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Approximation des Phases Aléatoires Self-Consistante
dans le Modèle de Hubbard
Steffen Schäfer
To cite this version:
Steffen Schäfer. Approximation des Phases Aléatoires Self-Consistante dans le Modèle de Hubbard.
Physique mathématique [math-ph]. Université Joseph-Fourier - Grenoble I, 1998. Français. �tel00002990�
HAL Id: tel-00002990
https://tel.archives-ouvertes.fr/tel-00002990
Submitted on 12 Jun 2003
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G CJ· eDGK ·^aC G™N ä 2 c G C ã NOC BÕ^a`fC w+º N ‰ ^ e C j ^ KDK G E g¹NB_^ HJHue c e g eiL GsE jDG ·NOgz·I`Dg G C H `DC g GH ^aC jDeiK N LJG `DC H"jDG g º ì KDHuLJeiL ` L 87`D C
`DN KMLdGKfG g G 6 L CJ^ KDe 6 jfG gzN 2 â `D C e · ã»ÿ
2 K )DK ` K aCJN KDj wyG CJ· e’N ä w"GIH BªNOC GKMLdHI„ w"GIH N w"eiHˆGILN ä NÞC ezG=$ ^ H GE G ì gzN w ^6O^ „ñ¶ ` e
wº ^ KML NOBDB_^aC L GhE L ^a` L · G¤j ^ KML } º N ‰ N ezH c GH ^ ezK B_^a`DCbCFG E NOg ezHJG Cb· G8L CJN ‰ N e g ÿ
G
‰Feie
‰Fezeze
0 á N KDH g GH{jDG C KDe Gä C GH j G E · GKDKDeiGHI„ g G ! " Nx}u^a`{G E ` K CÚ^av g G ·ICJ`D· e NOg_BÕ^O`DC gzN
·^ w BDCFGIE ãDGKfHJe ^ K"jDGIH,H ! HuLAGä wyGH j¿º G E g G · L Cu^ KDH 8ü^aC LJGwyGKML ·^aCuCFG E g¸GE H,H `DC,CFGE HuG NO` ÿ*#{ezGK'¶ ` GxL Cu^aB
HJezw BDg ezHuLJG BÕ^a`fC ` KDG NOBDBDg e ·N Lde ^ K"jfe C G · LJG NO`(BfCJ^acDg Gä wyG G E g G · L Cu^ KDei¶ ` G#j»º ` KsH ^Og ezjfG CFG E NOg ezHuLJGO„
e gûBDC GKDj gzNyBfg¹NO· G¬j¿º ` Kw ^ j»Gä g G 9GE K G E C ei¶ ` GO„ªLdG g H g GH w ^ j»Gä g GIH jfG ì HJeiK ^O` jDG ~GIezHJGIK c G C
BÕ^a`fCbg GHxH ! H LAGIä w"GIH~jDG¤H B ezKDHI„ C GIH B G · LJei‰OGwyGKMLA„ ·gzN HJHuez¶ ` GIH ^a` ¶ `ªN KMLJez¶ ` GHIÿ
1ûGyw ^ jûGä g G jDG `DcDcªNÞC j´H · ã GE w N LJezHJG gzN+BDCu^aBªNaN Lde ^ KGIL g º¼ezKMLdG CdNO· Lde ^ KjDGH G E g G · L CJ^ KfH
j N KDH ` KH ^ag eijDG ·C eiHuL NOgzg ezK´N"ä cDN KDjDGIH#j»º GE KDG C eiG G¸E L Cu^ eiLJGHÿ á G E } N ä H N ‰aG C HJe ^ K gzNyBfgz` H HJeiw BDg GO„
g Ghw ^ j»Gä g G#jfG `DcDcDNOC j­Nˆä ` KDG cªN KDjDG N ‰aG · jfGHxj G E Bfg¹NO· GwyGKMLdH G E g G · L CJ^ KDez¶ ` GIH g eiw"eíL GE H NO` ½
H NO` LJHQGKML C G BDC GwyezG C Hñ‰ ^ ezHuezKDHñj `ˆCFGE HJG NO` ·C ezH L NOgig ezK¿„ BÕ^ HuHGIä jDG ` K jDe NaCJN w"wyG jDG B ã N HJG{L C GIä H
C e · ãfGO„ ·^ KMLJGK N KML jfGH B ã N HJGIHQw N K G¸E Ldez¶ ` GIH„ËezH ^ag¹N K‹LJGHI„Úw GIE L NOgzg ez¶ ` GIHûGILQH `DBDCJNO·^ Kfj `D· L C e · GHIÿ
ì g Hº¼GH L CFG¸E ‰ G E g¸G E BªNOC Lde ·I`Dg e Gä C GwyGKML)ezw BÕ^aC L N KML)j N KfH g¹N¬·^ w BDCFGE ãDGIKDHJe ^ K(jDG g¹N L CdN KDHJeíLde ^ K
w G¸E L NOg $ ezH ^ag¹N K‹L GK GKDj CFGE G BªNOC jDGH~ezKMLdG CdNO· Lde ^ KDH G E g G · L CJ^ K $ G E g G · L CJ^ K»ÿ?> GILuLdG$# %&(') # ) & ) # *#+#,¼‡¦„ " NOBDBDNOCdNv LA„ gi^aC Hu¶ ` º ^ K 8éN eiL'‰ NOC ezG C'gzN˜BDC GHuHJe ^ K ^a`gzN LJGw B{G E CJN L `fC GO„,j N KDH
BDgz` HuezG `DC H ^ ½ ! jfGHˆjDGIH’w GIE L NO` ½jfGyL CdN KDHueiLJe ^ K»„¿K ^ L N w"wyGKML¬j N KDH g G HuGHu¶ ` e ^ ½ ! jfGsjDG ‰ N $
K N jDe ` w t -.0/ Š1 ƒ" „Éj ^ KML g¹N w ^ j G E g ezH N Lde ^ K C G · ¶ ` ezG C L ` K­w ^ j»Gä g G'jDG/ `DcDcDNOC j.N ä jDG ` ½
cªN KDjDGIH2 | "óÿ
1 N j G E ·I^a` ‰OG C LJGˆjDG g¹N H `fBDCdNO·I^ KDj `f· Ldeí‰FeiL GsE N ä ã NO` LJG¬LJGw B G E CJN L `DC G ·C eíLdei¶ ` G t436587 O† #¬Š
j N KDHhjfGH G E · ã N K‹LJe gigz^ KDH~w `Dg Lde B ã N H GIE H#jDG 1 N # N > `9. „ BªNÞC ÿ # GIjDK ^aC ! GIL # ÿ ; ÿ `D gzg G C GK
€a‚ ': '*" „ N HuLdeiw `Dg¸G E gzN8C G · ãfG CJ· ãfG8H `DC jDGH{w N L G E C e NO` ½N¬ä ^ ½ ! jDG#jfG ·` ei‰ C G N ‰aG ·~g¹N H ! KMLdã»GIä HJG
GIL g º GIE L ` jDG)jDG ·I^ w BÕ^ H GE HûLdG g HÓ¶ ` G 1 N*-<;>= µ C%= > `[email protected] „ N ‰OG ·CB 7 †(D  | „ ^a`0E # NF- > `?/G.0H%;>I „ N ‰OG ·
J 7 †(D  DDKDJ†LD | ÿ > GHbH `DBDCdNÞ·^ KDj `D· LJG `DC H~w ^ jfG C KfGH H ^ KMLbL ^a` H 8ü^OC w GE H jfG BDg¹N KfH > `9.A- ¶ ` e
HJG j G E ·C ei‰aGIKML BªNOCñg G c e N eiHÉj ` w ^ j»Gä g G)jDG `DcDcDNOC jN ä L CJ^ eiH cDN KDjDGIHM „ *"ó„ G E MNOg GwyGKML ·I^ KDK `
H ^a` H g G8K ^ wUjDGON1 P N GQ ‚F"óÿ

‡
Ó1 G"w ^ jûGä g G"jDG `DcfcªNOC j­eiKMLdG C ‰FezGKML NO` HJHue{j N KDH gzN jDGIH ·C e B LJe ^ KÍj»º NO` L C GHˆw N L G E C e NO` ½
Nä G E g G · L CJ^ KfH 8ü^aC LJGwyGKML ·I^aCJCFG E g¸GIE H„ ·^ wyw"G g GIH 8ü`Dgzg G C Gä KDGIH ^a`g GIH NOgig e N GHˆN(ä cªN HuG jDGˆLdG CJC GH
CdNOC GHI„ NOBDB G g¸GE H G C wye ^ KfH 1 ^a`DC jDHf„9GK CJN ezH ^ KSjDG gzN aCJN KDjDG(w N HuHJG(G G · Ldeí‰aGsjDG g G `DC H
BÕ^OC LdG `DC HxjDG · ã NÞC G: „ª€F"óÿ
2,K­j G E B eíL jDG(H N HJezw BDg e · eíL G'E HuL CJ`D· L `DC G gig G G¸L w NOg aCFG E g GHˆG ¿^OC LdH™j GIE ‰OG gz^OBDB{GE H j N KDH g GIH
L Cu^ ezHxjDG C KDe Gä C GIH#j G E · GIKDKDezGIH„ g¹N"·I^ w BDCFGIE ãDGKDHue ^ KjDG™HuGH BfCJ^aBDC e G¸E L GE H C GIHuLJGˆezK ·I^ w BDg G¸ä LdGOÿ KDG
H ^agi` Lde ^ KG½ NO· LJGK»º G½¦ezHuL N KML¶ ` ºüN­ä ` KfGjDezwyGKDHue ^ K ø † " „{e g g¹N K · G ` K[j G5E )DL GIE K ^aC wyG NO` ½
w G¸E Ldã ^ jDGIH'N ä ·^OCJB HIÿ á G¬K ^ w cfC G ` ½ 8ü^aC w NÞg ezHuw"GIH ^ KML G¸E L G’E j GIE ‰OG gz^OBDB{GE H#GIL NOBDBDg ez¶ `{GE H NO`
w ^ jûGä g G jDG `DcDcDNOC jÍGILsN ä j¿º NO` L C GIH’H ! HuLAGIä w"GIH™jDG 8 G C wye ^ KDH 8ü^OC LdGIw"GIKML ·I^aCJCFG E g¸GE H„ · ã NO·` K
N ‰aG · HuGH B_^ eiK‹LJH 8ü^aC LdH™G¸L HJGIH 8éN e cfg GHuHJGIH „ a" ÿ þ G E N KDw ^ ezKDHI„ NÞ` $ jDG g N ä jfG g¹N Ldã G E ^aC eiG j `
· ã N w B w ^(! GK»„ ` KDG NOBDBfCJ^F· ãDG5N ä ·^OCJB H ·I^ wyw ` K GIE w"GIK‹L NO·· G B L GE Gw N KD¶ ` G }u` Hu¶ ` ºüN ä
BDCFGIE HJGIK‹Lÿ
á N KDH · GILuLdG´LJã»Gä HuGSK ^a` H BDCu^aBÕ^ H ^ KDH ` KDGÍK ^a` ‰OG gzg GÍw GIE LJã ^ jDG´¶ ` e¬Hº¼GKMLdGIKDj ·^ wywyG
ñGE K G E CdNÞg ezH N Lde ^ K+jDe C G · LJG™jDG g¹N LJã G E ^aC ezG NOC L C GIG5$ ^F·76+NO` ½ 8ü^ K · Lde ^ KDH#jDG ·^aCuCFG E gzN Lde ^ KDH N ä ·^aCuB Hÿ > GILuLdGhw GIE LJã ^ jfGO„Fj ^ KML g GH cDN HJGIH ^ KML GIE L G E GIE L NOcDg ezGH)j N KfH g GIH N KDK GE GH ' † ˇ¦„_ „_AƒF"ó„
HJGhHuG C LbjDG g º32 ¶ `ªN Lde ^ K jfG á ! H ^ K ·^ w"wyG B_^ eiK‹L jDGhj G E BªNOC LAÿ 2 K C G¸LdGIK N KML ` KDei¶ ` GwyGKML g GIH
·^ KML C e cf` Lde ^ KDH#eiKDHuL N KML N K GE GH#jDG g º ^aB{G E CdN LJG `DC jDG w N HJHuGO„¿G gzg G’K ^O` H B G C wyGILhj»º ^Oc LdGIKDe C~` KDG
G¸½ BDC GHJHue ^ K+G¸½ BDg e · eiLdG B_^a`DCbg G BDCJ^OBªNMN LJG `DC N ä ·^aCuB Hÿ
> G 8ü^aC w NOg ezHJwyG NÍGIE L G E NOBfBDg ei¶ `{G#E GIL GIE LdGIKDj ` eiKDj G E B GKfj N wyw"GIK‹L BªNOC jDe G E C GKMLdH OCJ^a`DB GHI„
H `DC L ^a` LÍGIK B ã ! Huez¶ ` G K `f·g¸G E N e C GOÿ > G g¹N G¸½ BDg ez¶ ` G ¶ ` º¼e g H ^ eiL ·I^ KDK ` H ^a` HjDGH NOBDB G gig¹N $
Lde ^ KDH NÞ` HJHue»jDe xEG C GKMLdGIH ¶ ` G Ô Ì<; µ G g"8 $ > ^ KDHuezHuL N KMLdG D„ > gz` H LdG C G N K eiG g j ^a` GIK ·^OC G
[G¸E Ldã ^ jDG¤jDG g º32 ¶ `DN Lde ^ Ksj ` ^a` ‰OGwyGKML øˇF„» „û | „û('¦„û F„û‚f„¿€F"óÿ
á N KDH g Gxw ^ j»Gä g GbjDG GezHuGK c G C™N KMLde 8 G CJCu^ w N K G¸E Ldez¶ ` G™N¤ä ` KDG jDeiw"GIKDHJe ^ K ‡ † "_e gDC G BDCu^ $
j ` eíL g GbH B G · L C GyN8ä ` K w N K ^ K'jDG g¹N H ^agi` Lde ^ K G¸½ NO· LdGބ + "!$#&- % Huez"K ' % ÿ ; jfG ` ½yjDeiw"GIKDHJe ^ KDHI„
· G 8ü^OC w NOg eiHJwyGO„¶ ` GbK ^a` H NOBDB G gzg G Cu^ KDH9j ^aCFGIE K N ‰ N K‹)L (+*-,/. 02123 &(')' # & # t µ > Ô Ì<; Š¸„aGIHuL
GK cÕ^ K NÞ··^OC j N ‰OG · jDGH ·ANÞgz·`fg H jDG ^ K‹LJG5$ > NOCJgi^ `ªN K‹LJez¶ ` GÞÿ
á G · G 8éN eíLA„ g º NOBDBfg e ·N Lde ^ K NO` w ^ j»Gä g GjDG `DcfcªNOC jžNÍä ` KDG cªN KDjfG N ‰aG · ezKMLdG CdNO· Lde ^ K
CFG E BD`Dg Huei‰OG BªNOCdNv L BªNÞC Lde ·`Dg e Gä C GwyGKML N LJL CJN!MN KMLdGÞÿ 2 K B ã ! HJei¶ ` G"jfGH¤H ^ag ezjDGIH„»jfGH NOBDBDCu^ ½fe,$
w N Lde ^ KDH Huezwye g¹N e C GIH ^ K‹L G¸E L G E ` Lde g eiH GE Gj N KDH g G BªN HJH G E N )ªK jDGj GE j ` e C G g GH BDCu^aBDC e GIE L GE HSN ä
` KDG BªNOC Lde ·`Dg GO„¿GIK ·I^a`DBDgzN KML g G BDCu^aBªNMN LdG `fC N+ä ` K G E g G · L CJ^ K˜jfGyw N KDe Gä C G NOBDBDCu^ ½¦eiw N LJei‰OG
NO` ½ 8ü^ K · Lde ^ KDH¬jDG4 C GIGK.N ä L CJ^ ezH ·^aCuB H ¼‡¦a„9‡a‡F„9‡ " ÿ Ì ^a`fC™g GIH BDCu^aBªNaN LdG `DC H’jDG BªN e C GIH
jDG 8 G C w"e ^ KDHI„¿G gig G K ^O` H ·^ Kfj ` e CJN „ ·I^ w"wyGˆK ^O` H g Gˆ‰aG CJCu^ KDH BªNOC8gzN H ` eíLdGO„ Nsä ` KH ! H LAGIä w"G
j»º GE ¶ `ªN LJe ^ KDHxeiK‹L G E aCJNOg GIH K ^ K $ g eiK G E N e C GIHÿ
; `­BDC GwyezG C™NOc_^aC j»„ñK ^a` H K ^a` H g eiw"eíLdG CJ^ KDH NO` w ^ j»Gä g G"jDG `DcfcªNOC j¯Nä ` KDGsjDezwyGK $
HJe ^ K»ÿ þ ^O` H BÕ^O`DCJCu^ KDH N ezKfHJe ·I^ w BDNOC G C K ^ H CFGIE H `Dg L N LJHˆN'ä gzN H ^agi` Lde ^ KsG¸½ NO· LdGބ BDCJ^ ‰aGIK N KML~jDG
g º N KDH N L ! jDG #{GILdãfG  † "óÿ ; jfezwyGKDHue ^ K BDgi` H G E g GI‰ GE Gބ ^ K'KDGbjDeiH B_^ HJG BDN H j»º ` K LdG gªC G B Gä C G NOc $
H ^agi` ÿ á G BDgi` H g¹Nˆ·^ w BDg G¸½¦eiL G8E K ` w G E C ei¶ ` G¬jfGH GE ¶ `ªN LJe ^ KDH µ > Ô~Ì?; NÞ` wyGKMLdG CdNÞB ezjDGIw"GIKML
N ‰OG ·yg¹N jDeiw"GIKDHJe ^ KH BDN Lde NOg GÞÿ ì g HIº N ‰ÕGä C G"j ^ K · ezKDjDeiH B GKDH NOcDg Gˆj»º NO· ¶ `{G E C e C¤` KfGyG½ B G E C ezGK · G
GKL CdN ‰ N e gzg¹N K‹L j N KDH gzNˆg ezwyeiLJG ` KfezjDeiw"GIKDHJe ^ KDKDG gzg GÞÿ
1 º¼ezK ·I^ KM‰ GE KDeiGKML jfG · GILJLJGH L CJN L G E ezGGH L jfG ·I^ w"wyGK · G C'BªNÞCyg G ·AN H ^ ` ä g G ·ANÞCdNO· LAGä C G
¶ `DN KMLdez¶ ` G GHuL g G Bfgz` H’w N Kfe 8 GH LdGބñG¸L ^ `Sä g GH ·I^aCJCFG E gzN Lde ^ KDH¬H ^ KML g GH Bfgz` H BfCFGE HuGKMLdGIHÿ > G=$
B GIKDj N KMLA„ÓK ^ L C G'w GIE LJã ^ jfGyHuGyL CJ^a` ‰aGyGK NO··I^aC j N ‰OG ·yBDgz` HJezG `DC H8LJã G E ^aC Gä wyGH¬ezw BÕ^OC L N K‹LJH„
BªNOC wye g GIHJ¶ ` G g H K ^a` H · eiL ^ KDH · G gz` e jDG 1 ` LJLdeiK G C ¼‡ÞƒD„¦‡ | " ^O` g¹N8C Gä ag G~jDG~H ^ wywyG B_^ KDj G E CFGIE G
BªNOCbg º GE KfG C ezGOÿ
2 K · G+¶ ` e ·I^ K · G C KDG g¹N H ^Ogz` LJe ^ K K ` w G E C ei¶ ` GO„,K ^a` H"K ^O` H C GHuL C GIe K ^ KDH„,j N KfH · GILJLJG
Ldã»GIä HJGO„ N ä g¹N ("* , & N )' ÿ 2 K¯K G E ag e G N K‹L g GH ·^aCuCFG E g¹N LJe ^ KDH´N ä jDG ` ½ ·I^aCJB Hj N KDH
g º ^aB{G E CJN LdG `DC jDG,w N HJHuG jDG g¹N µ > Ô Ì<; „ · GILuLdG NOBDBDCu^ ½feiw N Lde ^ K 8éNO· e g eíLdG gzN H ^agi` Lde ^ K8K ` w G E C ei¶ ` G
BªNOC9CJNOBDB_^aC LxN#ä gzN µ > Ô Ì<; „ÚL ^a` LQGIK MNOC j N KML9HuGH N H B G · LJHQHJG g"8 $ ·I^ KDHuezHuL N KMLdHIÿ þ G BDC GIK N KML BDN H
GK ·^ KDHJeij G E CJN Lde ^ K g GIH ·^OCJCFG E g¹N LJe ^ KfH#jDeíLdGH, N # N & #é„ · º¼GHuL%$N ä $ jDe C G · G gzg GH~L CdN KfH 8AG E CJN KML
` KDG™eiw Bf`Dg Hue ^ K '! ‡ '9„_K ^a` HhKfG BÕ^O` ‰ ^ KfH a` Gä C G™GIH B{G E C G C8C G BfCJ^ j ` e C G’jfGH B ã GE K ^ w+Gä KfGH
L !¦B ez¶ ` GH B_^a`DC{g GIH w ^ j»Gä g GIH ` KDeijDezwyGKfHJe ^ KDKDG g H„aLJG g H ¶ ` G g G ·^ w BÕ^OC LdGIw"GIKML jDG g ez¶ ` eijDG jDG
1 ` LJLdeiK G C ¶ ` e NOBDBªNOCJNËv L j N KDH g¹N H ^agz` LJe ^ KsG½ NO· LJG’jDG gzNˆ· ã NËv KfG¬j ^aB{GE GÞÿ
> G B GIKDj N KML„fw Gv wyG gzN Ô Ì<; C GIK ^aC w NOg ezH GE GxK ^a` H 8ü^a`DC KDeiL ` K · G C L N eiK K ^ w cDC G jDG CFGIE H `Dg $
L N LJH ezKML G E C GIHJH N K‹LJH„ H `fC L ^O` Lxj N KDH gzNˆg eiw"eíLdGxj `·^a`fBDg¹N G 8ü^aC L jDG g¹Nˆ· ã Nv KDG8jDG `DcDcªNÞC jSN ä
w ^ eíLde G E C Gw BDg eiGOÿ á N KDH · GILuLdG g eiw"eíLdGxK ^a` H C GIL CJ^a` ‰ ^ KDHI„9N™ä ` K 8éNO· LdG `fC BDC GIä H„ gzN jDezH L C e cD` LJe ^ K
jDGHxezw BD`Dg HJe ^ KDH G¸L g º GIE KDG C: ezG 8ü^ KDj N wyGKML NOg G8jDG gzN H ^agi` Lde ^ KsG¸½ NO· LdGOÿ
þ ^a` H H ^a`Dg e K ^ KfHQGK ·I^aC G ` KfG 8ü^ ezHI„O¶ ` G · G¸LJLdG LJã»Gä HuG C G BfCFGE HuGKMLdG ` KDG BDC Gwye Gä C G NOBfBDg e ·AN $
Lde ^ K"j»º ` KDG K ^a` ‰aG gig G w G¸E Ldã ^ jDG N¬ä g º ` K"jDGIH w ^ j»Gä g GIH,j»º G E g G · L Cu^ KDH 8ü^aC LJGwyGKML ·^aCuCFG E g¸GE H)¶ ` eÕHuG
L CJ^O` ‰aG NO`(· GIKML C G#j»º eiKML G E CGIv L jDG g¹N¤·^ w"w ` K NO` L G#E jDGIH B ã ! HJe · ezGIKDH jDG g¹N w N LJe Gä C G ·^ KDjfGKDH GE G
jDG BD` ezH)jDG~K ^ w cDC G ` HuGH N KDK GE GIHÿ µ `DC L ^a` L8N¤ä ` KDG~jDezwyGKDHue ^ K¿„ae g KDG Hº N eíL BªN H)jDG w"G¸LJL C GxGK
·ANO` HuG g GH CFGIE H `Dg L N LJH ^ac LdGIK ` H BDNOC j»º NÞ` L C GH 8ü^aC w NOg eiHJwyGHñH ^a` ‰OGKML wyezG ` ½ N j NOB L GE H„‹w N ezH9jDG
LdGH LdG Chg¹N'B G C:8ü^aC w N K · G’GIL gzN 8éN eiH NOc e g eíL GhE jfGˆK ^ L C G NÞBDBDCJ^F· ãDGÞÿ 2 ` G E MNÞC j NÞ` ½ NOBDBDCu^ ½¦eiw N $
Lde ^ KDH eiK ·^aCuBÕ^OCFGE GIH„DHuGH BDCFGE jDe · LJe ^ KDH ·^aCuC GH BÕ^ KDjDGKML BDgi` L ^ v L¬Nˆä ` KH ·_GE K NOC e ^ ñGIE K G E C ez¶ ` GބD¶ ` G
g º ^ KsGH B Gä C G C G¸L Cu^a` ‰aG C GK+jDezwyGKDHue ^ KfH BDgz` H G E g GI‰ GIE GHÿ
Î õøÅ ÈËÑ_ÏDÒñèéïÏfùËèüÅ_Ò¥ÙÓÊÝaÊMùAùÊùËØ{ÊFî ïÊïÊ Ïªèúù"ïÊFôéÅ_Ò¥ôüÊ+ð9ôüϪқïAßÉèüÐfϪÒÕù ; `›BDCu^¦· ã N ezK · ã NOB eiL C GO„ K ^a` Hsj ^ KfKDG Cu^ KDH ` KDG cfC GIä ‰OGezKML Cu^ j `f· Lde ^ KNSä gzN Ldã G E ^aC eiGjDG gzN
CFG E B_^ KDHuG g ezK G E N e C GhG¸L NÞ` ½ 8ü^ K · Lde ^ KDHxjDG C GGIK»ÿ
ƒ
Ó1 G · ã NOB eíL C G GIHuL ·I^ KDH NO·CFG˜E Nä gzN+BDCFGE HuGKML N LJe ^ K­jfG g¹N Ô Ì<; µ G g"8 $ > ^ KDHuezHuL N KMLdGÞÿ þ ^a` H
·^ wywyGK · G CJ^ KDH BªNOC H N j G E C ei‰ N Lde ^ KNSä BªNÞC Lde C jDG g º 2 ¶ `ªN LJe ^ K›jfG á ! H ^ K»ÿ KDG BDC Gwye Gä C G
NOBDBDg e ·N Lde ^ K NO`(BDCu^aBªNMN LdG `fC N™ä ` KDG BªNOC Lde ·I`Dg G~K ^a` H N wGIä KDG CJN N™ä gzN Ldã G E ^aC ezG NOC L C GIG5$ ^F·76 ÿ
2 KDH ` eíLdGބÕK ^O` Hhj G¸E LdG C w"eiKDG Cu^ KDH g G BDCJ^aBDNMN LJG `DC~BªNOC LJe ·I`Dg G=$ L Cu^a` jfG g¹N µ > Ô Ì<; GIKLdG C wyGH
j»º ` KSH ! H LAGä wyG 8 G C w G E j»º GE ¶ `ªN LJe ^ KfH™ezKML G E aCJNOg GIH’K ^ K g eiK G E N e C GIHÿ þ ^O` H™G½ B_^ HuG CJ^ KDH H ^ K g ezGIK
N ‰aG · ` K BDC ezK · e B G)‰ NOC e N LJe ^ KDKfG g GIL9jDezH ·` LJG CJ^ KDH g GIH9jDe G E C GIK · GIHQGKML C G g¹N µ > Ô Ì<; G¸L g¹N Ô Ì<;
C GIK ^aC w NOg ezH GIE GO„ N eiKDHJe_¶ ` G g º NO··I^aC j N ‰OG ·8g¹N’C Gä Og G#jfGhH ^ wywyG B_^ KDj G E CFGE G BDNOC g º GE KfG C ezG#G¸L g GIH
BDCFGIE jDe · Lde ^ KDH B_^a`DCbg G BfCJ^aBªN MN LdG `DC N'ä ` KDG BªNOC Lde ·I`Dg GÞÿ
; ` j G E cD` L9j `™· ã NOB eiL C G)ƒ~K ^a` H BfCFGE HuGKMLdG CJ^ KfH g G,w ^ j»Gä g G,jDG `DcDcªNOC j»„ËHJGIH BDCu^aBDC e GIE L GE HÉGIL
H ^ K CÚ^Ov g G GK B ã ! HJez¶ ` G{jDGHÉH ^Og ezjfGHÿ þ ^a` H BªN HuHJG CJ^ KDHñGK C GI‰ ` G)¶ ` G g ¶ ` GH ` KDGHÉjDGIHñw GIE LJã ^ jfGH
NOBDBDCu^ ½feiw N Ldei‰OGH g GH BDgz` H ` Lde g eiH GE GIH„ N ‰ N KML8j¿º G¸E L ` jDeiG Chg GHhH ` H · G B LJe c e g eíL GE H~jDG · ã NÞC G G¸L8jDG
H B eiKGK µ > Ô~Ì?; GIL GK Ô~Ì?; C GK ^aC w NOg eiH GIE GOÿ
; `S· ã NÞB eiL C G | „ñK ^O` H™j ^ KDKDG Cu^ KDH ` KDG cDC GIä ‰OGsezKML Cu^ j `f· Lde ^ K NO` ½Íw ^ j»Gä g GH ` KDeijDezwyGK $
HJe ^ KDKDG g ÿ þ ^a` H BDCFGIE HJGKMLJG CJ^ KDH g G `fC H BªNÞC Lde ·`DgzNOC eíL GIE H’G¸L g GIH BDCu^acDg GIä w"GIHˆH ^O`Dg G¸‰ GIE H gz^aC HJ¶ ` º ^ K
LdGIK‹LJG ` KDGxjDGH ·C e B LJe ^ K'j N KfH g G ·AN j C GxjDG g¹N LJã G E ^aC ezGbjDGIH g ez¶ ` eijDGH9jDG G C wye ÿ2,KfH ` eíLdGO„MK ^a` H
jDezH ·` LJG Cu^ KDH gzN H ^agi` Lde ^ K jfG g¹N Ô Ì<; C GK ^aC w NOg eiH GIE G B_^a`DC gzN™· ã Nv KDGhjDG `DcDcDNOC j»„fGK 8ü^ K · $
Lde ^ K j `sC Gw BDg eiHJH N G#G¸LbjDG gzN’·I^ KDHuL N KMLdG8jDG ·^a`DBfg¹N GÞÿ þ ^a` H ·I^ w BDNOC G CJ^ KDH{HJGIH CFGE H `Dg L N LdHI„
K ` w G E C ei¶ ` GH,GIL N K NOg9! Ldez¶ ` GIH„¿N¤ä g º N KDH N L !bjDG # G¸LdãDG GIL NO` ½ ·NOgz·I`Dg H jfG ^ K‹LJG5$ > NOCJgi^ `ªN K $
Ldei¶ ` GOÿ 1 N(g ezwyeiLdG¤j `·I^a`DBDgzN G 8ü^aC L8K ^a` H B G C wyGILJL CdN jfG j G E C ei‰aG C jDGH¤G½ BfC GHuHJe ^ KDH N K NOg9! $
Ldei¶ ` GH BÕ^O`DC L ^a` H g GIH G E g¸GIE w"GIK‹LJH¤jDG"K ^ L C G'LJã G E ^OC ezGÞÿ á G Bfgz` HI„ · GILuLdG g eiw"eíLdG™K ^a` H¬j ^ KDKDG CdN
` KDG BDC Gwye Gä C G8ezj GIE G8j ` w G E ·AN KDeiHJwyG8ezw BDg ei¶ ` G#E j N KDH g¹N L CJN KDHJeíLde ^ K ^ LuL:$ `DcDcªNÞC j»ÿ
0 á N KDH · G · ã NOB eiL C Gބ_K ^a` H CFGE H ` w ^ KDH gzN LJã G E ^aC ezG¬jDGH 8ü^ K · Lde ^ KDH~jfG C GGIKGIK ·^ wywyGK· N KML
BªNOC#` KDG cDC G¸ä ‰aG w ^ Ldeí‰ N Lde ^ K BfCJ^ ‰aGIK N KML¤jDG g º NOBfBDCJ^ ½¦ezw N Lde ^ K+jDG g¹NyCFG E BÕ^ KfHJG g eiK G E N e C GÞÿ?2,K $
H ` eíLdGhK ^O` H CdNOBfB G gzg G CJ^ KfH g GIH G E g¸GE wyGKMLdH · GKML CJNO` ½j `48ü^aC w NÞg ezHuw"GxjDGIH 8ü^ K · Lde ^ KDH jDG C GIGK
GIL CdN HuHJGIw cDg G CJ^ KfH{¶ ` G g ¶ ` GH BDCJ^OBDC e GIE L GIE H N K NOg"! LJez¶ ` GH,¶ ` e K ^a` H{HuG CJ^ K‹L ` Lde g GIH j N KDH g GIH · ã N $
B eiL C GH{H ` eí‰ N KMLdHÿ ; g¹N )ªKsjDG · G · ã NOB eíL C GO„fK ^a` H jDeiH ·I` LdG CJ^ KDHbGIKs¶ ` G g ¶ ` GHbw ^ LdH ·^ wyw"GIKML
g G 8ü^aC w NOg eiHJwyG j ^ eíL GIv L C G N j NOB L G E B_^a`DC GIv L C G NOBDBfg e ·NOcDg GN ä jDGHˆH ! HuLAGIä w"GIHN ä LdGIw B{G E CdN L `DC G
)ªKDezGÞÿ Ì ^a`fC jDGIH’¶ ` GIHuLJe ^ KDH BDgi` H’j GIE L N e gzg¸GE GIH„»K ^O` H C GKM‰ ^(!O^ KDH g G g G · LdG `DC Nä g¹Nsg eiLJL G E CJN L `DC GO„
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j C G(N ä · G¸LJLdG¬¶ ` GH Lde ^ K»„ªG¸½ N wyezK ^ KDH ` K+H ! HuLAGä wyG’j G E ·C eiL BªNÞCb` Kã N w"e g L ^ KDezGIK
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jDG & C KDe Gä C G BDCJ^ ‰FezGIKML{j¿º ` K · ã N w B G½FL G E C ezG `DC „ t Š¸„O¶ ` e_j G E B GKDj"j ` LJGw B H,GIL{¶ ` eªGH L ·I^a`DBDg¸G E
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aÿ 1 N 8ü^ K · LJe ^ Ksj»º ^ KDjfG % t Š H N LJezH 8éN eiL™N ä g º GE ¶ `ªN Lde ^ K+jfG µ · ã C(^ jfezK G C ¾
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; )ªK¥jfGK ^a` H"j G E cDNOCJCJN HJHuG C jDG g¹NÍBfCFGE HuGK · GjDG & j N KDH t ‡Fÿ Š„,K ^a` H BDN HJH ^ KDHÍN´ä gzN
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L `DCuc{GOE ÿ 2 ‰FeijDGwyw"GIKMLA„ g GIHQ‰ NOg G `DC H w ^(! GKfKDGH ! KDGxH ^ KML BªN H N G · L GE GIH BªNOC,· G¸LJLdGbL CdN KDH $
8ü^aC w N LJe ^ K»ÿ % t Š bH N LJezH 8éN eiL’N'ä ` KDG GIE ¶ `DN Lde ^ Kj ` L !FB G µ · ã C(^ jfezK G C
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g G 8ü^ Kfj N wyGKML NOg K ^ K B G C L `DCuc GOE „ % & I„û¶ ` e,GH L™eiKDj G E B GKfj N KMLˆj ` LdGIw B H ·NOC GIK g º NOc HJGK · G
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> GILuLdG ·I^ KDjDeíLde ^ K NO` ½ g ezwyeiLJGHbK ^a` H B G C wyGIL~j»º¼ezKML G E aC G C 8ü^aC w"G gzg GwyGKML g º GE ¶ ÿÕjfG µ · ã C^ jDeiK $
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Ldei¶ ` GHxGIL ¶ ` G g G · ã N w B t Š H ^ eíL CFGE G g ÿ ; `BfC GwyezG Cb^aC j C GO„ g G · ã N K GwyGKML jDG gzN ‰ NOg G `DC
w ^(! GKDKDG¤jDG ]BªNOCbCJNOBDB_^aC LˆN ä H N ‰ NOg G `DC j N KDH g º G¸E L N L K ^ K B G C L `fCJc{GÞE „
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(PT # J JE R^ 6)]% ;/< $)]+"( Ya
? c d F&[ '
? cId [email protected] e(&=6< '
%&3( )]+"(/0 . -e/42 "H(6 7( D^ (@Yah:;5S< %&e I - `(J)g => X\ Y5-
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g (:)&@/)IIJ>
? 0.5
0.4
0.35
ch
.
#
1
pure RPA (analytical)
pure RPA (numerical)
ren. RPA
ren. RPA, strong coupling
0.45
-Im χ (q,ω)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ω
(PT # 3 R^HD)]% ;/< O),+"( Ya
? c d F&D^ ?'c d [email protected] :(&_6< %&3( ),+"(/0 . I: 4
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1
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& +) 7' ," 'Ka& $0 T > _
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^"=" 1
? 0.5
0.4
0.35
sp
#
1
pure RPA (analytical)
pure RPA (numerical)
ren. RPA
ren. RPA, strong coupling
0.45
-Im χ (q,ω)
.
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
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(PT $
^R 6)]% ;/< S =*Y5 !aJ"(& ? c d F 3O ? c d
[email protected] e(&=6< %&$(L),+"(/0 . I 24H(6 7( ^! (@YaO9;H< \ I I ``(J)g >
X (4ZY5- fg`@ )`(&S:%I5< $^J G5 e(@Y5-( ? c d F&>
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0.4
0.35
sp
$: > A L
> , +6
L
3.5
4
pure RPA (analytical)
pure RPA (numerical)
ren. RPA
ren. RPA, strong coupling
0.45
-Im χ (q,ω)
O 5
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
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(PT ,
^R D)]F /< / ^ Y5;!aa"(& ? c d F&D3$ ? c d
[email protected] e(&_6< [%&$(1)]+"(/0 . -S 2 "H(6 7( D^ (@YaO: H[< %&S - I ``(J)g >
X\ e(@Y5-PF& K;fg \ZY">OV5> U6Q*(Pa =(@ h( ?'cId & fI3;(/H>FX(P)&@/
)I 5 ( ? c d [email protected] :(&_6< - ` JgY-5 6G/54= 9 ) `(BL I
)=(@Yab @@SKZY5- ]Qg>
#
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{ 1 .:L.! ren. RPA, charge
pure RPA, charge
reference
5
S1(q)
4
3
2
1
0
0
+π/2
+π
q
( ,
1
7( : H<
? 7 Y5 L:O% `< 6< =( ]< -` Y5 O%& (:)]+"(/0 . -h/2 "H(6 1
h^ 7( < fD(& 6< 7( =(g gHD)]% ;/< */)]+"( Ya ( ? c d F *(&"!
G5 9( ?'c d [email protected] e(&=6< >FX( ) @/ )- 59D] _< `I 9: & / 5; (
7 Y53'`e:
K;fg ]g< G/>\K V5>ji i6Q QP< fD(& O<(Hfgg)'(gHD)]% ;/< 3)]+"( Ya'( ? c d
B
[email protected] e(&_6< >\X ( ) @/ JD)-I5e / -3 < f(&NH(&- ) :7 Y5 :`L:9(Hfgg)9(
gHD)]% ;/'< ( ?'c d F&>JS(&$ /& *)`(& / :9 OYJ(/a)]+-3 ,g< G/> K V5>ji i6Q `I
`(J)6< ZY5- \-H6< >/X(L/7 Y54/`L: < ;(&-$ ( b (&; )'ZY5-S/_< ba`< ^)
)b IF(Hfgg)S(:)&3)I 539JD)-I5)@ % a(&I >
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> K & : " $ ; ? " !6 ! D " =
C
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-
7L
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.
#
1,
ren. RPA, spin
pure RPA, spin
reference
S1(q)
1
0.5
0
0
+π/2
+π
q
( ,
? 7 5Y '`L: % `< 6< $=(O ]< -`!Y5 \ (L),+-(/0 . -3 2 "H(6 h7( 9;H<
D^B7( 1< fD(& O<6P7( =(g3g D)]F /< /S ^*Y5;!J"(& 3( ?'cId F&
(&!%G54S( ? c d [email protected] e(&=6< >"X(*)&@/4)- 53] < `I hS99&D 15; 4
(37 Y5/ `e:
K;fg O ,g< G>\K V5>ji gQ QP< fD(& < (Hfgg)(@ D)]% H;< / =*/ ( ? c d
B
[email protected] e(&=6< > X ( )&@/ aD)IIJL -3 < fD(& H(&I ) :/7 Y5 :`L:9(5fgg):(
@ D)]% H;'< ( ? c d F&D>5S(&$'& )`(& : &S YJ(/a)]+-3 ,g< G> K V5>ji gQ `I
(J)_6< 3 ZY5-%I56< g> [ ?'cId [email protected] e(&_6< I(47 Y54`L:hF (!b (& > , (& @ L%! JY-5`L( Y5-*9< ba`< D ) \-H< 99( )&@/ )I 5>
[ ?'c d F& \( /7 Y5*LL:9hf56< *a(& e(&I9 fgg)g && 9 ^L ;!O<
(/ &S 6 > $ fg (&!^(4ZY5-3J6)- 5' ]g< )`(g ` `(4ZY5-'< ba`< ) %I5< g>
#
##
1
4
{ ren. RPA, charge
pure RPA, charge
reference
S1(q)
3
2
1
0
0
( +π/2
q
+π
,.,
? 7 Y5:4L:% `< 6< 3_(3 ,< -` Y5S%&3( ),+"(/0 . Ie 2 "H(6 7( DeC
^! (@Yae GH(g 7( :< fD(& O<6 h7( _(g hg D)]%N ;< 3L),+"( YaL( ? c d &
(&"!G/5/( ? c d [email protected] :(&_6< >3D)! %*(&"(&gY-J 1
7( ( /Y- 3V5>H>
S1(q)
1
ren. RPA, spin
pure RPA, spin
reference
0.5
0
0
( ,
+π/2
q
+π
? 7 Y5:4L:% `< 6< 3_(3 ,< -` Y5S%&3( ),+"(/0 . Ie 2 "H(6 7( DeC
^! (@Ya: G/5(g*7( e< f(& 6< 1
7( =(g SgHD)]% ;/< 3/N^Y5;!J"(&Fh(
? cId F&h(& G5Lh( ? c d [email protected] e(&=6< > S6)! % (&-(&gY-5P7( ( /Y-&1V5> e(&
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(5fgg) 6 >
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L
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1
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large U limit
U=4
U=5
0.8
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n(k)
0.6
0.4
0.2
0
0
+π/2
+π
k
( ,
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0
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mean field
ren. RPA
ren. RPA large U expansion
Bethe large U expansion
-1.2
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0
1
2
3
4
5
6
7
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0.3
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mean field
ren. RPA
ren. RPA large U expansion
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0.25
doubly occupied sites
0.2
0.15
0.1
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0
0
1
2
3
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5
6
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3! /L< 1H @=( %
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