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Journal of Physics A Mathematical and General

Boundary conditions and inversion indentities for solvable lattice models with a sublattice symmetry

Je-Young Choi, D Kim, Paul A. Pearce

Published May 21, 1989• 8 citations
Publisher

Abstract

The transfer matrices of solvable lattice spin models are considered under general boundary conditions associated with the symmetry of the models. It is shown how the full symmetry can be exploited for systems having a sublattice symmetry by considering an extended model where the states distinguish the sublattices. In this extended scheme, the parity of the number of sites is tied to the boundary conditions. The inversion identities can be extended and be used to obtain the full operator content of the critical models. As an application the complete spectra of the transfer matrices for the eight-vertex model at the decoupling point are calculated.

DOI
10.1088/0305-4470/22/10/020

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The transfer matrices of solvable lattice spin models are considered under general boundary conditions associated with the symmetry of the models. It is shown how the full symmetry can be exploited for systems having a sublattice symmetry by considering an extended model where the states distinguish the sublattices. In this extended scheme, the parity of the number of sites is tied to the boundary conditions. The inversion identities can be extended and be used to obtain the full operator content of the critical models. As an application the complete spectra of the transfer matrices for the eight-vertex model at the decoupling point are calculated.
We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions.Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method.Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the Bn,B_{n},Bn​, DnD_{n}Dn​ and CnC_{n}Cn​ Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.
A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.
Abstract The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K -matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.
We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper [H. Meyer, J.-C. Angl`es d'Auriac, and J.-M. Maillard, Phys. Rev. E 55, 5261 (1997)]. We show that the statistical properties of these spectra can serve as a criterion of integrability. It also provides an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality, considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra, which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.
Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.
Abstract The square-lattice eight-vertex model with vertex weights a , b , c , d obeying the relation ( a 2 + a b ) ( b 2 + a b ) = ( c 2 + a b ) ( d 2 + a b ) and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2 n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue Θ n = ( a + b ) 2 n + 1 . This proves a conjecture by Stroganov from 2001. The proof uses the supersymmetry of a related XYZ spin-chain Hamiltonian. The eigenstates of the transfer matrix corresponding to Θ n are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights Θ n is the largest eigenvalue of the transfer matrix.