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

Physical review. B, Solid state

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

Theodore W. Burkhardt, Erich Eisenriegler

Published Oct 1, 1977• 124 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.16.3213

Related papers

Renormalization-group calculation of the critical properties of a free magnetic surface

Nenad M. Svrakic, Michael Wortis

A semi-infinite Ising model is studied by renormalization-group techniques in the position-space formulation due to Niemeijer and van Leeuwen. We show how to set up the calculation of thermodynamic properties (both critical and noncritical) of a sample with a free surface by iteration of a set of recursion relations for spatially inhomogeneous couplings and magnetic fields. Surface and bulk properties are clearly distinguished. The method is illustrated by a calculation (using a two-cell cluster approximation) of the magnetic-surface critical exponent in two dimensions, where exact results are available for comparison.

Physical review. B, Solid state•Jan 1, 1977•67 citations

Renormalization-group approach to surface critical behavior in the Ising model

Theodore W. Burkhardt, Erich Eisenriegler

A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The $d=2$ square lattice and the $d=3$ bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for $d=3$ are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.

Physical review. B, Condensed matter•Jan 1, 1978•41 citations

Anisotropic Heisenberg surface on semi-infinite Ising ferromagnet : renormalization group treatment

U.M.S. Costa, Ananias M. Mariz, Constantino Tsallis

We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.

Journal de Physique Lettres•Jan 1, 1985•5 citations

Renormalization-group approach to surface critical behavior in the semi-infinite spin- 32 Blume-Emery-Griffiths model

A. Bakchich, M. El Bouziani

Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-$\frac{3}{2}$ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.

Physical review. B, Condensed matter•Jan 19, 2001•17 citations

Mean-field renormalization-group approaches to the surface on a semi-infinite Ising model with a random surface field

Gang Sun

We study the critical behavior of the surface on a semi-infinite simple-cubic-lattice Ising model with a bimodal random surface field by a large-cell mean-field renormalization-group method. Our results show that a surface ferromagnetic phase exists at weak random fields above the bulk critical temperature, and no signs of the tricritical point are found until the surface critical temperature decreases to that of the bulk.

Physical review. B, Condensed matter•Feb 1, 1995•1 citation

Real-space renormalization-group transformation for semi-infinite Ising systems

Lawrence G. Dunfield, Jaan Noolandi

We use a real-space renormalization transformation with a cumulant expansion, modified to include free-surface boundary conditions, to study the critical behavior of semi-infinite ferromagnetic systems. We present first-order calculations for the Ising model on the semi-infinite square and simple cubic lattices. Our results serve as the basis of a critical discussion of previous real-space and momentum-space renormalization-group calculations.

Physical review. B, Condensed matter•Sep 1, 1980•12 citations
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.
A semi-infinite Ising model is studied by renormalization-group techniques in the position-space formulation due to Niemeijer and van Leeuwen. We show how to set up the calculation of thermodynamic properties (both critical and noncritical) of a sample with a free surface by iteration of a set of recursion relations for spatially inhomogeneous couplings and magnetic fields. Surface and bulk properties are clearly distinguished. The method is illustrated by a calculation (using a two-cell cluster approximation) of the magnetic-surface critical exponent in two dimensions, where exact results are available for comparison.
A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for d=3d=3d=3 are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.
We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.
Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-32\frac{3}{2}23​ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.