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Journal of Physics Conference Series

Eight-vertex model over Grassmann algebra

T.K. Kassenova, Pyotr Tsyba, Olga Razina

Published Nov 1, 2019• 5 citations
Publisher

Abstract

Abstract An eight-vertex model on a square lattice over a Grassmann algebra is investigated using an equation that is a three-dimensional generalization of the Yang-Baxter equation. Anticommuting quantum spin systems are studied, where the quasiclassical limit leads to some abstract classical physics with anticommuting variables. The solution of the quantum Yang-Baxter equation is the R - matrix, which corresponds to the transfer R - matrix of the eight-vertex model of statistical mechanics.

DOI
10.1088/1742-6596/1391/1/012035

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Abstract An eight-vertex model on a square lattice over a Grassmann algebra is investigated using an equation that is a three-dimensional generalization of the Yang-Baxter equation. Anticommuting quantum spin systems are studied, where the quasiclassical limit leads to some abstract classical physics with anticommuting variables. The solution of the quantum Yang-Baxter equation is the R - matrix, which corresponds to the transfer R - matrix of the eight-vertex model of statistical mechanics.
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 Uq(sl(2)ˆ). 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.
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator L0{\mathrm{L}}_{0}L0​ is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.
In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n -state spin model and of a certain specific n -state vertex model coincide for finite lattice sizes. The equivalent vertex model has n 3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.
A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for nonintegrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappear, and the Poisson distribution be recovered on (non Yang-Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion, we tested the generic calculability of the model on these algebraic varieties.