Superconductivity and integrability

The paper is a review of studies of integrability of the BCS Hamiltonian with discussion of some its integrable generalization which present an interest for a number of physical problems.

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Published in:	Вопросы атомной науки и техники
Date:	2001
Main Author:	Belokolos, E.D.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2001
Subjects:	Quantum fluids
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/80047
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Cite this:	Superconductivity and integrability / E.D. Belokolos // Вопросы атомной науки и техники. — 2001. — № 6. — С. 343-347. — Бібліогр.: 7 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Belokolos, E.D.
author_facet	Belokolos, E.D.
citation_txt	Superconductivity and integrability / E.D. Belokolos // Вопросы атомной науки и техники. — 2001. — № 6. — С. 343-347. — Бібліогр.: 7 назв. — англ.
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container_title	Вопросы атомной науки и техники
description	The paper is a review of studies of integrability of the BCS Hamiltonian with discussion of some its integrable generalization which present an interest for a number of physical problems.
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fulltext	SUPERCONDUCTIVITY AND INTEGRABILITY E.D. Belokolos Institute of Magnetism, National Academy of Sciences, Kiev, Ukraine e-mail: bel@imag.kiev.ua The paper is a review of studies of integrability of the BCS Hamiltonian with discussion of some its integrable generalization which present an interest for a number of physical problems. PACS: 74.20.Fg 1. INTRODUCTION In 1957 J. Bardeen, L.N. Cooper and J.R. Schrieffer have introduced the BCS Hamiltonian which was very successful in description of the superconductivity. In 1958 N.N. Bogoljubov et al. proved an equivalence of the BCS Hamiltonian to the quadratic one in the thermodynamic limit. At a finite number of particles R.W. Richardson (1965) proved an integrability of the BCS Hamiltonian [1] and M. Gaudin (1976) built an appropriate mathematical theory [2,3]. Recently an interest to the integrability of BCS Hamiltonian was renewed in connection with different applications. 2. INTEGRABILITY OF THE BCS HAMILTONIAN The BCS Hamiltonian is , , , , , ∑∑ ↑↓ + ↓ + ↑ + −= ji jjiii i iiBCS ccccgccH σ σ σε where are the annihilation and creation operators of electrons. In this Hamiltonian the pairing interaction does not act on singly occupied levels. As a result we may study these levels separately. By means of the operators ),cccc(:S ,cc)S(:S,cc:S jjjj z j jjjjjjj 1 2 1 −+= === ↓ + ↓↑ + ↑ + ↓ + ↑ +−+ ↑↓ − which obey to the commutation relations we can present the BCS Hamiltonian in a form .)2/1(2 111 − = + == ∑∑∑ −+= k L k j L j z j L j jBCS SSgSH ε The BCS Hamiltonian has the integrals of motion [4] ,H,gHSR L ik,kj ki ki ii z ii ∑ ≠= = ε−ε ⋅ =−= 1 SS which commute with each other. The number of pairs N and the Hamiltonian BCSH are linear and quadratic forms of these integrals of motion respectively, ∑ = += L i iRN 1 ),2/1( ∑ ∑∑ = == −++= L i L i i L i iiiBCS gRgRH 1 1 22 1 .)()2/1(2 Sε 3. THE GAUDIN ALGEBRA, THE RICHARDSON EQUATIONS, EIGENSTATES AND EIGENVALUES OF THE BCS HAMILTONIAN 1. The Gaudin algebra. Given a set of complex numbers { }Ljj ,...,1, =ε and a set of independent spin operators { }LjSSS z jjj ,...,1,,, =−+ , satisfying the commutation relations ,2],[,],[ z kjkkjkjkk z j SSSSSS δδ =±= −+±± we define the operator rational functions ∑ = ±= − = L j j j z S S 1 .,,)( α ωε ω α α The operators )(ωαS are the generators of the Gaudin algebra and obey to the commutation relations [ ] [ ] [ ] [ ] .)()(2)(),( ,)()()(),( ,0)(),(,0)(),( ωω ωωωω ωω ωωωω ωωωω ′− ′− =′ ′− ′− ±=′ =′=′ −+ ±± ± ±± zz z zz SSSS SSSS SSSS The Gaudin algebra is an infinitedimensional extension of the )2(sl algebra. We construct the representation of the Gaudin algebra, fixing the highest weight vector 0 by means of the following relations, ,00)( =+ ωS and define the representation space as a linear hull of vectors PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 343-347. 87 011 )(S)...(S,...,N NN ωω=ωω= −− with arbitrary N complex numbers .,...,1 Nωω 2. The generating function of the integrals of motion. The operator rational function ( ) ( ) ( ) ∑ ∑ − + − = =+= j j j j j j z SgR gSF 2 2 2 )(2 2/)( ωεωε ωωω S is a generating function of the integrals of motion of the BCS Hamiltonian since ).(ωεω FresR jj =−= It is easy to prove that values of the operator )(ωF at a different values of ω commute with each other, [ ] .0)(),( =′ωω FF We can prove that the NN ωω ,...,1= are eigenstates of the operator )(ωF if the quantities Nωω ,...,1 satisfy the Richardson equations ∑ ∑ = α≠ββ αβα =α= ω−ω − ω−ε L j L ,j j .N,...,, g s 1 111 Eigenvalues of the )(ωF are of the following form ( ) ( ) .1 2 1 2 1)( 1 1 22 2 1 1 1 1         − − − +        − + − + − + − = ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = L j N j j L j N j j L j N j j sg sg s F α α α α α α ωωωε ωωωε ωωωε ω 3. Eigenstates and eigenvalues of the BCSH . The function NN ωω ,...,1= with parameters ,,...,1 Nωω satisfying the Richardson equation, is an eigenstate of integrals of motion jR with the eigenvalue .1 ,1 1         − + − += ∑ ∑ ≠= = L jkk N jjk k jjj s gssR α αωεεε The same function NN ωω ,...,1= with the same parameters Nωω ,...,1 is an eigenstate of the Hamiltonian BCSH which is a quadratic form of the integrals of motion jR . In order to calculate the eigenvalues of the BCSH we ought to put the expressions for eigenvalues of the integrals of motion jR in the formula ( ) . 2/12 1 2 2 1 1 ∑∑ ∑ == = −     ++= L i i L i i L i iiBCS gRg RH S ε We can calculate the eigenvalues of the BCSH also by means of the asymptotic expansion of the generating function )(ωF at ,∞→ω .1 2 1 1)( 3 1 2 1 2 11 )(     +    − −    −== ∑∑ ∑∑ == = ∞ = − ω ε ω ω ωω OSgR RFF L j j L j jj L j j m mm Since the numbers of pairs N and the Hamiltonian BCSH are expressed in terms ,, )2()1( FF ( ) ,2 ,)2/1( 1 2)1()2( )1( ∑ = ++−= +−= L j jBCS FgFH LFN ε and since we know the eigenvalues of the operator )(ωF , we can obtain the following expression for the eigenvalues NE of the BCS Hamiltonian, ( )( ) ( )( ).12 ,2/2 1 0 0 1 ∑ ∑ = = −+= +−−= L j jjj N N gssE EgE ε ω α α 4. SOLUTIONS OF THE RICHARDSON EQUATIONS. CLASSIFICATION OF EIGENSTATTES The Richardson equations ,,...,1,11 , 1 N g sN L j j j == − − −∑ ∑ ≠ = α εωωωαββ αβα admit different interpretations. We can interpret the Richardson equations as conditions of local equilibrium for a set of charges on a plane (actually lines of charge perpendicular to the plane) which interact with each other by means of a logarithmic potential and with a uniform external field. Indeed if we assume that there are the N free charges of unit strength at points Nzz ,...,1 and the L fixed charges with a charge of strength jb located at a point ja of the real axis where Lj ,...,1= and a uniform external field g/1− then the energy of such a system of charges is 88 ( ) .1ln ln 2 1,..., 11 1 1 ,1 1 ∑∑ ∑ ∑ ∑ == = = ≠= +− +−= NN L j jj N N N z g azb zzzzW α α α α α αββ βα These charges are in equilibrium if their energy is stationary with respect to coordinates of free charges, i.e. ( ) .,...,1,0,...,1 NzzW z N == ∂ ∂ α α These equations coincide with the Richardson equations if we put Nz ,...,1, == αω αα and ( ) .,...,1,2/1, Ljsba jjjj =−== ε The energy ( )NzzW ,...,1 of the N free charges on a plane does not have a global extremum since it is not bounded from above and below but it has a number of local extrema described by solutions of the Richardson equations.These solutions of the Richardson equations correspond different quantum states of the BCS Hamiltonian. Since ( ) ,,...,1,lim 0 Ng g == +→ αεω αα we may label these quantum states by the quantum numbers of the free Hamiltonian corresponding to .0=g In such a way we come to conclusion that there are N LC states with N pairs for the BCS Hamiltonian. We can interpret the Richardson equations also as the equations for zeros of a polynomial satisfying a special ordinary differential equation of the second order with polynomial coefficients. To this end let us consider the polynomial ( ) ( )∏ = −= N zzzf 1β α with zeros .,...,1, Nz =ββ It is easy to show that ( ) ( ) ( )∑ ≠= − = ′ ′′ N zzzf zf αββ βαα α ,1 .1 2 1 If we insert this expression into the Richardson equations we obtain ( ) ( ) .012 1 =′        + − −′′ ∑ = ∂ αα ε zf gz s zf L j j j Since the polynomial ( )zf of the order N and the polynomial ( ) ( ) ( )         ′        + − −′′− ∑∏ == zf gz s zfz L j j j L k k 11 12 ε ε of the order N+L-1 have the same zeros Nz ,...,1, =αα there must exist such a polynomial ( )zC of the order L-1 that ( ) ( ) ( ) ( ) ( ) .0 12 11 = +         ′        + − −′′− ∑∏ == zfzC zf gz s zfz L j j j L k k ε ε Therefore the polynomial ( )zf with zeros Nz ,...,1, =αα must satisfy the written above differential equation of the second order ( ) ( ) ( ) ( ) ( ) ( ) 0=+′+′′ zfzCzfzBzfzA with polynomial coefficients ( ) ( ) ( ).,, zCzBzA There are several polynomials ( )zC with this property and their number is equal to the number of solutions of the Richardson equations. A dependence of the quantities N,...,1, =αω α on the interaction constant ∞<≤ gg 0, has the following properties: A. If there exist such a 00 ≠g that ( ) ( ) agg K === 001 ... ωω then we have 1) a= ;jε 2) ;1+= jsK 3) ( ) ( ) KpggCg K p ,...,1,/1 0 =−=ω in small neighborhood ;0gg − 4) 0g is a solution of the algebraic equation of the K-th degree. B. Ata ∞→g we have ( ) Pg ,...,1, =→ βεω ββ or ( ) ,,...,1, Qig =∆+→ γεω γγγ where .NQP =+ It means that ( ) Ng ,...,1, =αω α are N branches of the algebraic function ( ).gω C. At 0+→g we have ( ) ( ) ,sg dg dlim,glim gg αα +→ αα +→ −=ωε=ω 00 N,...,1=α 5. THERMODYNAMIC LIMIT FOR THE BCS HAMILTONIAN Now let us consider according to the paper [5] the thermodynamic limit .lim,lim,, G L gconst L NNL LL ==∞→∞→ ∞→∞→ Let us assume that there exist the density of states ( )ερ and the density of pairs ( )ξr satisfying conditions ( ) ( )∫ ∫ Ω Γ == .,2/ NdrLd ξξεερ Here Ω is a support of unperturbed spectrum and  K k k 1= Γ=Γ is a support of spectrum of pairs and they are symmetrical with respect of the real axis. In the thermodynamic limit the Richardson equations 89 ∑ ∑ = ≠= == − − − L j N j L g1 ;1 ,,...,1, 2 111 2 1 αββ αβα α ωωωε are transformed to the singular integral equation ( ) ( ) ., 2 1 Γ∈= −′ ′′ − − ∫∫ ΓΩ ξ ξξ ξξ ξε εερ G dr Pd According to a theory of singular integral equations this equation has a solution ( ) ( ) ( ) ( ) ( ) ( ) ( ) , R dRh,hr ∫ Ω ξ−εε εερξ=ξξ π =ξ 1 where kk ba , are initial and final points of the line kΓ . The following conditions must be satisfied ( ) ( ) ( ) ( ) .0,0 , 2 1! Kkd R G d R k K <≤= = ∫ ∫ Ω Ω − εε ε ερ εε α ερ The support  K k k 1= Γ=Γ is defined by the equation ( ) .,dh k ak Γ∈ξ=ξ ′ξ ′ℜ ∫ ξ 0 Example. Let us consider a simple example when Γ consists of one segment with limit points ., 00 ∆+=∆−= ibia εε Applying the theory presented above we obtain the following results: (1) the density of spectrum for pairs ( ) ( ) ( ) ( ) ( ) ( ) ,d hr ∫ Ω ∆+ε−ξξ−ε εερ ×∆+ε−ξ π =ξ π =ξ 22 0 22 0 11 (2) the gap equation ( ) ( ) , 2 1 22 0 G d = ∆+− ∫ Ω εε εερ (3) the Fermi energy equation (4) the following expression: ( ) ( ) ( ) ,1 2 1 22 0 0 εερ εε εε ξξ π d dh i N ∫ ∫ Ω Γ         ∆+− − − == (4) the ground state energy ( ) ( ) ( ) .1 22 1 22 0 0 2 εεε ρ εε εε ξξξ π d G dh i E ∫ ∫ Ω Γ         ∆+− − − +∆−== 6. NORMS OF THE EIGENSTATES CORRELATTION FUNCTIONS The normalization factor of eigenstate N is expressed in terms of the Jacobi matrix ∆ of the Richardson equations: ( ) ( ) ( )   21 1       −ξ−ξ=ε ∏ = K k kk baR ( ) ( ) ( ) .1 1,! 22 2         − − − + − =∆∆= ∑∑ ν ανα α β βα α β ωωωε δ ωω j j js NNN The correlation functions of variables jS are . 1 2 ∑ = ∂ ∂         − − −=⋅ N kj kj jkjkj sss α α α ε ω ωε εε SS Sklyanin has developed mathematical means to calculate different other correlation functions [6] 7. THE INTEGRABLE GENERALIZATIONS OF THE BCS HAMILTONIAN The generalized BCS Hamiltonian , , , ∑ ∑∑ ′↑↓ + ↓ + ↑ +−= ji ji iiijiiiiij i ii nnUccccgnH σσσε is integrable at a special form of the interaction functions ijg and ijU . Let us consider an integrable Hamiltonian ,2 , ∑∑∑ ++= i ii ki ki i iiN AH Sβτττε with the integrals of motion jτ of the form ., ,1 ααατ kj jkk jkjj z jj SSwS ∑ Ω ≠= =ΞΞ+= The operators jτ are called isotropic when z jk y jk x jk www == otherwise we call them anisotropic. The operators jτ commute with each other if [ ] [ ] [ ] ,0,,,0, =Ξ+Ξ=ΞΞ z jij z ikj SS or, in other words, if ., y ji x ijjkikikjijkij wwwwwwww −==+ βαγβγα Furthermore we impose an additional condition 90 { },,...,1,0, 1 Ω==    Ξ∑ Ω = jS j i z i which is equivalent to the equations .:,: , z ijij y ji y ij x ji x ijij jkikikjijkij wvwwwww wwvwvw =−==−== =+ These equations for the quantities ijw and ijv have solutions ( ) ( ),coth , sinh kjjk kj jk uuqqKv uuq qKw −= − = where ju are arbitrary complex parameters such that the quantities jkjk wv , are real. The parameter q can be real or imaginary. If q is real then we have hyperbolic functions, if ,, iKGiq == and juK , are real then we have trigonometric functions. The eigenfunctions of integral of motions jτ are of the form ( ) ( ) .0...... 0...... 1111 11 ' ... 11 ' ... 1 +++ − ++ −− ∑ ∑ +=Ψ jjjjj N jjjj Nj SSSjjc SSjjc NN NN Here ↓↓= ,...,0 is the vacuum, the primes at the sums mean that the indices run in the range { } { }j\,...,1 Ω . The eigenvalues of jτ are defined by the equalities ( ) ( ) ,12/1 jjjj h Ψ−=Ψτ where jh are solutions of the equations ( ) ( ),coth2 coth/ 1 1 ∑ ∑ = Ω = − −−= L j l ljj uqq uuqqKh α αω and αω fulfill the equations ( ) ( ).coth2 coth/1 ,1 1 ∑ ∑ ≠= Ω = − −−= L l l qq uqqK αββ βα α ωω ω Thus we obtain the integrable BCS Hamiltonian ,2 ∑∑∑ +−= −+ jk z k z jjk jk kjjk j z jjN SSUSSgSH ε with the following interaction functions ( ) ( ) ( ) ( ) ., ,,coth ,,sinh/ jjjjjj kjkjjk kjkjjk AUg kjuuqqKAU kjuuqqKg ββ εε εε +=−= ≠−−+= ≠−−−= Here parameters KA j ,, β are arbitrary real constants, while q can be real or imaginary. The eigenfunctions NΨ and eigenvalues NE of the Hamiltonian H are ( ) ( ) ( ).coth2 4 ,0 sinh 1 1 1,1 1 1 ∑ ∑ ∑∑ ∏ ∑ Ω = = Ω = Ω = = Ω = + ↓ + ↑ − +Ω−++= − =Ψ j N jj kj jk j jN N j j jj N uK KNAKNUE u cc α α α α ωε ε ω At limit case 0→q we come to the isotropic case, i.e. to the BCS Hamiltonian with the following constants ( )( )./ ,,,/ DjDjj jD EEu gAgEgK −= =−== εθε β Diagonal elements jjjj Ug , are arbitrary since they renormalize jε . There are generalizations to integrable quantum models with arbitrary Lie algebras, in particular, with )(NO and )2( kSp . 8. CONCLUSION We have presented above a review of the integrability of the BCS Hamiltonian. Further studies show that it has deep connections to integrable vertex models, conformal field theory, Chern-Simons theory, Bethe ansatz and quantum groups (see e.g. [7]). Since the integrability of the BCS Hamiltonian have been used essentially in description of superconductivity of nuclei, ultrasmall metallic grains and quantum dots at low temperatures it presents an interest also from point of view of applications. REFERENCES 1. R.W. Richardson. Exact eigenstates of the pairing-force Hamiltonian // J. Math. Phys. 1965, v. 6, p. 1034-1051. 2. M. Gaudin. Diagonalization d'une classe d'hamiltoniens de spin // J. de Physique. 976, t. 37, № 10, p. 1087-1098. 3. M. Gaudin. La fonction d'onde de Bethe. Paris: Masson, 1983, p. 352. 4. M.C. Cambiaggio, A.M.F. Rivas, M. Saraceno. Integrability of the pairing hamiltonian // Nucl. Phys. 1997, v. A624, p. 157- 167. 5. M. Gaudin. Étude d'un modèle à une dimension pour un système de fermions en interaction. Thèse, Univ. Paris, 1967. 6. E.K. Sklyanin. Generating function of corre- latoors in the sl_2 Gaudin model. 1997, solv- int/9708007. 7. G. Sierra. Integrability and Conformal Symmetry in the BCS model. 2001, hep- th/0111114. 91 REFERENCES
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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
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spelling	Belokolos, E.D. 2015-04-09T16:35:46Z 2015-04-09T16:35:46Z 2001 Superconductivity and integrability / E.D. Belokolos // Вопросы атомной науки и техники. — 2001. — № 6. — С. 343-347. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 74.20.Fg https://nasplib.isofts.kiev.ua/handle/123456789/80047 The paper is a review of studies of integrability of the BCS Hamiltonian with discussion of some its integrable generalization which present an interest for a number of physical problems. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum fluids Superconductivity and integrability Сверхпроводимость и интегрируемость Article published earlier
spellingShingle	Superconductivity and integrability Belokolos, E.D. Quantum fluids
title	Superconductivity and integrability
title_alt	Сверхпроводимость и интегрируемость
title_full	Superconductivity and integrability
title_fullStr	Superconductivity and integrability
title_full_unstemmed	Superconductivity and integrability
title_short	Superconductivity and integrability
title_sort	superconductivity and integrability
topic	Quantum fluids
topic_facet	Quantum fluids
url	https://nasplib.isofts.kiev.ua/handle/123456789/80047
work_keys_str_mv	AT belokolosed superconductivityandintegrability AT belokolosed sverhprovodimostʹiintegriruemostʹ

Superconductivity and integrability

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