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Journal of Physics A Mathematical and General

Two-dimensional semi-infinite multicritical Ising models

Per Fröjdh

Published Jan 7, 1994• 0 citations
Publisher

Abstract

The order-parameter correlation functions of the Z 2 -invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions.

DOI
10.1088/0305-4470/27/1/004

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The order-parameter correlation functions of the Z 2 -invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions.
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Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of $p\ifmmode\times\else\texttimes\fi{}1$ commensurate ordered phases for all integers $p\ensuremath{\ge}2$. The other sequence appears to consist of "generic" multicritical points terminating lines of nnn-phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.
We show how the apparent power-law scaling of classical two-dimenional structure factors in bounded domains at criticality is related to fixed and free boundary conditions on the order parameter. This enables us to predict the behavior of the corresponding apparent scaling dimensions based on a simple symmetry analysis. With the use of correlation functions from conformal field theory, our predictions are confirmed with exact calculations for multicritical Ising models. Effects of random boundary fields are also discussed.
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The authors consider m two-dimensional semi-infinite planes of Ising spins joined together through surface spins and study the critical behaviour near to the junction. The m=0 limit of the model-according to the replica trick-corresponds to the semi-infinite Ising model in the presence of a random surface field (RSFI). Using conformal mapping, second-order perturbation expansion around the weak- and strong-coupled planes limits and a differential renormalization group, they show that the surface critical behaviour of the RSFI model is described by Ising critical exponents with logarithmic corrections to scaling, while at multiple junctions (m>2) the transition is of first-order. There is a spontaneous junction magnetization at the bulk critical point.