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arXiv:cond-mat/9412022 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Soluble free-fermion model inddimensions

Fa Yueh Wu, H. Y. Huang

Published Feb 1, 1995• 4 citations
arXivHTMLPDF

Abstract

We consider a vertex model in d dimensions characterized by lines that run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity of -1 is associated with each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We also determine the critical point and the singular behavior of the free energy.

DOI
10.1103/physreve.51.889

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We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the freefermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions. Key words: Odd eight-vertex model, free-fermion model, exact solution. 1 2 1

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We consider a vertex model in d dimensions characterized by lines that run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity of -1 is associated with each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We also determine the critical point and the singular behavior of the free energy.
Abstract It is shown that Dimers is Yang–Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter λ = π 2 . A one-to-many mapping of vertices onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 ∘ compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for Dimers with periodic boundary conditions on the cylinder. 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 (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity β = 2 cos ⁡ λ = 0 . At the isotropic point, the exact solution allows for the explicit counting of 45 ∘ rotated dimer configurations on a periodic M × N rectangular lattice. We show that the modular invariant partition function on the torus is the same as that of symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge c = − 2 , minimal conformal weight Δ m i n = − 1 / 8 and effective central charge c e f f = 1 .
A model describing an ensemble of disjoint clusters on a square lattice is introduced. In addition to the usual surface energy terms, in x and in y , the model includes a bending energy in c per corner in the interface. It is formulated as a seven-vertex model which is in general not solvable. For beta in c =ln(2)/2, however, this seven-vertex model becomes free fermionic one and an exact expression for its free energy is derived. In the extreme anisotropic ('Hamiltonian') limit the operator content coincides with that of the 1D Ising model in a transverse field. Three critical lines with Ising-like behaviour are found.
An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent $\ensuremath{\alpha}=\frac{1}{2}$ and the system is frozen below a unique transition point. Conditions for these special cases are given.
We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the freefermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions. Key words: Odd eight-vertex model, free-fermion model, exact solution. 1 2 1