Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

Abstract

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two-and three-state vertex models. These models include the six-vertex model, the 15-vertex A(1) 2 model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices do not coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -including an O(n) model on the square lattice with open boundaries -is of relevance to the surface critical behaviour of two-dimensional polymers.Physics and Astronomy Classifications. 05.50.+q

Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions

“…where In particular, we found that the known diagonal reflection matrices for the Izergin-Korepin model [24] lead to two integrable sets of boundary weights which preserve the O(n) symmetry [22,25]. For each case t b = tb = 2 cos λ.…”

“…The two integrable sets of boundary weights are [22] (A) t s = sin 2λ sin λ and (B) t s = cos 2λ cos λ .…”

“…On the other hand, the thermal scaling dimension X ǫ , corresponding to the energy-energy correlation, is related to an excitation in the largest sector of the transfer matrix. Given X 1 and X ǫ , the surface critical exponents follow from [5,4]In terms of the more standard parameter g, where 4λ + πg = 2π, the bulk exponents γ andFor case (B), at the special transition (x * , y * a ), we have obtained the Bethe Ansatz solution for the eigenvalues of the transfer matrix [22],5 where the u j follow as roots of the Bethe Ansatz equationssinh(u j − u k + i λ) sinh(u j + u k + i λ) sinh(u j − u k − i 2λ) sinh(u j + u k − i 2λ) sinh(u j − u k − i λ) sinh(u j + u k − i λ) sinh(u j − u k + i 2λ) sinh(u j + u k + i 2λ) . (14) Here N is the width of the strip (e.g., N = 8 in Fig.…”

“…In order to do this, we adapted the Sklyanin construction of commuting transfer matrices [20] to the present geometry [21][22][23].…”

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)Vieira1, Santos2 2017Physics Letters A2030View full textAdd to dashboardCite

“…Email addresses: rsvieira@df.ufscar.br (R. S. Vieira), dals@df.ufscar.br (A. Lima Santos) had been diagonalized before, through the fusion procedure [26], in [27] and, with inhomogeneities, in [28]) and coordinate Bethe Ansatz of the IKvm was obtained in [29]; the algebraic Bethe Ansatz for the ZFvm and IKvm were derived respectively in [25] and [30]. Considering now nonperiodic boundary conditions, the diagonal K-matrices for the ZFvm were first derived in [31] and the general Kmatrices were deduced in [32] and [33]; for the IKvm, the diagonal K-matrices were found in [34] and the general K-matrices were derived in [33] and [35]; besides, the coordinate Bethe Ansätze for both models were presented in [36], while their algebraic versions were performed in [37] (although the ZFvm had been considered before in [31] with the help of the fusion procedure [26]).…”