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Quantum group invariant integrable n-state vertex models with periodic boundary conditions

Karowski, M, Zapletal, A

Published Jan 1, 1993• 24 citations
OpenAlex

Abstract

An $U_q(sl(n))$ invariant transfer matrix with periodic boundary conditions is analysed by means of the algebraic nested Bethe ansatz for the case of $q$ being a root of unity. The transfer matrix corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. the 3-dimensional interior of the torus. By means of finite size analysis we find the central charge of the corresponding Virasoro algebra as $c=(n-1) \\left[1-n(n+1)/(r(r-1))\\right] $.

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arXiv:nlin/0602003•Journal of Statistical Mechanics Theory and Experiment•Jul 12, 2006•3 citations
An Uq(sl(n))U_q(sl(n))Uq​(sl(n)) invariant transfer matrix with periodic boundary conditions is analysed by means of the algebraic nested Bethe ansatz for the case of qqq being a root of unity. The transfer matrix corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. the 3-dimensional interior of the torus. By means of finite size analysis we find the central charge of the corresponding Virasoro algebra as c=(n−1)left[1−n(n+1)/(r(r−1))right]c=(n-1) \\left[1-n(n+1)/(r(r-1))\\right] c=(n−1)left[1−n(n+1)/(r(r−1))right].
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.
It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter (q+q−1)/2(q+q^{-1})/2(q+q−1)/2 is evaluated at roots of unity qN=1q^{N}=1qN=1. These new symmetries have been linked to an U(A1(1))U(A^{(1)}_1)U(A1(1)​) invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups Uq(hatg)U_q(\\hat g)Uq​(hatg) into account. Here hatg\\hat ghatg is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of U(hatg)U(\\hat{g})U(hatg). For the simplest case hatg=A1(1)\\hat g=A_1^{(1)}hatg=A1(1)​ the discussion is also extended to even primitive roots of unity.
Abstract The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N . In the continuum scaling limit, in sectors with magnetization S z , we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q -Narayana numbers or, equivalently, skew q -binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L 0 . This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.