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Physical review. B, Condensed matter

Derivation of extended scaling relations between critical exponents in two-dimensional models from the one-dimensional Luttinger model

M. P. M. den Nijs

Published Jun 1, 1981• 175 citations
Publisher

Abstract

The extended scaling relations between the critical exponents of the 8-vertex model can be derived by a mapping of this model onto the Luttinger model. The equivalence of this method to the one that connects the 8-vertex model to the Gaussian model is discussed. The Luttinger model is equivalent to the Gaussian model. Its operators are identified as vortex and spin-wave operators. The spin-wave operator $cos4\ensuremath{\varphi}$ is present in the critical 8-vertex Hamiltonian via an umklapp process. This explains the Kosterlitz-Thouless transition in the 6-vertex model, and resolves questions concerning the validity of the lattice continuum limit in the treatment by Luther and Peschel.

DOI
10.1103/physrevb.23.6111

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The extended scaling relations between the critical exponents of the 8-vertex model can be derived by a mapping of this model onto the Luttinger model. The equivalence of this method to the one that connects the 8-vertex model to the Gaussian model is discussed. The Luttinger model is equivalent to the Gaussian model. Its operators are identified as vortex and spin-wave operators. The spin-wave operator cos4\ensuremathφcos4\ensuremath{\varphi}cos4\ensuremathφ is present in the critical 8-vertex Hamiltonian via an umklapp process. This explains the Kosterlitz-Thouless transition in the 6-vertex model, and resolves questions concerning the validity of the lattice continuum limit in the treatment by Luther and Peschel.
It is widely believed that the critical properties of several planar lattice models, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective Quantum Field Theory obtained as formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones by Kadanoff, [13], and by Luther and Peschel, [16]. 1 Introduction and main results Integrable models in statistical mechanics, like the Ising or the Eight vertex (8V) models in two dimensions, provide conceptual laboratories for the understanding of phase transitions. Integrability is however a rather delicate property requiring very special features, and it is usually lost in more realistic models.
We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.
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 Tc{T}_{c}Tc​ exists below which the susceptibility is infinite but without long-range order. At Tc{T}_{c}Tc​, the correlation-function exponent $\ensuremath{\eta}$ takes the values 181\sqrt{8}18​, 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 Tc{T}_{c}Tc​. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.
We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for ${U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|$ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.