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

Monte Carlo studies of the critical free energies for the three-dimensional Ising model with surfaces, edges, and corners

Pik-Yin Lai, K. K. Mon

Published Jun 1, 1989• 5 citations
Publisher

Abstract

We present an extensive Monte Carlo study of the three-dimensional Ising model with free surfaces, edges, and corners at bulk criticality. Finite-size scaling properties of the excess free energies of free surfaces, edges, and corners are estimated by use of the recently proposed applications of the parametric integration method. The corner excess free energies are obtained for the first time and are consistent with the logarithmic finite-size scaling predictions. Results of the simplecubic and body-centered-cubic lattices provide support for the proposed universality of the finite-size scaling amplitudes for the excess free energies of free surfaces, edges, and corners.

DOI
10.1103/physrevb.39.12407

Related papers

Surface free energies of three-dimensional Ising models and universality of finite-size scaling amplitudes by direct Monte Carlo sampling

K. K. Mon, M. P. Nightingale

Direct Monte Carlo sampling is employed to obtain estimates of excess surface free energies of three-dimensional Ising models at criticality. Results for simple and body-centered-cubic lattices provide strong evidence for the universality of finite-size scaling amplitudes and in particular the related interaction per unit area, mediated via the incipient long-ranged correlations, of two free surfaces a finite distance apart.

Physical review. B, Condensed matter•May 1, 1985•16 citations

Monte Carlo studies of universal critical amplitudes for the three-dimensional Ising model: Correlation length and renormalized coupling

Pik-Yin Lai, K. K. Mon

We report Monte Carlo studies of the finite-size second-moment correlation length and renormalized coupling at criticality for the three-dimensional Ising model on body-centered-cubic lattices. The related finite-size scaling amplitudes are estimated and compared with the simple-cubic lattice results of recent extensive simulations. The amplitudes agreed within statistical uncertainty and are consistent with finite-size-scaling arguments for universality.

Physical review. B, Condensed matter•Dec 1, 1989•6 citations

Monte Carlo estimates of the corner critical exponents of the three-dimensional Ising model

Pik-Yin Lai, K. K. Mon

The authors use Monte Carlo simulations to estimate for the first time the corner critical exponents for the three-dimensional Ising model with a free boundary condition at bulk criticality. Finite-size dependences of the spin-spin correlation functions are used to obtain η bulk-corner = 1.75 ± 0.2.

Journal of Physics A Mathematical and General•Dec 7, 1989•2 citations

Monte Carlo studies of critical free energies and the simple-cubic Ising model

K. K. Mon

This paper addresses the limitation for routine applications of the proposed Monte Carlo method of direct calculation of absolute free energy by sampling of finite-size dependences and discusses new extensions. The need for multistage sampling is removed and the method is now simpler to apply in practice. To exhibit this, the critical free-energy density for the simple-cubic nearest-neighbor ferromagnetic Ising model is studied for a wide range of lattice sizes, up to 32\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}32. The finite-size scaling is demonstrated and the scaling amplitude calculated with higher accuracy than before. These results indicate the possibility that very accurate free-energy density and related quantities such as surface free energy, can be routinely obtained for lattice models from Monte Carlo simulations, even at criticality.

Physical review. B, Condensed matter•Jan 1, 1989•25 citations

Monte Carlo study of surface phase transitions in the three-dimensional Ising model

David Paul Landau, Kurt Binder

We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling ${\mathit{J}}_{\mathit{s}}$. The surface-bulk multicritical point is located with improved precision, ${\mathit{J}}_{\mathit{s}}$/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}}_{1}^{\mathit{m}}$=1.41\ifmmode\pm\else\textpm\fi{}0.14, ${\ensuremath{\gamma}}_{11}^{\mathit{m}}$=0.96\ifmmode\pm\else\textpm\fi{}0.09, and along with critical amplitude ratios these are compared with theoretically predicted values.

Physical review. B, Condensed matter•Mar 1, 1990•146 citations

Finite-size behavior of the simple-cubic Ising lattice

David Paul Landau

A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large $N$ as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.

Physical review. B, Solid state•Jul 1, 1976•169 citations
We present an extensive Monte Carlo study of the three-dimensional Ising model with free surfaces, edges, and corners at bulk criticality. Finite-size scaling properties of the excess free energies of free surfaces, edges, and corners are estimated by use of the recently proposed applications of the parametric integration method. The corner excess free energies are obtained for the first time and are consistent with the logarithmic finite-size scaling predictions. Results of the simplecubic and body-centered-cubic lattices provide support for the proposed universality of the finite-size scaling amplitudes for the excess free energies of free surfaces, edges, and corners.
This paper addresses the limitation for routine applications of the proposed Monte Carlo method of direct calculation of absolute free energy by sampling of finite-size dependences and discusses new extensions. The need for multistage sampling is removed and the method is now simpler to apply in practice. To exhibit this, the critical free-energy density for the simple-cubic nearest-neighbor ferromagnetic Ising model is studied for a wide range of lattice sizes, up to 32\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}32. The finite-size scaling is demonstrated and the scaling amplitude calculated with higher accuracy than before. These results indicate the possibility that very accurate free-energy density and related quantities such as surface free energy, can be routinely obtained for lattice models from Monte Carlo simulations, even at criticality.
We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling Js{\mathit{J}}_{\mathit{s}}Js​. The surface-bulk multicritical point is located with improved precision, Js{\mathit{J}}_{\mathit{s}}Js​/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}}_{1}^{\mathit{m}}$=1.41\ifmmode\pm\else\textpm\fi{}0.14, ${\ensuremath{\gamma}}_{11}^{\mathit{m}}$=0.96\ifmmode\pm\else\textpm\fi{}0.09, and along with critical amplitude ratios these are compared with theoretically predicted values.
A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large NNN as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.