr (x)
ψr (x) = √
e
.
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ƒU iNOLGR JV€KÂWGKMFNOÆÇUÇJ!NOÆEKxC›»,‚ Ê A$Ê@GH 6;K€GU1ÆTK ∂xφr (x) = 2πρr (x) J!€GKML'7KÂ}MUTL:F!KHF!NOJ!K
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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) ,
JV€GK&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 JV€GK ³ CEDGPOCEÁÂIÕNOLqJVKMF!UT}&J!NCELp°IK&J*7KMKML:J!€GK>K&PKM}¶JVFCELGpMʃ„€GK>J!CTJ UÇP Ô UTÁ•NOPOJ!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 ) ,
H€GKMFK½J!€GK°K&PKM}¶JVFCELvWGK&LGp!N Jµ•CTB K&FVUÇJ!CEF ρ(x) }&CELqJ UTNOLGpQUrp!POC ÆÇUTFNUÇJ!NCEL ρ+(x) + ρ− (x) UTLGW UTp¬J
p HN JV€eUÂLGCEL´µPNOLGKjUÇF;WGK&B K&LGWGKML}MKrCEL φ ʱ 7KLGKMRTPKM}¶J„JV€GK&p!KE UTp¬JHCEp!}&NPOPUÇJ!NCELGp%
2kF CEp}MNPOPUÇJ!NCEL=
JV€GKÉNOLqJVKMF!UT}&J!NCEL Ô UTÁNP JVCELGNUTLI K&}MCEÁKMpłiD4UTWFVUÇJ!N}vNOL±JV€GKÉICEp!CTLGN} GKMPW9Ê U NOp>JV€GK 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
UTLGWÉJV€KJ!CTJ UÇP Ô UTÁNP JVCELNUTL•F!KjUÇWGp
vF
H=
8πg
Z
L/2
1
dx (∂x φ(x))2 + g(∂x ϕ(x))2
g
−L/2
AÊ Ã7G ,
H€GKMFK g = 1/p(1 + 2U0/πvF ) NOprJ!€GK ³ CEDGPCTÁÂINOLiJ!KMF!UT}&J!NCELB4UTF!UTÁ•K¶JVK&FMÊ K;4LGWJV€4UÇJ>»,‚ Ê
AÊ Ã7G ›NpH‚iD4UTWGF!UÇJVNO}J!€GKMLlJV€GKSJVCÇJ UTP Ô UTÁNPOJ!CELGNUTL NOp„KUT}&J!POÉpCEPOÆÇUTIPKUTLGWlNOJ!p„KMNOREKMLGp¬J UÇJ!KMp©UTFK
p!NOÁ•NOPUTFÐJVCÅJV€KrK&NREK&LGpJVUÇJVK&pHC €GUTÁ•CTLGN}"CTp!}MNOPPUÇJVCEF&Ê
4 *,1 .!- . .23 .! *,%01 .# !, 3 23 !" ƒ„€GK Ô UTÁNP JVCELNUTL H€GN} €lWGKMp}MFNIKMp„JV€GKK&WGREKÁ•CWGK&p©NOp„p!NOÁ•BGP vUÇLlK&PK&}&JVFCEpJVUÇJVNO}"JVK&F!Á' HH)(|~
1
H=
2
Z
L
dxV (x)eρ(x) ,
AÊ ÃEÃ 0
H€GKMFK x NOp°U}MDGF¬ÆNPNOLGKjUÇF„}MCCEF!WGNOL4UÇJ!KxUÇPCELGRÅJ!€GK>K&WGREK L NpHJV€KxPOKMLGRTJ!€zC/JV€GKrKMWREKÂUTLW V (x)
NpQJ!€GKS}MCEL74LGNLR•BCTJVK&LqJVNUÇP9F!K&PUÇJ!KMWÉJVCÅJ!€GKrUTBGBGPONKMWeK&PKM}¶JVFN} 4K&PW w ' HEÄF6Iq¿ (|~ V (x) = Eh(x) =
Ô UTÁNPOJ!CELGNUTL}jUÇL IK
vF Bρ(x)/ns HNOJ!€ ns NpÅJV€GKl*J ;CWGNÁKMLp!NCTL4UTPQK&PKM}¶JVFCEL WGK&LGp!N JµEʛƒ„€GK
F!K HF!NOJJVK&LzUÇp
Z
πvF L
A$Ê Ã H H=
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Î;KM}MUTDGpKeJ!NÁKMpÂCTL2J!€GKePC 7KMFÅ}MCTLiJ!CEDGF–UTFK bPUTF!REK&F JV€GUTL JVNÁKMp>CEL JV€GKeDGBGBKMFÅ}&CELqJVCEDF /JV€K
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 ,
H€GKMFK>JV€K>NLqJ!KMREF!UTPC ÆTKMF°J!€GK Ó K&PWp!€Õ}MCELqJVCTDGF K RECKM>p hF!CEÁ −∞ JVC +∞ UTLGWzJV€GK&L:I4UT} ÉJVC
−∞ ÊG±µLlJ!€GNp„}MUTp!K ;K€4UjÆEK
SK (−∞, −∞) = TK exp −i
(
Z
dt1 Hint (t1 )
K
X Z
= TK exp −i
η
η=±
+∞
dt1 Hint (t1 )
−∞
)
.
AÊ qÄ ½² CTJ!KJV€GUÇJNL2REK&LGKMF!UTP8MJ!€GK/JVNOÁ•K&p[UTBGBKjUÇF!NLRQNL"J!€GK,CEBKMFVU3JVCEF+BF!CWGDG}&J AI (t0 )BI (t1)CI (t2 )DI (t3 )
}jUTLeIKPC}jUÇJ!KMWeKMN JV€GK&F„CELlJ!€GKSDGBGBKMF©CEF„CELÉJ!€GKPC;K&FH}MCELqJVCTDGFMÊ
ƒ„€GKŸSF!K&KM)L 1 "p hDGL}&JVNOCELlNp½UTLÕK GUÇÁ•BGPOKC ÐUÅJVNÁKCEF!WKMF!K&W`BGFC$WGD}&JjÊ ƒ„€GK¸FKMK&L)1 p>hDGLG}¶JVNCTL
UTp!pC$}&NUÇJ!KMW HNOJ!€JV€GKÉJ*7CÇ´µIFVUTLG} €KMp Ó K&PWp€5}&CELqJVCEDGFÅNOpÂJV€GK&F!KhCEF!KlU 2 × 2 Á•UÇJ!F!N IKM}jUÇDGp!K
JV€GK&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 H€GKMFK G0 (t) }jUTLlIKr}&CEÁBGDJVK&WChFCEÁ J!€GKSJV€GK&F!Á•UTP¸SF!K&KML)1 p hDGLG}&J!NCELeDGpNLGRU N}vFCTJ U3JVNCTL+Ê
²½CTJ!KQJV€4UÇJ,J!€GK„DGpK©C Ó K&PWp!€•¸SF!K&KML)1 p hDGLG}&J!NCEL–UÇPPCHp/DpÐJ!CEHF!N JVK„WGNOF!KM}¶JVP xJ!€GK„CEBKMFVU3JVCEFp
[email protected] TK°J!€GK}MDGFF!KMLqJ IqÉ} €C$CEpNLGR•UTBGBF!CEBGFNUÇJ!KMP •J!€GKSJVNÁKMp;CELlJ!€GK}MCELqJ!CEDGFMÊ Ë4CEF;KUTÁBGPK
n
o
AÊ ià hψ2† (t)ψ1 (t)i = hTK ψ1 (t+ )ψ2† (t− )SK (−∞, −∞) i .
±µL:JV€GNOpSKM‚iD4UÇJ!NCE)L JV€GK–CEBKMF!UÇJVCEFpSNL:PO%K #JS€4UÇLGWpNWGK–UÇF!KÅNL:J!€GK Ô K&Np!K&LiI K&F!RlBGF!K&p!KMLqJVUÇJVNOCELUÇLGW
CELGK&p;NLJV€GK°F!NRT€iJ7€4UTLWvp!NOWGK"UTFK°NOL•JV€K°NLqJ!KMFVUÇ}&JVNOCEL BF!KMpKMLqJ U3JVNCTL HNOJ!€ JV€GK©JVNOÁ•KHCEFWGKM=F hCEPPOC Hp
JV€GK Ó K&PWp€:}MCELqJVCTDGFMÊ ƒ„€GK–UjÆEKMF!UTREKÂCÐCEB K&FVUÇJ!CEF!p°BGF!CWGDG}¶JSCELÕJ!€GKÂF!NORE€qJ"€4UÇLGW:p!NOWGKÂNp°U•JVNOÁ•K
CEF!WKMF!K&Wz‚iD4UÇLiJ!NOJµ H€GN} €l}jUTLlIKS}jUTPO}MDGPUÇJVK&WlIqÉDp!NLR NO} .1 p„JV€GK&CEF!K&ÁzÊ
4 ) 1.!,- 1!
±µLvJV€GNOpQp!K&}&J!NCEL)F;KS}&CELGpNWGK&F;JV€GKSp¬J UÇJ!NCELGUTF}MDF!F!K&LqJ 4CHNLGRÅIK&J*7KMK&LlJ*7C–KMWGREKSp¬J UÇJ!KMp4H€GNO} €
NpÐ}jUÇPPK&W–I4UT} ip!}MUÇJ!J!KMF!NOLGR>}&DGF!FKMLqJjʃ„€GK½KMWGREK©pJVUÇJVK&p7NOp7NOLÅJV€GK>hFVUT}¶JVNCTL4UTPG‚iD4UTLqJ!DGÁ Ô UTPOPGF!K&RENÁKTÊ
IEÄ
+
K
PSfrag replacements
t0
×
t
×
−∞
+∞
−
,n' 0G($%
'(! - ; 3A1 1'#'Z
„ƒ €GK Ô UTÁ•NOPOJ!CELGNUÇLEH€GN} €ÅWGK&p!}&F!NIKMp/J!€GKQp¬$p¬JVK&Á™Np H = H1 +H2 +HB HNOJ!€ H1/2 WGKMp}MFNIKMp/JV€K
PK#J FNRE€qJHKMWREKxp¬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!€GK•J!DGLGLGK&PNOLGR Ô UTÁNPOJ!CELGNUTL HB WGKMp}MF!NOIGNLRzJV€K
}MCEDBGPNOLGRÅI K¶*J ;K&KMLlJV€GKS*J 7C•K&WGREKMp 1 UTLGW 2 ~
A$<Ê H HB = Γ(t)ψ2† (t)ψ1 (t) + Γ∗ (t)ψ1† (t)ψ2 (t) ,
H€GKMFK
M1/2 i√νφ (t)
AÊ ψ1/2 (t) = √
e
,
2πα
HNOJ!€GNL M1/2 NpHU Ó POKMNL UÇ}&JVCTFH€GNO} € HNPP I KSCTÁ•N J!JVK&WɀGKMFKjU#J!KMF„IqÉLGCÇJVN}&NLGRÂJ!€4UÇJ M1/2
2
= 1Ê
¹½BCELÂJV€K„RqUTDGREK7JVFVUÇLG/p hCEFÁ U3JVNCTLxJV€KQJVDGLGLKMPNOLGRSUTÁBGPONOJVDWGK7IK&*J 7KMK&L•J!€GKQKMWREK„pJVUÇJVK&p›NOp Γ(t) =
K hFVUÇ}&JVNOCEL4UTP} €4UTF!RTKTʃ„€GK½REUTDGREK hDGLG}¶JVNOCEL χ WGKMBKMLGWp;CELGP ÅCEL
Γ0 e−ie χ(t)/c H€GKMFK e∗ = νe Np,JV€G?
JVNOÁ•K hCEFSUv}&CELGpJVUTLqJWG}ÅIGNUTp V0 NÁB CEpKMW`IK&*J 7KMK&L*J 7CeKMWGRTKMpM~ χ(t) = cV0t 9p!CvJV€GUÇJSNLzJV€GNOp
}jUTpK Γ(t) = Γ0 eiω t HNOJ!€ ω0 = e∗ V0 Ê
ƒ„€GKÂI4UT}ip}jUÇJJVKMFNLGRÉ}&DGF!FKMLqJCEBKMF!UÇJVCEF½}jUTLÕI KÂWKMF!N ÆEK&W'hF!CEÁ JV€K Ô K&Np!K&LiI K&F!RÉKM‚iD4U3JVNCTL:C
Á•CÇJVNCTL hCEFQJV€KrWKMLGpNOJµÉCEBKMF!UÇJVCTF GCEF©UTP JVK&F!L4UÇJ!NOÆTKMPO•IqÉ}jUÇP}MDPUÇJ!NLGR IB = −c∂HB /∂χ GJV€GK&L
1/2
∗
0
IB (t) = ie∗ Γ(t)ψ2† (t)ψ1 (t) − Γ∗ (t)ψ1† (t)ψ2 (t) .
AÊ ? „ƒ €GK•U1ÆTKMF!UTREK IGUT}ip!}jU3J!JVK&F!NOLGRÕ}&DGF!FKMLqJÂNOprK $BGF!K&p!p!K&W DGp!NOLGR Ó K&PWp!€}MCELqJ!CEDGF H€GN} €UTPPOCHp"J!C
JVFKjUÇJHLGCTLGKM‚iDGNOPNIF!NDÁ pNOJVDGUÇJVNOCEL+~
hIB (t)i =
n
o
R
1X
hTK IB (tη )e−i K dt1 HB (t1 ) i .
2 η
ƒC–POC7KMp¬J©CEF!WGK&FHNLÉJV€GKSJ!DGLGLGK&PNLR–UTÁ•BPNOJ!DGWGK
hIB (t)i =
Γ0
AÊ A 7K€4UjÆEK
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¿ ƒ„€GK©}MCEFF!KMPUÇJVCTF/NpÐWGN 9K&F!KMLqJ3hF!CTÁ MK&F!C>CTLGPOH€GKML 1 = − iJ!€4UÇJ›ÁKjUTLGp,J!€GK½‚iD4UÇp!NBGUTFJ!N}MPOKMpÐUTFK
}MCELp!KMF¬ÆEK&W:NLeJV€GKrJVDGLGLKMPNOLGRBGF!C}MK&p!pMÊ ±µLzJ!€GK>}MUTP}&DGPUÇJ!NCEL;K>UTF!KrPK&WzJ!C NLqJ!F!CWGDG}MKJV€Kx} €GNOFVUTP
¸FKMK&)L 1 p hDGLG}&J!NCELC QJV€KvI CTp!CELGNO
} 4K&PW U3JÂBCEp!N JVNCTL x = 0 H€N} € WGCKMpÂLCTJÂWGK&B K&LGW CELJ!€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 ƒ„€GNp¸FKMK&L)1 phDGLG}&J!NCELNpÂUÇp!p!C}MNUÇJVK&W HNOJ!€JV€K•J*7CÇ´µIGF!UTLG} €GK&p Ó KMPW$$p€ }MCELqJ!CEDGFÅUTp(;KɀGU1ÆTK
WGNp}MDGpp!K&WI%K hCEFKTÊ[ƒ„€GK•UjÆEKMF!UTREK}MDGFF!K&LiJx}jUÇLLGC¡IK–K$BGFKMp!pKMW2UTprUTLNLqJ!KMREF!UTP/C1ÆEK&FrJ!NÁKÅC
Ó KMPW$$p€`¸SF!K&KML)1 p hDGLG}¶JVNCTL
H€GKMFK
AÊ D Z +∞
ie∗ Γ20 X
η−η
dτ sin(ω0 τ )e2νG (τ ) ,
η
hIB (t)i = − 2 2
4π α η
−∞
=4LGN JVK„JVK&Á•BKMF!UÇJVDGFKMp%iJ!€GK°¸FKMKML1 p=hDGLG}&J!NCELNpÐREN ÆEK&L•Iq pKMK½»Ð‚ Ê AÊ H ? ¶~
τ = t − t1 Êi®QJ

Gη−η (τ ) = − ln 
sinh
π
(ητ
β
sinh

AÊ ?TÄ + iτ0 )
 ,
iπτ0
β
H€GKMFK τ0 = α/vF Êi®½BGBGPONOLGRSJV€GNOp;¸SF!K&KML)1 p hDLG}&J!NCEL)7K©}jUTL•CEIJ UÇNLÅJV€GK½UTL4UÇPOiJVNO}jUTPGK $BGF!K&p!p!NOCEL
hCEFQJV€KxUjÆTKMFVUÇREK}MDGFF!KMLqJ°UTp
e ∗ Γ2
hIB (t)i = 2 2 0
2π α Γ(2ν)
H€GKMFK
4 Γ
α
vF
2ν 2π
β
2ν−1
sinh
ω0 β
2
ω0 β
Γ ν +i
2π
AÊ ?7G 2
,
NpQJV€KrREUTÁ•Á•U:hDGLG}¶JVNOCEL+ÊG®QJ &KMF!C–JVK&Á•BKMF!UÇJVDF!KGJ!€GKrUjÆEKMF!UTREK}MDF!F!K&LqJ°NOp
a
vF
2ν
pREL
AÊ ?EÃ (ω0 )|ω0 |2ν−1 .
) 1.!,- 1! ! #.
±µL JV€GNOpÅp!KM}¶JVNOCEL ;Ke}MCTLGp!NOWGKMFÂJ!€GKepÁ•ÁK&J!F!NMKMW2LGCTNp!KÉIKM}MUTDGpKlPUÇJ!J!KMFÂCTL LCENpKeUÇJ ω = 0 NOp
}MCELp!NWKMF!K&W)4p!CJV€GKNOp!p!DKxCpÁÁ•K¶JVFN M K&WlÆTKMFp!DGp°UTpÁÁ•K¶JVF!NM K&WlWC$K&p½LGCÇJ½Á•UÇJJVKMF&Ê4¹½pNLGR–JV€K
e∗ Γ20
hIB (t)i =
2πα2 Γ(2ν)
pÁÁ•K¶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
JVC–PC7KMp¬J°CEFWGKMFHNOLeJ!€GKSJVDGLLGKMPONLGRUTÁBGPN JVDGWK Γ0 4NOJHNOp©LCTJHLGKM}&KMppVUTF¬zJ!CK $B4UTLGWlJV€GK Ó KMPW$$p€
K&ÆTCEPDJ!NCELÉCTB K&FVUÇJ!CEF„I K&}jUTDp!KJV€GK}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 JV€K>p!BKM}¶JVF!UTPWKMLGpNOJµzC,I4UT}ip}jUÇJJVKMFNLGR }MDGFF!KMLqJSLCENpKxNOp½CEI$J UTNOLGKMWzIi
}jUTPO}MDGPUÇJVNOLGR>J!€GKrË CTDGF!NOKMF;J!FVUTLGphCEF!ÁlÊ®QJ?4LNOJVKSJ!KMÁB K&FVUÇJ!DGF!KJV€GKLGCTNp!KU3J &KMFCÇ´ hFKM‚iDGKML}&eNp
(e∗ )2 Γ2
S(ω = 0) = 2 2 0
π α Γ(2ν)
α
vF
2ν 2π
β
2ν−1
IqÃ
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%- ' -$;<+q- %+*
1' q<jq#* 2C- θ
D*82- <I*
νL = 4 θ
UTLGWeNLÉJ!€GKPNÁNOJMK&F!C–JVK&Á•BKMF!UÇJVDGFK
(e∗ )2 Γ20
S(ω = 0) =
πα2 Γ(2ν)
α
vF
2ν
|ω0 |2ν−1 .
AÊ ?GA ±µL }&CELG}&PDGpNCEL);UÇJ MK&F!C VJ K&Á•BKMF!UÇJVDF!K ;K:FKM}MC1ÆTKMFeJV€GKÀ$} €CTJ!J iq´ PN ÇK:F!K&PUÇJ!NCEL HN JV€ VJ €K
e∗ = νe ~
A$EÊ ?E¿ S(ω = 0) = 2e∗ hIB (t)i .
®QJ 4LGN JVKSJVK&Á•BKMF!UÇJVDGFK J!€GKp!€GCÇJ J!€GKMFÁ UTP9LGCENpK}MF!CTp!p!C1ÆTKMF©Np„F!K&}MC1ÆEK&F!K&W
A$EÊ ?I S(ω = 0) = 2e∗ hIB (t)i coth(ω0 β/2) .
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KMPOKM}&J!F!CEL´µ€GCEPOK¶´ B4UTNFÐ}&F!KjU3JVNCTLÉNLvJV€KEHNF!K°Á•KMUTp!DGFKMp H€GNO} €lBGUÇJV€vJ!€GKSKMPOKM}&J!F!CELÉJ!C$C•UTFCEDGLGWÉJV€K
F!NOLGR 3UTLGWÅpCS}jUTDGpKMp/J!€GKQB4UÇJ!€GpJ!C"WGKM}&CE€GK&F!KML}MKTÊÀKM}MCTLG)W TÆNF¬JVD4UÇP$K&PKM}¶JVFCEL$´µ€CEPK ´µB4UTNOF}MF!KMUÇJVNOCELÂNOL
JV€G3K HNFKÐWGKM}&F!KjUÇp!KMpJV€GKÐJVF!UTLGp!ÁNpp!NOCEL>UÇÁ•BGPONOJ!DGWGKJ!€GF!CEDRE€xJ!€GK7‚iD4UÇLiJ!DGÁ WGCTJ 3POKjUTWNLGR„JVC½B C 7KMFb´
P!U WGKMBKMLGWKMLG}&K`C °J!€GKe®°€4UÇF!CELGC1Æq´bÎ7CE€GÁ CEp}MNPOPUÇJ!NCELp>CEL2JV€GKÉJ!KMÁB K&FVUÇJ!DGF!KeCEFÅJV€Kz}&DGF!FKMLqJ
Iq¿
JV€GFCEDGRE€vJ!€GK<HNFKTʃ„€Np„F!K&p!DGP J©UTPpC QUÇp©CTIJ UTNOLGKMWÉNOLeK$B K&F!NOÁ•K&LiJ ' GG!I)()H€GK&F!KSJV€GKSpKMLGpNOJ!NOÆNOJµ
C JV€GK‚iD4UTLqJVDÁ B CTNLqJH}MCELqJVUT}&J½U KM}¶JVp„JV€GKSÆNpNIGNOPN Jµ•CJ!€GKCEp!}&NPPUÇJVCTFNLqJVK&F/hK&F!KML}MKrp!NRTL4UTP Ê
®QJ„JV€GK"p!UTÁK"JVNÁK $Í[K&ÆNLp!CELe€4UTp;}jUTPO}MDGPUÇJVK&WÉNLGWGK&B K&LGWGK&LiJ!PO WGKMBG€GUTp!NOLGR–FVUÇJ!K"C UÂpJVUÇJVKSNOL
‚iD4UTLqJVDGÁ WGCTJNLWGDG}MK&WÂIq>N JVp }jUTBGUT}MN JVNOÆTK;}&CEDGBGPONLGRHJVC°U°‚iD4UTLqJVDGÁ BCENLqJ}MCELqJ UÇ}&J,UÇLGWÂNLqJ!KMF!K&pJ!KMW
Á UÇNLGP eNL:JV€GK–UÇWGWGNOJ!NCELGUTP}&CELqJVFNIGDJ!NCEL`J!CvJV€GK–WGK&BG€4UTpNLGReF!UÇJV:
K H€N} €Np"WGDK–J!CÉJV€GKÅ}&DGF!FKMLqJ
NLvJ!€GK‚DGUTLqJVDGÁ BCENLqJH}&CELqJ UT}¶
J ' Ã I ( Ê
ƒ„€GK°‚iD4UTLqJ!DGÁ WCTJQNOp;UTpp!DGÁKMW•JVC>IK°NOp!CEPUÇJVK&W hFCEÁ¡JV€GK½PKMUTWGp7UTLGWvCELPO–CELGK½pJVUÇJVK½K $Np¬JVNLR
HNOJ!€KMLGK&F!RT 0 ÊQƒ„€GK Ô UTÁNPOJ!CELGNUTL2CJV€GK`‚iD4UÇLiJ!DGÁ WGCÇJÉNp HQD = 0 c† c =H€GKMFK c NOpÉUTL
CEBKMFVU3JVCEF–F!K&Á•C1ÆNLRCELGKzKMPOKM}&J!F!CEL hF!CEÁ J!€GKՂiD4UTLqJ!DGÁ WCTJ p¬J UÇJ!KTʛƒ„€GK`NOLiJ!KMF!UT}&J!NCEL I K¶*J ;K&KML
JV€GK•‚iD4UTLqJVDGÁ«WGCTJ>UTLGWJV€GK•‚DGUTLqJVDGÁ«B CTNLqJ>}&CELqJ UT}¶!J UTp!pDGÁKMWJVCzI K ;KMU 6NOpxWGK&p!}&F!NIKMp>Ii
€4UTÁNP JVCELGNUTL+~ Hint = c† cW )Ê K•}MUTLpKMKJV€4UÇJ W NOpSJV€GK} €4UTLGRTK•C ;JV€GK–‚iD4UTLqJVDÁ«WCTJ>p¬J UÇJ!K
KMLGK&F!RT WGDKlJ!C‹J!€GKeNLqJVK&FVUT}¶JVNOCEL2C ½J!€GKlKMPK&}&J!F!CEL2NL J!€GKz‚iD4UÇLiJ!DGÁ WGCTJ HN JV€2JV€GKeK&PKM}¶JVFCEL
WGKMLp!NOJµvN0LÉJV€GKBCENLqJ„}&CELqJ UT}¶JjÊ
ƒ„€GKv}MCT€GKMFKMLG}&KvC „JV€K ‚iD4UTLqJVDÁ·WGCÇJÂpJVUÇJVK•NpxWGKMp}MF!NOI K&W IqJ!€GKÉUjÆEK&FVUTRTKÉUTÁBGPN JVDGWGK hci H€GN} €ˆ}MCELqJVUTNLGpeNOL hCEF!Á•UÇJ!NCEL UÇI CED$JlJ!€GKBG€4UTpKC xJV€GK‚iD4UTLqJVDGÁ WCT$
J ;UjÆE0
K hDGLG}&J!NCEL9ʄƒ„€GK
‚iD4UTLqJVDGÁ WGCTJH€4UTp©UÅ}MCT€GKMFKMLqJ°BGUTFJHN hci 6= 0 ÊGƒ„€GKJVNOÁ•K"K¶ÆECEPODJVNOCELÉC JV€GK}&CE€GKMFKMLG}&KxNOp„RENOÆTKML
Iq hc(t)i H€GKMFK c(t) = eiHt ce−iHt NL Ô KMNOp!K&LIKMFRF!KMBF!KMpKMLqJ U3JVNCTL;HN JV€ÂJ!€GK;JVCÇJ UTP Ô UTÁNP JVCELGNUTL
Ô
H = HQP C + HQD + Hint HNOJ!€ HQP C Ô UTLGW HQD UTFK UTÁNPOJ!CELGNUTL>C 9‚iD4UTLqJVDGÁ¡BCENOLiJ›}&CELqJ UT}¶J
UTLGWɂiD4UTLqJ!DGÁ WGCTJ F!K&p!BKM}¶JVNOÆTKMP EÊ4ƒ„€GK UÇÁ•NOPOJVCTLGNUTL Hint WGK&p!}&F!NINLGRÂJ!€GKSNLqJVK&FVUT}¶JVNOCELÉI K¶*J ;K&KML
‚iD4UTLqJVDGÁ¤WGCTJ„UTLGWɂiD4UTLqJVDÁ BCENLqJQ}&CELqJ UT}¶J„Np;UTp!pDGÁKMWvJVCÅIK 7KjU ÊGƒ„€K"Á•CÇJVNCTL KM‚iD4U3JVNCTLÉC JV€GKCEBKMF!UÇJVCTFQRENOÆTKMp
dc(t)
¿$Ê ? = i[H, c(t)] = −i[0 + W (t)]c(t) ,
dt
HNOJ!€ W (t) = eiHt W e−iHt Np„WGK&Á•CTLGpJ!FVUÇJ!KMWlNLeJV€GNOp©}MUTp!KrUTpHJV€KJ!NÁKSWGKMBKMLWGNLGRÁC$WGDPUÇJ!NCEL
C [JV€K"KMLGK&F!RT PK¶ÆEKMP 0 ÊG»,‚ Ê |¿$EÊ ? ›REN ÆEKMp7JV€GK"K $BGF!K&p!p!NOCELeC c(t) 7H€GCEpKUjÆEKMF!UTREKUTÁBGPONOJVDWGK½}MUTL
I EK HFNOJJVKMLzUTp
R
¿Ê A hc(t)i = hc(0)e−i t Tt e−i dtW (t) i ,
H€GKMFK Tt ÁKjUTLGpvJ!NÁKÕCEFWGKMFNLGRÊQÎ7UTp!pDGÁ•NOLGRJV€GUÇJÉJV€GK:POK&ÆTKMP"ÁC$WDGPUÇJ!NCEL5NpÉU ¸UTDGpp!NUÇL
BGF!C}MK&p!p–WGK&p!}MFNIKMW Iq U:‚iD4UTLqJVDGÁ }&CEF!FKMPUÇJVCEF K(t) = [hW (t)W (0)i + hW (0)W (t)i]/2 &;K
}jUTLeWGK&}MCEDBGPKSJV€KxUjÆTKMFVUÇREKxNOLl»Ð‚ Ê |¿$Ê A QUTp
¿Ê ¿ hc(t)i = hc(0)ie−i t e−Φ(t) ,
HNOJ!€
Z t Z t
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dt0
dt00 K(t0 − t00 ) .
0
0
®°pp!DGÁNLGR"J!€4UÇJ,J!€GK„‚iD4UTLqJVDGÁ¡WCTJ,WGCKMpÐLGCTJÐBKMF¬JVDGFIÅJV€GKH‚iD4UTLqJVDÁ¦B CTNLqJ,}MCTLiJVUT}&J EJV€GKHUjÆTKMFVUÇREK
NL hW (t)W (t0)i F!K&WGDG}&KMp;J!C>JV€GK°UjÆEKMF!UTREK HN JV€vFKMpB K&}&JQJ!CxJV€K"pJVUÇJVK½C +JV€K"B CTNLqJ;}&CELqJ UT}¶Jjʃ„€GK
}MCEFF!K&PUÇJ!CEF
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K HNOJ!€‹p!CEÁK>JVNOÁ•K>p!}jUÇPK H€GNO} €NOp°JV€K}&CEF!FKMPUÇJVNOCELÕJVNOÁ•K>C JV€GK–‚iD4UTLqJ!DGK(t)
Á WGCTJrpJVUÇJVKKMLGK&F!RTÁC$WDGPUÇJ!NCEL9Ê9Ë4CEF t ττCC +JV€GKp!K&}MCELGWNLqJVK&REFVUTP,C1ÆTKMF t00 HNOPP
pVUÇJ!DGFVU3JVK"JVC•UÅ}MCELpJ UÇLiJ ;Kr€4UjÆEK
¿Ê D hc(t)i = hc(0)ie−i t e−t/τ ,
HNOJ!€–UTp!pDGÁ•NOLGRSJV€4U3!J hCEF,POCELGR"J!NÁKMp ;KH}MUTL•FKMBGPUT}MKQJ!€GKHp!K&}MCELGWNLqJ!KMREF!UTPGC t00 C1ÆEK&F›J!€G4K H€GCEPOK
JVNOÁ•K"WCEÁ UÇNL+~
Z
1 ∞
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(τϕ ) =
dtK(t) .
t
0
0
0
0
2
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−∞
ϕ
„ƒ €GK;WKM}jUj>C hc(t)i HN JV€xJ!€GK;J!NÁK›}&CELGp¬J UTLqJ τϕ NOpWGK&BG€4UTpNLGRÇLGCÇJ/KMLKMF!Rǁ>FKMPUGU3JVNCTLxCEFKMp}jUTBK
hF!CEÁ J!€GK‚DGUTLqJVDGÁ«WGCTJ$' ÃI)( ʃ„€GKvUTDJV€CEFxp€GC;K&WJV€4U3JxJ!€GK }&CELqJVF!NOIGDJ!NCEL‹JVClJV€GK•WGKMBG€GUTp!NOLGR
FVUÇJ!K½WDGK½J!CxJV€K°IGNUTp7WKMBKMLGWGp;CEL•J!KMÁB K&FVUÇJ!DGF!K θ UTLW IGNUTp eV NLJV€GK½pVUTÁK?QUj UTp7p€GCTJ;LGCENpK
NLlJV€GKÂBCENOLiJ½}MCTLiJVUT}&JSUÇJ &KMF!C3´ hF!K&‚iDGKMLG}¶ +IDJ"WGCÉLCTJ hCEPOPC J!€GK T (1 − T ) pDGBGBGFKMp!pNCEL9Ê9ƒ„€GK
LGCELGK&‚iDGNPONIGFNDGÁ }MCTLiJ!F!NOIGDJVNOCELvJVCWGK&BG€4UTpNLGR–FVU3JVKNp„K&pJVNOÁ U3JVKMWlUTp
|¿[email protected]Ê GG (τϕ−1 )V ' λeV hCTF„€GNRE€ÉINUTp eV θ ,
|¿[email protected]Ê G1à (τϕ−1 )V ' λ(eV )2 /θ hCEF„POC ˆIGNUTp eV θ ,
H€GKMFK λ NpJV€K }MCTDGBGPNOLGRl}MCELGp¬J UTLq
J H€GN} €WKMp!}&F!NOI K&p>JV€GKNLqJ!KMFVUÇ}&JVNOCELIK&*J 7KMK&LJV€GK•‚iD4UTLqJVDGÁ
WGCTJ©UTLWlJ!€GK‚DGUTLqJVDGÁ BCENOLiJH}&CELqJ UT}¶JjÊ
ƒ„€GKeBGF!CEIPKMÁ«C ©WGK&BG€4UTpNLGR‹FVU3JVKÉC ½UTL2KMPOKM}¶JVF!CTL pJ U3JVKÉNL2U:BGNOLG} €GK&W ‚iD4UÇLiJ!DGÁ WGCÇJUTPpC
pJ!DGWGNK&:
W HNOJ!€ÂU©LGKjUTFIqxÆTCEPOJVUTREK¶´ IGNUÇp!KMWxI4UTPOPNp¬JVNO}/L4UTLGCTpJVFDG}&J!DGF!K ' GG!D (|ÊDZµL>J!€GNp ;CTF 63J!€GKQUÇDJV€GCTF
BGF!K&p!K&LiJ!KMW UREK&LGKMF!UTPN jUÇJ!NCELC ½J!€GKÉJV€GK&CEFRENOÆTKML NL ¯©K bÊ ' à I („JV€GUÇJÅJ U ÇKMpNLqJVCUT}&}MCEDLiJJV€K
p!BKM}&@N 4}"K 9K&}&J!p„UTBGBKjUTFNLGRxWGDGK°J!C>JV€K"}MCEÁBGPON}jU3JVKMW•REKMCTÁ•K¶JVF UÇLGW JV€K"} €GNF!UTPN Jµ–C +J!€GK"pJVUÇJVK&p
NLvJ!€GKL4UTLGCEp¬JVFDG}&J!DGF!KÇÊ
ID
$ ®°p 7KS€4UjÆEKSNOLqJVF!CWGDG}&KMWÉNOL JV€GK°BGF!K¶Æ$NOCEDGpQ} €GUTBJVK&FJ!€GK"WGK&}MCE€KMF!K&LG}MKC[J!€GKSKMPOKM}¶JVF!CTL JVF!UTLGp!BCEF¬J
JV€GFCEDGRE€vJ!€GK‚iD4UTLqJVDGÁ WGCTJHNp;WGDGKSJVCÂJ!€GK} €4UTF!RTKE4DG}&J!D4UÇJ!NCELGp„NOLvJV€GKS‚iD4UÇLiJ!DGÁ BCENLqJ„}&CELqJ UT}¶J
'IGà NI7UÇp„à C D J!G€GGKD7‚i D4G ÃÇUTÄLq(|JVÊqDG±µÁ LJ!B €GCEKMNOpLqK½JHF}MKMCEp!LqDJ POJVUÇp%}&qJ©JVN€Gp„KHNOLGWG}MK&FBGKj€4UTUTppKMNW+LGÊ RrFVUÇJ!K„Jµ$BN}jUÇPPO>NLG}&F!KMUTp!K&p3H€KML•J!€GKHÆECEP J UTREK
ƒ„€GKÂBGDGFB CTp!K>C /JV€GK>BGF!K&p!KMLqEJ 7CE/F zNOp©J!CvWGNp}MDGpp°JV€K>}jUÇp!K>C ÐWGKMBG€GUTp!NOLGR hF!CTÁ U ‚iD4UTLqJVDGÁ
B CTNLqJH}MCELqJVUT}&J½NLÉJV€K(hF!UT}&J!NCEL4UÇP+‚iD4UTLqJVDGÁ Ô UTPP9K% KM}&J°F!K&RENÁK#' GjÄqà (|Ê ="D4UTLqJVDGÁ BCENOLiJ©}&CELqJ UT}¶J
JVF!UTLGp!ÁNpp!NOCELe}jUTLlJV€GK&LÕIKxWGK&p!}&F!NIKMWzIqlJ!DGLGLGK&PNOLGR–I K¶*J ;K&KML:KMWREKxp¬J UÇJ!KM
p ' HEÄ (84J!€GKx‚iD4UÇLiJ!N &KMW
UTL4UTPOCERSC [}&PUTpp!NO}jUTP4/p NOBGBGNOLGRxCTF!IGN JVpÐC +K&PK&}&JVFCELGp&ʱµLJV€GNOpÐpJVFCELGREP Å}MCTF!F!K&PUÇJ!KMW•KMPK&}&J!F!CELF!K&RENÁK KMWGRTKp¬J UÇJ!KMpF!K&BGF!K&p!K&LiJx}MCTPPK&}&J!NOÆEK>K$}MN J UÇJ!NCELGpSC7J!€GK–‚DGUTLqJVDGÁ Ô UTPP 4DGNW9~+WGKMBKMLWGNLGReCEL‹JV€K
BGNL} €GNLGRrC +J!€GK°‚iD4UÇLiJ!DGÁ¤B CENOLqJ;}&CELqJ UT}¶!J NOJ›Np›KMN JV€GK&F hFVUÇ}&JVNOCEL4UTP4‚iD4UÇLiJ!DGÁ Ô UTPOP ‚DGUTp!NOB4UTFJ!N}&PKMp
CEFSKMPOKM}¶JVF!CTLG<p H€GN} €‹JVDGLLGKMP|Ê+± JrNpSB4UTF¬JVNO}MDGPUTF!P zNOLqJVKMFKMp¬JVNLRlIKM}MUTDGpKJ!€GK–}MDGFF!KMLqJ *iÆECEP J UTRTK•UÇLGW
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hCEF–JV€G,
K 7KjU 5I4UÇ} p}jUÇJJVK&F!NLR }jUTpK 7J!€GKÕ}&DGF!FKMLqJlUÇJ MK&F!CJVK&Á•BKMF!UÇJVDF!KÕÁ•Uj2NLG}&F!KMUTp!,
K H€KML
JV€GKÉÆTCEPOJVUTREKÉIGNUTpÅNp–PC 7KMFKM)W H€GNOPKÉNOL2J!€GKepJ!F!CELGRI4UT} ip}jUÇJJVKMFNLGR}jUTpKlJ!€GK I(V ) NOpŀGNRT€GPO
LGCELGPONLGKMUTFMʱ J©Np;J!€iDGp½NOÁ•BCEF¬J UTLqJ;J!CUTWGWGFKMp!p„J!€GKNp!pDGKSC WGKMB€4UTp!NOLG;
R hF!CEÁ U–ÍD$J!JVNOLGREK&F„PN‚iDGNOW+Ê
Ô K&F!K)7K–}MCELp!NWKMFSJV€GK–}MUTp!K–C7p!NOÁ•BGPOKÅÍUTDGRE€PNL$hFVUT}¶JVNCTLGpHN JV€ 4POPNLR UT}&J!CEF ν = 1/m m
C$WW>NLqJ!KMREK&F ¶Ê3®°pNL>¯©K bÊ ' à I)(93J!€GK;WKMBG€4UÇp!NLR"C U°p¬J UÇJ!K;NOLxJV€K;WGCTJNp NLGWDG}MK&WÂIq>N JVp }jUTB4UÇ}MNOJ!NOÆTK
}MCEDBGPNOLGRxJ!C>JV€GK°IGNUTpKMWv‚iD4UÇLiJ!DGÁ BCENOLiJ;}MCELqJVUT}&J GUTpp!DGÁNLGRxJV€4UÇJ7JV€GK°PK&ÆTKMP Á•CWGDGPUÇJVNOCEL•NOL JV€K
WGCTJHNOpHU•¸UTDGpp!NUTLeBGF!C}MK&p!p½UTLGWlLGKMREPOKM}¶JVNLRÅI4UT} E´bUT}¶JVNOCELe%K KM}&J!pMÊ
±µL J!€GNpe} €4UTB$JVKM%F ;K HNPOP 4FpJl}MCEÁBGDJ!K:JV€GKWGKMB€4UTp!NOLGR FVUÇJ!K:NL JV€GK 7KjU UTLWˆp¬JVF!CTLGR
I4UT}ip!}MUÇJ!J!KMFNLGRPNÁNOJHNOJ!€ J!€GKlUTp!pDGÁ•B$JVNCTL J!€4UÇJ–JV€GK ³ CEDGPOCEÁÂI2NLqJ!KMFVUÇ}&JVNOCEL Np–p!}&F!KMK&LGKMW Ii
LGKjUÇF!IqSÁK&J UÇPPNO}REUÇJVKÇ~MJV€GK ³ CTDGPCEÁ>ISNLqJVK&FVUT}¶JVNCTLSNp+JV€KMLrF!KMWDG}MK&WrJ!C½U„WGK&POJVU hDGLG}&J!NCELBCTJVK&LqJVNUÇP Ê
²½K $J 7K HNPOPK $J!KMLGW‹CEDF"F!K&p!DGP JVp°J!C JV€GKÅ}MUTp!KÅC ›UTF!IGN JVF!UTFlI4UT} ip}jUÇJJVKMFNLGR 9DGpNLGRvJ!€GKÂK UT}¶J
p!CEPODJVNOCELÕC ›J!€GKÅICEDGLGW4UÇF‹ÀNLGK–¸SCEFWGKML:ÁC$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!CÕJV€K }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
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0>'(! O- '
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$ !#"%$'&(*),+.-0/'1324$5276*),839':06;27$3),<=9356>(*9':%?#6*@)A6>(*2'?*BC"%?D?*"093E"0F6>@),GH(*2'<=6*"%932':JIK&L27K6*&BNMO2':%:KPL&"0Q
6>@(D93&[email protected]*@)SIT&2'56>&BVUW93"%56X<=9356;2'<Y6Z-\[[email protected])S?*"0$3:0)4:%)Y83)]:^MO2'B_"0:06*93"`27CGH93(A6*@)SQ9'6a(*)]2'Q?
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*@)~$5276>)SUW9'6>)=56>"`27:9'GhIK&L2'56*&B€Uw93"056<]9356;27<=6Z1\93.)‚<]2'ƒ? b "06*<;@„GH(*93B 2 b )Z2'†
pL2'<;†K?*<]276*6*)](D"%$ƒ?*"06*&L276*"%93f1 b @.)](*)C6>@)_Mm27:%:g:%"0IK&"%Qz(*)=B‡2'"0?S"%z93)CU"%)=<]) #"0$3&(D)_+J-0/ˆ2 16>9ƒ2
?D6*(*93$!pL27<;†T?D<Z276D6>)=(*"%.$‡?*"n6>&L2c6>"%9' b @)](D)~6*@)EMO2':%:‰:%"%IK&"0Qo"%??DU:%"n6X"%{6 b 9 Š"%$3&(D)S+.-n/Zp -.ƒ6>@.)
GH93(*BC)](!<]2'?*)71k6>@.)„)]56*"06>"0)]? b @"%<;@‹6*&)=:42'(D)„)=Q$3)„IK&L27?*"%U2'(D6*"%<]:0){)YŒJ<]"n6;276*"%93?=-ky6>@)ƒ:`276D6>)](
<Z2'?D)'1wpw)Y6 b )=)]6>@)‚6 b 9oPL&."%Q?=1w93.:0iu)]:0)]<=6*(*93.?4<Z2'ƒ6>&.)]:Ž-wMO)=(*)71 b )e<]93?D"%Q)=(OxL(D?D646*@) b )Z2'†
pL2'<;†K?*<]276*6*)](D"%$<]2'?*)71X2'Q6*@)] b )‘&?*)’2“Q.&L2':%"n6i|6>(>27?DGH93(DB‡2c6>"%9' •” /c1,–3+ 6>9“Q)=?*<=(*"%pW)‘6>@.)
?D6*(*93$!pL27<;†T?D<Z276D6>)=(*"%.$_<]2'?*)7-W[[email protected])S6*&)=:%"0$CMm2'BC"%:n6>93"%2'‡pW)=6 b )])=„)=Q$3)]?~/~2'Q„rF(*)Z27Q?]—
'
Ht = eiω0 t Γ0 ψ2† (0)ψ1 (0) +
M‚-˜<'-
,
(
+.- ”
b @ )](D) b )[email protected])m&.?*)]QC6>@)O™Š)="%)=(*:%?d?D&p?D6*"06*&.6>"093e6>9~"%<=:%&Q)X6*@)a8'93:06>2'$3)7—'GH93(g6>@) b )Z2'†epL27<;†T?D<Z276jš
6>)=(*"%.$1
1
"0?v6*@)d)=›w)]<Y6>"08')k<;@2'(*$3)q2'.Q 6>@.)dx:%:%"0$,Gœ2'<Y6>93( 1 b @"0:%)
GH9'(
ω0 = e? 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"0ƒ6>@.)E<Z27?*)F9'G b )Z27†ƒpL27<;†T?D<Z276jš
√
√
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"%$R<]2'?*)~6>@)‚)]:%)=<=6*(*93‘9'Uw)=(>276*93("%?a93p.6;27"%)=Q b "06*@„6>@.)
?*&p.?D6>"n6>&.6*"%93
[[email protected])aMm2'BCν"%:n6>→
93"%1/ν
2'~Q)]?D<](*"0p"%.$S6*@)"%56>)=(>2'<Y6>"093f1327?*?*&.B_)=QC6*9~pW) b )Z2'†\1Kpw)Y6 b )=)]C6>@)Q9'6q27Q
IK&L2'56>&BUw93"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_)=Qƒ6*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>9‡6*@)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"%$9'GO2')=:%)=<=6>(D93?D6>276>)R"02‘Q9'6!<=93&U:0)]Q“6*9’2„PL&<=6*&L276*"%$<=&(*(D)]56!"%?F<]2'&?D)]Q
p5i 6>@.)R)]:%)=<=6*(*93Q)=?*"n6i“PL&<Y6>&L2c6>"%9'?]1 b @"%<;@|$3)=)](*276>){2‘PL&<Y6>&L2c6>"%.$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)YŒJU(D)]?*?D)]Q‘"0R6>)](DB_?X097G#"0(*(*)=Q&<="%p:0) <;@L2'(*$')SP&<=6*&L276>"093?X"0{6*@)~2'Q *2cš
<])=K6 b "0(*) "%R6*@) GH(>2'<Y6>"%9'L2':‰IK&L27K6*&B€Mm27:%:f)Y›‰)=<=6Z1L"0? b (*"n6*6>)=„27? r'+J1/3/ˆ–
'
τϕ−1
1
=
4
Z
∞
dt
−∞
Z
Z
dxf (x)
(
A
+.-
dx0 f (x0 )hhρ1 (x, t)ρ1 (x0 , 0) + ρ1 (x0 , 0)ρ1 (x, t)ii .
‘93(DB‡27:Š27Q?*&UW)](D<]93.Q&<=6*"%$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<ƒ"056>)](*2'<=6*"%93‰1k"n6‡"0?o<]9358')]"0)]56R6>9“&.?*)‘6>@.
) )]:%[email protected] 2'UU(D952'<;@ ˜”3” 1k+ -, 6‡"%Jš
Q&<=)]?,6*9E$3)=)](*276>)a6>@)O?DiTB_BC)=6*(*" )]Q‡<;@L2'(*$')mQ)=?*"n6i Q)]?D"06i_<]93(D(*)=:`276*93(]156;27†T"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')YŒJ<]"n6;276*"%93 b @"%<;@.)])]Q.?46>9oU(D93UL2'$3276>)~6>9
x = 0
6>@)E:%9T<Z276*"%93u9'Gg6>@.)FQ9'6]-‰[[email protected])F)=IK&"%:0"%p(D"%&B
)](D9RUw9'"%56 <]9356*(*"%p.&.6>"093„6>9_6>@)FQ.)][email protected]?*"%.$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
) ŒJ"%?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>@) )=(*97š 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" $_6*@)eIK&L2'?D"%UL2'(j6>"0<]:%)‚<]93.?*)](j87276>"093f1‰6*@T&.?46>@)e)=IT&."nš
γ = −γ
:%"0p(*"0&B Q)][email protected]'?*"0$_(*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$3)
"%?
i i = 1, 2 Gηi 1 η2 (x−x0 , t1 −t2 ) = hφi (x, tη11 )φi (x0 , tη22 )−
-5[[email protected])O<=9J) <]"%)=56>? 1
"0Q)]56>"nG iE6*@)O&.UUw)=(q2'Q‡:09 b )=(qp.(>2'<;@.)]?,97Gw6*@) )=:%Q.iT?*@C<]9'Jš
φ2i i
η η1,2 = ±
6>93&.(]\9'(m?*)=<]93.Q 93(DQ)]( <=9356>(D"%p&.6*"%93ƒ6>9o6>@.)eQ)][email protected]'?*"0$R(>276*)'1w?*"%.<])
2'(*)E"%Q)=Uw)=Q)=K6~"0
ψ1 , ψ2
6>@)~2'p.?*)].<])~9'GŠ6*&)=:%"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$3)=93B_)Y6>(jio9'GŠ6>@)S?*)Y6>&U{8T"`2F6>@.)‚:0)]$'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)]QƒpKiu(D)]U:%2'<]"0$
ν → 1/ν
)YŒT6α6>9e6*@
) S(D)])]
•?GH&<Y6>"%9' Q&L2':0"06i [[email protected]
) S(D)])]
•?GH&<Y6>"%9'ƒ2c6XxL"06*)46>)=B_UW)](*276>&(D)S"%?X$3"n83)]u2'?

0
G1ηη (x, t) = − ln 
sinh
h
π
β
(x/vF − t)
η+η 0
sgn(t)
2
sinh
−
η−η 0
2
+ iτ0
i 
iπτ0
β
η + η0
η − η0
sgn(t) −
2
2
+.-n/ ”
 .
ƒ6*@) )](D9_6>)=B_UW)](*276>&.(*)~:%"%BC"06]1 b )E) ŒJUL2'Q„6*@)
HG &<=6*"%93ƒ"%56>9_2![#2ZiT:%93(a?*)=(*"0)]?O2'Q
6;2'†7)]?O"%56>9‡2'<]<=93&56493.:0io6*@)E:09 b ]) ?j6493(DQ)](a<]9'K6*(*"0sinh(·
p&.6>"093f· ·1 )b F
) 93p.6>2'"%{6>@.) (*)])=˜?OGH&<Y6>"093„2c6
)=(*97šŽ6>)]BCUW)](>2c6>&(D)S2'?"%
)=G - ˜”3” —
' (
0
Gηη
1 (x, t)
= − ln τ0 + i(t − x/vF )
, *, '!- % # * : #% .!,- ! #!- 1!'!
!
1# !,-
+.-n/
.
)
[[email protected])R2'?D?*&BCU.6*"%93“97G?D6>(D93$?D<](*)=)]"0$
"0?FB‡27Q)'1v6>@L2c6FBC)Z2'?‚6>@) ,9'&:%93BEp
0
"%56>)=(>2'<Y6>"093
2'? b )[email protected]')4<;@93?D)]{2'pWλ9ˆs83)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 )oxLQ6>@L2c6 −|x|/α Q)]<=(*)]2'?*"0$‘Gœ2'?D6E6>9 )=(*9 b "n6>@“6>@.)
f (x)
λs ∼ α b
872':%&)E9'G "%?m93Jš "%.xL."06>)=?*"0B‡2': -\’6*@"%? <]2'?*)71"0646*&e(*.? 93&.6
2'Q6>@K&? √ 2
x
xd
x + d2 ' d
[[email protected]) ,93&:093BFp{"%56>)=(>2'<Y6>"093R"%?X(D)]Q&<=)]Qu6>9e6>@)SQ)=:06>2eGH&.<=6*"%93f1
α
f (x) ' 2e2 δ(x) .
d
?
+J-0/
M 9 b )=83)=(]1 b )!xLQ’6>@276S6>@"0?‚27?*?*&.B_U.6*"%93’"0? 9'6~)]<=)]?D?>2'(ji31Š2'Qz"06 b "%:0:hpW)C(D)]:%2cŒJ)]Qz:`276*)]( 39 fO
?*)=(D6*"%$6>@.) S(D)])]•?eGH&<Y6>"093|"%|lgI\- +.-n/ ” "0“6>@){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'.$3)‚97GŠ872'(D"`2'p:0)]?
1
1\27Q
1 )~9'p.6;2'"0f—
τ = t 1 − t 2 τ1 = t − t 1
τ2 = t 2 b
(τϕ−1 )(2)

sinh2ν πβ iτ0
sinh2ν βπ iτ0
h
h
i+
i
dτ cos[ω0 τ ]
=
2ν π
2ν π
−∞
sinh β (ητ + iτ0 )
sinh β (−ητ + iτ0 )
η
Z ∞ π
π
× dτ1 sgn(τ1 ) coth
[−ηsgn(τ1 )τ1 + iτ0 ] + coth
[ητ1 + iτ0 ]
β
β
−∞
Z ∞ +.-n/
π
π
[ηsgn(τ2 )τ2 + iτ0 ] + coth
[ητ2 + iτ0 ]
.
× dτ2 −sgn(τ2 ) coth
β
β
−∞
e 4 ν 2 Γ2
− 2 2 20 2
4β π vF d
XZ

∞
‚6*@)k"056>)]$'(>2':59ˆ83)=( 1 b )k<;@2'$3)q8'27(*"`27p:%)=?v6*9
GH93(6>@)dxL(*?j6 ?*)]<=93Q 6>)=(*Bu1
τ
t = −τ ∓iτ0 ±iβ/2
2'Q‡6*@)m"056>)]$'(>2':\9 b (*&.?A"%C6>@)O<]93BCU:%) Œ_U:`2'.)aGH93(DB
6*9
−∞ ∓ iτ0 ± β/2 +∞ ∓ iτ0 ± β/2
)Ep(*"0$R"06mpL2'<;†ƒ6*9
pKi„Q)=GH9'(*BC"%$‡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/ˆ+
+.-n/ˆ–
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/ˆ+ b "06>@!6>@.)OpL27<;†T?D<Z276jš
6>)=(*"%.$e<]&(D(*)]56O93"0?*) "%ulgI\- - 2':0?*9!<]93BCUL2'(D"%$!lgI\- +.-n/ˆ– b "n6>@ƒldI\- - 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]?*"%.$e(>276*) "%?AU(D93UW93(D6*"%93L27:W6*9F6>@) pL27<;†T?D<Z276jš
6>)=(*"%.$R<]&(D(*)]56‚93"0?*)'1 b @."%<;@z<Z2'zpw)!&.Q)](D?D6>9T9TQ p5i976>"%<="%$o6*@)!<]9356*"%K&"06i‘)]IK&L276*"%93 b "06*@
2'?*?D&BCU.6>"093C6*@L276q6>@)O)]Q$')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$3)=?D6k&?g6>9~)YŒT6>)=Q‡9'&(dU(*93p.:%)]B 6>9S6>@)<]2'?*)a9'G‰2'(*p"n6>(*2'(DiEpL2'<;†K?*<]276*6*)](*"0$
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>)=(*"%.$E<Z27?*)'—56*@)O)=IK&"%:0"%pJš
(*"0&B€<]9356>(D"%p&J6>"%9'‘"%‘6*@)FQ)[email protected]'?D"%${(>2c6>)F"0?O6>@.)e?>2'BC)e2'? "0„6>@) b )Z27†„pL2'<;†K?D<Z276D6>)](D"%${<]2'?*)7[[email protected])e93.)]IK&"%:0"%p(D"%&BV<=9356>(D"%p&.6*"%93„<Z2'pW)e93p.6>2'"%)=Q„GH(*9'B 6*@) b )]2'†„p2'<;†K?*<Z2c6*6>)=(*"0${<Z2'?D)epKi
(*)=U:`2'<="%$
(D)]<]2':%:‰6*@L276
"0?XQ)=xL.)]Q„"0u2!Q."0›w)](*)=56B‡2'.)]("%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':ˆ6>9X6>@) )](D97š 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^+ J1v–3+ -v[[email protected])!6>@) š
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]ˆ8')FpW)])]‘8')]( š
"0xL)=Q "0’Uw93"056S<]9356;27<=6‚)YŒJUw)=(*"0B_)=K6*?E2c6~x:%:%"0$oGœ2'<Y6>93(
.1 /31 r -[[email protected]"%?S"%?S&Jš
ν = 1/3, 1/5
Q)](D?D6*9J9TQ GH(D93B 6>@.)ƒ<=9356>"%K&"n6i|)]IK&L276*"%93‰1 b @"%<;@(D)]:%276>)=?!6*@)u<]&(D(*)]56‡93Uw)=(>276*93(e6>9z6>@)uQ)]Tš
?*"n6iR93UW)](>2c6>93( / r - A6 )](*9C6>)=B_UW)](*276>&(D)'16*@)‚.93)]IK&"0:%"0p(*"0&B Q.)][email protected]?*"%.$_(*276>)~9'GglgI\- +.-n/ˆ– GH93( b )]2'†pL2'<;†K?D<Z276D6>)](D"%$„Q)]UW)]Q.?E93’6>@)CIK&L2'56>&.B UW93"0K6~<=9356;2'<Y6Ep"%2'? b "06*@z6*@)_) ŒJUw93.)]56
-v[[email protected]"%?~"%?~"% ?*@L27(*U“<=9356>(>27?D6 b "n6>@ a)YG - r'+ 1 b @)](D)_6>@.)‡IK&L2'56>&.B UW93"%56‚<]9356>2'<=6
2ν − 1 < 0
p"`27?4Q)]UW)].Q)]<=)‡"0?4:%"0)Z2'(=- )!2':%?D9{<Z2':0<]&:%276>)FK&BC)](D"%<]2':%:niu6*@"%?m<]9356>(D"%p&J6>"%9'’276 xL"n6>)F6>)=B!š
Uw)=(>276*&(*)=?a2'Qu<]93?D"%Q)=("06a2'?a2FGH&<Y6>"%9'ƒ9'G^6>@.)Sx:%:%"0$FGœ2'<Y6>93(X93(A6*@)~IT&2'56>&BVUW93"%56<=9356;2'<Y6
8393:n6;2'$')ap."`2'?=-5o93&(qK&B_)=(*"0<Z2':L<]2':%<=&:`276*"%93.?]1 b )m<;@9T93?D)O6>@.)O"0K8')](D?*)4<=&.6>9'› −1 2'?q6>@)a)])=(*$'i
?*<]2':%)R27Q“6>@)R.93)]IK&"0:%"0p(*"0&B <]9356>(D"%p&J6>"%9' GH93(E6*@)oQ)[email protected]'?D"%$’(>2c6>)o"0?FU:%τ9706*6>)=Q"%“&."06*?e9'G
e4 Γ20 τ0 /(π 2 ~4 vF2 d2 )
#"0$3&(D)e+.- rJ1 b )CU.:%9'646*@)!Q)]UW)].Q)]<=)‡9'Gd6*@"%? <=9356>(*"0p&.6*"%9393’6>@.)FxL:%:0"%$‡Gœ27<=6>9'(
GH9'(
ν
pw976>@ b )]2'†|2'Qy?D6>(D93$zpL2'<;†K?*<]276*6*)](D"%$“<]2'?*)=?CGH9'(C?D)=8')](>27:a6*)]BCUw)=(>276*&(*)=?
276
β = 5, 10, 50 x.ŒJ)]Q‘IK&L27K6*&B Uw93"056m<=9356;2'<Y64p"%2'?]- "0?O<=93?*"0Q)](D)]Q‘@.)](*)F2'?42C<]9356>"0K&93&?a8'27(*"`27p:%)71 b @"%:0)‚"n6
@L2'[email protected]?D"%<Z27:fB_)]2'"0$F93:nio2c6tv2'&[email protected]ν:0"%CGH(>2'<Y6>"%9'? / 5r -LL93(q6>@) ?j6>(D93$epL2'<;†K?D<Z276D6>)](D"%$!<]2'?*)71
6>@) Q.)][email protected]?*"%.$e(>276*) "%<=(*)Z27?*)]? b @.)]{6>@)4xL:0:%"0$‚Gœ2'<Y6>93(,"%<=(*)Z27?*)]?=- A6?*B_2':0: 1."06A"0? )=(*9 6>@)=f1
"06E"%<=(*)Z27?*)]?!(*2'U"%Q.:0i3-^[[email protected]){@"[email protected])](‚6>@)o6*)]BCUw)=(>276*&(*)71#6>@)oGœ27?D6>)=(F6>@)R"0<](D)Z2'ν?D)'-gL93(‚6>@) b )Z2'†
pL2'<;†K?*<]276*6*)](D"%$’<Z27?*)'1h6*@)R?*@2'Uw)R97GO6*@)RQ)][email protected]'?*"0$’(>276*)oQ)=Uw)=Q?C93“6>@)R(*276>"099'GaIK&L2'56>&B
Uw9'"%56,<=9356;2'<Y6Ap"%2'?k2'Qo6>)=B_UW)](*276>&.(*)'- A6X:09 b 6>)=B_UW)](*276>&(D)
1K6>@)OQ)][email protected]'?*"0$F(>2c6>)
eV GH&<Y6>"%9'[email protected]?,2~:%9T<Z2':LB_2cŒJ"%BF&.BN276
136>@)OUw9'?*"06*"%93!9'G b 1/β
@"%<;@‡
Q)]UW)]Q.?,93C6>)]BCUW)](>2c6>&(D)'—
ν < 1/2
b @)] 6>@)!6>)=B_UW)](*276>&(D)_"0<](D)Z2'?D)]?=1Š"n6‚$')=6>?~<]:093?*)=(~6*9 ν = 1/2 2'.Q "06*?‚@.)]"%$'@K6~Q)=<](D)Z2'?D)]?]1Š6>@)
(>276*){276
"0?e?*B_2':%:0)](E6*@L2'6>@276C276
-#[[email protected]"%?!(*)=?*&:n6!Q)]BC93?j6>(>2c6>)]?e6*@L276!GH93(E6 b 9
ν = 1
ν = 1/3
Q"0›w)](D)]56E
x:%:%"0$RGœ2'<=6*93(*?=1 b )o<Z27“@L2Z83)R<=93B_
U2'(>2'p.:%)CQ)][email protected]'?*"0$‘(>2c6>)]?=- O(*93&.Q 6>@)‡<](*9'?*?*9ˆ8')](
"%„6*)]BCUw)=(>276*&(*)
1W6>@)e:09J<]2':vB‡2 Œ."0BF&B€"%ƒ6>@)eQ)[email protected]'?D"%$R(*276>)Ep(*9527Q)]?=- A6 @"[email protected]
'
' ( !
'
(
(
' (
' (
"!
!
#!
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)
1‰6>@.)_Q)[email protected]'?D"%${(*276>)!"%.<](*)]2'?*)=? b @.)] 6*@)exL:%:0"%$_Gœ2'<=6*93( "%<=(*)]2'?*)=?] )xL.Q 6*@L276R1/β
6>@.)z>
Q.)][email protected]?*"%.$y(*276>)=?u)Y872':%&L2c6>)]Q 276{Q."0›w)](*)=56u6*)]BCUw)=(>276*&(*)=?{<]93"0<]"0Q)276{6>@.)
&[email protected]?D"%<]2': 872':0&)
1,pW)]<Z27&?*)6>@.)@KiTUW)](Dpw93:0"%<‘<=93?*"0)BF&:n6>"0U:%"0)]Q pKi‹6>@)?DIT&2'(*)=Q
ν = 1/2
B_9TQ&:0&?,97GŠ6*@) ~27B_B_2‚GH&<Y6>"093R"%{lgI\- +.-n/ˆ+ Q9J)=?9'6Q)=Uw)=Qƒ9'o6>)]BCUW)](>2c6>&(D)'1 b @"0:%) 2c6
6>@)‚?>2'BC)‚6*"%BC)~6*@)F) Œ.UW93)=56
"%? )](D9—w6*@"%?O"%?a†K9 b ’GH9'(mUW)](D6*&(*p276>"n83)E<]2':%<=&:`2c6>"%9'?
(2ν − 1)
9'G^6>@)SpL2'<;†K?D<Z276D6>)](D"%$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$3)"%?dU:%976*6>)=QeGH93(d?*)=8')](*2':W6*)]BCUw)=(>276*&(*)=?]-K!6>@)a<Z2'?D)
9'GŠ?D6>(D93$!pL2'<;†K?D<Z276D6>)](D"%$.16>@)SQ)[email protected]'?D"%$C(>276*) "%<=(*)]2'?*)=? b @)=ƒ6*@) p"`27?
"0<](D)Z2'?D)]?=- @)]
6>@)S6>)=B_UW)](*276>&.(*)~"%?a:%9 b )].93&[email protected]
1 L6>@)‚Q)][email protected]'?*"0$‡(*276>)‚?>276*&(>eV
2c6>)]?=-wƒ6>@)~<Z2'?D)F9'G
@"%$'@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 2„1/β
xL"06*)‡
872':%&.) 976e?*@9 b 1 b @"%<;@|"0?FU(*9'Uw93(j6>"093L2':g6>9‘eV
6*@)o6>)=B_UW)](*276>&.(*)'[[email protected]"%.$3?S2'(*)!IK&"n6>)eQ"n›‰)=(*)]56‚276 b )Z2'†p2'<;†K?*<Z2c6*6>)=(*"0$- [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*93B‡2c6>"%<e97GX<]&(D(*)=K6F2'Q 93"%?D)
<;@L2'(*2'<=6*)](*"0?D6*"%<a"%R2‚tv&J6*6>"0$3)=(,:%"0IK&"%QfeV
-5_6>@)4:09 b š 6>)=B_UW)](*276>&.(*)a<Z2'?D)
1TGH9'(k?DB‡2':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'Q‡6*@)mGœ2'?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)]?*&.:069'G a)YG - r7+ -
!
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!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>"nš
K&"06iu)]IK&L2c6>"%9'‘(*)=:`276*)]?O6*@)E<=&(*(D)]56 93UW)](*276>9'(O6>9_6>@)‚Q)]?D"06iƒ93UW)](>2c6>93(O2'?4"[email protected]':%Q?a"%‘?D)]<=93Q
–
A
IK&L2'56>" )]Q{GH93(DBƒ—
ρ(x, ω) =
+J-srJ/
1
∇.J(x, ω) .
iω
L(*93BVlgI\- +.- rJ/ 1 b )E<]2' b (*"n6>)‚2_<=93)=<=6>"093„GH93(DBF&:%2!pW)=6 b )])=6>@)‚)]Q$3)‚<]&(D(*)=K6m93"0?*)E<=93(jš
(*)=:`276*93(X2'QR6*@)SQ)]?D"06i5šQ.)]?D"06i{<]9'(*(*)=:`276*93(276XxL."06>)4GH(D)]IK&)=<=iƒ2'?
hhρ(x1 , ω)ρ(x2 , −ω)ii =
Z
+∞
−∞
dt iωt
e ∂x1 ∂x2 hhI(0)I(t)ii .
ω2
+.- r3r
{93&(<Z2'?D)'16*@)SQ)]?D"06i5šQ.)]?D"06i{<]9'(*(*)=:`276*"%93.?A"%?X<=93?*"0Q)](D)]Q b "06*@
-[#27†T"0$e6>@)SQ)=(*"n8'2 š
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"%$‡2‡xL"n6>)E<=9356>(D"%p&.6*"%93u6>9o
6*@)FQ)=?*"n6iƒPL&.<=6>&276>"093?]-w"%2_Uw)=(D6>&.(*pL276*"08')F<Z2':0<]&Jš
:`276*"%93 9'Ga6>@)o6*&)=:%"%.$’Mm2'BC"%:n6>93"%2'f
- O976>)o6>@276C276 )=(*96>)=B_UW)](*276>&.(*)'1Š6>@)R<=&(*(D)]56o27:%93$
6>@)S)=Q$3) b "06*@93&.6pL27<;†T?D<Z276D6>)=(*"%.$_Q9T)]?976B‡2'†7)[email protected]'6a93"0?*)716>@)=u6*@) PL&<=6*&L276*"%93?9'G^6>@.)
<]&(D(*)=K6*?e2':093$ƒ6*@)‡)=Q$3)=?!2'(*)‡2':%?D9„"%Q.)]56>"0<Z2':g6>9ƒ6>@)‡P&<=6*&L276>"093?‚9'GA6>@.)‡6>&.)]:0"%$ƒ<]&(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>@.)Yš 27?>276
6>)=<;@"%IK&)=? /3/ 1f/ˆr3rJ1f/ r ” MO)](D)'1^pw)YGH93(*)C?*@.9 b "0$ƒ6*@)C(*)]:%276>"093’pw)Y6 b )=)]“6*@)‡Q)[email protected]'?D"%$ƒ(>276*)‡27Q 6*@)_<=&(*(D)]56F.93"%?D)
GH93(2'(Dp"06*(>2'(ji‡pL27<;†T?D<Z276D6>)=(*"%.$1\ b "0:%:‰(D)]Q)=(*"n83)S6>@)S$3)=)](*2':)YŒJU(D)]?*?D"%93ƒ9'G^6>@)S<]&.(*(*)=56O.93"%?D)‚27?
?*@9 b _"% a)YG - /3/ -KC6>@"0? b 93(D†W1K6>@.)m2'&J6>@93(D?k<=93?D"%Q)=(q6*@)aGH(*2'<=6*"%932':IK&L2'56>&.B MO2':%:LBC9TQ)]:
√ -L[[email protected])
"%R6>)=(*BC?9'G#)Y83)=2'Qu9JQ.Qƒ:0)=G 6Dš B_9ˆ8T"%.$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"%$3"0L2':T:%)=G 6jšŠ2'QC(*"[email protected]šBC9ˆ8T"%$
√
√
)]Q$')]?4p5i
2'Q
"0?a6*@)E6>976;2':^<;@L2'(D$3)E93„pw9'6*@
∆Q = Q1 − Q2 = 2Qo
Q1 + Q2 = 2Qe Qe
)]Q$')]?k2'QC"%?d<]93.?*)](j83)=Qo)=8')]o"0C6*@)U(*)=?*)].<])m97G‰6>@.)O"0K6*)](*2'<=6*"%93f-'[[email protected])Op2'<;†K?*<Z2c6*6>)=(*"0$E<=&(*(D)]56
6>@K&?eQ)=Uw)=Q?!93:ni 9'|6*@)R9TQQpw9'?*936>@.)]93(ji3-h[[email protected]){Q)=?*<=(*"%UJ6>"%9'9'Ga93&(FBC9JQ.)]:,"0“6>)=(*BC?e9'G
IK&L2'?*"0UL2'(j6>"%<=:%)=?O27:%:%9 b ?&?a6>9‡<]2':%<=&:`2c6>)S)YŒ.2'<Y6m6>(*2'?DUw93(j6mU.(*93UW)](j6>"%)=?]-W56>)]$'(>2'p"0:%"n6io)].?*&(D)]?46>@.)
)YŒJ"%?j6>)=<])R9'GO2‘IK&L2'?D"%UL27(D6>"0<]:0)_pL27?*"%? b @)](D){6*@)o?D<Z276D6>)](D"%$"0?F93)‡pKi 93)7-h[[email protected])]?D){(*)]?D&:06*?e2'(D)
†K9 b ‡"% )=GH?=- / r 1L/ r 1L/ˆr -KL93(d2'5i 156>@.)a?DUw)=<=6*(*&B <]9'K6>2'"%."%$S2S†K"%†e27Qo2'_2'56>"0†K"%†
ν
b "06*@„6>@)E<;@L2'(*$')]? Qo = 1/√2 2'Q −1/√2 (*)=?*UW)]<Y6>"08')]:ni31‰"0?m<;@L27(>2'<Y6>)](D" )]Qp5i„6*@)F(>27U"%Q"n6i θ
Q)=x)]Qup5i
1 @)=(*)
"0?2'u2'(*p"n6>(*2'(Di‡?*<Z27:%)'E = −pvF = M eθ /2 b
M
@)]‡2~UW93?*"n6>"n83)X8393:n6;2'$')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-Š GA6*@)=izQ9ƒ9'6~"%56*)](>27<=6Z16*@)‡†K"%.†T? b "%:0:hxL:%:d27:%:gBC93B_)=56>&B ?j6;276*)]? b "06>@
p<
276 )](D9ƒ6>)=B_UW)](*276>&.(*)'-[[email protected])_UW93?*"n6>"093’9'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'.$3)]Q Q&)ƒ6>9“6>@.)‘"%56>)=(>2'<Y6>"093f-d G b )Q)YxL)
"0?_6*@)‘Q)=?*"n6iy6*@)]
ρ(θ)
GH93(
"%?A6*@)S?*@"nG 69'G^6>@)~L)](DB_"f:0)=8')]:Žb "06>@
ρ(θ) = 0
"06>@.93&.6SθpL2'><;†KA?D<Z276D6>)](D"%A$„"%56>)=(>2'<Y6>"%9'f1‰6>@.)_<=&(*(D)]56F2c6 )](D9ƒ6>)=B_UW)](*276>&.(*)C2'(*"0?*)=?~GH(D93B 6>@.)
†K"%†K?B_9ˆ8T"%.$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$‡<=&(*(D)]564"0?X6>@)~(>276*)~276 b @."%<;@ƒ6*@)~<;@L2'(*$')‚9'GŠ6>@)~:%)YG 6Dš B_9ˆ8T"%$e)=Q$3)~"%?Q) š
√
U:%)Y6>)=Q
-L{6>@.)‚IK&L2'?D"%UL27(D6>"0<]:0) pL2'?*"0?]16*&)=:%"%.$C<=93(*(D)]?DUw93.Q?
IB = ∂t [e∆Q/2] = ∂t [(e/ 2)Qo ]
6>9e6>@.)~U.(*9T<])]?D?9'Gg2e†K"%†‡?*<Z2c6*6>)=(*"0$!9'›‘6>@.)~<=9356;2'<Y6a"%56>9C2'u2'56>"0†K"%†\ )4<=93?D"%Q)=(A6>@)4"0B_U.&(*"n6i S B_276>(D"nŒ_)]:0)]BC)]56 Sjk(p/TB ) b @"%<;@{Q)=?*<=(*"%pW)]?2F?D"%$':%)mIK&L27?*"nš
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'<Y6ƒ"056>9|2
IK&L2'?*"0UL2'(j6>"%<=:%)F97Gk6jiTUW) -f[[email protected])C)])=(*$'i p <;@L2'(*2'<=6*)](D" )=?~6*@)!<]9356>2'<=6
UW)](j6>&( š
k
TB
TB ∝ λ1/(1−ν)
pL276*"083)S27(*$3&BC)]56X97GŠ6*@) (*)=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
B‡2c6>(*" ŒS)]:0)]BC)]56>?#2'(*)qGH&<Y6>"%9'?
θB
TB = M e /2
' (
–
S
9'G
θ − θB
' ) (
-.[[email protected])E}k)=6>@.)Yš 27?>276
6>)]<;@."%IK&)~$3"08')]?X6>@.)S6*&)=:%"0$eU(*9'pL2'p"0:%"06i‡2'? / 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)].Q‘93{6>"%BC) ?*"0<])S6>@.)
I I(V, TB ) = I0 (V ) + IB (V, TB )
?DiT?D6*)]B€"%?a"%ƒ2!?D6>)]2'Q.iu?D6>276>)7-wuGœ2'<=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(*)=B‡2'"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- ) <Z27ƒQ)=?*<=(*"%pW) 6>@)SQ<[email protected]'693"0?*)4GH(D93B 6*@)SIT&2'?*"0UL2'(D6*"%<=:%) 2'U.U(*9527<;@ƒpW)]<]2'&?*)S6>@.)
?*<]276*6*)](*"0$‡97›z6*@)FUW93"%56a<]9'K6>2'<=64"%?a)]:`27?D6>"0<E27Q„93.)Ep5i„9')':0)=G 6OB_9ˆ83)=(mpL27<;†T?D<Z276D6>)=(*? "%56*9
2C(D"%[email protected]BC9ˆ83)](a<]9'(*(*)=?*UW93Q?a6>9_2'u9JQQTšpW93?*93u†K"%†‡?*<Z2c6*6>)=(*"0$_"056>9_2'„27K6*"%†K"%.†W-L G b )‚Q)=x)
@)] 2„†K"0† 97GB_9'B_)=K6*&B
?*<]276*6*)](*?e"056>92'“2'56>"%†K"0†\1#2'.Q
"0GX"n6F?*<]276*6*)](D?
f = 1 b
"%56>9‚2‚†K"0†e6>@)=‡6>@)m2Z83)](*2'$3)O9 8')](AB_2'p5i!)=8')]56>?X"0?
-2 Tf‡=6>@0)aIK&L2'?D"%UL2'(j6>"0<]:%)
2'UU(D952'<;@f1v6>@.)o93"0?*)‡"0?~6*@)]U.(*93UW93(D6*"%932':h6>9„6*@)_hf
P&i<==
6*&L27|S6>"0+−
93“(p/T
9'G B— )|
f C(0) ∝ (hf 2 i − hf i2 )
J"%.<])
"%?)]"n6>@)=( 93( 1 2
1 b )[email protected])
'” 1
!
f
0
1 hf i = hf i
Z
2
C(0) = e vF
L93:%:09 b "0$‚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
) <Z27U&:%:6*@)
93&.6497Gg6>@)F"056>)]$'(>2': - a?*"0$‡6>@.)
ρ
A
θB b
∂ θB
)YŒJU(D)]?*?D"%93.? +.-sr ” 2'Q J+ -sr GH93(
'2 .Qƒ976>"%<="%$‚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*@L276BC)Z2'?
' ?(
V
∂ I(V, TB )
V 2 ∂ I(V, TB )
=−
,
∂TB
V
TB ∂V
V
6>@)= b ) xLQƒ2'9'6*@)](XGH93(DBF&:%2EGH93(
)=(*97šŽGH(*)]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)~"058393†7)~<=&(*(D)]56O<]93?D)](j8'2c6>"%9'ƒ2c6X6>@)SUW93"%56<]9'K6>2'<=66*9CQ.)](*"n83)~2!$3)])=(>2':
S
GH93(*BE&:`2GH93(!6>@)ƒQ)]<[email protected])](D)]<=)(>276*)'1 b @"%<;@ Q.)]?*<=(*"0pw)=?‡6>@)ƒ<](D93?*?D9 8')](oGH(D93B 6>@) b )]2'† 6*9 6>@.)
?D6*(*93$!pL27<;†T?D<Z276D6>)=(*"%.$_(D)]$3"0B_)71
(τϕ−1 )(2) =
+J- ” r
e3 τ02 ν
(V Gdif f − I) ,
d2 1 − ν
lgIT&276>"093 +.- ” r 27:%:%9 b ?a&?m6*9‡Q)=?*<](D"%pW)E6>@.)E<](D93?*?D9ˆ83)]( "0ƒ6*@)FQ)[email protected]'?D"%$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
$3)].)](>27:^(D)]?D&:06m276xL"n6>)S6>)]BCUW)](>2c6>&(D)‚pW)]<]2'&?*)~6>@)‚<]&(D(*)=K6m"%{6>@)‚2'p?D)]<=)F9'G#pL2'<;†K?*<]276*6*)](*"0$
PL&<Y6>&L276*)]?a276xL"n6>)46*)]BCUw)=(>276*&(*)7-
,*, '!-1) .!/*( # % , :1!.!,-
)‡9 b <=93?*"0Q)](F2'$52'"0 6>@.)RUw)=(D6>&.pL276>"n83)o<]2':%<=&:`2c6>"%9' 9'G6*@)oQ)[email protected]'?D"%$(*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‘$39pW)=i39'Q6>@)R?j6>(*9'$
?*<=(*)])="%.$_:0"%BC"06Z1T2'.Q b )~<Z2'u<]93BCU&.6*)SlgIW- +.-0/ˆr GH93(2'ƒ2'(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:%) š 6*"%BC)Š"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.(*93UJš
τ0 ω0−1 , β b i
h
i
h
)](j6>"%)=?g9'G‰GH&.<=6*"%93
2'QeGH&(j6>@)=(d?*"%BCU:0"nš
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'›Š-, G
6>@) ,93&:093BFp "%56>)=(>2'<Y6>"093y†7)](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':‰) ŒJU(*)=?*?*"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)=&B‡27‘GH&<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"%<
$5276*)]? 2': b 2ZiT? "%BCUW93?*)=? 2‡xL"n6>)F?D<](*)=)]"0$u:%)].$'6>@fQ)=<](D)Z2'?D)]? b "06*@
2'Qz2'UU(D952'<;@)=?
F
d/λs
1
b @)] λ "%?m<=:%93?D)‚6*9o6>@)‚?*UL276*"`2':^<]&J6>9'› α ?j6>(D93$o?D<](*)=)]"0$ -‰[[email protected])FQ.)][email protected]?*"%.$R(>276*)E"%.<](*)]2'?*)=?
s
b @)]{6>@.)~?D<](D)])]."%$CQ)]<=(*)]2'?*)=?]-
$ [Š9S?D&B_B_2'(D" )71 b )[email protected]')O)=?D6;27p:%"0?*@)=Q‡2 $3)=)](*2':GH93(*BE&:`2mGH93(h6*@)Q)][email protected]'?*"0$~(>276*)a97Gf2SIK&L2'56>&B
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