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arXiv:1307.3002

Correlation Critical Exponents for the Six-Vertex Model

Pierluigi Falco

Published Jul 11, 2013• 0 citations
arXivHTMLPDF

Abstract

The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by Bethe Ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are in general not available. In this paper we consider the isotropic "zero-field" six-vertex model at small |Δ|. We derive the large-distance asymptotic formula of the arrow-arrow correlation, which displays a power law decay and an anomalous exponent. Our method is based on an interacting fermions representation of the six-vertex model and does not use any information obtained from the exact solution.

DOI
10.48550/arxiv.1307.3002

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The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by Bethe Ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are in general not available. In this paper we consider the isotropic "zero-field" six-vertex model at small |Δ|. We derive the large-distance asymptotic formula of the arrow-arrow correlation, which displays a power law decay and an anomalous exponent. Our method is based on an interacting fermions representation of the six-vertex model and does not use any information obtained from the exact solution.
The six-vertex model on a square lattice is "exactly solvable" because an exact formula for the free energy can be obtained by the Bethe ansatz. However, exact formulas for the correlations of local bulk observables, such as the orientation of the arrow at a given edge, are, in general, not available. In this Rapid Communication, we consider the isotropic "zero-field" six-vertex model at small Δ. We derive the long-distance asymptotic formula of arrow-arrow correlations, which display power law decays with one anomalous exponent. Our method is based on an interacting fermion representation of the six-vertex model and does not use any information obtained from the exact solution.
Some features of integrable lattice models are reviewed for the case of the six-vertex model.By the Bethe ansatz method we derive the free energy of the six-vertex model.Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears.We discuss the connection of the six-vertex model to the conformal field theory with c = 1.We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models.We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group.The graphical approach is emphasized in this review.We explain the meaning of the Yang-Baxter equation by its diagram.Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation.We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models.As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Some features of integrable lattice models are reviewed for the case of the six-vertex model. By the Bethe ansatz method we derive the free energy of the six-vertex model. Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears. We discuss the connection of the six-vertex model to the conformal field theory with c=1. We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models. We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group. The graphical approach is emphasized in this review. We explain the meaning of the Yang-Baxter equation by its diagram. Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation. We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models. As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Some features of integrable lattice models are reviewed for the case of the six-vertex model.By the Bethe ansatz method we derive the free energy of the six-vertex model.Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears.We discuss the connection of the six-vertex model to the conformal field theory with c = 1.We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models.We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group.The graphical approach is emphasized in this review.We explain the meaning of the Yang-Baxter equation by its diagram.Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation.We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models.As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.
Abstract In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Δ<1\Delta <1Δ<1 Δ<1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a=b=1a=b=1a=b=1 a=b=1 and c≥1c\ge 1c≥1 c≥1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.
A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.