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, Condensed matter

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

Lawrence G. Dunfield, Jaan Noolandi

Published Sep 1, 1980• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.22.2586

Related papers

Real-space renormalization-group analysis for finite ferromagnetic systems

Lawrence G. Dunfield, Jaan Noolandi

The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) and slabs ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with $n<~8$, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.

Physical review. B, Condensed matter•Nov 1, 1980•9 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

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

Study of the cumulant expansion for renormalization transformations on two- and three- dimensional Ising models

Shin-chieh Hsu, J. D. Gunton

The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.

Physical review. B, Solid state•Mar 1, 1977•25 citations

Real-space dynamic renormalization group. II. Simple examples

Gene F. Mazenko, Michael J. Nolan, Oriol T. Valls

The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent $z=1.70$. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order $z=2.22$, together with a critical coupling ${K}_{c}=0.251$ and a thermal critical index $\ensuremath{\nu}=0.950$.

Physical review. B, Condensed matter•Aug 1, 1980•19 citations
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.
The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) and slabs ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with n< 8n<~8n< 8, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.
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 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.
The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.
The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent z=1.70z=1.70z=1.70. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order z=2.22z=2.22z=2.22, together with a critical coupling Kc=0.251{K}_{c}=0.251Kc​=0.251 and a thermal critical index $\ensuremath{\nu}=0.950$.