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

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

Thomas A. Larsson

Published Jun 21, 1986• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/19/9/039

Related papers

Critical behavior of a semi-infinite Ising model with an edge

V. K. Saxena

A semi-infinite Ising model on a simple-cubic lattice, bounded by two perpendicular planes which meet at an edge, is studied with the Migdal-Kadanoff renormalization-group approach. It is found that the edge spins can order at a temperature higher than the bulk transition temperature if one of the surfaces is also ordered, whereas no edge ordering is possible if none of the surfaces nor the bulk are ordered. A complete phase diagram is presented in the parameter space of the model considered. Correlation-length critical exponents \ensuremath{\nu} and the crossover exponent \ensuremath{\varphi} are calculated and various universality classes are established.

Physical review. B, Condensed matter•Mar 1, 1987•5 citations

Kadanoff's variational renormalisation group method: the Ising model on the square and triangular lattices

B. W. Southern

The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.

Journal of Physics A Mathematical and General•Jan 1, 1978•5 citations

A real-space renormalisation group study of a semi-infinite Ising model

H. C. Tseng, H H Chen

A modified Migdal-Kadanoff renormalisation group scheme, called the cluster-decimation approximation, is generalised and applied to study surface critical properties of the Ising model with a free surface. The authors determine the fixed points and calculate surface critical exponents of this model. Comparisons of their results with other real-space renormalisation group estimates are given.

Journal of Physics A Mathematical and General•Aug 21, 1989•0 citations

Critical properties of the site-diluted Ising ferromagnet

Julia M. Yeomans, Robin B. Stinchcombe

The site-diluted Ising ferromagnet is treated by a position space renormalisation group method. The method is briefly outlined and results obtained by considering three different two-dimensional clusters are presented. The results include the dependence of transition temperature T c on concentration p and the thermal exponents for the percolation and Ising fixed points. Corresponding results are also presented for the bond-diluted Ising ferromagnet.

Journal of Physics C Solid State Physics•Jul 14, 1978•28 citations

Critical behaviour at an edge

John Cardy

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 .

Journal of Physics A Mathematical and General•Oct 21, 1983•97 citations

Renormalization-group approach to the Ising model with a free surface

Theodore W. Burkhardt, Erich Eisenriegler

We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.

Physical review. B, Solid state•Oct 1, 1977•124 citations
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.
A semi-infinite Ising model on a simple-cubic lattice, bounded by two perpendicular planes which meet at an edge, is studied with the Migdal-Kadanoff renormalization-group approach. It is found that the edge spins can order at a temperature higher than the bulk transition temperature if one of the surfaces is also ordered, whereas no edge ordering is possible if none of the surfaces nor the bulk are ordered. A complete phase diagram is presented in the parameter space of the model considered. Correlation-length critical exponents \ensuremath{\nu} and the crossover exponent \ensuremath{\varphi} are calculated and various universality classes are established.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
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 .
We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.