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Journal of the Physical Society of Japan

Integral Representation of the Internal Energy of theFModel

Таичиро Такаги

Published Apr 15, 1996• 3 citations
Publisher

Abstract

We give an integral representation of the internal energy of the F model, a special case of the anti-ferroelectric six-vertex model. This representation is obtained naturally as a trace function of the vertex operators. We also take the critical limit of our expression and find that it reproduces Lieb's result of the internal energy at T c .

DOI
10.1143/jpsj.65.921

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We give an integral representation of the internal energy of the F model, a special case of the anti-ferroelectric six-vertex model. This representation is obtained naturally as a trace function of the vertex operators. We also take the critical limit of our expression and find that it reproduces Lieb's result of the internal energy at T c .
We study exact solutions of the four-point correlation functions around a vertex of the anti-ferroelectric six-vertex model. After showing that an expression for the correlations is readily obtained by differentiating the known exact solution of the free energy, we present a new expression for the correlations derived through the recently developed vertex operator approach. It has the form of a double contour integral of a meromorphic function. To give a new verification of the vertex operator approach beyond the case of the one-point function, we prove the equality of the two expressions by calculating the contour integrals.
We study analytic expressions for the averaged fraction of each vertex type of the six-vertex model extensively. Upon the critical line the expressions of the quantity in the anti-ferroelectric phase and in the gapless phase get simplified. This enables us to see more easily the equivalence of the two quantities of which one is derived by differentiating the free energy, and the other is by calculating a trace of the vertex operators in the theory of a quantum affine algebra. We also prove an extra liner relation satisfied by the fractions on the critical line.
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.
We perform a numerical study of the F model with domain-wall boundary conditions. Various exact results are known for this particular case of the six-vertex model, including closed expressions for the partition function for any system size as well as its asymptotics and leading finite-size corrections. To complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size, up to L=512. We compare the energy to its exactly known large-L asymptotics. We investigate the model's infinite-order phase transition by means of finite-size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with Duine and Barkema. In addition we analyze local properties of the model. Our data are perfectly consistent with analytical expressions for the arctic curves. We investigate the structure inside the temperate region of the lattice, confirming the oscillations in vertex densities that were first observed by Syljuåsen and Zvonarev and recently studied by Lyberg et al. We point out "(anti)ferroelectric" oscillations close to the corresponding frozen regions as well as "higher-order" oscillations forming an intricate pattern with saddle-point-like features.