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Journal of Physics A Mathematical and General

Parafermions and surface exponents of self-dual Z(N) spin models from conformal invariance

Francisco Castilho Alcaraz

Published Jul 11, 1987• 17 citations
Publisher

Abstract

The author studies the critical behaviour of a family of self-dual two-dimensional spin models with Z(N) symmetry (N<or=8) in its Hamiltonian formulation. Using the relations between the mass-gap amplitudes of the Hamiltonian in a finite strip and the critical exponents he estimates the anomalous dimensions of the parafermions occurring in the underlying field theory as well as the surface exponents.

DOI
10.1088/0305-4470/20/10/001

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The author studies the critical behaviour of a family of self-dual two-dimensional spin models with Z(N) symmetry (N<or=8) in its Hamiltonian formulation. Using the relations between the mass-gap amplitudes of the Hamiltonian in a finite strip and the critical exponents he estimates the anomalous dimensions of the parafermions occurring in the underlying field theory as well as the surface exponents.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.
A quantum spin model with three-spin coupling is studied by finite-lattice methods. The critical indices nu and alpha are obtained by standard finite-size scaling analysis. Conformal invariance is used to estimate the exponents nu and eta , the central charge or conformal anomaly c and the anomalous dimension of a spin 1/4 parafermion. The authors conclude that the model belongs to the same universality class as the four-state Potts and Baxter-Wu models, although the convergence of the various finite lattice estimators is extremely slow.
We calculated the spectral properties of two related families of non-Hermitian free-particle quantum chains with NNN-multispin interactions (N=2,3,…N=2,3,\ldotsN=2,3,…). The first family have a Z(N)Z(N)Z(N) symmetry and are described by free parafermions. The second one has a U(1)U(1)U(1) symmetry and are generalizations of XXXXXX quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent z=1z=1z=1. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase while the XXXXXX multispin models show multiple phases with distinct conformal central charges in the periodic case. The critical exponents of the models are calculated for N=3,4,5N=3,4,5N=3,4,5 and 666.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
The spinor operator in the two-dimensional Ising model can be readily generalized to other self-dual models as the product of the order parameter and its dual image, the disorder operator. Recently, the exponent of this and other operators in the qqq-state Potts model received renewed attention, as the theory of conformal invariance produces complete lists of critical exponents for some of these models. In this paper we calculate the critical and tricritical indices of the spinor operator in the two-dimensional qqq-state Potts model, via a well-known mapping into a solid-on-solid model. Our results give a physical identification for the exponents predicted by the conformal theory.