Critical properties of a two-dimensional planar model

Abstract

Recently, critical exponents in the continuum Baxter model have been calculated using an equivalence of this model to the Luttinger model of a one-dimensional spinless fermion gas. We discuss here the extension of this method to the two-dimensional planar model, using the Stoeckly-Scalapino representation of the transfer matrix. This representation is shown to be equivalent to the spin-1/2 fermion gas in one dimension, which further separates into a Luttinger model and the quantum-mechanical sine-Gordon equation. We emphasize the role that soliton bound states play in determining the critical properties. We find that a critical temperature ${T}_{c}$ exists below which the susceptibility is infinite but without long-range order. At ${T}_{c}$, the correlation-function exponent $\ensuremath{\eta}$ takes the values $1\sqrt{8}$, and decreases as the temperature is lowered, consistent with low-temperature results. As in the Baxter-model calculations, we argue that these results are exact for the continuum limit of the transfer matrix, and therefore provide a solution for the asymptotic properties near ${T}_{c}$. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.