Quantum spin chains and integrable many-body systems of classical mechanics
aa r X i v : . [ m a t h - ph ] J a n Quantum spin chains and integrable many-bodysystems of classical mechanics
A. Zabrodin ∗ September 2014
ITEP-TH-29/14
Abstract
This note is a review of the recently revealed intriguing connection betweenintegrable quantum spin chains and integrable many-body systems of classical me-chanics. The essence of this connection lies in the fact that the spectral problem forquantum Hamiltonians of the former models is closely related to a sort of inversespectral problem for Lax matrices of the latter ones. For simplicity, we focus on themost transparent and familiar case of spin chains on N sites constructed by meansof the GL (2)-invariant R -matrix. They are related to the classical Ruijsenaars-Schneider system of N particles, which is known to be an integrable deformation ofthe Calogero-Moser system. As an explicit example the case N = 2 is consideredin detail. In this paper we present some results of [1]-[4] in a short compressed form and in the simplestpossible setting. First of all let us explain what we mean by “quantum spin chains” and“integrable many-body systems of classical mechanics”.The best known example of integrable quantum spin chain is the isotropic (XXX) homoge-neous Heisenberg model with spin on an 1D lattice with coupling between nearest neighbours.Throughout the paper, we use the words “spin chain” in a broader sense, not implying exis-tence of any local Hamiltonian of the Heisenberg type. In fact integrable local Hamiltoniansin general do not exist for inhomogeneous spin chains which are closely involved in our story.However, such models still make sense as generalized spin chains with long-range interactionand a family of commuting (non-local) Hamiltonians. We call them inhomogeneous XXX spin ∗ Institute of Biochemical Physics, 4 Kosygina, 119334, Moscow, Russia; ITEP, 25 B. Cheremushkin-skaya, 117218, Moscow, Russia; National Research University Higher School of Economics, Interna-tional Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya Ulitsa, Moscow101000, Russia hains. Alternatively, one may prefer to keep in mind inhomogeneous integrable lattice mod-els of statistical mechanics rather than spin chains as such. In either case, the final goal ofthe theory is diagonalization of transfer matrices which are generating functions of commutingconserved quantities. This is usually achieved by one or another version of the Bethe ansatzmethod.The integrable model of classical mechanics we are mainly interested in is the N -bodysystem of particles on the line called the Ruijsenaars-Schneider (RS) model [5]. It is oftenreferred to as an integrable relativistic deformation of the famous Calogero-Moser (CM) modelwith inversely quadratic pair potential [6, 7].As is common for integrable models, the classical dynamics can be represented in the Laxform, i.e., as an isospectral deformation of a N × N matrix called the Lax matrix. Matrixelements of this matrix are simple functions of coordinates and momenta of the particles whilethe eigenvalues are integrals of motion. In a nutshell, the essence of the quantum-classical (QC)duality Quantum integrable models ←→ Classical many-body systems . (1)lies in the fact that spectra of quantum Hamiltonians of a model from the left hand side appearto be encoded in the algebraic properties of the Lax matrix for a classical system from the righthand side.In the case of the inhomogeneous XXX spin chain, a refined version of (1) isQuantum XXX spin- chain on N sites ←→ Classical N -body RS model . (2)More precisely, the spectral problem for the quantum Hamiltonians of the inhomogeneous XXXspin chain on N sites is reduced to a sort of an inverse spectral problem for the N × N Laxmatrix for the classical RS system. Given its spectrum and the coordinates of the particles, theproblem is to find possible values of their momenta compatible with these data. In general thisproblem has many solutions which just yield different eigenvalues of the quantum Hamiltonians.In a special scaling limit, the XXX spin chain turns into the Gaudin spin model [8]. On theright hand side of (2), this corresponds to the non-relativistic limit of the RS system:Quantum Gaudin model ←→ Classical CM model . (3)The QC duality is traced back to [9], where joint spectra of some finite-dimensional operatorswere linked to the classical Toda chain. The existence of an unexpected link between thequantum Gaudin and the classical CM models was first pointed out in [10], see also [11]. In amore general set-up, the correspondence between quantum and classical integrable systems wasindependently derived [1, 3, 12, 13] as a corollary of an embedding of the commutative algebraof spin chain Hamiltonians into an infinite integrable hierarchy of soliton equations known asthe modified Kadomtsev-Petviashvili (mKP) hierarchy. Namely, the most general generatingfunction of commuting integrals of motion of the spin chain (the “master T -operator”) wasshown to satisfy the bilinear identity and the Hirota bilinear equations for the tau-function ofthe mKP hierarchy [14].Although only a limited number of examples are available at the moment, the very phe-nomenon of the existence of hidden non-standard connections between quantum and classicalintegrable systems seems to be rather general. Presumably, it can be thought of as a new kindof a correspondence (or duality) principle in the realm of integrable systems. In [4], the QC du-ality (2), (3) was checked directly using the Bethe ansatz solution of integrable spin chains. Therole of this duality in the context of supersymmetric gauge theories and branes was discussedin [15, 16, 4]. t is worthwhile to stress that the both sides of the correspondence, i.e. quantum andclassical integrable systems, participate in the game as two faces of one entity on an equal-rights basis. In the theory of quantum models, there are some fundamental relations, exact forany ¯ h = 0, which assume the form of classical equations of motion for some other system. (Oneof such examples is the classical integrable dynamics naturally realized in the space of conservedquantities of quantum integrable models, see [1] and earlier works [17, 18].) At the same time,given a many-body problem of classical mechanics, one may extract from it, by addressing somenon-traditional questions about the system, the spectral properties of a quantum model. Thispicture becomes valid and meaningful if the systems from both sides are integrable. It mightbe interesting to combine the hypothetical “correspondence principle” based on the QC dualitywith the standard correspondence principle of quantum mechanics.Let us outline the contents of the paper.In section 2, we start with the most familiar example of integrable spin chain: the Heisenbergmodel with spin and periodic boundary conditions (the XXX magnet) solved by H.Bethe in1931 [19]. The “spin variables” are vectors from the spaces C at each site. However, thismodel itself is too degenerate to be directly linked to a classical many-body system. To thisend, we need an inhomogeneous version of the model with twisted boundary conditions. Such ageneralized XXX model has N + 2 free parameters which are N “inhomogeneity parameters” oneach site and 2 eigenvalues of the twist matrix which is assumed to be diagonal. The generalizedXXX model can be naturally constructed in the framework of the Quantum Inverse ScatteringMethod (QISM) developed by the former Leningrad school [20, 21]. In the inhomogeneousmodel, the locality of spin interactions does not take place. Instead, there are N non-localcommuting Hamiltonians (which are cousins of the Gaudin ones). They can be simultaneouslydiagonalized using the algebraic Bethe ansatz.In section 3, the necessary formulae related to the classical RS model are presented, includingthe Lax matrix. The rules of the quantum-classical correspondence between the integrablemodels are explained in section 4. As an example we consider the case N = 2, where allcalculations can be done directly by hands (section 5). Finally, in section 6 we give someremarks on the scaling limit to the Gaudin model which corresponds, on the classical side, tothe non-relativistic limit of the RS system. Some generalizations and perspectives are brieflydiscussed in the concluding section 7. The Hamiltonian of the isotropic Heisenberg spin chain (also called the XXX-magnet) withperiodic boundary condition is H xxx = 2 N X j =1 (cid:16) s ( j ) x s ( j +1) x + s ( j ) y s ( j +1) y + s ( j ) z s ( j +1) z − I (cid:17) , N + 1 ≡ , where the spin operators ( s x , s y , s z ) = ~ s are expressed through the Pauli matrices as s x = 12 (cid:18) (cid:19) , s y = 12 (cid:18) − ii (cid:19) s z = 12 (cid:18) − (cid:19) and I = ⊗ N is the identity operator. (Hereafter stands for the identity matrix in C ). Wewill also use s + = s x + i s y = (cid:18) (cid:19) , s − = s x − i s y = (cid:18) (cid:19) , s = + s z = (cid:18) (cid:19) nd s = − s z = (cid:18) (cid:19) . The operator ~ s ( j ) = ⊗ ( j − ⊗ ~ s ⊗ ⊗ ( N − j ) acts non-trivially atthe j th site of the chain. Clearly, they commute for any j ′ = j . The Hamiltonian acts in the 2 N -dimensional linear space V = ⊗ Nj =1 V j , V j ∼ = C . Basis vectors in this space can be constructedas tensor products of local vectors with definite z -projection of spin, i.e., eigenvectors of s z .Note that P ij = (cid:0) I + 4 ~ s ( i ) ~ s ( j ) (cid:1) is the permutation operator of the i th and j th spaces, andso the Heisenberg Hamiltonian can be written in the form H xxx = P j P j j +1 − N I .The Hamiltonian commutes with the operator M = 12 N X j =1 ( I − s ( j ) z ) = N X j =1 s ( j )2 (4)which counts the total number of spins in the chain with negative z -projection. Namely, thestates in which M spins look down (and so the rest N − M spins look up) are eigenstates forthe operator M with the eigenvalue M . The space of states V is decomposed in the direct sumof eigenspaces for the operator M : V = N M M =0 V ( M ), M V ( M ) = M V ( M ). It is clear thatdim V ( M ) = (cid:18) NM (cid:19) = N ! M !( N − M )! . In particular, V (0) and V ( N ) are one-dimensional spaces generated by the states in which allspins look up or down respectively.The common spectral problem for the operators H xxx and M , H xxx Ψ = E Ψ, M Ψ = M Ψ,has the famous Bethe ansatz solution [19]. The eigenvalues E for 0 ≤ M ≤ [ N/
2] are given bythe formula E = M X α =1 ε ( v α ) , ε ( v ) = −
41 + 4 v , (5)where the auxiliary quantities v α (the Bethe roots) are to be found from the system of algebraicequations v α + i v α − i ! N = M Y β =1 ,β = α v α − v β + iv α − v β − i (6)(the Bethe equations). Different solutions to this system give energies of different eigenstates.The exact solution of the Heisenberg spin chain is possible due to the fact that the modelis integrable. This means that there is a sufficiently large family of independent commutingoperators, one of which is the Heisenberg Hamiltonian. The other operators of this family arehigher integrals of motion. A general prescription how to construct models possessing higherintegrals of motion is provided by the Quantum Inverse Scattering Method (QISM) [20].We start by reformulating the XXX spin chain in the framework of the QISM, following[21]. Such a reformulation makes integrability of the model explicit and, what is even moreimportant, it suggests natural integrable generalizations of the XXX chain.Let V ∼ = C be another copy of the complex linear space C (the auxiliary space). Thequantum Lax operator at the j th site acts non-trivially in V ⊗ V j . It is L j ( x ) = x ⊗ I + η P j = (cid:16) x + η (cid:17) ⊗ I + 2 η ~ s ⊗ ~ s , (7) r, in the block-matrix form, L j ( x ) = x I + η s ( j )1 η s ( j ) − η s ( j )+ x I + η s ( j )2 . (8)The variable x ∈ C is called the (quantum) spectral parameter. The extra parameter η intro-duced here for the reason clarified below is not actually essential because it can be eliminatedby a rescaling of the spectral parameter (unless one tends it to 0 as in the limit to the Gaudinmodel [8]). The Heisenberg Hamiltonian does not depend on η which is usually put equal to i = √− L -operator satisfies the “ RLL = LLR ” intertwining relation R ( x − x ′ ) L j ( x ) ⊗ L j ( x ′ ) = L j ( x ′ ) ⊗ L j ( x ) R ( x − x ′ ) , where the quantum R -matrix R ( x ) acts in the tensor product of two auxiliary spaces V ∼ = V ′ ∼ = C . In the natural basis in C ⊗ C it is R ( x ) = x + η η x x η
00 0 0 x + η = η ⊗ + x P ′ . (9)Note that in this particular case the R -matrix is almost the same object as the quantum L -operator: they differ only by a permutation operator of the two spaces, so that the intertwiningrelation is equivalent to the Yang-Baxter equation for the R -matrix. The quantum transfermatrix is defined as T ( x ) = tr h L ( x ) L ( x ) . . . L N ( x ) i = 2 I x N + J N − x N − + . . . + J x + J . (10)The intertwining relation implies that the transfer matrices with different spectral parameters(and the same η ) commute: [ T ( x ) , T ( x ′ )] = 0 for any x, x ′ . In its turn, this implies thatthe operators J k in (10) all commute with each other. At the same time, the operator J isproportional to the cyclic permutation of the chain: J = T (0) = η N P P P . . . P N − N P N while the Hamiltonian of the spin chain is given by H xxx = η ddx log T ( x ) (cid:12)(cid:12)(cid:12) x =0 − N I = η J − J − N I . The operators J − J k are then the higher integrals of motion. The operator J − J is local dueto the special property of the quantum Lax operator L j (0) = η P j and the homogeneity ofthe chain. The operator M (see (4)) commutes not only with H xxx but with the whole one-parametric family T ( x ), and the Bethe states are common eigenstates for the T ( x ) and M : T ( x )Ψ = T ( x )Ψ, M Ψ = M Ψ.The transfer matrix T ( x ) can be diagonalized by means of the algebraic Bethe ansatzmethod. The eigenvalues T ( x ) are given by the formula T ( x ) = ( x + η ) N M Y α =1 x − u α − ηx − u α + x N M Y α =1 x − u α + ηx − u α . (11) he Bethe roots u α are to be found from the system of Bethe equations (cid:18) u α + ηu α (cid:19) N = M Y β =1 ,β = α u α − u β + ηu α − u β − η , (12)where is implied that 0 ≤ M ≤ [ N/ E = M X α =1 η u α ( u α + η )which is equivalent to (5) under the substitution v α = iu α η + i R ( x − x ′ ) L j ( x − x j ) ⊗ L j ( x ′ − x j ) = L j ( x ′ − x j ) ⊗ L j ( x − x j ) R ( x − x ′ ) . The latter is due to the GL (2)-invariance of the R -matrix (9): g ⊗ g R ( x ) = R ( x ) g ⊗ g for any g ∈ GL (2). This property implies that commutativity of the transfer matrices still holds if oneinserts a matrix g ∈ GL (2) in the auxiliary space before taking trace. For simplicity, we assumethat g is diagonal: g = (cid:18) w w (cid:19) . (13)The generalizations a) and b) can be applied simultaneously, which leads to the most generalone-parametric family of commuting operator-valued polynomials in x : T ( x ) = T ( x ; g, η, { x j } ) = tr h g L ( x − x ) L ( x − x ) . . . L N ( x − x N ) i . (14)These operators commute for different x ’s and the same η , g and x j :[ T ( x ; g, η, { x j } ) , T ( x ′ ; g, η, { x j } )] = 0 . Similarly to (10), one can expand T ( x ) = I tr g x N + J N − x N − + . . . + J x + J , (15)the J k ’s being commuting integrals of motion. Note, in particular, that J N − = η P i g ( i ) ,where g ( i ) is the operator acting as the matrix g at the i th site: g ( i ) := ⊗ ( i − ⊗ g ⊗ ⊗ ( N − i ) .In general there is no way to construct local Hamiltonians from the J k ’s. Instead, assumingthat all the x j ’s are distinct and in general position (meaning that x i − x j = ± η for all i, j ),one can define non-local Hamiltonians as residues of T ( x ) / Q j ( x − x j ) (cf. [22]): T ( x ) Q Nj =1 ( x − x j ) = tr g · I + N X j =1 η H j x − x j . In general, the Hamiltonians H j = H j ( η, g, { x i } ) imply a long-range interaction involving allspins in the chain. Their explicit form is H i = −−−→ N Y j = i +1 (cid:18) I + η P ij x i − x j (cid:19) g ( i ) −−→ i − Y j =1 (cid:18) I + η P ij x i − x j (cid:19) , (16) here we use the notation −→ m Y j =1 A j = A A . . . A m for the ordered product. It follows from thedefinition that N X j =1 H j = N X j =1 g ( j ) .The operator M (4) still commutes with T ( x ) and all the H j ’s, so, again, all these operatorsare diagonalized simultaneously: T ( x )Ψ = T ( x )Ψ, H j Ψ = H j Ψ, M Ψ = M Ψ. The algebraicBethe ansatz gives the following result. The eigenvalues T ( x ) and H j are given by the formulae T ( x ) = w N Y k =1 ( x − x k + η ) M Y α =1 x − u α − ηx − u α + w N Y k =1 ( x − x k ) M Y α =1 x − u α + ηx − u α , (17) H j = w N Y k =1 , = j x j − x k + ηx j − x k M Y α =1 x j − u α − ηx j − u α . (18)The Bethe roots u α are to be found from the system of Bethe equations w w N Y k =1 u α − x k + ηu α − x k = M Y β =1 ,β = α u α − u β + ηu α − u β − η , (19)where it is implied that 0 ≤ M ≤ [ N/ The RS model [5] is an integrable model of classical mechanics. It is an N -body system ofinteracting particles on the line with the Hamiltonian H RS1 = η − N X i =1 e − ηp i N Y k =1 , = i x i − x k + ηx i − x k . (20)For some reason it is often called the relativistic deformation of the Calogero-Moser model,the parameter η being the inverse “velocity of light”. The Hamiltonian equations of motion (cid:18) ˙ x i ˙ p i (cid:19) = (cid:18) ∂ p i H RS1 − ∂ x i H RS1 (cid:19) give the following connection between velocity and momentum˙ x i = − e − ηp i N Y k =1 , = i x i − x k + ηx i − x k (21)and the equations of motion¨ x i = − X k = i η ˙ x i ˙ x k ( x i − x k )(( x i − x k ) − η ) , i = 1 , . . . , N. (22)The RS model is known to be integrable, with the higher integrals of motion in involutionbeing given by H RS k = η − tr ( Y RS ) k , where Y RS = Y RS ( { x i } ; { ˙ x i } ) is the Lax matrix of the odel. Its matrix elements are Y RS ij = η ˙ x i x i − x j − η , i.e., Y RS ( { x i } ; { ˙ x i } ) = − ˙ x η ˙ x x − x − η η ˙ x x − x − η . . . η ˙ x x − x N − ηη ˙ x x − x − η − ˙ x η ˙ x x − x − η . . . η ˙ x x − x N − η ... ... ... . . . ... η ˙ x N x N − x − η η ˙ x N x N − x − η η ˙ x N x N − x − η . . . − ˙ x N . (23)Equations of motion (22) are equivalent to the Lax equation ˙ Y RS = [ B , Y RS ], where B ij = X k = i ˙ x k x i − x k − X k ˙ x k x i − x k + η δ ij + ˙ x i x i − x j (1 − δ ij ) . The Lax equation implies that all eigenvalues of the Lax matrix are integrals of motion.Let X = diag( x , x , . . . , x N ) be the diagonal matrix with the diagonal entries being coor-dinates of the particles. It is easy to check that the matrices X , Y RS satisfy the commutationrelation [ X , Y RS ] = η Y RS + η ˙ XE , (24)where E is the N × N matrix of rank 1 with all entries equal to 1. Note also that the Lax matrix Y RS can be represented in the form Y RS = ˙ X C , (25)where C is the Cauchy matrix C ij = ηx i − x j − η . Consider the Lax matrix (23) of the N -particle RS model, where the x i ’s are identified with theinhomogeneity parameters x i at the sites of the spin chain and the inverse “velocity of light”, η , is identified with the parameter η introduced in the quantum L -operator (8). Let us alsosubstitute ˙ x i = − H i : Y RS ( { x i } ; {− H i } ) = H ηH x − x + η ηH x − x + η . . . ηH x N − x + ηηH x − x + η H ηH x − x + η . . . ηH x N − x + η ... ... ... . . . ... ηH N x − x N + η ηH N x − x N + η ηH N x − x N + η . . . H N . (26)The decomposition (25) for the matrix (26) acquires the form Y RS ( { x i } ; {− H i } ) = − HC , (27) here H = diag( H , H , . . . , H N ).The claim is that if the H i ’s are eigenvalues of the Hamiltonians of the spin chain inthe invariant subspace V ( M ), then the first N − M eigenvalues of this matrix coincide witheigenvalues of the twist matrix w while the rest M eigenvalues coincide with w :Spec ( Y RS ) = (cid:16) w , . . . , w | {z } N − M , w , . . . , w | {z } M (cid:17) . (28)This means that the values of the higher RS Hamiltonians are η H RS k = ( N − M ) w k + M w k . (29)In general, the matrix Y RS with multiple eigenvalues is not diagonalizable and contains Jordancells.To put it somewhat differently, one can say that the eigenstates of the quantum spin chainHamiltonians correspond to the intersection points of two Lagrangian submanifolds in the phasespace of the RS model. One of them is the hyperplane defined by fixing all the coordinates x i while the other one is the Lagrangian submanifold obtained by fixing values (29) of the N integrals of motion in involution H RS k . In general, there are many such intersection pointsnumbered by a finite set I , with coordinates, say ( x , . . . , x N , p ( α )1 , . . . , p ( α ) N ), α ∈ I . The valuesof p ( α ) j give, through equation (21), the spectrum of H j : H ( α ) j = e − ηp ( α ) j Y k =1 , = j x j − x k + ηx j − x k . However, we can not claim that all the intersection points correspond to the energy levels ofthe spin chain Hamiltonians. The example of N = 2 considered below in detail suggests thatsome intersection points do not correspond to the energy levels of a given spin chain. Theirmeaning is to be clarified.Anyway, the spectral problem for the non-local inhomogeneous spin chain Hamiltonians H j in the subspace V ( M ) appears to be closely linked to the following inverse spectral problem forthe RS Lax matrix Y RS of the form (26). Let us fix the spectrum of the matrix Y RS to be (28),where w , w are eigenvalues of the (diagonal) twist matrix g . Then we ask what is the set ofpossible values of the H j ’s allowed by these constraints. The eigenvalues H j of the quantumHamiltonians are contained in this set.A similar correspondence between quantum and classical integrable systems was suggestedin [10], see also [11]. In a more general set-up, this assertion was derived [1, 3, 12, 13] as acorollary of the embedding of the spin chain into an infinite integrable hierarchy of non-linearPDE’s. In [4], it was checked directly using the Bethe ansatz solution.In order to find the characteristic polynomial of the matrix (26) explicitly, we use the wellknown fact that the coefficient in front of λ N − k in the polynomial det N × N ( λ I + A ) equals the sumof all diagonal k × k minors of the matrix A . All such minors can be found using decomposition(27) and the explicit expression for the determinant of the Cauchy matrix:det ≤ i,j ≤ n ηx i − x j − η = ( − n Y ≤ i 0) = N ! w n n !( N − n )! . N = 1 and N = 2 The case N = 1 is trivial. The only quantum Hamiltonian H is diagonal in the standard basisof C and coincides with the twist matrix, so we have two eigenvalues: H = w or H = w .The one-particle RS model is the model of a free particle on the line, the Lax “matrix” is justthe number − ˙ x . Fixing it to be w or w , as required by the QC duality, we obtain the twoeigenvalues of H by the identification H i = − ˙ x i , see (26). he case N = 2 is meaningful and instructive. First, let us find the spectrum of the quantumHamiltonians directly. The transfer matrix is: T ( x ) = tr "(cid:18) w w (cid:19) ( x − x ) I + η s (1)1 η s (1) − η s (1)+ ( x − x ) I + η s (1)2 ! ( x − x ) I + η s (2)1 η s (2) − η s (2)+ ( x − x ) I + η s (2)2 ! A simple calculation gives the following explicit form of the Hamiltonians: H = w s (1)1 + w s (1)2 + ηw x − x ( s (1)1 s (2)1 + s (1) − s (2)+ ) + ηw x − x ( s (1)2 s (2)2 + s (1)+ s (2) − ) , H = w s (2)1 + w s (2)2 + ηw x − x ( s (1)1 s (2)1 + s (1) − s (2)+ ) + ηw x − x ( s (1)2 s (2)2 + s (1)+ s (2) − ) . We see that H + H = g (1) + g (2) , as it should be. The space C ⊗ C is decomposed intothe direct sum of the one-dimensional space V (0) generated by the vector | ++ i ( M = 0), two-dimensional space V (1) generated by the vectors | + −i , |− + i ( M = 1) and one-dimensionalspace V (2) generated by the vector |−−i ( M = 2). We have: H | ++ i = w (cid:16) ηx − x (cid:17) | ++ i , H |−−i = w (cid:16) ηx − x (cid:17) |−−i , H | + −i = w | + −i + ηw x − x |− + i , H |− + i = w |− + i + ηw x − x | + −i . Here we use the usual notation for the basis vectors in C ⊗ C : | ++ i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) , | + −i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) , and so on.The vectors | ++ i and |−−i are eigenvectors of H . The rest part of the spectrum is found bydiagonalizing the 2 × (cid:18) w ηw x − x ηw x − x w (cid:19) . The two eigenvalues are (cid:16) w + w ± √ R (cid:17) ,where R = ( w − w ) + 4 η w w ( x − x ) . The final result for the joint spectrum of the operators H i is as follows:( H , H ) = (cid:18) w + ηw x − x , w − ηw x − x (cid:19) , M = 0 , w + w + √ R , w + w − √ R ! , M = 1 , w + w − √ R , w + w + √ R ! , M = 1 , (cid:18) w + ηw x − x , w − ηw x − x (cid:19) , M = 2 . (35)Note that in the case of the periodic boundary condition w = w = 1 the eigenvalue H =1+ ηx − x becomes 3-fold degenerate as it should be due to the GL (2)-invariance of the R -matrix. ow consider the Lax matrix of the 2-particle RS model, where we substitute ˙ x i = − H i : Y = H ηH x − x + ηηH x − x + η H The characteristic equation det( Y − λ I ) = 0 reads λ − ( H + H ) λ + x H H x − η = 0, where x ≡ x − x and the two eigenvalues are12 H + H ± s ( H + H ) − x H H x − η ! . In the subspace with M = 0 the eigenvalue of H + H is 2 w and the Lax matrix has thedouble eigenvalue w . This implies that the expression under the square root vanishes, i.e., wearrive at the system H + H = 2 w H H = w (cid:16) − η x (cid:17) which is a particular case N = 2 of the general system (33). There are two solutions:( H , H ) = (cid:18) w ± ηw x − x , w ∓ ηw x − x (cid:19) , M = 0 . The choice of the upper sign corresponds to the first line in (35). The meaning of the othersolution is to be clarified. In a similar way, for M = 2 we obtain two solutions( H , H ) = (cid:18) w ± ηw x − x , w ∓ ηw x − x (cid:19) , M = 2 , of which the one with the upper sign corresponds to the last line in (35). Finally, at M = 1 wehave the system H + H = w + w H H = w w (cid:16) − η x (cid:17) . There are two solutions which coincide with the second and the third lines in (35). In the limit η → GL (2) Gaudin model [8] is the η → T ( x ; e ηh , η, { x j } ). The expansion as η → T ( x ; e ηh , η, { x j } ) = 2 I + η tr h + N X i =1 x − x i ! I + η 12 tr h I + N X i =1 H Gi x − x i ! + O ( η ) , here h = (cid:18) ω ω (cid:19) is the Gaudin analogue of the twist matrix, and H Gi = lim η → H i ( η, e ηh , { x j } ) − I η = h ( i ) + X j = i P ij x i − x j = X j = i I x i − x j + h ( i ) + 2 X j = i ~ s ( i ) ~ s ( j ) x i − x j (36)are the Hamiltonians of the GL (2)-invariant Gaudin model. Here h ( i ) = ω + ω I + ( ω − ω ) s ( i ) z is the twist matrix acting in the space V i ∼ = C at the i th site. In the context of the Gaudinmodel, the parameters x i (in general, complex numbers) are often called marked points of theRiemann sphere. Since the first two terms in the η → T ( x ; e ηh , η, { x j } )are proportional to the identity operator and thus commute with everything, commutativity ofthe transfer matrices implies commutativity of the Gaudin Hamiltonians: [ H Gi , H Gj ] = 0. TheGaudin spectral problem consists in the simultaneous diagonalization of these operators andthe operator M which has the same form as above: H Gi Ψ = H Gi Ψ, M Ψ = M Ψ. The Betheansatz solution is the η → H Gj = ω + X k = j x j − x k + M X α =1 u α − x j , (37)where the Bethe roots u α satisfy the system of equations ω − ω + N X k =1 u α − x k = 2 M X β =1 , = α u α − u β . (38)An alternative solution is achieved via the QC duality with the classical CM model withthe Hamiltonian H CM = 12 N X i =1 p i − X i Discussions with A.Alexandrov, A.Gorsky, V.Kazakov, S.Khoroshkin, I.Krichever, S.Leurent,M.Olshanetsky, A.Orlov, T.Takebe, Z.Tsuboi, and A.Zotov are gratefully acknowledged. Someof these results were reported at the International School and Workshop “Nonlinear Mathe-matical Physics and Natural Hazards” (November 28 - December 2 2013, Sofia, Bulgaria). Theauthor thanks the organizers and especially professors B.Aneva and V.Gerdzhikov for the in-vitation and support. This work was supported in part by RFBR grant 12-01-00525, by jointRFBR grants 12-02-91052-CNRS, 14-01-90405-Ukr and grant NSh-1500.2014.2 for support ofleading scientific schools. References [1] A. Alexandrov, V. Kazakov, S. Leurent, Z. Tsuboi and A. Zabrodin, Classical tau-functionfor quantum spin chains , JHEP (2013) 064 [arXiv:1112.3310].[2] A. Alexandrov, S. Leurent, Z. Tsuboi and A. Zabrodin, The master T -operator for theGaudin model and the KP hierarchy , Nucl. Phys. B883 (2014) 173-223 [arXiv:1306.1111].[3] A. Zabrodin, The master T-operator for inhomogeneous XXX spin chain and mKP hier-archy SIGMA (2014) 006 (18 pages), [arXiv:1310.6988].[4] A. Gorsky, A. Zabrodin and A. Zotov, Spectrum of quantum transfer matrices via classicalmany-body systems , JHEP (2014) 070, [arXiv:1310.6958].[5] S.N.M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation tosolitons , Ann. Phys. (1986) 370-405;S.N.M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and el-liptic function identities , Commun. Math. Phys. (1987) 191-213. 6] F. Calogero, Solution of the one-dimensional N -body problems with quadratic and/or in-versely quadratic pair potentials , J. Math. Phys. (1971) 419-436;J. Moser, Three integrable hamiltonian systems connected with isospectrum deformations ,Adv. Math. (1976) 354-370.[7] M. Olshanetsky and A. Perelomov, Classical integrable finite dimensional systems relatedto Lie algebras , Phys. Reps. (1981) 313-400.[8] M. Gaudin, Diagonalisation d’une classe d’hamiltoniens de spin , J. de Phys. (1976),no. 10 1087-1098.[9] A. Givental and B.-S. Kim, Quantum cohomology of flag manifolds and Toda lattices ,Commun. Math. Phys. (1995) 609-641 [arXiv:hep-th/9312096].[10] E. Mukhin, V. Tarasov and A. Varchenko, Gaudin Hamiltonians generate the Bethe alge-bra of a tensor power of vector representation of gl N , St. Petersburg Math. J. (2011)463-472 [arXiv:0904.2131];E. Mukhin, V. Tarasov and A. Varchenko, Bethe algebra of Gaudin model, Calogero-Moser space and Cherednik algebra , Int. Math. Res. Not. (2014) Issue 5 1174-1204[arXiv:0906.5185].[11] E. Mukhin, V. Tarasov and A. Varchenko, KZ characteristic variety as the zero set ofclassical Calogero-Moser Hamiltonians , SIGMA (2012) 072 (11 pages) [arXiv:1201.3990];E. Mukhin, V. Tarasov and A. Varchenko, Bethe subalgebras of the group algebra of thesymmetric group , [arXiv:1004.4248];E. Mukhin, V. Tarasov and A. Varchenko, Spaces of quasi-exponentials and representationsof the Yangian Y ( gl N ), [arXiv:1303.1578].[12] A. Zabrodin, The master T -operator for vertex models with trigonometric R -matrices asclassical tau-function , Teor. Mat. Fys. (2013) 59-76 (Theor. Math. Phys. (2013)52-67) [arXiv:1205.4152][13] A. Zabrodin, Hirota equation and Bethe ansatz in integrable models , Suuri-kagaku Journal(in Japanese), Number 596 (2013) 7-12.[14] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equa-tions , in “Nonlinear integrable systems – classical and quantum”, eds. M. Jimbo and T.Miwa, World Scientific, pp. 39-120 (1983);M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras , Publ. RIMS, KyotoUniv. (1983) 943-1001.[15] N. Nekrasov, A. Rosly and S. Shatashvili, “Darboux coordinates, Yang-Yang functional,and gauge theory”, Nucl. Phys. Proc. Suppl. (2011) 69-93 [arXiv:1103.3919].[16] D. Gaiotto and P. Koroteev, On three dimensional quiver gauge theories and integrability ,JHEP (2013) 126 [arXiv:1304.0779].[17] I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum Integrable Models andDiscrete Classical Hirota Equations , Commun. Math. Phys. (1997) 267-304 [arXiv:hep-th/9604080];A. Zabrodin, Discrete Hirota’s equation in quantum integrable models , Int. J. Mod. Phys. B11 (1997) 3125-3158;A. Zabrodin, Hirota equation and Bethe ansatz , Teor. Mat. Fyz., (1998) 54-100 (En-glish translation: Theor. Math. Phys. (1998) 782-819. 18] V. Kazakov, A. S. Sorin and A. Zabrodin, Supersymmetric Bethe ansatz and Baxterequations from discrete Hirota dynamics , Nucl. Phys. B (2008) 345-413 [arXiv:hep-th/0703147];A. Zabrodin, B¨acklund transformations for difference Hirota equation and supersymmetricBethe ansatz , Teor. Mat. Fyz. (2008) 74-93 (English translation: Theor. Math. Phys. (2008) 567-584) [arXiv:0705.4006].[19] H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atom-kette , Zeitschr. fur Physik (1931) 205-226.[20] L. Faddeev, E. Sklyanin and L. Takhtajan, The quantum inverse problem method. I , Theor.Math. Phys. (1980) 688;V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering methodand correlation functions , Cambridge Monographs on Mathematical Physics, CambridgeUniversity Press, Cambridge U.K., 1997.[21] L. Faddeev and L. Takhtajan, The spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model , Zap. Nauch. Semin. LOMI (1981) 134-178.[22] K. Hikami, P. Kulish and M. Wadati, Construction of integrable spin systems with long-range interaction , J. Phys. Soc. Japan (1992) 3071-3076.[23] I. Macdonald, Symmetric functions and Hall polynomials , 2nd ed., Oxford University Press,1995.[24] K. Sawada and T. Kotera, Integrability and a solution for the one-dimensional N -particlesystem with inversely quadratic pair potential , J. Phys. Soc. Japan, (1975) 1614-1618.[25] S. Wojciechowski, New completely integrable Hamiltonian systems of N particles on thereal line , Phys. Lett. (1976) 84-86.[26] P. Kulish and N. Reshetikhin, Diagonalization of gl N invariant transfer matrices andquantum N -wave system (Lee model) , J. Phys. A (1983) L591-L596.(1983) L591-L596.