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Journal of Physics A Mathematical and General

Mirror theory of spin systems with a surface

Kaoru Ohno, Yutaka Okabe

Published Jun 21, 1985• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/18/9/012

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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 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 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 .
We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings JS{\mathit{J}}_{\mathit{S}}JS​ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For JS{\mathit{J}}_{\mathit{S}}JS​\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For JS{\mathit{J}}_{\mathit{S}}JS​=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.
The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for nnn-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.