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M. Corgini, Huang–Yang–Luttinger model: Gaussian dominance and Bose condensation, Teoret. Mat. Fiz., 1999, Volume 121, Number 2, 347–352 DOI: http://dx.doi.org/10.4213/tmf814 Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use http://www.mathnet.ru/eng/agreement Download details: IP: 186.37.130.22 April 16, 2016, 06:13:55

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L = {Lx : x ∈ };

HL

= L2(L)

¨ ¯ãáâì FB (HL) { ᨬ¬¥âà¨ç­®¥ 䮪®¢áª®¥ ¯à®áâà ­á⢮, ¯®áâ஥­­®¥ ­  HL. Ž¯¥à Ä â®à S L = −
§ ¤ ¥â á ¬®á®¯à殮­­ë© £ ¬¨«ìâ®­¨ ­ ­  HL á ¤¨áªà¥â­ë¬ ᯥªâ஬ 0 = L−2E0 < L−2E1 6 L−2E2 6 · · · ; ¯®¤áç¨â ­­ë¬ á ãç¥â®¬ ªà â­®á⥩, ¨ ᮡá⢥­­ë¬¨ äã­ªæ¨ï¬¨ {k (x)}, 㤮¢«¥â¢®àïÄ î騬¨ ãà ¢­¥­¨î 1 −
k (x) = L−2 Ek k (x) 2 ¯à¨ ¯®¤å®¤ïé¨å £à ­¨ç­ëå ãá«®¢¨ïå.  ¡®à ä㭪権 {k (x)} ®¡à §ã¥â ¡ §¨á ¢ ¯à®áÄ âà ­á⢥ L2(L). ‹î¡ ï äã­ªæ¨ï (x1 ; : : : ; xN ) ∈ L2(NL ) ¬®¦¥â ¡ëâì § ¤ ­  á ¯®¬®Ä éìî ¬­®¦¥á⢠ ä㭪権 (

{ k x1 ; : : : ; x N ∗

)=

N O j=1

k (xj )}; j

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=

N Y

j=1

hkj ; lj iL2 (L ) ;

£¤¥ hk ; l iL2 ( ) { ®¡ëç­®¥ ᪠«ïà­®¥ ¯à®¨§¢¥¤¥­¨¥ ¢ ¯à®áâà ­á⢥ L2(L) = HL. Ž¡®§­ ç¨¬ç¥à¥§ HBN;L ¯®¤¯à®áâà ­á⢮ HN;L, á®áâ®ï饥¨§á¨¬¬¥âà¨ç­ëåä㭪権 (¡®§¥-á¨á⥬ ) ¨ á­ ¡¦¥­­®¥ ®àâ®­®à¬ «ì­ë¬ ¡ §¨á®¬, ¯®áâ஥­­ë¬ ¨§ ¡ §¨á  { k } á ¯®¬®éìî ᨬ¬¥âਧ æ¨¨. …᫨ ¯®«®¦¨âì HB0;L = C , ⮠䮪®¢áª®¥ ¯à®áâà ­á⢮ FB (HL ) ®¯à¥¤¥«ï¥âáï ᮮ⭮襭¨¥¬ ∞ M N;L : L F B (H ) = HB j

j

L

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‚ í⮩ áâ âì¥ ¨§ãç ¥âáï â ª ­ §ë¢ ¥¬ ï ¬®¤¥«ì •ã ­£ {Ÿ­£ {‹ â⨭¦¥à  [1], ®¯¥Ä à â®à í­¥à£¨¨ ª®â®à®© ¨¬¥¥â ¢¨¤   X a b2 X 2 † L H = L (j )aj aj + (1) 2V 2N − nj ≡ H0 + HI ; j> 1

j> 1 2 £¤¥ V { ®¡ê¥¬ ®¡« á⨠L, L(j ) = L Ej , H0 § ¤ ¥â ᢮¡®¤­ë© FB (HL ), ¯®áâ஥­­ë© ®¡ëç­ë¬ ᯮᮡ®¬ ¨§ S L . Ž¯¥à â®à −

Nb =

X

j> 1

nj =

X

j> 1

£ ¬¨«ìâ®­¨ ­ ¢

a†j aj

¥áâì ®¯¥à â®à ç¨á«  ç áâ¨æ, aj , aj { ¡®§¥¢áª¨¥ ®¯¥à â®àë, 㤮¢«¥â¢®àïî騥 ª®¬¬ãâ Ä 樮­­ë¬ ᮮ⭮襭¨ï¬ [ai; a†j ] = aia†j − a†j ai = Æi;j : DZ®áâ®ï­­ ï a > 0  áá®æ¨¨àã¥âáï á ª®íää¨æ¨¥­â®¬ ¯à¥«®¬«¥­¨ï ¯à¨ ®¯â¨ç¥áª®©  ­ «®Ä £¨¨, ¨á¯®«ì§ã¥¬®© ¢ 䨧¨ç¥áª®© ¨­â¥à¯à¥â æ¨¨ à áᬠâਢ ¥¬®© ¬®¤¥«¨. Žç¥¢¨¤­®, çâ® Nb kB (x1 ; : : : ; xN ) = N kB (x1 ; : : : ; xN ): ‚¢¥¤¥¬ £¨¡¡á®¢áª®¥ á®áâ®ï­¨¥ â¥à¬®¤¨­ ¬¨ç¥áª®£® à ¢­®¢¥á¨ï b Tr : : : e− (H −N) (2) h: : : iH = b Tr e− (H −N) b i ¨ á।­îî ¯«®â­®áâì ç¨á«  ç áâ¨æ ¢ ª®­¥ç­®¬ ®¡ê¥¬¥ V = hN=V niH , £¤¥ nb H = hb { ®¯¥à â®à ¯«®â­®áâ¨. ‚ â¥à¬®¤¨­ ¬¨ç¥áª®¬ ¯à¥¤¥«¥ ¯«®â­®áâì (3)  = lim V ; L→∞  ¨ ¯à¥¤áâ ¢«ïîâ ᮡ®© 娬¨ç¥áª¨© ¯®â¥­æ¨ « ¨ ®¡à â­ãî ⥬¯¥à âãàã, ᮮ⢥âÄ á⢥­­®. †

L

L

L

L

L

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=

∞ X

N=0

L

−

∞ X b N) = TrH N=0

N;L B

e N TrH

N;L B

e− H

N;L

e− (H

L

−

b N) =

(4)

;

£¤¥ H N;L = H L|H , Z ;[H L]N=0 = 1, [H L]N=1 = H0. ‚¥«¨ç¨­  TrH (: : : ) § Ä ¤ ¥â ®£à ­¨ç¥­¨¥ á«¥¤  ­  ¯à®áâà ­á⢮ HBN;L.  áᬮâਬ ᥬ¥©á⢮ ®¯¥à â®à®¢ N;L B

HL



hj √ V



N;L B



   h∗j  hj hj = L(j ) aj − √V aj − √V + HI ≡ H0 √V + HI ; j> 1 X

†

£¤¥ L (j ) = L−2Ej ¨ ⮫쪮 ª®­¥ç­®¥ ç¨á«® ¯ à ¬¥â஢ hj ∈ C ®â«¨ç­® ®â ­ã«ï. ˆ¬¥¥â ¬¥á⮠ᮮ⭮襭¨¥      hj L b Trexp − H √V − N = =

∞ X

N=0

e N TrH

N;L B

exp

   hj √ − H0

V

+ HI



:

(5)

Ÿá­®, çâ® ®¯¥à â®à HI |H ¯®«®¦¨â¥«¥­. ‚¢¥¤¥¬ ¢ à áᬮâ७¨¥ â ª ­ §ë¢ ¥¬®¥ ®¡®¡é¥­­®¥ ­¥à ¢¥­á⢮ ƒ®«¤¥­ {’®¬¯á®­ . DZãáâì −A ¨ −B { ¯®«®¦¨â¥«ì­ë¥, á ¬®á®¯à殮­­ë¥ ®¯¥à â®àë ¨ ¯ãáâì ®¯¥à â®à A + B áãé¥á⢥­­® á ¬®á®¯à殮­ ­  D(A) ∩ D(B). ’®£¤  Tr eA+B 6 Tr(eAeB ): ‘®®â­®è¥­¨¥ (5) ¨ ­¥à ¢¥­á⢮ ƒ®«¤¥­ {’®¬¯á®­  ¢¥¤ãâ ª ­¥à ¢¥­áâ¢ã N;L B

TrH exp N;L B



−



H0





TrH exp ª®â®à®¥ ¬®¦­® ¯¥à¥¯¨á âì ¢ ¢¨¤¥ 6

Tr exp

N;L B

hj V

√



+ HI

  hj √ − H0

V



6 



(6)

e− H ; I

     hj L b √ − H − N 6

6

=

∞ X

N=0 ∞ X

N=0

V

e N Tr

N;L

HB

e V n TrH



N;L B

exp

    hj − HI √ − H0 e

=

V      exp − H0 √hVj e− H : I

(7)

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„«ï ª ¦¤®£® N áãé¥áâ¢ã¥â ã­¨â à­ë© ®¯¥à â®à UN â ª®©, çâ® TrH

N;L B





exp

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e H −

I



= TrH (UN+ e− H0 UN e− H ) 6 +k TrH (e− H0 UN e− H ):(8) 6 k UN H I

N;L B

N;L B

I

N;L B

ˆ§ 横«¨ç­®á⨠᫥¤  ¢ë⥪ ¥â, çâ® TrH (e− H0 UN e− H ) 6 kUN kH TrH (e− H0 e− H ); I

N;L B

N;L B

I

N;L B

£¤¥ k · kH ¥áâì ®¡ëç­ ï ­®à¬  ¢ HN;L. “ç¨â뢠ï, çâ® kUN+ kH = 1, Nb ª®¬¬ãâ¨Ä àã¥â á H L ¨ H0 ª®¬¬ãâ¨àã¥â á HI , ¨§ ãà ¢­¥­¨ï (5) ¨ ­¥à ¢¥­á⢠ (8) ¯®«ãç ¥¬ N;L B

Tr exp

N;L



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HL



hj V

√



b − N



6

∞ X

N=0

e V n TrH

N;L B

= Tr e− (H

L

−

e− H

N;L

=

b N) :

’ ª¨¬ ®¡à §®¬, ¨¬¥¥â ¬¥áâ® á«¥¤ãîé ï ⥮६ .

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Z ; H L {hj }

6 Z ;

[H L]:

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X

j> 1

w(j ) < ∞:

Žá­®¢­ë¬¨ á«ãç ©­ë¬¨ ¯¥à¥¬¥­­ë¬¨ ïîâáï ç¨á«  § ¯®«­¥­¨ï {nj : j = 1; 2; : : : }. DZਠí⮬ ®â®¡à ¦¥­¨¥ nj : → N ®¯à¥¤¥«¥­® â ª, çâ® nj (!) = !(j ) ¤«ï ¢á¥å ! ∈ . ‚ â ª®¬ ¯®¤å®¤¥ £ ¬¨«ìâ®­¨ ­ ¬®¤¥«¨ •ã ­£ {Ÿ­£ {‹ â⨭¦¥à  ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥ X a L (j )nj (!) + H L (!) = 2V j> 1





X 2N 2(!) − n2j (!) ≡ H0 (!) + HI (!): j> 1

(10)

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„ ¢«¥­¨¥ pL() ¨¤¥ «ì­®£® ¡®§¥-£ §  ¢ ®¡ê¥¬¥ L ¯à¨ 娬¨ç¥áª®¬ ¯®â¥­æ¨ «¥  ¬®¦­® § ¯¨á âì ¤«ï  < 0 ª ª i −1 X h (11) PL () = ( V )−1 ln 1 − e (− (j)) : L

j> 1

Ž¯à¥¤¥«¨¬ ¯ à樠«ì­®¥ ¤ ¢«¥­¨¥ P ( | ) = −1 ln[1 − e (−) ]−1 ¨ äã­ªæ¨î à á¯à¥¤¥«¥­¨ï FL () = V −1 max{j : L (j ) 6 }: ’®£¤  ä®à¬ã«  (11) ¯¥à¥¯¨á뢠¥âáï Z¢ ¢¨¤¥ P ( | ) dFL (): PL () = [0;∞) „«ï ®¡¥á¯¥ç¥­¨ï á室¨¬®á⨠¯®á«¥¤®¢ â¥«ì­®á⨠{PL()} ¢ à ¡®â¥ [3] ®¯à¥¤¥«ï¥âáï äã­ªæ¨ï Z L( ) = e−  dFL () [0;∞) ¨ ¢¢®¤ïâáï á«¥¤ãî騥 ãá«®¢¨ï: 1) ( ) = limL→∞ L( ) áãé¥áâ¢ã¥â ¤«ï ¢á¥å ¨§ ¨­â¥à¢ «  [0; ∞), 2) ( ) ®â«¨ç­  ®â ­ã«ï, ¯® ªà ©­¥© ¬¥à¥ ¤«ï ®¤­®£® §­ ç¥­¨ï ¨§ (0; ∞) ¯à¨ ¢ë¯®«Ä ­¥­¨¨ ãá«®¢¨ï 1. DZਠ¢ë¯®«­¥­¨¨ ãá«®¢¨ï 1 áãé¥áâ¢ã¥â ¥¤¨­á⢥­­®¥ à á¯à¥¤¥«¥­¨¥ F â ª®¥, çâ® Z ( ) = e−  dF () [0;∞) ¨ FL() → F (), ¯® ªà ©­¥© ¬¥à¥ ¢ â®çª å ­¥¯à¥à뢭®á⨠ä㭪樨 F . ”ã­ªæ¨ï F ï¢Ä «ï¥âáï ¨­â¥£à «ì­®© ¯«®â­®áâìî á®áâ®ï­¨©. …᫨ ãá«®¢¨ï 1 ¨ 2 ¢ë¯®«­¥­ë, â® ¯à¥¤¥« P () = lim PL () L→∞ áãé¥áâ¢ã¥â ¤«ï  < 0 ¨ § ¤ ¥âáï ä®à¬ã«®© Z P ( | ) dF (): P () = [0;∞) Šà¨â¨ç¥áª ï ¯«®â­®áâì c ®¯à¥¤¥«ï¥âáï á«¥¤ãî騬 ®¡à §®¬: ¥á«¨ äã­ªæ¨ï  → 0 P (0 | ) ¨­â¥£à¨à㥬  ­  [0; ∞) ¯® ®â­®è¥­¨î ª F , â® Z c = P 0 (0 | ) dF (): [0;∞) DZãáâì E (Y ) ®¡®§­ ç ¥â ¬ â¥¬ â¨ç¥áª®¥ ®¦¨¤ ­¨¥ á«ãç ©­®© ¯¥à¥¬¥­­®© Y ¢ ¡®«ìÄ è®¬ ª ­®­¨ç¥áª®¬  ­á ¬¡«¥ á® á।­¥© ¯«®â­®áâìî . ã¤¥¬ £®¢®à¨âì, çâ® ¯à®¨á室¨â ¬ ªà®áª®¯¨ç¥áª®¥ § ¯®«­¥­¨¥ ®á­®¢­®£® á®áâ®ï­¨ï, ¥á«¨ ¯à¥¤¥« lim E [V −1n(1)] V →∞  áãé¥áâ¢ã¥â ¨ áâண® ¯®«®¦¨â¥«¥­.

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ˆá¯®«ì§ãï áâ ­¤ àâ­ãî ¯à®æ¥¤ãÄ àã [5] ¨ ¯®«ã祭­ãî ¢ à §¤¥«¥ 2 ®æ¥­ªã, ¬®¦­® ­ ¯¨á âì á«¥¤ãî騥 ­¥à ¢¥­á⢠ ¤«ï E (nj ) ¢ ®¡ê¥¬¥ V : 3.2. ‚¥àå­ïï ¨ ­¨¦­ïï ®æ¥­ª¨ ¤«ï

E (nj ).

"s

 q 
1 RL(j; ) E (nj ) 6 ch 2 (j ) RL(j; ) − 1 2 (j ) 1 E (nj ) > R (j;) ; e −1

#

+ V1



(12)

;

(13)

L

£¤¥



RL (j; ) = L (j ) −  + 2a E



N V



+ E



nj V



:

DZ।¯®« £ ï ¨­â¥£à¨à㥬®áâì ä㭪樨 P 0(0|) ¯® ®â­®è¥­¨î ª ¬¥à¥ dFL(), á ãç¥â®¬ ¯à¨¢¥¤¥­­ëå ®æ¥­®ª ¨¬¥¥¬ ¢ ¯à¥¤¥«¥ V → ∞ à ¢¥­á⢮     Z N n(1) dF () E = lim ; (14) − lim E V →∞ V →∞ V V e [0;∞)  − 1 £¤¥  = 2a = limV →∞ E(N=V ). ˆ­ë¬¨ á«®¢ ¬¨, ¬®¤¥«ì •ã ­£ {Ÿ­£ {‹ â⨭¦¥à  ¤®¯ã᪠¥â ­ «¨ç¨¥ ¡®§¥-ª®­¤¥­á æ¨¨ ¤«ï ®¡à â­ëå ⥬¯¥à âãà > c , £¤¥ c ®¯à¥¤¥«ï¥âáï ¥¤¨­á⢥­­ë¬ à¥è¥­¨¥¬ ãà ¢­¥­¨ï Z dF () (15) c = e [0;∞)  − 1 ¨ c = 2ac. DZ®«ã祭­ë© १ã«ìâ â ¤«ï §­ ç¥­¨ï 娬¨ç¥áª®£® ¯®â¥­æ¨ «  c = 2ac ᮢ¯ ¤ ¥â á १ã«ìâ â®¬ áâ âì¨ [3], £¤¥ ¨á¯®«ì§®¢ ­ ¬¥â®¤ ¡®«ìè¨å 㪫®­¥­¨© ¤«ï ¤®ª § â¥«ìá⢠ áãé¥á⢮¢ ­¨ï ¡®§¥-ª®­¤¥­á æ¨¨. Ž¤­ ª® ¢ ®â«¨ç¨¥ ®â ¬¥â®¤  ¡®«ìè¨å 㪫®­¥­¨© à Ä ¡®âë [3] ­ è ¯®¤å®¤ ­¥ ¤ ¥â ¨­ä®à¬ æ¨¨ ® ¯®¢¥¤¥­¨¨ á¨áâ¥¬ë ¤«ï §­ ç¥­¨© 娬¨ç¥áª®£® ¯®â¥­æ¨ « , ¬¥­ìè¨å ªà¨â¨ç¥áª®£®. « £®¤ à­®áâ¨. „ ­­ ï à ¡®â  ç áâ¨ç­® 䨭 ­á¨à®¢ « áì DZ१¨¤¥­â᪮© ª ä¥¤Ä னáâ®å áâ¨ç¥áª®£® ­ «¨§ (—¨«¨)(CatedraPresidencialenAnalisisEstocastico,Chile). c

‘¯¨á®ª «¨â¥à âãàë [1] [2] [3] [4] [5]

K. Huang, C. N. Yang, J. M. Luttinger . Phys. Rev. 1957. V. 105. P. 776. M. van den Berg, J. T. Lewis . Helv. Phys. Acta. 1986. V. 59. P. 1271. M. van den Berg, J. T. Lewis, J. V. Pule . Commun. Math. Phys. 1988. V. 118. M. van den Berg, J. T. Lewis, P. de Smedt . J. Stat. Phys. 1984. V. 37. P. 697. M. Corgini, D. P. Sankovich . Int. J. Mod. Phys. B. 1997. V. 11. ò 28. P. 3329.

P. 61.

DZ®áâ㯨«  ¢ । ªæ¨î 3.III.1999 £.