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TRANSCRIPT
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y L y L w L# | CJ: |> 8L L L L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L yDvy L y L y L*.(<8DGEF" BJ8D; F 8D:<8 L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L yDy
y L I?L L L LF(A;~;BF; $L L LLL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L y Hy L x L !A A;=?8>~F~?L L L;;FL L LLL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L y Oy L H?L L L LF9;C D| ~F; 8>/L+LL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L y My L O?L1+CDG < /. | ~F~? | | B =L L L N;8>CJ=DE;@L L L LL+L+LL+L+LL+L+LL+L+LL+L+LI y
y L O?L w L10 | .bE L L L L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LI yy L O?L y L10 | .bE L L L L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LIDK
y L , L 4$E;G~;CJ=DBFE 8 CDJCJ=DCJ: | ~;~; LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L x I2 nkri hnbm# khm l"! s3 p(j*l4 'I;L w L 4$G A;9 L+LL+L+LL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L x,I;L y L50UCD:<8@?A;~FCDBB?CD: | FF; F~;E;@(A;~;BF; L+LL+L+LL+L+LL+L+LL+L+LL+L+L x M
I;L y L w LoL L LF(A;~;BF; L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L x MI;L y L y LoL L LF(A;~;BF; L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LH yI;L y L I;LoL L LF(A;~;BF; L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LH x
I;L I?L1+CDG < /. | ~F~?L L L/6+C<C C-. FEF=DCJC A =DCJ: | ~;~;L L LLL+L+LL+L+LL+L+LH ,I;L x L 4$E;G~;CJ=DBFE 8 CDJCJ=DCJ: | ~;~; LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LO y
7 fghij*lmi8 9):x L w L 4$E?@; <~;-G9F =D=J F~FC+* | ~;~; L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LODHx L y L ,cCJ~F8kL L LUL+LL+L+LL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LO ,x L I?L L L LF(A;~;BF; L+L+LL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L ,?wx L x L "E;G | )(D~ | FCJG < . | ~F~? 8L L L"L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L ,>x
x L H?L 4$E;G~;CJ=DBFE L+L+LL+L+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LKDO: ikjYnr&n"f p(hp s$i s+ikhh
H;L w L L L LF9;:FE:FC>>:F8>@?A;~;B;AL L L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LKDKH;L y L'(>8D 8>(J~ | ~;~;~<8>DFE/. | ~;~; EF9?AL L L;~<8+=JE;9<8<CDBL L L C | <&L+LL+L+LMDIH;L I?L "E;G | )(D~ | FCJG < . | ~F~?=D9;FEF=>AL L LF~F8L L L;=>c8>G E;=JCDG -GE;G D| E L+L+LM xH;L x L 4$E;G~;CJ=DBFE 8 CDJCJ=DCJ: | ~;~; LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+LL+L+L wvJy
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< FG~F (JG | : | (L L L' ~F8D= | | ~;EF+"CD: AD<<L L LD4GE?FA CJCFN 6+C+L L LA =DCJ:; (DG L L L;# 8>BFE;L L L N;= G=DCJ | :FA?NcL L L4 : | AJ( 8 "9F:;CJ= | | ~;CJC L L L$8D~F8>F AL L LDAD<C =DG 8D~;CJ=> | ~FC;N&6+C L L L$# 8DB?C-.N
L L L$G9FCDG D| :F 8><8DG ( = | B?G9 | :FE | ~ 8>@ L L L w L$# 8DB;E ;EF~;C N$9FCDG 8<C 9;E 8>~F~?9;:FC~ | CJ>@; <~; G (9FCDDAD<CD=JE ~;CJ=DE;@ | C< = D| CD: | EF;~FCDC<CDG < /. | ~F~?L L L N 6+C9;:FE; 8<E+<C+A;=?8>EL L L4Y9 | :/*+E;@ D| CD: | EF;~FE?@ :FCDJC 8>@ M Fw I&9FC<CDG < /. | ~F~;L L L GE;G D| N =FBF
L L L;CD9FE;GA;=J8D=DG k"<CD9FC CJCJoL L LF0 :FE 8D~;-: | AD)( 8 E9;:FC |> CD~FG :;A;=J8>FE-N6+CkL L L C-. | =DEF~;EFBJ8 E>8k:F8>@?A;~;CJB L L L":;C D| N-G F =J ;>~F8FE E-N 6+C=:F8 BJ8@ L L L 8DB;EF: | AJ( 8 GA;9 | : | ;E ( | B?G9 | :FE | ~ 8>(J~;E <8D~;E K L1;=J:F8>@?A;=J8D~;~;=J9?<E;=DA L L L=&:FCDJC 8>@+78 GA/*+ E wx N w H?>AJFC3D8>9F:;CJ9;CJ~;CJ=J8D~;C
C | (FN<=kFB; D8>9F:;CJ9;CJ~;CJ=J8D~;CL L L(:FCD= | | ~FL L L<:FC>>:F8>@?A;~;BFEcNc=k:<8 B?8@L L L N<9F:;C z |> CD~FG :;A;=J8>FE-N6+CL L LF=D < :<8> ( =?8-. FEF=>A:;C)(+AL L L N?>CDBF: |> 8?NF=DCJ~;EL L L;# 8DB;E ;EF~;C ND9F:;E+<CDG < /. | ~F~;cL L LD>AJFC"C :;E 8D~;C: | AJ( 8 E-NDFB;?FBF G~FC&A?C/. A; ( zG + | B?G9 | :;E | ~ 8)(D~FE E <8D~;E LJ0(F~<8>B-ND~ | <CF B?C D8>9F:;CJ9;CJ~;CJ=J8D~;CJL L L C | F
"~ | =D:<8@?A;=?8>~F~?L L L(BDAD<C 9;CJBJ8>>8D~;C =b:;CJDC 8>@ w-, N w K L L L "=DE;~<8F8>(J~;CJ9F:;EFCJG F . | ~; L L LL L LF8D~F8>F bL L L;DAD<C+F FG~ | ~FC = :;CJDC w M LF4 : | AJ( 8 -9;:FCD= | | ~FCDCb8>~<8< A
>AJFCG"CJ: AJ(JCD=?8>~FC"=J F9FCD=J F~;AL L L C | (FND:FC>>:F8>@JCJ=J8D~;C L L LJ;9F:;C?8>~<8< CJ=J8D~;C L L L0 :;E 8>~F-: | AJ( 8 E=D <9;CJ=D c8> (+c8>~FE C :FE 8D~;E =b: | AJ( 8 L L L O L ~/*+EF E;9GEFG D|> NF;FCD9FE;GAFBFE?@ C-. | >A E=DEFB?CD:FE;G 8>~FE; L L L N GE;G D| E
CDGCDDFEF=D G >AJFCJ=DEkFBFE?@ (~<8F=J~; G (L L L N/6+C C-. | >A E/. | : | <C L L L (:FE;B;<8 zFC * 8DB;E;@ GEFG D|> L L L N 6+C~F8> | .8 (FC L L L!$9; )(D~FCD =>c8>G E;=J G L L L G @;E z(J~; G (FC L L L yJv L(B 9F:F8D=DE;FCFN 8>BF+L L L A CD=D | ~;L L L N6+C<C>>=DCF =D=?8-.+8 EL L L! | : | /L L L ~F8D;J (* | 8)(D~FC FCJG F . | ~; L&,c= ;CJB . L L L$< ;G~; (DG
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| : | AJ( 8 FCJD: | A?C. A (JG c8>~FE E | B?G9 | :FE | ~ 8>(J~;E;@:;CJD 9;CL L L y K?N y M N? | 9;CJBJ8D8>~FC;N6+CL L LFGA;9F:;CJ=DC. A (DG L L L4 :FCDJC I v -9FC+<CDG F . | ~;~FYL L L N;>AJFCbC :FE 8D~;CL L L 8B?CJ:;EFG ( =?8-. <E;=JC :;C< L L L;CJ=DCJ:J (c8>~F L L L I w IDI N;9;:FCD= | | ~Fc=kL L L;5&= C z
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• L L L;CJ9;EFGA; (DG C | FFYL L L N/6+C+<C>>=DCF +L L LFc=J:F8>@?A;=J8 EL L L
• =D=JCFE (DG L L L
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L L L 8 "~F8DG A;9F~;EF=DEF ;;U
H =∑
i
Hi +∑
ijσ
tij+iσjσ, w L w
Hi = Uni↑ni↓ − µ∑
σ
niσ + g∑
σ
niσSzi − hSz
i − ΩSxi ,
C-. | DA EkC :;E 8>~FE;9;:FEk:;CD F;F 8>Bb>=J8D~;CJCkL L L x , ?=D ~kDA;= 8DBJC-.YD8>9F:;C z9;CJ~;CJ=J8D~;EF= :FCDJC 8>@ x K;N x M L | :*+3<CDG < /. | ~F~? 8DB?C C | F(=JE;B?CJ~?A;=?8<E;G-(= :<8 B?8@ ~<8>DFE/. | ~/( FB;
=D F9FCD=J <8D (=DEF9F8>FB;A L L L4o:FCDJC x H3DAD<C9F:;CJ= | | ~;C:;CD:<8@?A;~FCDB L L L =JG 8 z~;CJ=> | ~FC;N 6+Ck9;:FE9 | =D~FE?@~<8 | ~;~;;@9<8>:<8 | :; =FNFFBFU=D <9;CJ=D c8> (L L L N<=DEF~;EFBJ8 L L L-4:FCDJC x M N<b=DEFB?CD:FE;G 8>~F~? L L L Nc>AJFCkF FG~ | ~FCL L Lc~<8kCJG~FCD=JUFB?CDC>AJFC9;CJ>AJFCJ=J8D~;C L L L .(:F CDC;N =:F8 BJ8>@c8>~FC C | F N 9F:;CJ=DC<E?<E;G-(FCJG F . | ~;~; | C<C L L L< =JE;B?CJ:;EFG 8>~F~? L L L H v N H w Nc8 8>B?C-. CJDCD=JCD:F&=?8c8>G-( C-. FEF=D G (= ;>B?AL L L H y N HDI L'=DEF9F8>FB;AL L L 6+C=J F9FCD=J <8 &: | AD)( 8 8 :FCDJ x H;N H y CD9FE;G+L L LJDAD<C9F:;CJ= |z
| ~;C+=:FCDJC HDO HDM<(=DEFB?CD:FE;G 8>~F~? | CFAL L L?4 :<8>~FE;FUL L LDFCJG F . A;=J8><8DG (
9;CJ= | F ~;BJ8 L L L =D8 | .b~FCDG &=D *L L L4G 8>~FCD=D | ~;CFN 6+C L L L9;:FE;=JCFE (FC L L L &9;C zF=JE L L L AJFC:FCDD | ~;C=D <9;CJ=D <~;$=JE;G~;CJ=DBFE 6+C<C | | B E;=J~;CJC>=>A . | ~F~?>CD~9;:FCD=J <~;CJG 89FCD:;A/* | ~F~? | | B :;CJ~ z < :;B?CJ=DC>3GE | :; FBJ8=J F9FCD=J <8 >8 ~;n→ 2−n >8:<8@?A;~FCDB=D>8 CF -"9;G | =DFCJG9F ~;CJ=DCJ 9F FGE;G D| CJL "CDBJ8>>8D~;CFN 6+C 8 G | 9 | : | ~FCD: A;=J8D~;~;9<8>:<8 | :F8 68>JJ8D:JF =JG (JB?C B?CJ: | ;FF U9;:FE9 | =J~;E;@A CD=?8@ | | B EF=D~<83B;AD<CD~F =DG-(DB?83=>D8 C< | | B :FCD~F = | ~* | ~;ADF 6+CFN=&G=JCD | :;AJNc=J F9FCD=J <8 +9;:FE F 8>~F~; . | | B :FCD~<8 E L-4:FCDJC HDK = | .+8>@ CJC./G>8 CJC(~<8>DFE/. | ~;~; 9F:;CJ=DC<E;=JG +:FC>> ;; G 8 EF;~FC<F | | B :FE;F~;C><G9;:FE;F~? >zFEF=DCJG L*;;@?C :;CD:<8@?A;~FB?A ~FE D8>G | | ~FCDG D| 9; =J9?<E;=JC 9;CJ9 | : | ;~FCDC| | B :FE;F~;CJC9;CF ~F8D9;:J | ~FCDC"9 | :;9 | ~;FEFB?AJ;F:F~;C"<C* 8D:; = A =JE;9<8<B?A 8 D| :; 8< = E;9;A DAD<CC :FE 8D~;C C-. FEF=D G (= | FEFB;E;@>~F8 | ~(G 8 z EF;~FC(< | | B :;EF;~FC"G 8>FC>&A =DEF9F8>FB;A F8DG B?CJ=DCJC>8D9;CJ=D~ | ~F~? | | B :;CJ~;~FE?@9; ;>CD~-L | >=J8 .+8>&FE ~F8&>~F8;~F?AJGE;;F " EFB; = D| CJ: | EFB; =FN><CDG F . | ~;~; C | </6+C
CD9FE;GA (JG 8 ( CJ~; 8D~;C w L w (<8> | B; =J D8>= | :* | ~F~?-L ":;EF;EF~;CJ 8>B?CJC9;CJ=D (J~;CJC"9;:FCD: | GA D| CJ:; $GB?c8<~; G ( C | < N>FBJ8(=JCFC< $G9; )(D~FE E :FE;G>8 EFB C | <<68>JJ8D:J<8;N 8>B-9;G | =DFCJG9F ~;CJ=DE;@ C | | E;9;A C | <- ~F 8A9;CJ9 |z: | ;~FC Ak9FC< N? NJFBk~<8>G < FCJBcN?=J FGA ~; G (+9;:FC>>CD:FCU: | AJ;F:F~;C>-G8 C>A?>C. | ~FCD| CD:F DA;: | ~/(FN)6+C <C>>=DCF;<8b <CDG < /. A;=?8 E=k9;CJ=D~F :;-FBFEF~F8 B?AGEFG zD|> E 8DBF D| : CFEF~F8 ;~F-=>c8>G E;=JCDG L!38 8 (JB;E C | )(+6"8DD?8>:Jc8 O v N;~ | >=J8 .+8>(;E~F8 U" EF;~;A9;:FCDG C A 8
~ | = | FEFB?AB; )(DBF G (k9F8D:F8 | :F =;N ~F8>;>=DEFF8D;~FCkGB;<8>F~<8+ CDFB;ECJ:?Ak 8 D|z 8 EF;~FC CDJ:;CJDBFEcL;8 ( CD~F 8>~F C | <
H = U∑
i
ni↑ni↓ +∑
ijσ
tijc+iσcjσ w L y
G E (FE* | <=J8"FCc8>~FB;E-NJFB;<CD9FE;GA; (+9 | : | ~FCDG | | B :;CJ~; =(=>A?<8~F8=>A?>C
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, =DEFB?CD:FE;G 8>~F~? F C | < =D<8 (DG CJ9;EFG8 EU<CDG < FE E:;C(=DC<~ | =DE;@= ;>B; =~F8 | | B :;CJ~;~F$=D<8DG EF=DCJG 3B?C 9? | B?G =9 | : | @; F~FE E | 8c8 E-NU>C zB;: | 8;N~<8A =DCJ: | ~;~; >8D:J;FCJ=DCJA?G EF~;~FC(@;=JE?<&3 "8>CD=JCDC=J9;CJ:J;FB;A;=J8D~;~;=k8<CD | ~F~;C G B?CJ=DE;@ G9;CFA;BJ8@c~<8 | | B :;CJ~;~;A9F:;CJ=D <~; G (k=9 | :;CJ=DGB;E >z~;CJC E;9;A CJB?GE;<8>@cL?# 8>B?C-.N;=JE;=JF8D&FE 8 D| :F 8<E E;9;Ab(<CD9FC C zCJ 9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | < ND=D<8 (DG bCD9FE;G>8 E | FB;FG9FCDG D| : | . A;=?8)(D~FG :?A;B A;:;~Fc8>~FC 8>F N 8>BF;FB | | B EbJ G 8DD )(D~FCDG 9F:;Eb ~F D|> 9 | :F8 A;:;Ek;EB?CD~F | ~ :<8>F c=J8DBJ8D~;G NJ~ | G 8DD )(D~FCDG ;G | ~ | C | | B :FE;F~;CJC EF9?A?NJ8>CD= | :;CD z* 8D:?A;=?8>~F~?-N C+6+CFL":;E D| CJ: | E;F~;C A/<CDG < /. | ~F~;k9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | FkCF~FE
~F8D;J )(+* F BJ8D=DE JGB;<8>F~FE =JE;9<8<B?C N=JG | 6 | D8<E/* 8 (DG =JE;9<8<CDB~ | ~;AD)(DC z=DCJC | | B :;CJ~;~FCDC9 | : | ~FCDGA . =>A?<8 EcL "BDAD<C=JBJ8>>8D~;C=JE/6 | N-CJG~FCD=J~;E | C<C 9;:FE"<CDG < /. | ~F~; C | <?BJC<E+ G~;A 9 | : | GB?CJB | | B :FCD~F =3$=DA?c83~<83=DA zC | C' &6()kNFFBF "9FCDG < FCJ=D~FE :;CD=JE;~ | ~F~? ~F8D><E . | ~F~?68>JJ8D:J z L,-8>9F:;CJ9;CJ~;CJ=J8D~ | F;+<CDG < /. | ~F~?+F | | B :FE;F~;C><G9F:;EF;~; <E;=JCDG ND~F8D><E . | ~F~?'" (6() ~ | DNC<~F8DBcN>G8 C>A?>C. | ~FCD D| CJ:; J:F8 B?CD=JJ~ | G FB?CDG DFCJG F . A; z (JG bFBb: | AJ( 8 :FC>>D .b~FCDG D| bB?CJ: | ;FF ;~FE?@b(A;~;BF; -N6 C"~ | C-.b~<8"=D=?8-.+8 E
>8>FCJ=D (J~;E 9;:FE ~<8F=J~;CJG 3= C | F$8>CJ=DE;@ 9 | : | @JCF =FL 4 <9;CJ=D <~;C=J FBF:;E z E >8>FE* 8<CDG-("9FE 8D~;~;k:;CD:<8@?A;~FB?A D| : CFEF~F8 ;~FE?@k(A;~;BF; -L;49 | :* | 9F:;EFCJG F . | ~;~F?9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | <JA:<8>~FE;F~;C A=JE;9<8<B?A
Ω = 0$B;A z
FCJ~; =DG-(DBFE =J * CJ=>@?A;=?8>~F~? U =∞ NJ8 8DBJC-.o;F C | <?6"8DD?8>:Jc8& U =∞ N
< FG~ | ~FC A?D8>8)(D~ | ~F~?G @ | E' &6() ~<8b=JE;9<8<CDB:;CD:<8@?A;~FB?A D| : C<E;~<8 z~;E;@(A;~FB;F =:FCDJC KJI L AD<C9FCDBJ8>>8D~;CFN 6+C 8>B | ~F8D><E . | ~F~? 8>~<8<CDC ~F8D><E . | ~F~? G | : | F~(DCJC9;CF;FGE?)(D~FCDB?CJ: | (JCD=?8>~FE?@GE;G D| L 4/: | AJ( 8 9;:FCD= | | ~FCDC:;CD:<8@?A;~FB?A DAD<C=JG 8>~FCD=D | ~;CFN 6+CB?CD~FB?A;: | ~F; . 9;:J CJ 8| | B EF=D~FCD =>D8 C< E9F:;EF=DCFE (FC9;CF=JE =GE;G D| $G | ~ | C | | B :FE;F~;C>8>EcL' =DEF9F8>FB;A =D <GA ~FCDG (9;:J C> | | B :;CJ~ z | | B :FCD~F~;C>=D>8 CF
U = 0N$=
GEFG J| 3 G~?A (JB;E | | B E;=J~F8J8D 8 CD<8>G E;~FBJCJ=J8=>D8 C< . 9;G | =DFCDG9; z~F8 E | : | | | B :;CJ~;E9;:FCD=J F~FCDG N- C . | NU; BJ8D=DC<CDG < FE E=J9?<E;=~ | 9F:J C=>D8 C< cD8>9F ~F&(;CJC E;9;A~F8 D| : CFEF~F8 B?A C | F LF4$ ;>~F8;E C;N/6+C 8 z( CJ~; 8D~ 9;G | =DFCJG9F ~ z | | B :FCD~F~;C> C | < w L w A=DEF9F8>FB;A U = 0
NΩ = 0 8DB
=?8>~<8b9;G | =DFCJG9F ~ z | | B :FCD~F~F8 C | (b ~ | 9;:J CJ/=>D8 C< + ~;=?8>:F 8>~ ~;EF6+CFC9 | : | =JCD: | ~F~?
µ → −µ N h → 2g − hNn → 2 − n
NSz → −Sz | | B :;CJ~ z
F :;B?CD=?8GE | :; 8 ( CD~F 8>~;A N 6+CFCD=JC; 8DB?C-. CJ9;EFGA;=?8 E 9;G | =DFCJG9F ~ zF :;B?CD=DAGEFG D|> AJL, ~/*+C> CJ;BFECJ:?AJN 8 ( CJ~; 8D~9;G | =DFCJG9F ~ z | | B :FCD~F~;C> C | < ~ | 9;:J z
CD=>D8 C< <8 C-. FEF=D G ( CD9FE;G>8 E C | ( E;9;A C | <<J ~F8D:;~FCDC+G9;<8 z=>A?L":FE /(JC A?N>>:?A;F~;C=D= | G E CJ9 | :F8 CD:FEb9;:FC | B A;=?8>~F~?k~F8"9;G | =DFCJG9F ~;CJ=DFG 8>~FE
P±i = 12 ± Sz
i
C<c8 )( CD~F 8>~ C | FcJ ~F8D:;~FCDC+G9?c8>=DA C-.b~<8+C :FE 8 E=DE zB?CD:FE;G CD=DA;&FE9F FG 8>~FCD=JB?A
P+i = pi
NP−i = 1 − pi
|pi +B?CD~F | ~ :<8>F C<~;
kB?C 9FCD~ | ~ G9?c8>=DA1 − pi
=J F9FCD=J F~FCB?CD~F | ~ :<8>F ~*+C$B?C 9;CJ~ | ~ E-LU0"G z~;CJ=D~<8+=D ~;~F G ( . C | ; E9;CFF 8 (=F FG~ | ~F~;c9F:;CJ | ;A;:F<A?G | : | F~ | ~F~?9;C9;:FC | B;F ;~FE?@CJ9 | :F8 CD:<8@
P± D| 9 | :<8 A;:F~ | G 8 EFG EF;~ | AJG | : | <~ | ~;~;;F=DEF9F8>FB;A9;G | =DFCJG9F ~ z | | B :FCD~F~;C> C | <Ub~ | 9;:J CJo=>D8 CF o B?CJ~;" A;:F8 z
; ;~ | ;FJ ~F8D:;~FCDCG9;<8D=>A N 8 8>B?C-. A | C< G>8 CDA?C/. | ~F~? " B?GA;=J8D~;~F9;CD<CD=+.b~/(JCDC"9FC;
h;F 9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | FFF" BJGA;=J8D~;~FF= | FEF;E z
~;E | | B :FCD~F~;C> B?CD~F | ~ :<8>F pBJC 9;CJ~ | ~ ED ~<8>:F~;CJC G9?c8>=DA L
$ (* CDC;N>FB6+C&A =DEF:F8>?F;b 8 )( CD~F 8>~;A+9FG | =><CDG9; ~ z | | B :;CJ~;~FC C |zFF(~ | 9;:J CJ =>D8 C< N E ~F | CG9F ~;CJ=DF ~; | B?GEbF9 | : | 9FE/* |> C"8 )( C z~; 8>~~F8 CD=J-CJ9 | :F8 CD:F =k:?A?@?C E;@
di ciσ = di c<CDB?8< CJ=J8D~;E;@
fi P+i = f+i fi
N
P−i = 1 − f+i fi| | B :FCD~F =;N C=: | AD)( 8 $C :;E 8 C 8 ( CJ~; 8D~ C | F
) 8< B?CD=?8 z ." | <8(=(FB?C Ah=D < :<8 $:FC((@; F~;CJC"9;C D| ~F; 8>;A+F; FCJBJ8>F >C z
=J8D~;E;@fz | | B :;CJ~; =FL<0(G~;CJ=D~FE;G 8D~ C | <U) 8< B?CD=?8 z ." | <8A=DEF9F8>FB;A" B z
GCJ=J8D~;CJC+>~F8 | ~F~? | | B :;CJ~;~FC B?CD~F | ~ :<8>F UF;9; <GEFG D|> N "~ | CF~FCD:F F~FE F=>F (DG C-. FEF=D G (9;CF=JEFBG9; = :;~FCDC 8DB ~ | G9; = :;~FCDCk=J9;CJ:J;FB;A z=J8D~;~;FE8>CD=JCDC+:FC> * 8>:;A;=J8D~;~;=+>8> | .b~;CJG -=D B?CD~F | ~ :<8>F 8+= | <E;FE;~FE=>D8 C< M v L"CD= | : 8>(;EFG (k<C:FC>> F;;A C-. FEF=DE;@9F ;@?CF =9F:;ECD9FE;G C | | bGE;( z
~;CJ | | B :FCD~F~;CJB?CD: | FF; +N;8 8>B?C-. 9;G | =DFCJG9F ~ z | | B :FCD~F~;E;@ C | | -N?G < =D ;>~F8;E E | CFEF~F8 ;~FCDC G | : | <~/(JCDC 9;CF-L
| | Cb>AD<C(>8D9;:FCD9FCD~FCD=?8>~FC(=":;CJDC 8>@ KJH?N KJO?F; C | <?6"8DD?8>:Jc8 <E;=FL 8>B?C-. K , +9;CJGE;<8D~;~;*= ~; L&1+E;~<8 F~ | G | : | F~ 9FC | F=>F GCJDCJ 8D~F8 zF E;F~?A G @ |> A | =JE;B?CJ:;EFG 8D~;~; D| CD:F &DA;: | ~/(FN FBJ8<C>>=DCF G9F:;C | B A;=J8 E C | (6"8DD?8>:J<8~F8CF~FCFC *+B?CD=DA C | )(5&~; | :;GCD~<8?N 6+Cc8 C-. FEF=D G (C :;E 8 E CDF~;c: | AD( 8 Ek=+:<8>~FE;F< | | .b~FC-9;:FCDG CD:FCD=JCU:;CD :F~;CJG L; 8 z><E . | ~F~? FEF~F8 ;~FCDC G | : | F~(DCJC 9FC; A =DEF9F8>FB;A | | .b~FC+:;CD :F~;CJG 9;:FCDG CJ:?A
d = ∞ N$FB | >AJFC =JE;B?CJ:;EFG 8D~ | 9F:;E FCJG F . | ~;~F C | <(6"8DD?8>: z<8;NFCD9FE;GA 8>BF-=?8-. <E;=J-B?CD: | FF; F~; | | B EFB- C D>z @J8>JJ8D:JF =JG (JB;EF9 | : | @; | 8> z < | | B :;EFBcNJB;:FCDGCD= | :+=D bBJC | B E;=J~;CJC 8>~ | E? AFC 8D~ | E; A$FC zBJ8>F >CD=?8>~FE E 8>~; ~FE E C | ~ 8 EcN 9FCD:;A/* | ~F~? | : z :; <E;~F~;C>(9;CJ= | F ~FB;E9;CJ><E?A 9 | : | @JC;A | 8> z < | | B :;EFBcL&0(F~<8>B-N$C<~;CJ=>A?FCJ=J8>8><8<8;F C |zFU68>JJ8D:J<8 CDF~;C~ | :FC>>= ;D8>~<8 C A=JE;B?CJ:;EFG CJ=>A; (DG FCJ9;C .b~; | CFE
;F :;CD= ;B;AJNFCJB;: | 8bB;=?8>~ CD=JE; | C7CJ~ D|z ."8D:D<C 8k: | ~;CJ: :?A;9FCD=JE; | C K , KJM L-!< =J ;>~F8FE E-N 6+C G~?A :J;~ | =JE;:F * | ~FE?@9;:FCDD |> = | C<FEF~F8 ;~FCDCbG | : | F~(DCJC 9FC;-N;FBF-9;CJ= ;D8>~F<+=JE?@JC<C >8 :F8 B;E~ | GB; ~F | ~;~FC:;CD :F~;CJG -:<8 BFEcLc,cCJB;: | 8;N :;AD<~;C6+U=bC :FE 8D~;UB?CJ: | B ~FE?@:;CDB;<8>F = D8G J| 9 | ~; E
1/dL
"G | =><CDG9; ~ z | | B :FCD~F~F8 C | ((3~ | 9F:J CD =>D8 C< =DEF=D<8c8>G-((=":F8 BJ8>@~F8D><E . | ~F~?b<E;~<8 F~;CJCG | : | F~(DCJC9;C;k=":FCDJC K x L":;CJ= | | ~;EF J| : C<E z~F8 ;~FE; 8D~F8>F 9F = | :J<E;= CDG~;CJ=D~F&: | AD)( 8 E C :;E 8>~F(=:FCDJC 8@ ,+, N , K;F C | Fc+9F:J CD*=D>8 CF . | | B :;CJ~F8 E8> | 9;:FE=J FGA ~;CJG | | B z :FCD~F~;CJC9 | : | ~FCDGAJLU' :F8D~;EF;3~?AJ(JCD=JC | | B :FCD~F~;C>(B?CD: | FF; 9F:;E d = ∞ 9;G | =DFCJG9F ~ z | | B :;CJ~;~<8 C | )( CDF~;C:;CD= ;A;=?8>~FCDo=:<8 BJ8>@(DCJC9; F@JC z;AcU=: | .bE
µ = const 8 G | 8>CD=JE;9 | : | @; 9 | :*+CDC:FC;AJN CF FB=
: | .bE n = const
9F:;E 9 | =D~FE?@>~F8 | ~F~?;@+9<8>:<8 | :; = C-. | =D <>A;=?8 EFG +8>CJ= |:;CD-*b8>:;A;=J8D~;~;= | | B :;CJ~;~F 9; <GEFG D|> L ,-8A;=?8-.bE C;N 6+CFCJG F . | ~;~;G9F:;C z6 | ~;C>9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | < 9;:FCD=JCFE;FEFG ( | k=D:<8@?A;=?8>~F~? C-. FEF=DC>9;CF=JE= GE;G D| -~F8>G :?A;B A;:;E-L;1;=DE;F=> | ~F~?~ | G 8DD )(D~FCDG D| 9;CJ= ;D8>~FE?@k~ | ~?AJ(JCD=JE @;=DE;(JCD=JE = | B CD:FC NU~ | CJ>@; <~;C:FC>>:F8>@JCJ=>A;=?8 E(A;~FB;F 3=D FA;B?AJL5&~<8< EF;~;AG @ | A:FC>>:F8>@?A;~;B;AB?CD: | FF; F~;E;@(A;~;BF; F; GE?)(D~FCDB?CJ: | (JCD=?8 z~;E;@GEFG D|> =:F8 BJ8>@~<8>DFE/. | ~;~;FEF~F8 ;~FCDCG | : | <~/(JCDC9FC;>AJFCD8>9F:;C z9;CJ~;CJ=J8D~;C = :;CJDC 8@ K , N M v N M w L?0(F~<8>Bk<CDG F /. | ~;~;(A;~FB;F =D <A;B;Ak~ |> C-.b~F8=D=?8-.+8 E <CDG 8 ~F ;F FCJG F . | ~;~; D| : C<E;~<8 BFE C;AJ(JCD=?8>~FCDCG 8>~;A 8>CD=JE?@9 | : | @?CF = = C | F L
| :<8 =D F~<8;E C;N<~ | =?8-.+8D&;E~<8 D| N)6+C | C<E;~<8 F~;CJCkG | : | F~(DCJC9;CFC@JCJ9?< b?8> 8 CB;F(;CJ=DE;@:;EFGb9FCD= | < ~;BFEFCJG F . A;=J8D~;E;@GE;G D| ~F8D=D (;F GB; ~F | ~;C>&9;:FCDG CD:FCD=JC&=JE :F~;CJG N =D ~ 8 kCJ | =DE;F~F$$=J8 . FEF=D3CJ | . | ~ z~?UNFB?C<EcN<~<8>9F:;EFB?c8UN ~ | CJ>@; <~; G (bCD9FE;G>8 E8>EF=J~;Ck~ | FCJBJ8>(J~;E 9F8D:F8 z | :FC 9FCD:J;<B?AJLF0"B;:F CDC;N;~ | @ A;=J8D~;~;~ | FCJBJ8>(J~;E;@bFEF~F8 ;~FE?@kB?CD: | FF; =D9;FEF=J8 E U| ~F8D=D (~F88>E FCJBJ8>(J~;E 9<8>:<8 | :;C 9;CJ:J;FB;A M y L!$9F:;CJDE
:;CD-* E;:FE E | CUNU>8<CD9FC CJCJ =DB;F( | ~;~; ~ | FCJBJ8>(J~;E;@B?CD: | FF; -NU9F:;CJ=DC zFE;FEFG (=:;CJDC 8@ MJI MDH L(:FC D| Nc=JG =JCD~FE 8>FEG=JC~ | <CF BFEcNcFB8>~<8< E;F~;
>CDBF: |> 8;N-~ | 9;CJ=D~ | N-FE~ | : | ADFF:;~ | =J:F8>@?A;=J8D~;~;=DG @F 8D:<8 CF~<8>B?CJ=DC> CD9FC zFCJ &$9FCD:J;<B?A 9FC
1/d$FB ~F8DG F <CDB 9;CJ:?A/* | ~;~; 9F:F8D=DE; 9;:FE;FE;~F~;CJG 8>B
;EFG C D| @;~F ;~FcGB;<8>F~FCDG c9F:;E=JE;B?CJ:;EFG 8D~;~F | C;AkFC :;CD:<8@?A;~FB?Ac9;CJ>AJFCJ=D=D F9FCD=J F~FE?@8>CD=JE?@k< 8D:<8 L0"9;EFG8D~;F=DE6 |& | C<E =":<8 B?8@+FBFE?@ 9F:;CJ=DC<E?<E;G-((<CDG < /. | ~;~;b9;G | =DFCJG9F ~ z
| | B :FCD~F~;C> C | < NFB"A=JE;9<8<B?A9F:J CF9;G | =DFCJG9F ~;CJ=DC> | | B :;CJ~;~FC =D>8 C zF 8DBkFAk=DEF9F8>FB;AbCJ9;CJG | : | FB?CJ=J8D~;C>c?8>8 CJF8DG EF~;B?CJ=DC>cD8>9F ~F&(;C>-=>D8 C< N~ | =JE?@?C; ( D8 | .b-~;AD)(DCJ=DCJCb~<8>DFE/. | ~F~?9FCbCD | :F~ | ~;C A:<8< AJGA=>D8 C<
6+C=J F9FCD=J <8 k=D FJCD:;A< 8>:F8 CF~F GA CD 9FC@;=DE;(JCD=JC A = | B CD:;Ak L
# 8DB;E FE;~FC N =JE;B?CJ:;EFG 8D~;~; ;E;@ ~<8>DFE/. | ~( ~<8>DFE/. | ~;~; G | : | F~(DCJC9FC;-NB?c8>G D| :;~ | ~<8>DFE/. | ~;~;G | : | <~/(JCDC 9FC;-NFA?D8> 8>(J~ | ~ | ~<8>DFE/. | ~;~;@J8DC EF;~FE?@8?N ~<8>DFE/. | ~;~; FEF~F8 ;~FCDCG | : | F~(DCJC9FC; ;F =JE;9<8<B?A GB; ~F | ~;C>(:;CD z :;~FCDG $9F:;CJG CJ:?A c8 C-. <E;=J G (C :FE A;=?8 E B?CD: | B ~;3: | AD)( 8 E )(DBFE>8 | .+8 E CJB?CF =B;:FE EF;~FE?@ CJ;CJBcL34YCJB?CF&BF:;E E;F~;E;@ CDFCDB | | B EcN$9;C z= ;>8D~;- 3FA;B A;8>F EG | : | F~(DCJCk9FCF-NcG 8> (=DE;~<8F8>(J~;E EUF;CD9FE;GAB;:FE E;F~;E;@ =>c8>G E;=JCDG D| GE;G D| E ~ | CJ>@; <~;C 9;:FE; 8 E <CA;=?8>E =J~ | GBFE =D 3FA;B A;8>F G>8 CDA?C/. | ~FCDC+9FC;9FG | =><CDG9; ~F =;L
ikjYnr3nf p(hi p(j/i3 nkri hm8 khm l"!fq jonfp(ql rhp%$&jYnbu s rikt+jonflt u
4EF=D<8 (DG L L L FEF~F8 BJ8L L L C | < N)6+CCD9FE;GA kL L L F;CJ9;EFGAFB?C> <CDF (J~;C=DE zB?CD:FE;G CD=DA;=J8 EL L L;9F: | FG 8>=D | ~;~; L<0(GCDDFEF=D G oL L L C | F 6+C :FC>> ;;c8 (JG L L L C | )(+bL L L L | <C>>=DCF (~<8 <CDG < FE E=D9;FEF=L L L4 <8D~;C A:FC>F <c<E;G | : 8>F >8D9;:FCD9FCD~FCD=?8>~FCkG8 C>A?>C. | ~;EF | C:FC>>:F8 z
@?A;~;B;AL L L C | <<"~ | 9F:J CD=D>8 CF +L/0UCD:<8@?A;~FCDBkF FG~F (JG kAk=JE;9<8<B?AL L Lc=DEFBJCJ:;EFG 8D~;~; L L LU=FB?CDG J8>EFGAJN 6+Cc8 C-. FEF=D G ( CDF~;C=J:F8>@?A;=J8 E=>D8 C< oL L L4 CDG~;CJ=>A+D8>9F:;CJ9;CJ~;CJ=J8D~;CJC~F8D><E . | ~F~?k9;CJB?c8 | ~FC | CL L LJ=:F8 BJ8@ FB?CDC
< FG~F (JG L L L N?FBF-=DEF~;EFBJ8> (b=J~F8DG F <CDBL L L4$E;B?CJ:;EFG CJ=>A;(;EbC :;E 8>~F<8>~<8< E;F~;<=DEF:F8>E ;FL L L N>=CF~FCD:F F~FC A =JE;9<8 z
B?A9;:FCD=JCFE (DG "<CDG < /. | ~F~?kL L L NDL L L N C-. <E;=JCDG J9FCF=DEbL L L=&GEFG D|> L"CDDAD<CD=?8>~FC=D F9FCD=J F~F L L L L
j )
L L L 8 "~F8DG A;9F~;EF=DEF ;;
H = H0 +∑
ijσ
tijc+iσcjσ, y L w
H0 =∑
i
−µ∑
σ
niσ + g∑
σ
niσSzi − hSz
i
,
|c+iσNciσ
CD9 | :<8 CJ:;Ek~<8>:FC/. | ~F~?F~FE/6 | ~;~; | | B :FCD~F =+~F8=>A?Fi<G9; ~FC
σniσ = c+iσciσ
CD9 | :<8 CJ: FE;G c8&>8D9;CJ=D~ | ~F~?-Ltij ~ D| :F8>b9 | : | ~;CJGA | | B :;CJ~F8
=>A?<8~<8=>A?>C-L 0&<~;CJ=>A?FCJ=J8F8DG EF~F8 8 ( CJ~; 8D~F8 y L w + G E ( | ~ | :; 9;G | =DFCJG9F ~FCD=JC&9; <GEFG D|> EFBJ8>~F8>@JC<E (JG A 9FCD=D<CD=+.b~/(JC A 9;C< ∼ h
+ 8FCJBJ8>(J~?A=>D8 C< ∼ g
$| | B :FCD~F = 9F:;CJ=D <~;CJG c9FG | =><CDG9; ~<8 E-L"Bb>8>~<8F8>FCJG (+=+9 | :*+C Ak:FC>F <F<E;G | : 8D; N; 8 ( CJ~; 8D~ y L w N;9; G Fb |z
FBFE?@G9F:;C6 | ~(;N C-. | >A E9 | : | 9FE;G>8>~FE;kA=DEF ;;<c8 )( CD~F 8>~;A | | B :FCD~F~;C>9; FGE;G D| E C | FUD ~<8>:F~;CJCkG9;<8D=>A?Nc8 8>B?C-. A=DEF ;;<-8 )( CD~F 8>~;A C | F) 8< B?CD=?8 z ." | <8;L30(G~;CJ=D~<8=J ~F~; G ( . ;E E C | ; E 9FCFF 8 = |z CF&< ;G~ | ~;~; 9F:;CJ | ;A;:FE A?G | : | F~ | ~F~? D| 9 | :<8 A;:F~ | G 8 EFG EFF~ | A?G | : | z~ | ~;~;A =DEF9F8>FB;A9FG | =><CDG9; ~ z | | B :;CJ~;~FC C | <$ C | <3) 8< B?CD=?8 z ." | <8;NB?CD~F" A;:<8>F ;~ | A?G | : | F~ | ~F~? Ab=JE;9<8<B?A C | <FJ ~F8D:;~FCDCG9;<8D=>A L;!F 8>B?C-.=D ;>~F8;E EcN 6+C>8D9;EFG8D~;EF 8 ( CJ~; 8D~=DC<C< 9 | =J~;CJY=D~;A :; *+~(DCJYGE |>z:; ~;=?8>:F 8>~ ~; G (6+C<C9 | : | =JCD: | ~F~? µ → −µ N h → 2g − h
Nn → 2 − n
N
Sz → −Sz L | <C>>=DC; ~F8 =DEFBJCJ:;EFG CD=DA;=J8 E y L w F; CJ9;EFGA 9;G | =DFCJG9F ~ zF :;B?CD=JE?@GEFG D|> L | : | 9FE/* |> C( 8 ( CJ~; 8D~ 9;G | =DFCJG9F ~ z | | B :FCD~F~;C> C | F y L w FCD9 | :<8 CJ:;E-N
FBF =D <9;CJ=D c8> (" >E;F~;E = | <E;FE;~<8 NU=DEFB?CD:FE;G CD=DA;&FE9;: | <G 8>=D | ~;~;= C z:;EF~;~FCDC BF=J8D~ A;=?8>~F~?~<8 CD=JcCD9 | :<8 CJ:; = ~<8>:FC/. | ~F~? >~;E6 | ~F~? $| | B :;CJ~; == G 8>~Fc=JE?>~F8 | ~;CJCJ:; ~ 8>F *9;G | =DFCJG9F ~?Ac
aiσ = ciσP+i , a+iσ = c+iσP
+i , aiσ = ciσP
−i , a+iσ = c+iσP
−i . y L y
1 |P±i
"CD9 | :<8 CJ:F8 E9F:;C | B A;=J8D~;~;~<8+9FG | =><CDG9; ~FCD=JcG 8D~;E
P±i =1
2± Sz
i , (P±i )2 = P±i , P+i P
−i = 0.
4= | | ~F&=>A?FCJ=D&CD9 | :<8 CJ:;E ~<8>:FC/. | ~F~? ~FE/6 | ~;~; D8<CD=JC(J~?F (~<8>G A;9 z~;E B?C A 8D; F~;E G9; =J=D <~;C* | ~F~?
a+iσ, ajσ′ = δijδσσ′P−i , a+iσ, ajσ′ = 0,
a+iσ, ajσ′ = δijδσσ′P+i , a+iσ, ajσ′ = 0. y L I
4E;@; <~;EF8 )( CJ~; 8D~ y L w ~<8 CJ=D-;E;@CD9 | :<8 CJ:; = 8 (~<8>G A;9;~FE;=DEF ;;U
H =∑
i
Hi +Hint y L x
=∑
i
ε(ni↑ + ni↓) + ε(ni↑ + ni↓)− hSzi
+∑
ijσ
tij(a+iσajσ + a+iσajσ + a+iσajσ + a+iσajσ),
| =D= | | ~FC ~F8DG A;9F~;U9;CD~<8 | ~;~;
ε = −µ+ g/2, ε = −µ− g/2 y L H ;F | ~ | :F kCF~FCD=DA?<CD=JE?@kG 8>~F =
Hi C<~;CJ=>A?FCJ=DEF < 8>CD~<8)(D~FE; FCc8>~FCDB-N
Hint CJ9;EFGA "9 | : | ~;CJG | | B :;CJ~; = =DA?c8+~<8+=DA?CUL
' =DEF9F8>FB;A GE?)(D~FCc9FG | =><CDG9; ~ z | | B :;CJ~;~FCc=>D8 C< g À t :;A;;~FC"=JE;J:F8 z
ECF~FCD=DA?<CD=JE; 8 ( CJ~; 8D~Hi=+FB?CJG -~;AD)(DCJ=DCJC ~<8>DFE/. | ~;~;-
H0 =∑
i
Hi
CDCk=D<8DG~F (A;~FB;F $9;CJ>AJFCJ=J8D~; ~F8= | B CJ:F8>@ |ni↑, ni↓, Szi 〉A =JCD:F& (9FCD=J~;EF
J8>EFG"G 8D~; = | | | ~ 8D:;~FC B?C :FB;E-
|1〉 = |0, 0, 12〉, |2〉 = |1, 1, 1
2〉, |3〉 = |0, 1, 1
2〉, |4〉 = |1, 0, 1
2〉, y L O
|1〉 = |0, 0,−1
2〉, |2〉 = |1, 1,−1
2〉, |3〉 = |0, 1,−1
2〉, |4〉 = |1, 0,−1
2〉.
, | CD:;CD:<8@?A;~FB?A D| : CFEF~F8 ;~FE?@(A;~;BF; Ek=JE;B?CJ:;EFG 8 C G 8>~;<8D: z ~ | 9F: | FG 8>=D | ~;~;G 8 E;G E;F~FCDCbCD9 | :<8 CJ:F8;
e−βH = e−βH0σ(β), y L ,
σ(β) = Tτ exp
−β∫
0
Hint(τ)dτ
. y L K
#$A NF; D8>9FE;GA ~ | < 8>CD~<8)(D~FC F8DG EF~;E 8 )( CJ~; 8>~<8Y=DEFB?CD:FE;G CD=DA (DG 9;: | <G 8D=> | ~F~?=D>8 CF
A(τ) = eτH0Ae−τH0.
#UC< NJF;G | : | <~/(JCDC~<8 | ~;~;=D CJ9 | :F8 CD:<8+9;G | =DFCJG9F ~?A 〈Szl 〉C :FE A C ~F8 z
G A;9;~FE;=JE;:<8
〈Szl 〉 =
1
〈σ(β)〉0〈Sz
l σ(β)〉0 = 〈Szl σ(β)〉c0. y L M
'G | : | F~ | ~F~? 〈. . .〉0 9F:;CJ=DC<E (JG >8G 8 EFG EF;~FE :FC>>9;C< FC 8 )( CD~F 8 z~;C H0N$8GE =JC 〈. . .〉c0
C>>~F8<8 =JE?< | ~;~; = ;>8D~;E;@ F 8D:<8 ~ | >= ;D8>~FCDF 8>:F8 CJ E~<8>E;=?8 C 8DB?AJNFFB?8 G E (b9FCbBF:F8D;~F :F-CF~;AF8DG EF~?A?N)6+C~ | 9;:FE F~<8>~<8+F ~F E=>D8 C< -<C : | * 8+<8>G EF~F 8D:<8 E L
%
0UC>>B?c8 L L L-9F:;EF=DC<E (FNL L L Nc<C=DEF:F8>A?N 6+C 8 =JE; F; L L L 0UCD:<8@?A;~FCDB L L L C-. |>A E9F:;CJ= | | ~;EFkD8FCJ9;C CDCD*=D <9;CJ=D <~;C> D| CD: | EL L L ODH L& D| CJ: |> 8D8c8 8<CJ:;E J L L LF# 8>B | = | | ~F~?<8 C-. FEF=D G ( >8D9;EFG8 E: | AD)( 8 A=DEF F;F L L L0UCD FF~ |> CkL L L
〈Taα11 (τ1) . . . aαi
i (τi) . . . aαnn (τn)〉0.
1; >:?A;F~;CJG E DAD |> C=D=?8-.+8 E 6+CT L L L0"9 | :F8 CD:L L L C-. | 9;:FE; 8 E
~<8 | ~;~;-
aαi
i (τi) = (a+i (τi), a+i (τi), ai(τi), ai(τi)).
4= | | C ~<8>G A;9;~ | 9;CD~<8 | ~F~?F;L L L
Ψ = 〈aα11 (τ1) . . . aαm−1
m−1 (τm−1)aαo (τ)a
αm+1
m+1 (τm+1) . . . aαnn (τn)〉0.
4E;F | ~;EFL L L;9F:;EF;~;F=+*+E<C+A;=?8>E9;:FCDG D| 9;:<8>=JE?<CL L L
aαm−1
m−1 (τm−1)aαo (τ) = −aαo (τ)a
αm−1
m−1 (τm−1) + aαm−1
m−1 (τm−1)aαo (τ), y L wv
" G ;FCDC =JE;B?CD:FE;G 8 Cb~<8>G A;9;~ | G9; =J=D F~FC+* | ~;~;-
〈aαo (τ)aα11 (τ1) . . . aαnn (τn)〉0 = e±βεα〈aα11 (τ1) . . . a
αnn (τn)a
αo (τ)〉0,
| L L L 8DB?C . =J= | | ~;C9;CD~<8 | ~;~;-εα = ε
9F:;Eaα = (a+, a)
Nεα = ε
B?C<Eaα =
(a+, a)Lc1 8< N<>~;CJ=>AF Ck F~FC9;:<8>=JE?c8 y L w>v 3L L L-4: | AD)( 8 9;:FCD= | | ~FC
9;:FCD | FA;:;EC :;E 8 CF
Ψ =m−1∑
i=1
(−1)pi〈aα11 (τ1) . . . aαi
i (τi)aαo (τ) . . . aαn
n (τn)〉0
+e±βεαn∑
j=m+1
(−1)pj〈aα11 (τ1) . . . aαm−1
m−1 (τm−1) . . . aαj
j (τj)aαo (τ) . . . aαn
n (τn)〉0
+(−1)pne±βεαΨ,
|pn L L L N;CDGB; )(DBFEL L L;# 8>BFE FE;~FC NF=JE;:<8;FL L L 8 (=JE; F;-
Ψ =1
1 + e±βεα
m−1∑
i=1
(−1)pi〈aα11 (τ1) . . . aαi
i (τi)aαo (τ) . . . aαn
n (τn)〉0
+e±βεα
1 + e±βεα
n∑
j=m+1
(−1)pj〈aα11 (τ1) . . . aαm−1
m−1 (τm−1) . . . aαj
j (τj)aαo (τ) . . . aαn
n (τn)〉0.
| @J8>-N?F;=JE?>~F8 | ~FCDG NcL L L)0UC>> FF~ | C = | FEF;EF~?A1
1 + e±βεαaαi
i (τi)aαo (τ).
,c <~;C y L I L L L (JB;EoAo=DEF9F8>FB;AoB?CFE L L L 8 9F:;EY/(JC Ao=JE; F;-(1 +
e−βε)−1P+δoieε(τ−τi)
LF5&~F8>FCJ ;~FC C :;E A C;
e±βεα
1 + e±βεαaαj
j (τj)aαo (τ) =
P+
1 + eβεδoie
ε(τ−τj). y L wDw4= | | C ~ | >>A;: | ~;AkC<~;C | | B :;CJ~;~;A(A;~;BF; -:; ~<8?
gio(τ − τ ′) =〈Tτai(τ)a
+o (τ
′)〉0〈aia+o 〉0
= eε(τ′−τ)δoi
(1 + e−βε)−1 , τ > τ ′ ,
−(1 + eβε)−1 , τ ′ > τ . y L wy
0(~<8FE CbCD9 | :<8>F G9F8D:;(=J8D~;~;-
¾
ai(τ′)a+o (τ)= gio(τ
′ − τ)P+i ,
-
a+o (τ)ai(τ′)= −gio(τ ′ − τ)P+
i .
5&~<8<CD ;~FC ;F=DEF9;E;FB;Aaαo (τ) = a+o
>8D9;E* | CF
¾
ai(τ′)a+o (τ)= gio(τ
′ − τ)P−i ,-
a+o (τ)ai(τ′)= −gio(τ ′ − τ)P−i , y L w I
"
gio(τ − τ ′) =〈Tτ ai(τ)a
+o (τ
′)〉0〈aia+o 〉0
= eε(τ′−τ)δoi
(1 + e−βε)−1 , τ > τ ′ ,
−(1 + eβε)−1 , τ ′ > τ .":;EF;~;F=+*+EbFCA;=?8>Eb=JE;9FE;G>8>~F<"CD: AD<E y L wDw y L w I N?CDG 8 CJ;~FE;=JE;:<8(;FG | : | F~(DCJCFN;9F G ;=DEFB?CD~<8>~F~?CJ9;EFG8D~;C>U9F:;CJ | ;A;:FE>8D9;E* | C+A=JE; F;F
〈Tτaα11 (τ1) . . . aαm−1
m−1 (τm−1)aαo (τ)a
αm+1
m+1 (τm+1) . . . aαnn (τn)〉o = y L wx
=m−1∑
i=1
(−1)pi〈Tτaα11 (τ1) . . . a
αi
i (τi) . . .¾
aαo (τ) . . . aαnn (τn)〉o+
+n∑
j=m+1
(−1)pj〈Tτaα11 (τ1) . . .
-aαo (τ) . . . a
αj
j (τj) . . . aαnn (τn)〉o
0 . | N<;F=D= | | ~FE?@=DA?<CD=JE?@CJ9 | :F8 CD:F =~F8D:;C/. | ~F~? ~FE/6 | ~;~; =kG 8>~F=JE?>~F8 | ~FCD CD:F ~ 8D; 9FG | =><CDG9; ~;AJN D| CD: | A 4$ BJ8 C-.b~<8G"CJ: AD<&=?8 EFB- [d>_
T D] Ja ` d<]`J_ \J`?]a _] [>ab] [<_ d [c]_FZ \`Ja([c]F_ dkd aF_!"d<]#o] [<_ Fa$<_ #dD]F_ D] [<_ d#%& ('(_D](F`J_ d)(d";`JZ* d>ZF`?]F_&d(_+'"] \;Z<dD_% [#(d(,)%;\#%&D](>a `?]>d(kd [* d(- [<]a _! [a<]>dD]`
pidD[,'"_+ [#&d((.0/0a>ab]e,FZ1d>ZD[
FZ a&, " d>Z2 \,>_3 ('(Zd>_ 4D] Ja ` 5( [FZ*"6! F`$<_ 1"7<[<_ "_ [F_ *d<]`J_ \J`?]a _!D[c]`$F_ 1FZ*] [>ab] [<_ d.
m )8 9 : )
4*: | AJ( 8 =JE;B?CJ:;EFG 8>~F~? D| CD: | E4$ BJ8=~<8>G D8<E/* 8D (JG G | : | F~F =D FC z>A B?Ab9FG | =><CDG9; ~FCD=JE?@bCD9 | :<8 CJ:; =
Sz L;4 G=DCJ | :;A G | : | <~;<=D b<CDDA B; = 8>BFE?@CD9 | :<8 CD:F = :FC>>B?c8c8> (DG D8+B?A AJ;F~ 8 E-
〈Szi 〉0 = b, 〈Sz
i Szj 〉0 = b2 + b′δij, y L w H
〈Szi S
zjS
zk〉0 = b3 + b′b(δij + δik + δjk) + b′′δijδjk y L w O
LF9-L N;9F:;EF;C A
b = 〈Szi 〉0 =
Sp(Szi e−βH0)
Sp(e−βH0)=
1
2tanh
β
2h+ ln
1 + e−βε
1 + e−βε
,
b′ = 〈SzSz〉0c =∂
∂(βh)b, b′′ = 〈SzSzSz〉0c =
∂2
∂(βh)2b.
":;CJ F(G :;A C CD9FE;G>8>~;A 9F:;CJ | ;A;:;A CJ C =DEFBJCJ:;EFG 8 C D| CJ: |> A4 B?8 :FCDJE C :;CDB;<8> >8B;A ADFF~ 8 E ~F89;:FE;B;<8>F":;CD:<8@?A;~FB?A G | : | F~(DCJC=D CD9 | :<8 CD:<89;G | =DFCJG9F ~?A 〈Sz〉 N =DE;F FF&FE C<E;~ <C<8D~;BF =;N6+C9;CF=D; (JG A | = | : C A9;CJ:D;<B?A D| CJ:; U>>A;: | ~(
β∫
0
dτ1
β∫
0
dτ2
β∫
0
dτ3
β∫
0
dτ4∑
iji1j1
∑
i2j2i3j3
tijti1j1ti2j2ti3j3 y L w-,
×〈TτSzl a
+i (τ1)aj(τ1)a
+i1(τ2)aj1(τ2)a
+i2(τ3)aj2(τ3)a
+i3(τ4)aj3(τ4)〉0.
"CDG < FCJ=D~ | G9<8>:F&=?8>~F~? | FB?CDC =JE;J:F8D~;CJCb=DA?<CD=JCDC CD9 | :<8 CJ:F8 ~*+E E<8 C-. <E;=J G (= | G E =JE;:<8
(2.17)<CGA E G | : | F~F @ =D | ~*+C"BF (JB?CDG
CD9 | :<8 CD:F =
〈TτSzl a
+i (τ1)aj(τ1)a
+i1(τ2)aj1(τ2)a
+i2(τ3)aj2(τ3)a
+i3(τ4)aj3(τ4)〉0
= 〈TτSzl
-
a+i (τ1)aj(τ1)a+i1(τ2)aj1(τ2)a
+i2(τ3)aj2(τ3)a
+i3(τ4)aj3(τ4) 〉0
+〈TτSzl
-
a+i (τ1)aj(τ1)a+i1(τ2)aj1(τ2) a
+i2(τ3)aj2(τ3)a
+i3(τ4)aj3(τ4)〉0
= −gij3(τ1 − τ4)〈TτSzl P
+j3aj(τ1)a
+i1(τ2)aj1(τ2)a
+i2(τ3)aj2(τ3)a
+i3(τ4)〉0
− gij1(τ1 − τ2)〈TτSzl P
+j1aj(τ1)a
+i1(τ2)a
+i2(τ3)aj2(τ3)a
+i3(τ4)aj2(τ3)〉0.
"Cc8)(+* | =JE;B?CJ:;EFG 8D~;~;b9;:FCD | FA;:;EbG9F8D:;(=J8D~;~;b9;:FE;=JCFE ("FC~F8DG A;9F~;CJC: |zAJ( 8 A
−gij1(τ1−τ2)gi1j2(τ2−τ3)gi3j(τ4−τ1)gi2j1(τ3−τ2)〈TτSzl P
+j P
+j1P−j2P
+j3〉0
−gij3(τ1−τ4)gi1j2(τ2−τ3)gi2j(τ3−τ1)gi3j1(τ4−τ2)〈TτSzl P
+j P
+j1P−j2P
+j3〉0 y L w K
+gij3(τ1−τ4)gi1j2(τ2−τ3)gi2j1(τ3−τ2)gi3j(τ4−τ1)〈TτSzl P
+j P
+j1P−j2P
+j3〉0.
4= | | C+< 8>:F8 ~;cCJD:<8-. | ~F~?
− Szl ; 1 1′ − t11′;
1 1′ − (g11′P+1 + g11′P
−1 )
4EF:F8>A y L w K =+c8>~FC A=J F9FCD=J <8D (b=DEF9F8>FB;Ak< 8>:F8 Ec
2× ;
0UCDB;<8>G | : | <~; @A"CJ: AJF y L w K >8 B;A ADFF~ 8 E9F:;EF=DC<E (kFCk:;CD ~FC z. | ~;~; F 8D:<8 =:<8>" F~;C A>CDJ:F8 . | ~;~F3B;A ADFF~ 8 =J F9FCD=J <8D (CJ=J8>FE-N6+CbC@?CD9;F( (k=J F9FCD=J F~FU= | :/*+EF~;E 9;G | =DFCJG9F ~;CJ=DE ECD9 | :<8 CJ:F8 E G (
δz GE =JCFE.(:FCD~ | B | :F8=D =>A?FCJ=DE;@ ~? | B?G = LU "8D= | |> C9;:FE;B;<8>FE 8>BFE?@< 8 z:<8 N
2. 3.
4. 5.
1.
6.
;
; ;
; ;
.
4~<8 *+E;@+9FC<8>(*+E?@ :FC>>:F8>@?A;~;B?8@ E DAD | C"~ | @ A;=?8 E < 8>:F8 8 E EF9?A y NI?NFH?N CJ C E EkF 8D:<8 8 EcNJFB; G (+B;A ADFF~ E=DE6 | 9 | :*+CDC9;CJ:J;FB;Ab= | .+8>@CF~F 9 | F | CD~<8<8 DN)6+C <8D~;F&CD=JU":F8D | ~ E< 8>:F8 "CD: A; (CF~FC | | B :FCD~F~?A(A;~;BF; c:F ~F8b=k~<8>DFE/. | ~;~FU6"8DD?8>:J z N-8 8>B?C-. < 8>:F8 8 z E 6+C F~;A; (+<=Dc9 | F-D )(+* | ~F . CF~FE B?A AJ;F~ C | C>>~F8<8 DN/6+C E=D:<8@?CD=DA CG>8 CDA?C/. | ~ | 9;C | =~;AD(JCD=JC A~<8>DFE/. | ~;~F U~ | @ A;=J8D~;~; 8>BFE z EF 8D:<8 8 E= C | F$ > ~;8=D <9;CJ=D c8<C=D <B;E;<8D~;~F ?8> 8 C@;=DCJG CJBcNUFB;=D~FCDG (b9;CJ9;:<8>=JB;E<C G>8 CDA?C/. | ~FCDC+9FC;G9; ~F = L
2 8 39
| : | ; |> CkA=DEF:F8>>8>@L L LcFCL L Lc9F: | FG 8>=D | ~;~;-L-4$E?>~F8 | ~F$L L Lc=JCD~FE:FC>>B?c8c8> z (JG L L LF~F8DG A;9F~;E FE;~FC
g(τ) =1
β
∑
n
eiωnτ g(ωn), g(τ) =1
β
∑
n
eiωnτ g(ωn), y L w M
g(ωn) = −1
iωn − ε, g(ωn) = −
1
iωn − ε, ωn =
2n+ 1
βπ.
~ D| :;A;=J8D~;~;>8k<8>GCD=JE E ~;~FE EA=JE;:<8D8@cNcFB; =D <9;CJ=D c8> (F 8D:<8 8 N>8 :F8>@?A;~;CJBG9; =J=D <~;C* | ~F~?
∫ β
0
ei(∑
ωn)τdτ = βδ(∑
ωn) y L yDv 9;:FE;=JCFE (&FC&A CD=FN-6+C&C>>~F8<8> (&:F =D~F G ((GA 8D:;~FC;=D@; F~FC 83=JE?@; <~;C>;<8>G C AB?C-.b~F = | :/*+EF~< 8>:F8 D| G8 | 9F G ; (A;: k9 | : | @?CFA=J F~FCDGE (JG FC@;=DE;(JCD=JCDC = | B CD:<8 L68>:<8>B D| :F~;CJ/:FE;GCD+NFFB~<8>= | | ~FE?@< 8>:F8 N 8>BUF 8D:<8 ~*+E?@9FCD:J;<B; =
D| CD:F F>>A;: | ~(;N 3~F8>F=D~F G ("":<8> | ~ =;NDFBF 8D ("c8>~F;(CJ=>A G :?A;B A;:;A CJ z C 8>BFE?@cND=(FB;E;@+CDBF: |> | |> | ~ E <~?A; (DG 9;CJG F <CD=J~;C&@;=JE?FFG E E+F ~F E
tk L< "8D;9F:;CJG *+EF:J;F 8D:<8 c8>~F;&CD=JCDC EF9?A 8 "=JE; F;-
+ ++= . . . , y L y?w
|
= g(ωn) =〈P+〉iωn − ε
+〈P−〉iωn − ε
y L yDy U=D <9;CJ=D c8 DNFFB-F; C | F-68>JJ8D:J<8 FEF=;L MJO?N MJK NF~<8>DFE/. | ~;~F*68>JJ8D:J z ;F C<~;C | | B :;CJ~;~FF(A;~;BF; c:F ~F8;LJ!A ?< 8>:F8 ~;CJC(:J;FA y L y?w ;F C<~;CJ:; z~;CJC=DEF9F8>FB;A 〈Sz
i 〉 = 〈Sz〉 = 9?AJ(JG~FC z <8>G C ~FC A9F: | FG 8>=D | ~;~F=D <9;CJ=D c8 =DEF:F8>J
= Gk(ωn) =1
g−1(ωn)− tk. y L y I
"C<&GE(A;~FB;F c:F ~F8
Gk(ωn)<8D (kG9 | B :C<~;C | | B :;CJ~;~FE?@>>AD. | ~/(
εI,II(tk) =1
2(−2µ+ tk)±
1
2
√
g2 + 4tk〈Sz〉g + t2k . y L y>x "CD= | < ~;B?8 | | B :FCD~F~;E;@ CJ~FB (A;~FB;F (= | FEF;EF~;E 9FG | =><CDG9; ~ z | | B :;CJ~;~FC
=>D8 C< &9F: | FG 8D=> | ~<8~F8:;EFGA;~FB?A y L w L4/c8>~FC A ~<8>DFE/. | ~;~F3 FBFEεI(tk)
εII(tk)"CD: A; (<=D | | B :;CJ~;~FFCJ~;EbG9 | B :;AJN 6+C"D8>=+. FEk:;CD< | ~;6+ <E;~FCD+N
= | FEF;EF~F8 FB?CUD8 | .bE (b=J = | <E;FE;~FE9FCDG ;~FC =>D8 C< L /EF:;EF~F8 CJ~kD8 |z.bE ("=J G | : | <~/(JCDC&~<8 | ~F~? CJ9 | :F8 CD:<8(9;G | =DFCJG9F ~?A 〈Sz〉 ?A+=JE;9<8<B?A GE;(J~;C>=>D8 C< g À t
9F =+*+EF:;EF~F8CJ~:F =D~<8t(12 ± 〈Sz〉) | t *+EF:;EF~F89;CJF8 B?CD=JC
| | B :FCD~F~;C>UCJ~;E L
0.0 0.5 1.0 1.5 2.0-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5εε
g
0 EFGDL y L w Lc:<8>~FE;F | | B :;CJ~;~FE?@CJ~ tk=0 = 0.4N 〈Sz〉 = 0.2
L
1
,<A;9;EF~;E CDG ~<8:;CD F;FFE* | L L L N =FBF L L L , =D:<8@?A;=?8>~F~? ~F8D= | | ~;E;@=DE6 | :;B;A;=J8D~/(FN6+CFC B;<8DGEF" BJ8D; $L L L N;:F =D~;F~;~;L L L C-.b~F8+D8>9FE;G>8 EA=DEF ;;<EL L L
〈Szl 〉 = = - + _1
2!-
- _13!
+ ... .
y L y H
5&~<8< EF;~FE;=DEF:F8>";FL L L "~<8>G A;9;~FE
=2
N
∑
n,k
t2kg−1(ωn)− tk
(
P+i
iωn − ε+
P−iiωn − ε
)
= β(α1P+i + α2P
−i ), y L y O
| =D= | | ~FC 9;CD~<8 | ~;~;
α1 =2
Nβ
∑
n,k
t2k(g−1(ωn)− tk)
1
(iωn − ε), α2 =
2
Nβ
∑
n,k
t2k(g−1(ωn)− tk)
1
(iωn − ε).
4EFB?CD:FE;G CD=DA;&FEL L LFC :FE 8 C;
α1 =2
N
∑
k
tk[A1n(εI(tk)) +B1n(εII(tk))],
α2 =2
N
∑
k
tk[A2n(εI(tk)) +B2n(εII(tk))],
A
A1 =εI(tk)− ε
εI(tk)− εII(tk), B1 =
εII(tk)− ε
εII(tk)− εI(tk),
A2 =εI(tk)− ε
εI(tk)− εII(tk), B2 =
εII(tk)− ε
εII(tk)− εI(tk),
n(ε) = 1
1 + eβε ":;CD9FCF ) | : L
4 8D~F8>F EF;~F "CD: L L L C-. | DA E9 | : | 9;EFG8D~;C A=DEF F;F
〈Szl 〉 = 〈Sz
l 〉0 − 〈Szl β(α1P
+l + α2P
−l )〉c0
+1
2!〈Sz
l β2(α1P
+l + α2P
−l )2〉c0 − . . . = 〈Sz
l e−β(α1P
+l +α2P
−l )〉c0. y L y+,
# 8DB;E ;EF~;C N-L L L | B;=J =J8> | ~ ~FE L L L
Heff=∑
i
ε(ni↑+ni↓) + ε(ni↑+ni↓)− ySzi , y L y K
|y = h+ α2 − α1
=DE;~<8F8 L L L;5(~F8>F EF;~FE;=DEF ;;L L L?>8D9;E* | (JG -
y = h+2
N
∑
k
tkε− ε
εI(tk)− εII(tk)[n(εII(tk))− n(εI(tk))]. y L y M
0"G 8 CJ;~ | :; =J~?F~F~?;FL L L "~<8>G A;9;~FE
〈Szl 〉 = 〈Sz〉eff = b(y), y L I v
|b(y) =
1
2tanh
β
2y + ln
1 + e−βε
1 + e−βε
. y L I w0 ;kF;CD9 | :<8 CJ:F8 〈n〉 9F G ;=JE;B?CJ:;EFG 8D~;~; D| CD:F N D| CJ: |> E $ C-.b~<8>8D9;EFG8 EA=DEF ;;<
〈ni〉 = = - + _12!
-
,Σ++ ...α
α
α
- _13!
. y L I y
| =D= | | ~FC ~;CJ=D-9FC>>~F8 | ~F~?-
α =〈P α〉
iωn − εα, α =
1
iωn − εα, = ni,
P α = (P+;P−), εα = (ε; ε)
0"G 8>~F~; FCc8>~FCDB F=>F (DG kD8+:F8>@?A;~;CJBL L L5(~F8>F EF;~FE;=DEF:F8>"F; y L I y D8>9FE/* |> C+A=DEF ;;<
〈ni〉 = 〈ni〉eff +2
Nβ
∑
n,k,α
t2k(g−1(ωn)− tk)
〈P α〉(iωn − εα)2
. y L IDI
|
〈n〉eff =Sp(ne−βHeff)
Sp(e−βHeff)= 2
e−β(ε−h2+α1)(1 + e−βε) + e−β(ε+h
2+α2)(1 + e−βε)
eβ(h/2−α1)(1 + e−βε)2 + e−β(h/2+α2)(1 + e−βε)2 y L I x
"B/6+CL L L N;C :;E 8 CF
〈n〉 = 〈n〉eff +2
N
∑
~k
n(εI(t~k)) + n(εII(t~k)) − 2(n(ε) + n(ε)) y L IJH
4EFB?CD:FE;G CD=DA;&FE~<8>G A;9;~ | G9; =J=D <~;C* | ~;~;-
〈n〉eff − n(ε)− n(ε) = 2〈Sz〉(n(ε)− n(ε))
0"G 8 CJ;~FE;=DEF:F8>bL L L?D8>9FE/* |> C+A=DEF ;;<
〈ni〉 =2
N
∑
k
[n(εI(tk)) + n(εII(tk))]− 2〈P+〉n(ε)− 2〈P−〉n(ε). y L IDO
: ) 9
1;:;CD:<8@?A;~FB?AL L L N E=D= | | CL L L
Hλ = H0 + λHint, y L I , 8>B-N6+C
H → H0;F
λ = 0H → H0 +Hint
F;λ = 1
L
Zλ = Sp(e−βHλ) = Sp(e−βH0σλ(β)) = Z0〈σλ(β)〉0, |
σλ(β) = Tτ exp
−λβ∫
0
Hint(τ)dτ
,
=D F9FCD=J F~FC
Ωλ = −1
βlnZ0 −
1
βln〈σλ(β)〉0, y L IDK
∆Ωλ = Ωλ − Ω0 = −1
βln〈σλ(β)〉0.
#$A Ω0 |&D| : CFEF~F8 ;~FE;9;C D| ~F; 8>k:FC>>:F8>@JCJ=J8D~;EFb~<8"CF~FCD=DA?<CD=J b<8>G E;~F
=DE;@; <~;CJC 8 )( CD~F 8>~;AJL# 8DB;E FE;~FC 8 C;N
∆Ω =
1∫
0
dλ
(
dΩλ
dλ
)
. y L IDM
1;= | FEF;EF~;EdΩλ/dλ
N E C-. | C >8D9;EFG8 Ek< 8>:F8 ~;EF:J;A=DEF ;;<
βdΩλ
dλ= + + ...+ , y L xJv
|tλNU8k@;=JE?FFG 8F ~F | <E;~<8k< ~; =F 8D:<8 ~FC A:J;;A y L xJv J ;
FB?C&~ | G C (9F8D:F8 | :λL !< >8A;=J8 .bE E-N 6+C | 9<8>:<8 | : G C> ( 8>B?C-.
D ;>AJ( z FBJC>UF ~F N/6+C C :;E A (JG 9F G ;:;CD9FE;GA;=J8D~;~; 〈Sz〉 > <~;C+?
-= - + 12_
!
.+ . . .- 13 !_
4 8D~F8>F EF;~F "CD: -=DEF:F8> y L IDM >8D9;E* | C+FB
∆Ω =2
Nβ
∑
n,k
1∫
0
λt2kg2λ(ωn)
1
1− λtkgλ(ωn)dλ
= − 2
Nβ
∑
n,k
ln(1− tkg(ωn))−2
Nβ
∑
n,k
1∫
0
λtkdgλ(ωn)
dλ
1− λtkgλ(ωn)dλ. y L x;w
| :*+E;k<C<8D~;CJB y L x;w3 C-.b~F8 9;: | <G 8D=DE E= F 8D:<8 ~FC A=DEF ;;<
+ ...12_ 1
4_+1
3_+ .
y L xJy 1+ 8>:F8 ~;EF :J; y L xJy CJ9;EFGA | | B :FCD~F~;EF 8?N | ~ | :F | E;F~;EF G9 | B : FB?CDC=DE;~<8F8 (DG D8>8)(D~FE 9;C | 9FG | =><CDG9; ~F =;L1+:?A;E; <C<8D~;CJB = y L xFw b C-.b~F89;:FCD ~ D| :?A;=?8 E 3>8D9;EFG8 E A "CJ: 3< 8 z
:<8 ~FCDC :J;;Ac
13!−−12! +
−
− ... −
, y L x I
6+C+ F=>F (DG kD8+:<8@?A;~FCDB~F8>F=D~FCDG -9;G | =DFCJG9F ~;CJ=DC>U9; <GEFG D|> EcL0"G 8 CDF~;CFN F 8D:<8 ~FE; :J; F; D| : C<E;~<8 F~;CJC9;C D| ~F; 8>;A =FBFE; ~ |
=>@?CFE ( FCc8 B?CJ=DEF9F8D:F8 | :λ C-.b~<8+>8D9;EFG8 EkA=JE; F;F
∆Ω = + ...1_ + 13_
2 + −− −12!
+... , y L xDx
4 8D~F8>F EF;~FC A=DEF ;;<
Ωeff = − 2
Nβ
∑
n,k
ln(1−tkg(ωn))−2
Nβ
∑
n,k
g(ωn))t2k
g−1(ωn))− tk− 1
βln Sp(e−βHeff). y L x H
4E;F FEF=+*+Ek9;C<&GEk(A;~;BF; -N?9F:;CJGA A;=?8>=*+Eb9FC+F8DG C 8@kF>:;CJDEF=+*+Eb | FB;<~ |zGB;<8>F~FF9 | : | =DCJ: | ~;~;-N>D8>9FE/* |> C(CJG 8 CDF~;EFb=DEF:F8>3;F D| : CFEF~F8 ;~FCDC"9;C zD| ~;F 8FAc
∆Ω = − 2
Nβ
∑
k
ln(cosh β
2εI(tk))(coshβ2εII(tk))
(cosh β2ε)(cosh
β2 ε)
−1
βln cosh
β
2(h+ α2 − α1) + ln
1 + e−βε
1 + e−βε
+1
βln cosh
β
2h+ ln
1 + e−βε
1 + e−βε
+ 〈Sz〉(α2 − α1). y L x O 1 8< NCJGBF (JB;E D| : C<E;~<8 F~;EF+9FC D| ~;F 8 $(A;~;BF; =D G | : | F~(DCJC&~<8 | ~ z~?CD9 | :<8 CJ:F89;G | =DFCJG9F ~?A?NJ9 | : | =J :;E C"A?C/. | ~F G (+~F8D><E . | ~(+=DEFB?CD:FE;G 8>~FE?@9;:FEb:FC>>:F8>@?A;~;B;A G | : | <~/(JCDC"=J bCJ9 | :F8 CD:<8"9FG | =><CDG9; ~;A FB; )(DB?CJG <F8DG EF~;CJB 89;:FE:;CD:<8@?A;~FB?A D| : C<E;~<8 F~;CJCb9FC D| ~;F 8FA
dΩ
d(−µ) =2
N
∑
k
[n(εI(tk)) + n(εII(tk))]− 2〈P+〉n(ε)− 2〈P−〉n(ε),
dΩ
d(−h) =1
2tanh
β
2(h+ α2 − α1) + ln
1 + e−βε
1 + e−βε
.
# 8DB;E ;EF~;C EC :FE A CdΩ
d(−µ) = 〈n〉, dΩ
d(−h) = 〈Sz〉.0 . | 9F:;CJ= | | ~;EF:FC>>:F8>@?A;~;CJB D| : CFEF~F8 ;~FE?@(A;~;BF; < ;G~ | ~;Cb=k:F8 BJ8>@CF~FCDC CDC . ~F8D><E . | ~F~?-L
9 f ) ,8 / 8
":;E<CDG < /. | ~F~;"L L L E =DE;F F CF=?8: | .bE E =FBFE?@ C-. | G~?A;=?8 E ~<8 * 8GEFG J| 8?8 : | .bE L L L JB?CFEk=J=J8 .+8 (JG 46+CbL L L;4 (DC Ab=DEF9F8>FB;AL L L?=DE;~<8<8 &A CD=DAL L L : | .bE L L L | =DEF9F8>FCJB D )(+* L L L &C>>~F8<8 DN 6+C L L L3L L L :; =J~;CJ=J8 .b~;CJ
A CD=JCD*= (DC A=DEF9F8>FB;AJL
9
1;: | .bE A L L L-=DE;~<8F8 (DG A CD=JCD L L L(A CJ=J8 ( ∂Ω∂〈Sz〉
)
T,µ,h= 0
"CJ: A :; =D~;F~;~;~<8+G | : | <~ "=D CD9 | :<8 CJ:F8 9FG | =><CDG9; ~;A y L I v L":;CJ= | |> C+9FCDG < FCJ=D~ | FCJG F . | ~;~;9;CJ= | F ~FB;EL L L 8>8c8 C =DEF ;;:; =J~?F~F~?;FL L L
〈Szl 〉 =
1
2tanh
β
2(h+ α2 − α1 +
2
βln
1 + e−βε
1 + e−βε
. y L x, 1;=DEF9F8>FB;A
T → 08>JC
β →∞ L L L
Υ =2
βln
1 + e−βε
1 + e−βε=
−2g; µ >g2 , (ε < 0, ε < 0),
−2µ− g; −g2 < µ < g
2 , (ε > 0, ε < 0),
0, µ 6 −g2 ; (ε > 0, ε > 0),
y L x K
ε = −µ+g
2, ε = −µ− g
2,
α2 − α1 =2
N
∑
k
tkg
√
g2 + 4tk〈Sz〉g + t2k[n(εII(tk))− n(εI(tk))], y L x M
|n(ε) =
1
1 + eβε,-9;:FE
β →∞ 8 C
n(ε) = 1, ε < 0,
n(ε) = 0, ε > 0. y L H v
4EF:F8>"F;L L L 8 "=DEF ;;U
〈n〉 =2
N
∑
k
[n(εI(tk)) + n(εII(tk))]− (1 + 2〈Sz〉)n(ε)− (1− 2〈Sz〉)n(ε). y L H w
4EFB?CD:FE;G CD=DA;&FE~<8>G A;9;~ | G9; =J=D <~;C* | ~;~;
1
βln cosh
β
2X → |X|
2
9;:FEβ →∞, y L H y
C-.b~F8+D8>9FE;G>8 EL L L;9F:;ET → 0
FBc
∆Ω = Ω− Ω0 = |ε|+ |ε| − 1
N
∑
k
(|εI(tk)|+ |εII(tk)|) + 〈Sz〉(α2 − α1)
−1
2|h+ α2 − α1 +Υ|+ 1
2|h+Υ|, y L HDI
|
Ω0 = −1
βln(
eβh2 (1 + e−βε)2 + e−
βh2 (1 + e−βε)2
)
. y L H x 0UCD FF~ |> C+=JE;9<8<CDB-NFB?CFEL L L
µ < −g2− t,
9;:FE(DC AεNεINεNεII > 0
Nα2 − α1 = 0
NΥ = 0
L 0 =J~?F~F~? L L LU9 | : | ? | A~F8DG A;9;~FE;=JE;:<8?
〈Sz〉 = 1
2tanh
βh
2 y L HDH
0 . | =+c8>~FC A=JE;9<8<B?AL L L;~F8DG A;9F~;E ;EF~;C
9F:;Eh > 0 〈Sz〉 = 1
2, Ω = −h
2, 〈n〉 = 0; y L HDO
F;h < 0 〈Sz〉 = −1
2, Ω =
h
2, 〈n〉 = 0. y L H ,
1+CDG < /. | ~F~?=DEF9F8>FB;A
−g2− t < µ < −g
2+ t,
B?CFEL L L;:FC>> C ~<8<=J8 =DEF9F8>FBFEcL | :*+E;=JE;9<8<CDB L L L N;B?CFE
−g2− t < µ < −g
2
C . |, εI > 0, Υ = 0.
0UC>>= ;>CDB:F =D~;F~;~;L L L y L x , N-L L L C-. A (bDA E>8D9;EFG8D~;-= ~<8>G A;9;~FC A=DEF ;;< 9F:;E>~F8 | ~F~;
h > 0 〈Sz〉 = 1
2, Ω = −h
2, 〈n〉 = 0; y L HDK
h = 0, Ω = 0, 〈n〉 = 0,F;=JG @k~<8 | ~/( 〈Sz〉 > 1
2+µ+ g/2
t; y L HDM
h =2〈Sz〉3gt
[
(
µ+ g/2
1/2− 〈Sz〉
)3
+ t3
]
− 1
2t
[
(
µ+ g/2
1/2− 〈Sz〉
)2
− t2
]
, y L O v
Ω = (h− t/2)(1/2− 〈Sz〉)− (µ+ g/2)[
1 +µ+ g/2
2t(1/2− 〈Sz〉]
− h
2,
〈n〉 = µ+ g/2
t(1/2− 〈Sz〉) + 1,B?CFE 〈Sz〉 < 1
2+µ+ g/2
t;
9;:FEh 6 −(µ+ g/2)3 + t3
3gt− (µ+ g/2)2 − t2
2t, y L O w
8 C 〈Sz〉 = −1
2, 〈n〉 = µ+ g/2
t+ 1,
Ω =h
2− t
2− µ− g
2− (µ+ g/2)2
2t,
0 :;E 8>~F-: | AJ( 8 ECJD:<8-. | ~;cA=JE; F;F L L L
µ = −0.6, t = 0.2, g = 1=b:<8>~FE;F
T →∞,
~F8:FE;GA;~;BJ8>@ y L y" 8 y L I;L;4 : | .bE µ = const
~F A D| : CFEF~F8 ;~FCDC9FC D| ~ z; 8FA
Ω=JE?>~F8<8 $A CD=DA+:F =D~FCD=?8>E ="GEFG D|> LJ,kC :;E 8>~FCc9FCD= | < ~;BFEL L L NDG F FA DN
6+C 9F:;E9 | =J~;C AL L L N6+C = G=DCJ | :FAJNF9;:FE;=JCFE ( <CL L L1;F:;A;CJC =JE;9<8<B?A?NFB?CFE
−g2< µ < −g
2+ t, εI > 0, Υ = −2µ− g,
L L L N;:;CD= ;CJB:; =J~?F~F~? y L x , 3 C-. | DA E>8D9;EFG8D~;EF~<8>G A;9;~FE ;EF~;C
〈Sz〉 = 1
2, 〈n〉 = 0, Ω = −1
2max(h, 4|ε|−h) B?CFE
h >2t2
3g−Υ, y L O y
0.00 0.03 0.06
-0.5
0.0
0.5
h
Sz
0.00 0.03 0.06-0.03
-0.02
-0.01
0.00
0.01
h
ΩΩΩΩ
0 EFGDL y L y L-L L L?D8 | .b~;CJG L L L
0.00 0.03 0.060.0
0.1
0.2
0.3
0.4
0.5
h
n
0.00 0.03 0.06
-0.6
-0.3
0.0
0.3
0.6
h
εεεε
µµµµ
0 E;GJL y L I?L (CJ= | F ~FBJ8kL L LF=+>8> | .b~;CJG U=D L L L
h =4t2
3g〈Sz〉 −Υ, Ω =
1
2|h+Υ| − 〈Sz〉(h+Υ)− 1
2max(h, 4|ε|−h), y L OJI
〈n〉 = 1− 2〈Sz〉, F; 〈Sz〉 > 1
2− µ+ g/2
t,
h =2〈Sz〉3gt
[
(
µ+ g/2
1/2− 〈Sz〉
)3
+ t3
]
− 1
2t
[
(
µ+ g/2
1/2− 〈Sz〉
)2
− t2
]
−Υ, y L O x
Ω =1
2|h+Υ| − 〈Sz〉(h+Υ)− 1
2max(h, 4|ε|−h)
− t2(1/2− 〈Sz〉) + µ+
g
2− (µ+ g/2)2
2t(1/2− 〈Sz〉) ,
〈n〉 = µ+ g/2
t(1/2− 〈Sz〉) − 2〈Sz〉, ;F 〈Sz〉 < 1
2− µ+ g/2
t;
9;:FEh < −(µ+ g/2)3 + t3
3gt− (µ+ g/2)2 − t2
2t−Υ, y L ODH
8 C 〈Sz〉 = −1
2, 〈n〉 = µ+ g/2
t+ 1,
Ω =1
2|h+Υ| − 1
2(h+Υ)− 1
2max(h, 4|ε|−h)
− t2+ µ+
g
2− 1
2t(µ+ g/2)2.
8+:;EFGA;~FBJ8>@ y L x ky L , 9;: | <G 8>=D | ~;C D8 | .b~FCDG L L L?F;=JE;9<8<B?A
µ = −0.4, t = 0.2, g = 1FB= :F8D~;EF;
T →∞,
8>BcF;L L L4 CJ;F L L L 8 CL L L<, ~*+CDC DCJB?A?N;FYL L L"B=DE;F~FC :<8>" BF = ~<8+:FE;GA;~;B;A y L O;NcL L L C-. | F=D; EFG (L L L?D8+:<8@?A;~FCDBL L L# 8DB;E ;EF~;C N L L L N-=GEFG D|> 3L L L-=DEF~;EFBJ8 C-. <E;=J G (L L LU0(GB; )(DBFE L L L N 8 C
=D F9FCD=J F~;Ab ~?AL L L":;EL L L
−g2+ t < µ < 0, Υ = −2µ− g,
C-. |> C+D8>9FE;G>8 E
εI > 0, n(εI) = 0; εII < 0, n(εII) = 1; α2 − α1 = −4t2
3g〈Sz〉.
4 : | AD)( 8 -C :;E A CF
〈Sz〉 = 1
2, 〈n〉 = 0, Ω = −1
2max(h, 4|ε|−h), 9;:FE
h >2t2
3g−Υ; y L ODO
h =4t2
3g〈Sz〉 −Υ,
;F 〈Sz〉 6= ±1
2, 〈n〉 = 1− 2〈Sz〉, y L O ,
Ω =1
2|h+Υ| − 〈Sz〉(h+Υ)− 1
2max(h, 4|ε|−h);
〈Sz〉 = −1
2, 〈n〉 = 2, Ω = −1
2max(h, 4|ε|−h), B?C<E
h < −2t2
3g−Υ. y L ODK
4EF9;EFG8D~;c8D~F8>F EF;~F<: | AJ( 8 Ek9F: | FG 8>=D | ~;C+~F8":FE;GA;~;B?8@ y L K" y L M?L "Bb=JE?<~;C":FE;GA;~;BF = y L K;N?=L L L;=DEF9F8>FB;Ak G~?A +L L L N 6+C+<8 +L L L?=JBJ8> =JB?Ak~<8 C-. <E;=J G ( 9FCF=DEL L L
0.18 0.20 0.22 0.24 0.26 0.28-0.5
-0.3
-0.1
0.1
0.3
0.5
h
Sz
0.18 0.20 0.22 0.24 0.26 0.28-0.5
-0.3
-0.1
0.1
0.3
0.5
h
Sz
0 E;GJL y L x L"C(JCD=?8&D8 | .b~F G ( 〈Sz〉 tk=0 = 0.2Nµ = −0.4 N g = 1.0
F;k: | .bE Aµ T = 0.01
T = 0L
0.18 0.20 0.22 0.24 0.26 0.28-0.14
-0.13
-0.12
-0.11
-0.10
ΩΩ
h 0.18 0.20 0.22 0.24 0.26 0.28-0.14
-0.13
-0.12
-0.11
-0.10
ΩΩ
h
0 E;GJL y L H?L(C)(DCJ=J8 D8 | .b~F G ( D| : CFEF~F8 ;~FCDC9;C D| ~F; 8>;A tk=0 = 0.2Nµ =
−0.4 N g = 1.0 T = 0.01
T = 0
L
0.18 0.20 0.22 0.24 0.26 0.28
-0.6
-0.3
0.0
0.3
0.6
εε
µµ
h 0.18 0.20 0.22 0.24 0.26 0.28
-0.6
-0.3
0.0
0.3
0.6
εε
µµ
h
0 E;GJL y L O?L "CD= | < ~;B?8:F8D~;EF/( | | B :;CJ~;~FE?@ >CD~ =J = | FEF;EF~;E 9;CJ=>FCJ= .b~(DCJC9;CF
h tk=0 = 0.2Nµ = −0.4 N g = 1.0
T = 0.01
T = 0
L
0.18 0.20 0.22 0.24 0.26 0.280.0
0.3
0.6
0.9
1.2
1.5
n
h 0.18 0.20 0.22 0.24 0.26 0.280.0
0.3
0.6
0.9
1.2
1.5n
h
0 E;GJL y L , L "C(JCD=?8oD8 | .b~F G ( | | B :FCD~F~;C> B?CJ~; | ~ :F8D; n tk=0 = 0.2
N
µ = −0.4 N g = 1.0 T = 0.01
T = 0L
":;CJ= | | ~ | =DE6 | 8D~F8>F EF;~ | <CDG < /. | ~F~?L L L;< ;G~ | ~;C+A=DEF9F8>FB;AB?C<Eµ ∈
(−∞, 0)L<4EF9F8>FCJBB?C<EL L L? | B?C C :;E 8 EbL L L
µ→ −µ, h→ 2g − h, n→ 2− n, Sz → −Sz. y L ODM ,cCJD:<8>E CC :FE 8D~;c: | AD)( 8 Ek=bL L L N?8 8DB?C . : | AD( 8 Ek;EFG | (J~;E;@ :FC>>:F8 z
@?A;~;BF =L L L N?A=DEF ;;< L L L 89F: | FG 8>=D | ~;E;@k:FE;GA;~;B?8@ +L L L;4 | .8>@kFE?@CDD<8DG D| cN-L L L N 8D ( G | L L L, :;CJG C L L L NcFB | CJD:<8-. | ~FC~<8L L L ,-8 | .b~; G (L L L-=J L L L-9F:;CJ F(G :FCD=?8>~<8
~F8k:;EFGA;~FB?A y L wJw ;F<=DC@:; >~;E;@>~F8 | ~(L L L "B=DE;F~FC;N | | B A CD=D | ~;EFL L L CDD<8DG (+>~F8 | ~(L L L N/6+C 9FCD= ;>8D~;C+bL L L< "8>@;E?L L L;C>>~F8<8 C-. FEF=D G (L L L G~;A;=J8D~;~;L L L;=D9 | :* | >AJFCbC :FE 8D~;Cb=b:;CJDC ,+, cF;L L L C | F L L L5(~F8>F +9;CJ= | F ~FB;EL L L? |> CD~FG :;A DN6+CL L L;=D <9;CJ=D c8 L L L :FE;GA;~;CJB y L w>y L!< k>8A;=?8-.bE EcN/6+CkL L L NJAb=JE;9<8<B?AkB?CFEL L L N C-. | >A EL L L?4 CJ>c8>G F>~F8 | ~(
L L L; G~?A C-. <E;=J G (L L L
9
' CJ=DCJYL L L?;F: | .bE An = const
L L L(kA CJ=J8 "CD: A L L L y L IJO L0 | AJ( 8 E8>~<8< E;F~;CJCb:;CD:<8@?A;~FB?AL L L?9F: | FG 8>=D | ~;C+A=DEF F;Fc>8 | .b~FCDG z
D| L L LF~F8 :;EFGA;~FBJ8>@ y L wx L
0.95 1.00 1.05
-0.5
0.0
0.5
h
Sz
0.95 1.00 1.05-0.53
-0.52
-0.51
-0.50
h
ΩΩΩΩ
0 E;GJL y L K;LcL L L?D8 | .b~FCDG L L L;=J L L L
0.95 1.00 1.050.0
0.5
1.0
1.5
2.0
h
n
0.95 1.00 1.05
-0.6
-0.3
0.0
0.3
0.6
h
εεεε
µµµµ
0 E;GJL y L M?L "CD= | < ~;B?8 | | B :FCD~F~;C>&B?CD~F | ~ :<8>F n$:<8>~FE;(G9 | B :;A=>8> | . z
~;CJG -=D = | <E;FE;~FE9;CFh tk=0 = 0.2
Nµ = −0.0 N g = 1.0
NT = 0
L,~<8>= | | ~FE?@:FE;GA;~;BF =;N; | CJD:<8-. | ~FCb9;CJ= | F ~FB?A= | FEF;EF~;E@; ;~FCDC 9FC D| ~ z
; 8FA=D B?CD~F | ~ :<8>F | | B :;CJ~; =k9F:;E:F ~FE?@~<8 | ~F~?F@ D| 9 | :<8 A;:FEcNc=JE?<~;CCDD<8DG ( +=D ~;CJ*9;C@; <~;CJ
dµ/dn < 0= FBF G 8D~;E+CF~FCD:F F~FE :FC>>9;C< z
FC BF (JB?CDG J<8>G E;~FCDB ~ | G ;BFE E-L>4: | .bE J9FCDG ;~FCDC3~<8 | ~;~;"@; ;~FCDC9;C D| ~;F 8FA
µ = const=D8>~F8 | ~; kCJ>c8>G <=J FDA;=J8 (DG 8>CD=JE;k9 | : | @; 9 | : z
*+CJC":;C;A+>;G :FE;JB?C 9F8D:F8 | :<8(9FCD:J;<B?A ;B?CJ~; | ~ :F8D; cF8DG EF~;CJB CJ;B?8(8 zCJ=DCJC&9 | : | @?C;A=JE?>~F8<8 (JG "$A CD=JE ~F A A D| : C<E;~<8 F~;CJC(9;C D| ~F; 8>;A
Ω8DDC>89;:<8>=JE?<C 78>B?G= | <8;L?4 : | .bE c9;CJG F~;CJC~<8 | ~;~;kB; )(DB?CJG <<8>G E z
~;CJBn = const
=+F CDD<8DG c~<8 | ~/( | | B :;CJ~;~FC B?CD~F | ~ :<8>F N;=J (J~F8 | ~ | :F 8 $9F:;CJEF~+=JCJ:?AJN+6+C&c8 C-. FEF=D G ("9;:FCD= | G E<C EF;~;A< ~F N>FBJ83 | .bE ("9;
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
n=0
n=0 0<n<2
0<n<2
Sz=-1/2
Sz=+1/2
n=2
n=2h
µµ
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
h
µµn=0 0<n<2
0<n<2
n=0
n=2
n=2
Sz=+1/2
Sz=-1/2
0 E;GJL y L wv Lc) 8>CJ=J8bF 8D:<8 8µ − h
L<# CDFB?CD=?8b CD~FBJ8kGA;; )(D~<8 F ~F C@?CD9;F( (CDD<8DG -+~<8 | ~;~;
Sz = ±12
N<=J F9FCD=J F~FC;Lc 8D= | | ~F8 GA;; )(D~<8 F ~F CJD:<8-.+8 CJ;BFE8>CJ=DCJC9 | : | @JCFA9 | :*+CDC:FC;A?L =DEF9F8>FCJB~?AJ(JCD=JC D| 9 | :<8 A;:FE T = 0.002 tk=0 = 0.2
g = 1
L
0.20 0.22 0.24 0.260.000
0.005
0.010
0.015
0.020
0.025
h
Tc
t=0.15
t=0.2
0 E;GJL y L wJw L<) 8>CD=?8F 8D:<8 8Tc − h tk=0 = 0.2
Nµ = −0.4 N g = 1
L
-0.14
-0.13
-0.12
-0.11
0.00 0.01 0.02 0.03ΤΤ
ΩΩ
0.00 0.01 0.02 0.03 0.04-0.6
-0.4
-0.2
0.0
0.2
0.4
Sz
T
0 EFGDL y L wy LcL L L?D8 | .b~F G (L L L
0.016 0.017 0.0180.000
0.005
0.010
0.015
0.020Tc
h
0 E;GJL y L w I?L<) 8>CD=?8F 8D:<8 8Tc − h µ = −0.6 N g = 1
Ntk=0 = 0.2
LB;:FE;=JCD
F (n)LU4: | AD)( 8 C :;E A CFN 6+C D| : CFEF~F8 ;~FCD )(+* G FB;E EcN
6+CFCCF~FCD:F F~FCDC"=JE;9<8<B?AJN 3G 8>~FE+(:; >~;CJ B?CJ~; | ~ :F8D; | | B :FCD~F = 8>C z=DCb:FC> * 8>:FCD=?8>~F-G 8D~;E LF# CDFB;EFC E;B;AFCbBF:;EF=DC>
F (n)~F8 :FE;GA;~;B;A y L wx =JE?>~F8 z
F8D (>~F8 | ~F~? | | B :;CJ~;~FC-B?CD~F | ~ :<8>F -F=J F9FCD=J F~FCDC(>~F8 | ~F~?kG | : | F~(DCJC=D CD9 | :<8 CJ:F89;G | =DFCJG9F ~?AA:FC> * 8>:FCD=?8>~FE?@G 8D~F8>@ :;EFGA;~FCDB y L w H CDFBJCJ=DF z~; L (;~<8 | ~;~;-N 8>B?C-. C-.b~F8":;CD:<8@?A;=?8 E D8&FCJ9;C CDCD 9F:;EF=DE;<8+78DB?G= | c8~F8 :F8D( B?AkD8 | .b~FCDG c@; ;~FCDC 9;C D| ~F; 8>;A=J B?CJ~; | ~ :F8D;
µ(n)L
# 8DB;E FE;~FC N-9F:;E" B?GCD=?8>~FC A~<8 | ~;~F B; )(DB?CJG <8>G E;~FCDB n = const N
8>CD=JE; 9 | : | @; 9 | :/*+CJC:;CFA =: | .bE µ = const 9 | : | @JC<E (A 8>CJ= |
:;CD-*b8>:;A;=J8D~;~;-L0 | AJ( A;(;<8>CJ=D;< 8>:F8 8n−h 9F:;Ek:; >~;E;@+~<8 | ~;~;;@ D|>Uz
9 | :F8 A;:FE 9F: | FG 8>=D | ~;"~F8 :;EFGA;~FB?A y L w O | GA;; )(D~F&< ~; =J F9FCD=J <8D ( E ~<8 | ~;~; | | B :;CJ~;~FC-B?CJ~; | ~ :F8D; U~F8"FB;<:FC> * 8>:FCD=DA (JG kGE;G D| 8"9F:;Ek" B z