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International Journal of Modern Physics B

SIX-VERTEX MODELS AS FERMI GASES

Peter Orland A. B. C

Published Aug 20, 1991• 1 citation
Publisher

Abstract

The two-dimensional six-vertex model with a particular choice of external field is known to be equivalent to a theory of free Fermions. Two conditions are made on the six-vertex weights. A simple physical interpretation of the ordered and disordered phases is found. The Fermi sea is filled and the free energy, correlation length, order parameter, correlation functions and phase diagram are determined.

DOI
10.1142/s0217979291000936

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The two-dimensional six-vertex model with a particular choice of external field is known to be equivalent to a theory of free Fermions. Two conditions are made on the six-vertex weights. A simple physical interpretation of the ordered and disordered phases is found. The Fermi sea is filled and the free energy, correlation length, order parameter, correlation functions and phase diagram are determined.
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Abstract We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm. Particular attention is paid to the free fermionic point (Δ = 0), for which the correlation functions are obtained analytically in the thermodynamic limit. A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered (|Δ| < 1) phase and to monitor the logarithm-like behavior of correlation functions there. For the antiferroelectric (Δ < −1) phase, the exponential decrease of correlation functions is observed.
Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.
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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