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Extended scaling relations for planar lattice models

G. Benfatto, Pierluigi Falco, Vieri Mastropietro, Via Della Ricerca Scientifica

Published Jan 1, 2009• 21 citations
OpenAlex

Abstract

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.

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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 present the first rigorous derivation of a number of universal relations for a class of models with continuously varying indices (among which are interacting planar Ising models, quantum spin chains and 1D Fermi systems), for which an exact solution is not known, except in a few special cases. Most of these formulas were conjectured by Luther and Peschel, Kadanoff, and Haldane, but only checked in the special solvable models; one of them, related to the anisotropic Ashkin-Teller model, is novel.
believed to be true in a wide class of systems with continuously varying indices, among which are interacting planar Ising models, vertex or Ashkin-Teller models, quantum spin chains and 1D Fermi systems; by such relations one can predict several quantities in terms of a few measurable parameters without relying on the specific microscopic details. The validity of such relations can be checked in special solvable models but, despite several attempts, the proof of their general validity was up to now an open problem. A rigorous derivation of several of such relations (for solvable and not solvable models and without any use of exact solutions) has been recently obtained in [8] and [11] through Renormalization Group methods. The proof is based on the representation in terms of Grassmann integrals and the validity of the Adler-Bardeen property of the non renormalization of the quantum anomalies in the asymptotic Ward identities. Gauge invariance is exact only in the scaling limit but the lattice corrections can be rigorously taken into account.
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.
Lines of critical points which emerge from the two-dimensional planar model's multicritical point are analyzed. These lines seem to be in the same universality class as both the Ashkin-Teller and the eight-vertex (8V) models' critical lines, with the multicritical point being isomorphic with the 8V model at both the points $tanh2\ensuremath{\lambda}=\frac{\sqrt{2}}{2} \mathrm{and} \ensuremath{-}1$. Expansions of critical indices about these values of $\ensuremath{\lambda}$ are performed. If this analysis is right, then this work permits the evaluation of a whole class of interesting correlation functions at the critical point of the FFF model and that of the four-state Potts model.