Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Critical behaviour at an edge

John Cardy

Published Oct 21, 1983• 97 citations
Publisher

Abstract

The critical behaviour of a magnetic system with O(N) spin symmetry, bounded by two (d-1)-dimensional hyperplanes meeting at an angle alpha , is studied within mean field theory and in d=4- epsilon dimensions. New exponents emerge for correlation functions, and magnetisation and susceptibilities, for spins close to the edge. They can be expressed in terms of known bulk and surface exponents, together with a single new edge exponent which depends, however, on the angle alpha . This exponent is computed to first order in epsilon .

DOI
10.1088/0305-4470/16/15/026

Related papers

Critical behaviour of semi-infinite systems

Alan J. Bray, M A Moore

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 .

Journal of Physics A Mathematical and General•Nov 1, 1977•319 citations

Ising model at an edge: a position space renormalisation group approach

Thomas A. Larsson

The critical behaviour of a three-dimensional Ising model, bounded by two plane surfaces meeting at an angle alpha , is studied using the Migdal-Kadanoff and two-terminal cluster renormalisation methods. The authors obtain expressions for the edge fixed point and for the independent edge exponent y he at the various transitions. They find that y he is not completely universal but depends on the angle alpha , and that it shows some unexpected features.

Journal of Physics A Mathematical and General•Jun 21, 1986•6 citations

Critical phenomena in semi-infinite systems. II. Mean-field theory

T. C. Lubensky, Morton H. Rubin

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.

Physical review. B, Solid state•Nov 1, 1975•304 citations

Effects of surface, wedge, corner, and mixed boundary conditions on the local critical behaviour

Zheng-G Wang, A. M. Nemirovsky, Karl F. Freed +1 more

Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.

Journal of Physics A Mathematical and General•Jun 21, 1990•6 citations

Critical behaviour at an edge for the SAW and Ising model

Anthony John Guttmann, Glenn M. Torrie

The authors have studied the behaviour of two- and three-dimensional self avoiding walks confined to a wedge of wedge angle alpha . Series have been obtained and analysed for the (angular dependent) critical exponents characterising various edge susceptibilities. In terms of a general scaling form for the edge free energy, f e approximately t (d-2) nu psi e (ht -y0 nu , h 1 t -y1 nu , h 2 t -y2 nu ), they find for the two-dimensional case the following scaling indices: y0=91/48, y1=3/8, y2( alpha )=-5 pi /8 alpha . They argue that these results are exact, from which follow all exponents for the bulk, surface and edge problem. In three dimensions the authors obtain y0 approximately=2.488, y1=0.65+or-0.02, y2( alpha )= alpha +b pi / alpha where a=0.51+or-0.04, b=-0.847+or-0.017, which, for y2, is precisely of the functional form given by mean-field theory, y2( alpha )= 1 / 2 - pi /2. They argue that a= 1 / 2 for all three-dimensional O(N) models. This simple angular dependence of y2 is different from that suggested by Cardy's one-loop epsilon -expansion, (1983). For the square lattice, they have also studied the case in which the wedge is rotated through an angle of pi /4, and find that the various exponents are unchanged. For the three-dimensional Ising model in a wedge, analogy with the SAW results, plus mean-field results in conjunction with RG and series work yield y0 approximately=2.485, y1=0.71+or-0.02 and y2=a+b pi / alpha with a= 1 / 2 and b=-0.79+or-0.02.

Journal of Physics A Mathematical and General•Dec 21, 1984•59 citations

Corner critical exponents from the mean-field renormalisation group

Joseph O. Indekeu, G. Menu

The previously introduced method of mean-field renormalisation has provided a unified approach to bulk and surface critical behaviour. The method is used for the computation of the critical exponent associated with the order parameter at corners. Applications to the square and triangular Ising model are presented. The angle dependence of the corner critical exponent is qualitatively reproduced. The results for the bulk critical exponents are more accurate than in previously employed mean-field renormalisation schemes.

Journal of Physics A Mathematical and General•May 7, 1988•2 citations
The critical behaviour of a magnetic system with O(N) spin symmetry, bounded by two (d-1)-dimensional hyperplanes meeting at an angle alpha , is studied within mean field theory and in d=4- epsilon dimensions. New exponents emerge for correlation functions, and magnetisation and susceptibilities, for spins close to the edge. They can be expressed in terms of known bulk and surface exponents, together with a single new edge exponent which depends, however, on the angle alpha . This exponent is computed to first order in epsilon .
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.
Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.
The authors have studied the behaviour of two- and three-dimensional self avoiding walks confined to a wedge of wedge angle alpha . Series have been obtained and analysed for the (angular dependent) critical exponents characterising various edge susceptibilities. In terms of a general scaling form for the edge free energy, f e approximately t (d-2) nu psi e (ht -y0 nu , h 1 t -y1 nu , h 2 t -y2 nu ), they find for the two-dimensional case the following scaling indices: y0=91/48, y1=3/8, y2( alpha )=-5 pi /8 alpha . They argue that these results are exact, from which follow all exponents for the bulk, surface and edge problem. In three dimensions the authors obtain y0 approximately=2.488, y1=0.65+or-0.02, y2( alpha )= alpha +b pi / alpha where a=0.51+or-0.04, b=-0.847+or-0.017, which, for y2, is precisely of the functional form given by mean-field theory, y2( alpha )= 1 / 2 - pi /2. They argue that a= 1 / 2 for all three-dimensional O(N) models. This simple angular dependence of y2 is different from that suggested by Cardy's one-loop epsilon -expansion, (1983). For the square lattice, they have also studied the case in which the wedge is rotated through an angle of pi /4, and find that the various exponents are unchanged. For the three-dimensional Ising model in a wedge, analogy with the SAW results, plus mean-field results in conjunction with RG and series work yield y0 approximately=2.485, y1=0.71+or-0.02 and y2=a+b pi / alpha with a= 1 / 2 and b=-0.79+or-0.02.
The previously introduced method of mean-field renormalisation has provided a unified approach to bulk and surface critical behaviour. The method is used for the computation of the critical exponent associated with the order parameter at corners. Applications to the square and triangular Ising model are presented. The angle dependence of the corner critical exponent is qualitatively reproduced. The results for the bulk critical exponents are more accurate than in previously employed mean-field renormalisation schemes.