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Journal of Physics A Mathematical and General

Dynamic critical behaviour of semi-infinite systems: conformal invariance and mirror theory

Kaoru Ohno, Yutaka Okabe

Published Apr 21, 1986• 1 citation
Publisher

Abstract

The critical dynamics of the d-dimensional semi-infinite systems is studied at bulk critical temperature. As a result of the conformal invariance, the dynamic response and correlation functions depend only on the 'real' distance r, the 'image' distance r and the time t (mirror theory). Assuming the non-conserved order parameter with O(n) symmetry, the authors evaluate the response and correlation functions exactly in the large-n limit and also to first order in epsilon =4-d. The authors' results satisfy the mirror theory for the various types of the phase transitions including the ordinary, special, anisotropic special and extraordinary transitions.

DOI
10.1088/0305-4470/19/6/031

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The critical dynamics of the d-dimensional semi-infinite systems is studied at bulk critical temperature. As a result of the conformal invariance, the dynamic response and correlation functions depend only on the 'real' distance r, the 'image' distance r and the time t (mirror theory). Assuming the non-conserved order parameter with O(n) symmetry, the authors evaluate the response and correlation functions exactly in the large-n limit and also to first order in epsilon =4-d. The authors' results satisfy the mirror theory for the various types of the phase transitions including the ordinary, special, anisotropic special and extraordinary transitions.
The authors discuss the mirror theory of spin systems with a surface. In the semi-infinite system, the 2-point correlation function at bulk critical temperature depends only on the 'real' distance and the 'image' distance, as was shown by the present authors within the framework of 1/n expansion and more recently by Cardy with the use of the conformal invariance. According to this mirror theory, they directly show the scaling relation 2 eta perpendicular to - eta /sub /// eta . They also find a universal combination of amplitudes. They check the mirror theory by means of the epsilon (=4-d) expansion and present the explicit form of the correlation function in real space up to O( epsilon 2 ). The resulting surface critical exponents eta perpendicular to and eta /sub /// coincide with those obtained previously.
The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .
The critical behavior of a continuous spin system in a semi-infinite sample is studied for all values of the extrapolation length $\ensuremath{\lambda}$ using mean-field theory. A new transition, which we have called the extraordinary transition, is found for $\ensuremath{\lambda}<0$ in which the bulk orders at a temperature below the surface ordering temperature. In this paper we have calculated the magnetic susceptibilities $\ensuremath{\chi}(z)$ and $\ensuremath{\chi}(z, z)$ and the correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}})$ at all the phase transitions. We have used these results to compute the various $\ensuremath{\gamma}$ and $\ensuremath{\eta}$ exponents and to study the scaling relations introduced by Binder and Hohenberg.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!
The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.