The Ising Limit of the XXZ Heisenberg Magnet and Certain Thermal Correlation Functions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec The Ising limit of the XXZ Heisenberg magnetand certain thermal correlation functions
N. M. Bogoliubov † , C. Malyshev ‡ Steklov Mathematical Institute, St.-Petersburg Department, RASFontanka 27, St.-Petersburg, 191023, Russia † e-mail: [email protected] ‡ e-mail: [email protected] In Memorium of A. N. Vassiliev
Abstract
The spin-
XXZ
Heisenberg magnet is considered for the case of the anisotropyparameter tending to infinity (so-called,
Ising limit ). A thermal correlation functionof the ferromagnetic string is calculated over the ground state. The approach tothe calculation of the correlation functions is based on the observation that thewave function in the limit considered is expressed in terms of the symmetric Schurfunctions. It is demonstrated that at low temperatures the amplitude of the asymp-totical expression of the present correlation function is proportional to the squarednumbers of strict boxed plane partitions.
Keywords:
Heisenberg magnet, Ising limit, correlation functions
Introduction
The Ising model was one of the favorite models of A. N. Vassiliev. His studies wereconcentrated on the scaling properties of this model [1, 2]. It is remarkable that in aspecial limit of the anisotropy parameter tending to infinity, the Hamiltonian of the spin -1/2
Heisenberg
XXZ chain is transformed into the Hamiltonian of one-dimensional Isingmodel [3]. This limit is usually called as the
Ising limit of the
XXZ chain [4]. In our paperwe shall investigate the Ising limit of the
XXZ magnet using an effective Hamiltonianwhich is commutative with the Ising Hamiltonian and has a complete system of the eigen-functions which is common with that of the Ising Hamiltonian. This effective Hamiltonianis an appropriate object for studying of the Ising limit since it is related to the transfer-matrix of the four-vertex model, which arises in the limit of infinite anisotropy from thetransfer-matrix of the six-vertex model. In their turn, the trace identities relate the
XXZ
Hamiltonian and the transfer-matrix of the six-vertex model.The problem of calculation of the correlation functions of
XXZ
Heisenberg modelin the framework of the Quantum Inverse Scattering Method [5] has required seriousefforts [6–11]. We shall show that calculation of the correlation functions in the Isinglimit is closely related to the theory of the symmetric functions [12], and thus is closelyconnected with the problems of enumerative combinatorics, such as the random walks ofvicious walkers [13–17] and enumeration of boxed plane partitions [18, 19]. In this paperwe shall calculate the thermal correlation functions of the ferromagnetic string over theground state using the approach developed in [20–24].The
XXZ spin-
Heisenberg magnet is defined on the one-dimensional lattice con-sisting of M + 1 sites labeled by elements of the set M ≡ { ≤ k ≤ M, k ∈ Z } , M + 1 = 0 (mod 2) . The corresponding spin Hamiltonian is defined as follows: b H XXZ = − M X k =0 ( σ − k +1 σ + k + σ + k +1 σ − k + ∆2 ( σ zk +1 σ zk −
1) + h σ zk ) , (1)where the parameter ∆ ∈ R is the anisotropy of the model, and h is the external magneticfield. The local spin operators σ ± k = ( σ xk ± iσ yk ) and σ zk , dependent on the lattice argument k ∈ M , are defined standardly (see [6, 7]), and their commutation rules are given by therelations: [ σ + k , σ − l ] = δ k,l σ zl , [ σ zk , σ ± l ] = ± δ k,l σ ± l . The state-space is spanned over ofthe state-vectors N Mk =0 | s i k , where s implies either ↑ or ↓ . The spin “up” and “down”states ( |↑i and |↓i , respectively) provide a natural basis in the linear space C so that |↑i ≡ (cid:16) (cid:17) , |↓i ≡ (cid:16) (cid:17) . The periodic boundary conditions σ k +( M +1) = σ k are imposed. The Hamiltonian (1)commutes with the operator of third component of the total spin b S z : [ b H XXZ , b S z ] = 0 , b S z ≡ M X k =0 σ zk . (2)To represent N -particle state-vectors, | Ψ N ( u , . . . , u N ) i , let the sites with spin “down”states be labeled by the coordinates µ i forming a strict partition µ ≡ ( µ , µ , . . . , µ N ) ,2here M ≥ µ > µ > . . . > µ N ≥ . There is a correspondence between each partitionand an appropriate sequence of zeros and unities of the form: (cid:8) e k ≡ e k ( µ ) (cid:9) k ∈M , where e k ≡ δ k,µ n , ≤ n ≤ N . The Hamiltonian (1) is diagonalised via the following ansatz: | Ψ N ( u ) i = X { e k ( µ ) } k ∈M χ XXZ µ ( u ) M Y k =0 ( σ − k ) e k |⇑i , (3)where the sum is taken over C NM strict partitions µ , and |⇑i ≡ N Mn =0 |↑i n is the fullypolarized state with all spins “up”. It is proposed to use bold-faced letters as short-handnotations for appropriate N -tuples of numbers: for instance, u instead of ( u , . . . , u N ) ,etc. Therefore, the wave function χ XXZ µ ( u ) in (3) is of the form: χ XXZ µ ( u ) = X S p ,p ,...,pN A S ( u ) u µ p u µ p . . . u µ N p N , (4) A S ( u ) ≡ Y ≤ j
XXZ chain
Less studied limit of the
XXZ model is the
Strong Anisotropy (SA) limit ∆ → −∞ [4, 29–31]. This limit is called the Ising limit since the simplest Hamiltonian in this limitturns out to be the Hamiltonian of the one-dimensional Ising model [4]: lim ∆ → −∞ b H XXZ = b H Is ≡ − M X k =0 ( σ zk +1 σ zk − . (9)However we shall study the strong anisotropy limit using the effective Hamiltonian b H SA ,which is formally equivalent to the XX Hamiltonian (8), provided the latter is suppliedwith the requirement forbidding two spin “down” states to occupy any pair of nearest-neighboring sites: b H SA = − M X k =0 P ( b h k +1 ,k + h σ zk ) P , P ≡ M Y k =0 ( I − ˆ q k +1 ˆ q k ) . (10)The projectors P “cut out” the spin “down” states for any pair of nearest-neighboringsites. The local projectors onto the spin “up” and “down” states are of the form: ˆ q k ≡
12 ( I − σ zk ) , ˇ q k ≡
12 ( I + σ zk ) , ˇ q k + ˆ q k = I , k ∈ M , (11)where I is the unit operator in ( M + 1) -fold tensor product of the linear spaces C . Thepoint is that the Hamiltonians (9) and (10) are commutative. Indeed, the Hamiltonian(9) can be represented by means of (11) as follows: b H Is = b N − M X k =0 ˆ q k +1 ˆ q k , b N ≡ M X k =0 ˆ q k , (12)where b N is the operator of number of particles commuting with the Hamiltonians b H XX and b H SA . On another hand, the definition of the projector P yields: P M X k =0 ˆ q k +1 ˆ q k = M X k =0 ˆ q k +1 ˆ q k P = 0 . Then, [ b H SA , b H Is ] = 0 and so the Hamiltonians (9) and (10) possess a common system ofthe eigen-functions.The wave function (4) takes the form in the limit ∆ → −∞ : χ SA µ ( u ) = det( u µ k − N + k ) j ) ≤ j,k ≤ N Y ≤ n The solutions of the Bethe equations (14) constitute a complete set of the eigen-states,and therefore one can obtain the resolution of the identity operator: I = 1( M + 1)( M + 1 − N ) N − X { θ } (cid:12)(cid:12) V ( e i θ ) (cid:12)(cid:12) | Ψ N ( θ ) ih Ψ N ( θ ) | , (24)where the summation over { θ } implies summation over all independent solutions (15).Here the exponential parametrization for the solutions of the Bethe equations is used inthe compact form, e i θ ≡ ( e iθ , e iθ , . . . , e iθ N ) . We shall calculate the nominator of (20)using insertions of the resolution (24). Taking into account that h Ψ N ( v ) | e − β b H SA | Ψ N ( θ ) i = h Ψ N ( v ) | Ψ N ( θ ) i e − βE N ( θ ) , we obtain: h Ψ N ( v ) | ¯ Π n e − β b H SA ¯ Π n | Ψ N ( u ) i = 1( M + 1)( M + 1 − N ) N − X { θ } e − βE N ( θ ) × (cid:12)(cid:12) V ( e i θ ) (cid:12)(cid:12) P ( v − , e i θ ) P ( e − i θ , u ) , (25)where P ( v − , e i θ ) ≡ X e λ L S e λ L ( v − ) S e λ L ( e i θ ) , P ( e − i θ , u ) ≡ X e λ R S e λ R ( e − i θ ) S e λ R ( u ) . (26)The range of summation in (26) is taken as follows: e λ L , e λ R ⊆ { ( M − N − n + 1) N } .Then, using the relation (21) we obtain from (25): h Ψ N ( v ) | ¯ Π n e − β b H SA ¯ Π n | Ψ N ( u ) i == 1( M + 1)( M + 1 − N ) N − V ( u ) V ( v − ) X M − N ≥ I >I ··· >I N ≥ e − βE N ( θ ) × det (cid:16) − ( e iθ i v − j ) M − N − n +1 − e iθ i v − j (cid:17) ≤ i,j ≤ N det (cid:16) − ( u l e − iθ p ) M − N − n +1 − u l e − iθ p (cid:17) ≤ l,p ≤ N , (27)where summation goes over the ordered sets { I k } ≤ k ≤ N that parametrize θ (15), and E N ( θ ) is given by (17). Expression for T ( θ v , n, β ) (20) appears from (27) as follows: T ( θ v , n, β ) = 1( M + 1) ( M + 1 − N ) N − X M − N ≥ I >I ··· >I N ≥ e − β ( E N ( θ ) − E N ( θ v )) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) − e i ( M − N − n +1)( θ l − θ v p ) − e i ( θ l − θ v p ) (cid:17) ≤ p,l ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (28)where θ v implies the ground-state solution (16), and E N ( θ v ) is the ground-state energyin the limit of strong anisotropy. 7 Four-vertex model and boxed plane partitions XXZ Hamiltonian at ∆ → −∞ and the four-vertex model The six-vertex model on a square lattice is defined by six different configurations of arrowspointed both in and out of each lattice site. A statistical weight w k ( k = 1 , , . . . , isascribed to each admissible type of the vertices (Fig. 1). Representing the arrows pointingup or to the right by the solid lines one can get the alternative description of the verticesin terms of lines floating through the lattice sites. So far the bonds of a lattice may beonly in two states – with a line either without it, there is a one to one correspondencebetween the admissible configuration of arrows on a lattice and the network of lines – thenest of lattice paths. The four-vertex model is obtained in the limit when the weights w = w = 0 . w w w w w w Figure 1: The vertex configurations of the 6-vertex model.The L -operator of the six-vertex model is equal to [36]: L ( n | u ) = − ue γσ zn − u − e − γσ zn γ ) σ − n γ ) σ + n ue − γσ zn + u − e γσ zn ! , (29)where u ∈ C and γ ≡ Arch∆ , and it satisfies the intertwining relation: e R ( u, v ) (cid:0) L ( n | u ) ⊗ L ( n | v ) (cid:1) = (cid:0) L ( n | v ) ⊗ L ( n | u ) (cid:1) e R ( u, v ) . (30)Here, e R ( u, v ) is the (4 × -matrix: e R ( u, v ) = e f ( v, u ) 0 0 00 e g ( v, u ) 1 00 1 e g ( v, u ) 00 0 0 e f ( v, u ) , (31)where e f ( v, u ) = u e γ − v e − γ u − v , e g ( v, u ) = uvu − v (cid:0) e γ − e − γ (cid:1) , (32)and u, v ∈ C .Let us consider the following transformation of the L -operator (29): ˇ L ( n | u ) = e h σ zn e ( ω/ σ z L ( n | u ) e − ( ω/ σ z (33) = (cid:18) − ue ( h + γ ) σ zn − u − e ( h − γ ) σ zn γ ) e ω + hσ zn σ − n γ ) e − ω + hσ zn σ + n ue ( h − γ ) σ zn + u − e ( h + γ ) σ zn (cid:19) . R -matrix of the form: ˇ R ( u, v ) = (cid:0) ⊗ e − hσ z (cid:1) e R ( u, v ) (cid:0) ⊗ e hσ z (cid:1) (34) = e f ( v, u ) 0 0 00 e g ( v, u ) e h e − h e g ( v, u ) 00 0 0 e f ( v, u ) . If one puts h = ω = γ , then the limit lim γ →∞ e − γ ˇ L ( n | u ) = L ( n | u ) ≡ (cid:18) − u ˇ q n σ − n σ + n u − ˇ q n (cid:19) , (35)where ˇ q n = σ + n σ − n (see (11)), gives us the L -operator of, so-called, four-vertex model.Analogously, the limit γ → ∞ transforms the matrix e − γ ˇ R ( u, v ) into the R -matrix of thefour-vertex model [22]: R ( u, v ) = f ( v, u ) 0 0 00 g ( v, u ) 1 00 0 g ( v, u ) 00 0 0 f ( v, u ) , (36)where f ( v, u ) = u u − v , g ( v, u ) = uvu − v . (37)The monodromy matrix of the models is defined as the matrix product of L -operators: T ( u ) = L ( M | u ) L ( M − | u ) · · · L (0 | u ) = A ( u ) B ( u ) C ( u ) D ( u ) ! . (38)The transfer matrix is the matrix trace of the monodromy matrix: τ ( u ) = Tr T ( u ) .In [33] it was proved that the transfer-matrix of the six-vertex model commutes with theHamiltonian of the XXZ model (1): [ b H , τ ( u )] = 0 .The transfer-matrix of the four-vertex model τ ( u ) satisfies the property: P τ ( u ) = τ ( u ) P = τ ( u ) , (39)where P is the projector defined in (10). Indeed, using the explicit expression for theentries of the product of two L -operators, L ( n + 1 | u ) L ( n | u ) = (cid:18) A n +1 ,n B n +1 ,n C n +1 ,n D n +1 ,n (cid:19) , (40)one can show that C n +1 ,n L ( n + 1 | u ) L ( n | u ) = L ( n + 1 | u ) L ( n | u ) C n +1 ,n = L ( n + 1 | u ) L ( n | u ) , (41)where C n +1 ,n ≡ − ˆ q n +1 ˆ q n . It follows from (41) that C n +1 ,n τ ( u ) = τ ( u ) C n +1 ,n = τ ( u ) at n = 0 , . . . , M − . Invariance of the matrix trace with respect of cyclic permutation9mplies that C ,M τ ( u ) = τ ( u ) C ,M = τ ( u ) , and therefore the relation (39) is alsofulfilled.Further, one can advance the following Proposition: The transfer matrix of the four-vertex model commutes with the XXZ Hamiltonian taken in the Ising limit : [ b H SA , τ ( u )] = 0 . (42) Proof: In order to prove (42), it is suffices to consider b H SA (10) at h = 0 . We shalluse the method proposed in [33] and then developed in [34]. Generalizing this approachwith respect of the problem under consideration, we shall take into account the relation(subscripts are omitted): C n +1 ,n (cid:2)b h n +1 ,n , L ( n + 1 | u ) L ( n | u ) (cid:3) C n +1 ,n = C n +1 ,n (cid:0) Q ( n + 1 | u ) L ( n | u ) − L ( n + 1 | u ) Q ( n | u ) (cid:1) C n +1 ,n , (43)where the operators b h n +1 ,n and C n +1 ,n are introduced in (8) and (41), respectively. If so,we obtain the vanishing of the commutator in question, (cid:2) b H, τ ( u ) (cid:3) = − M X n =0 P (cid:2)b h n +1 ,n , τ ( u ) (cid:3) P = − M X n =0 Tr (cid:0) P (cid:2)b h n +1 ,n , T ( u ) (cid:3) P (cid:1) = 0 , (44)provided that (cid:2)b h n +1 ,n , T ( u ) (cid:3) = L ( M | u ) · · · L ( n + 2 | u ) (cid:2)b h n +1 ,n , L ( n + 1 | u ) L ( n | u ) (cid:3) L ( n − | u ) · · · L (0 | u ) . (45)Furthermore, the commutators of the entries from the right-hand side of (40) look asfollows: (cid:2)b h n +1 ,n , A n +1 ,n (cid:3) = − (cid:2)b h n +1 ,n , D n +1 ,n (cid:3) = ˇ q n +1 − ˇ q n (cid:2)b h n +1 ,n , B n +1 ,n (cid:3) = − uσ − n +1 + u − σ − n + uσ − n +1 ˆ q n − u − ˆ q n +1 σ − n , (cid:2)b h n +1 ,n , C n +1 ,n (cid:3) = − u − σ + n +1 + uσ + n + u − σ + n +1 ˆ q n − u ˆ q n +1 σ + n . (46)In turn, one obtains from (46): C n +1 ,n (cid:2)b h n +1 ,n , A n +1 ,n (cid:3) C n +1 ,n = C n +1 ,n (cid:0) ˇ q n +1 − ˇ q n (cid:1) C n +1 ,n , C n +1 ,n (cid:2)b h n +1 ,n , B n +1 ,n (cid:3) C n +1 ,n = C n +1 ,n (cid:0) − uσ − n +1 + u − σ − n (cid:1) C n +1 ,n , C n +1 ,n (cid:2)b h n +1 ,n , C n +1 ,n (cid:3) C n +1 ,n = C n +1 ,n (cid:0) − u − σ + n +1 + uσ + n (cid:1) C n +1 ,n , C n +1 ,n (cid:2)b h n +1 ,n , D n +1 ,n (cid:3) C n +1 ,n = C n +1 ,n (cid:0) − ˇ q n +1 + ˇ q n (cid:1) C n +1 ,n . (47)On the other hand, using the diagonal matrix Q ( n | u ) = (cid:18) u − σ uσ (cid:19) , we obtain the following relation for the L -operators (35): Q ( n + 1 | u ) L ( n | u ) − L ( n + 1 | u ) Q ( n | u ) = (cid:18) ˇ q n +1 − ˇ q n − uσ − n +1 + u − σ − n − u − σ + n +1 + uσ + n − ˇ q n +1 + ˇ q n (cid:19) . (48)10omparison of (47) and (48) demonstrates that the relation (43), as well as the statementof the relation (42) are valid indeed. (cid:4) The state vector of the four-vertex model is constructed in the framework of thealgebraic Bethe ansatz as follows: (cid:12)(cid:12) Ψ N ( u ) i = N Y i =1 B ( u i ) (cid:12)(cid:12) ⇑i . (49)This vector is the eigen-vector both of the transfer-matrix of the four-vertex model andof the Hamiltonian (10), provided that the parameters u l fulfill the Bethe equations (14).It was shown in [22] that this vector can be represented in the form (19).According to the representation (22) taken at n + 1 = 0 , the scalar product h Ψ N ( v ) | Ψ N ( u ) i is proportional to the determinant of the matrix, say, ( T kj ) ≤ k,j ≤ N .The Bethe equations (14) enable one to represent this determinant as follows: det( T kj ) ≤ k,j ≤ N ≡ det − ( u k /v j ) M − N +2 − u k /v j ! = N Y k =1 − v k u k ! N det( A kj ) ≤ k,j ≤ N , (50)where the entries A kj take the form: A kj = ( u k v j ) M +3 − N u k − v j N Y l =1 , l = k u l − N Y l =1 , l = j v l ! . (51)It is crucial that the determinant of the matrix ( A kj ) ≤ k,j ≤ N vanishes provided the par-ameters u and v are independent solutions of the Bethe equations (14). Really, thismatrix has a non-trivial eigen-vector χ j with zero eigen-value: N X j =1 A kj χ j = 0 , χ j = ( u j − v j ) N Y l =1 , l = j u j − v l u j − u l . (52)Validity of (52) is justified by two identities: N X k =1 y k − x k y k − x j N Y l =1 , l = k y k − x l y k − y l ! = 1 , N X k =1 y k − x k y k N Y l =1 , l = k y k − x l y k − y l ! = 1 − N Y l =1 x l y l . Therefore, the scalar product h Ψ N ( v ) | Ψ N ( u ) i vanishes (i.e., the state-vectors are ortho-gonal) provided the sets of the parameters u and v represent independent Bethe solutions.Completeness of the system of the state-vectors was proved in [4, 30]. Consider the four-vertex model [22]. There is one to one correspondence between thestrict plane partitions (i.e., the plane partitions that decay along each column and eachraw [12]) and the nests of the admissible paths on a square lattice of the size N × ( M + 1) N columns(counting from the left) are pointing inwards and the arrows on the top and bottom ofthe last N ones are pointing outwards (see Fig. 2).It was proved in [22] that if one puts v j = q − j and u j = q j − in the scalar product ofthe state-vectors (49), then h Ψ N ( q − , . . . , q − N ) | Ψ N (1 , . . . , q N − ) i = q − N ( N − Z spp q ( N, N, M ) , (53)where Z spp q ( N, N, M ) is the generating function of strict plane partitions placed into abox of size N × N × M : Z spp q ( N, N, M ) ≡ q N ( N − Y ≤ j,k ≤ N − q M +3 − j − k − q j + k − . (54)Being taken at q = 1 , this formula gives the number of strict plane partitions in a box ofsize N × N × M : Z spp q =1 ( N, N, M ) = Y ≤ j,k ≤ N M + 3 − j − kj + k − . (55)It is straightforward to obtain [19, 24] that the entry lim q → h Ψ N ( q − , . . . , q − N ) | ¯ Π n | Ψ N (1 , . . . , q N − ) i = Z spp q =1 ( N, N, M − n − (56)is equal to the number strict partitions in a N × N × ( M − n − box. Let us obtain the low temperature estimate of the correlation function (20). The survivalprobability of the ferromagnetic string is written as follows: T ( θ v , n, β ) = (cid:12)(cid:12) V ( e i θ v ) (cid:12)(cid:12) V X { θ } e − β ( E N ( θ ) − E N ( θ v )) (cid:12)(cid:12) V ( e i θ ) P ( e − i θ , e i θ v ) (cid:12)(cid:12) , (57)12here V ≡ ( M + 1)( M + 1 − N ) N − . Here P ( e − i θ , e i θ v ) is given according to (26), providedthe summation domain is given by e λ ⊆ { ( M − N − n + 1) N } .If the chain is long enough while the number of quasi-particles is moderate, i.e., N ≪ M , we replace the sums in (57) by the integrals as follows: T ( θ v , n, β ) ≃ π ) N N ! Y ≤ r XXZ Heisenberg magnetic chain in a specific limit of theanisotropy parameter: ∆ → −∞ . The, so-called, survival probability of the ferromagneticstring has been calculated over the N -particle ground state of the model. It was provedthat the Hamiltonian of the Ising limit, previously introduced in the papers [24, 29, 30],is correct as the limit ∆ → −∞ of the XXZ Hamiltonian. Finally, we have foundthe leading term of the low temperature asymptotics of the survival probability of the13erromagnetic string for the model in the finite volume and at fixed number of quasi-particles. We have demonstrated that the amplitude of the asymptotics is proportionalto the squared number of strict plane partitions in a box, while its critical exponent isproportional to the squared number of particles. Comparing this result with that obtainedin [24, 37], one can conclude that this type of the behavior is universal for a special classof the integrable models. 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