On the Yang–Baxter equation for the six-vertex model

Abstract

In this paper we review the theory of the Yang-Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra U q ( sl (2)). In the simplest case of the spin s = 1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators. and the general scheme of quantum SoV has been developed [29][30][31][32]. The integral Q-operator for the case of the XXX chain was first calculated in [33]. It is worth noting that taking the limit N → ∞ [34] in the Bazhanov and Stroganov construction [22] one can recover the results of [27] and [33].The main difference of the above approach from the original Baxter method is that the "quantum" representation space is infinite-dimensional. It has the structure of a tensor product of Verma modules with the basis chosen as multi-variable polynomials p(x 1 , . . . , x M ), where M is the size of the system. The Q-operators appear as integral operators with an explicit action on such a polynomial basis. A detailed construction can be found in [35,36] for the XXX case and its generalization to the XXZ case in [37,38]. The non-compact case and applications of the Q-operators to Liouville theory are discussed in [39][40][41]. It is worth mentioning that the representation of the Q-operator by an integral operator is known only for the XXX case [33]. The proper deformation of such integral operator for the case of the six-vertex model is still a challenging problem.Another problem arises when spins take (half-) integer values. In this case the quantum space becomes reducible and the action of the Q-operator on the polynomial basis becomes singular. This difficulty can be overcome by expanding near the limit 2s → Z + as shown in [35,36]. However, a removal of such a regularization is technically challenging and it is desirable to have an alternative approach which is free from this difficulty.In 1997 Bazhanov, Lukyanov and Zamolodchikov (BLZ) suggested another method to derive the Q-operators related to the affine algebra U q ( sl(2)) [42,43]. Based on the universal R-matrix theory [44] they showed that the Q-operators can be constructed as special monodromy operators with the auxiliary space being an infinite-dimensional representation of the q-oscillator algebra. Although their original approach was developed in the context of quantum field theory, the results of [42,43] can be easily adjusted to the spin s = 1/2 XXZ chain [45]. However, the derivation of the local Q-operators from...

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

“…Using the Sears transform (A.7) we can transform the sum in (3.22) to (5.8) in [19] by identifying n = i, a = q −2j ′ , b = λ 2 q −I−J , c = λ 2 q 2+I+J−2i−2j , d = λ 2 q 2+I−J−2i , e = q −2i−2j and f = λ 2 q 2+J−I−2j ′ . Let us note that (5.8) in [19] requires a regularization but the expression (3.22) is free from any divergences.…”

“…Previously a closed formula for the higherspin R-matrix of the 6-vertex model was obtained in [19] based on the positive solution of the tetrahedron equation [18]. It can be interpreted as the R-matrix of the higher-spin stochastic 6-vertex model [20][21][22][23].…”

“…However, most of the results in [26] were obtained using the machinery of quantum groups. Our strategy is to derive explicit formulas for the R-matrix related to symmetric representations of U q (A (1) n ) extending the method of [19]. The structure of the paper is as follows.…”

Matrix Factorization for Solutions of the Yang–Baxter Equation

“…There we demonstrated that the R-operator can be realized as an integral operator, and it factorizes in a product of four Faddeev-Volkov type R-matrices [20]. In [29,30] an explicit hypergeometric formula for the R-matrix of U q (sℓ 2 ) acting on a tensor product of two highest-weight representations has been discovered. In [24] a universal factorization formula for the trigonometric R-operator has been obtained by means of the universal R-matrix.…”

“…It would be interesting to relate the factorization formula (35) to explicit expressions for R-matrices from [29,30] as well as to the universal factorization formula [24].…”