We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.
We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.
A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.
We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschütz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.
Recently it was shown that the eigenfunctions for the the asymmetric exclusion problem and several of its generalizations as well as a huge family of quantum chains, like the anisotropic Heisenberg model, Fateev- Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc, can be expressed by a matrix product ansatz. Differently from the coordinate Bethe ansatz, where the eigenvalues and eigenvectors are plane wave combinations, in this ansatz the components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. In this work, we introduce a formulation of a matrix product ansatz for the six-vertex model with periodic boundary condition, which is the paradigmatic example of integrability in two dimensions. Remarkably, our studies of the six-vertex model are in agreement with the conjecture that all models exactly solved by the Bethe ansatz can also be solved by an appropriated matrix product ansatz.
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.
Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz, the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian defines the algebraic properties of the matrices defining the amplitudes. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. The matrix product ansatz we propose allows an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t – J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with z -magnetic fields at the surface and arbitrary hard-core exclusion among the spins.
A new scheme to study the XXZ spin chain in massive regime is introduced by Davies et al in [1]. In this scheme, which is called symmetry approach, the space of states is identified with some Uq(A\_ ) -module. Under this identification, the XXZ Hamiltonian can be constructed on this module as an operator which commutes with the action of Uq(A '). Furthermore, the row transfer matrix and creation (annihilation) operators can be constructed by using vertex operators. In a similar way to [1], the higher spin chain [2] and the models related to the vector representation ofUq(A^) [3], Uq(B^ U q ( D ( n } ) [5] are studied. In this paper, we consider the model related with the spin representation of Uq(Dn ). In physics the model that we consider here is explained as follows. It is a one dimensional quantum spin chain model constructed from the spin representation of Uq(Dn )• The Hamiltonian acts on the space of the infinite tensor product