Physical review. B, Condensed matter

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

David Paul Landau, Kurt Binder

Published Mar 1, 1990• 146 citations

Abstract

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.

DOI: 10.1103/physrevb.41.4633

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Crossover Scaling and Critical Behavior at the "Surface-Bulk" Multicritical Point

Kurt Binder, D. P. Landau

Physical Review LettersJan 30, 1984288 citations

Monte Carlo Study of the Surface Critical Phenomena of the Ising Model

Macoto Kikuchi, Yutaka Okabe

Monte Carlo technique is applied to the three-dimensional Ising model with free surface. The critical behavior of the layer magnetization, the layer susceptibility and the local susceptibility near the ordinary transition point is investigated. The universal properties of the surface critical exponents and the surface critical amplitude ratios are confirmed. The scaling of magnetization profile is studied, and the bulk amplitude ratio for the correlation length ξ0h / ξ0- is calculated.

Progress of Theoretical PhysicsJan 1, 198519 citations

Magnetic-Phase Transitions of Ising Surfaces with Modified Surface-Bulk Coupling: a Monte Carlo Study

Michele Vendruscolo, Mauro Rovere, Annalisa Fasolino

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Surface and bulk transitions in three-dimensional O(n) models

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Critical parameters for thed=3 Ising model in a film geometry

C. Ruge, Friedrich Wagner

Physical review. B, Condensed matterAug 1, 199529 citations

Monte Carlo study of surface critical phenomena: The special point

Martin Hasenbusch B

arXiv:1108.2425Physical Review BOct 7, 201131 citations

Physical review. B, Condensed matter

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

David Paul Landau, Kurt Binder

Published Mar 1, 1990• 146 citations

Publisher

Abstract

{\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.

DOI: 10.1103/physrevb.41.4633

Related papers

Crossover Scaling and Critical Behavior at the "Surface-Bulk" Multicritical Point

Kurt Binder, D. P. Landau

With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.

Physical Review LettersJan 30, 1984288 citations

Monte Carlo Study of the Surface Critical Phenomena of the Ising Model

Macoto Kikuchi, Yutaka Okabe

Progress of Theoretical PhysicsJan 1, 198519 citations

Magnetic-Phase Transitions of Ising Surfaces with Modified Surface-Bulk Coupling: a Monte Carlo Study

Michele Vendruscolo, Mauro Rovere, Annalisa Fasolino

We present the Monte Carlo phase diagram of Ising surfaces with intrasurface ( J s ) and surface-bulk ( J 1 ) interactions modified with respect to the bulk exchange interaction J . The new parameter J 1 strongly affects the position of the Binder-Landau multi critical point and the crossover from three-dimensional to two-dimensional behaviour. The surface critical exponents β 1 and ϕ are determined by crossover scaling analysis. A detailed comparison with the results of the mean-field equations is also presented. The Monte Carlo phase diagram shows that stronger modifications of the surface couplings, with respect to those obtained by mean field, are needed to produce an ordered surface phase on a paramagnetic bulk. This is particularly relevant at low values of J s ; in the Monte Carlo simulation the surface is disordered for all values of J 1 when J s < (0.4 ÷ 0.5) J , while the mean field predicts that the ordered phase is present even in the limit J s → 0.We present the Monte Carlo phase diagram of Ising surfaces with intrasurface ( J s ) and surface-bulk ( J 1 ) interactions modified with respect to the bulk exchange interaction J . The new parameter J 1 strongly affects the position of the Binder-Landau multi critical point and the crossover from three-dimensional to two-dimensional behaviour. The surface critical exponents β 1 and ϕ are determined by crossover scaling analysis. A detailed comparison with the results of the mean-field equations is also presented. The Monte Carlo phase diagram shows that stronger modifications of the surface couplings, with respect to those obtained by mean field, are needed to produce an ordered surface phase on a paramagnetic bulk. This is particularly relevant at low values of J s ; in the Monte Carlo simulation the surface is disordered for all values of J 1 when J s < (0.4 ÷ 0.5) J , while the mean field predicts that the ordered phase is present even in the limit J s → 0.

Europhysics Letters (EPL)Nov 15, 199214 citations

Surface and bulk transitions in three-dimensional O(n) models

Youjin Deng, Henk W. J. Blöte, M. P. Nightingale

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the $\mathrm{O}(n)$ models on the simple-cubic lattice with $n=1$ , 2, and 3, i.e., the Ising, $XY$ , and Heisenberg models. For the critical couplings we find ${K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659$ (10) and ${K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003$ (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be ${y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)$ , 0.781 (2), and 0.813 (2) for $n=1$ , 2, and 3, respectively. Then we vary the surface coupling ${K}_{1}$ and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are ${y}_{t1}^{(\mathrm{s})}=0.715$ (1) and ${y}_{h1}^{(\mathrm{s})}=1.636$ (1) for the Ising model, and ${y}_{t1}^{(\mathrm{s})}=0.608$ (4) and ${y}_{h1}^{(\mathrm{s})}=1.675$ (1) for the $XY$ model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the $\mathrm{O}(n)$ models on the simple-cubic lattice with $n=1$, 2, and 3, i.e., the Ising, $XY$, and Heisenberg models. For the critical couplings we find ${K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659$ (10) and ${K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003$ (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be ${y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)$, 0.781 (2), and 0.813 (2) for $n=1$, 2, and 3, respectively. Then we vary the surface coupling ${K}_{1}$ and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are ${y}_{t1}^{(\mathrm{s})}=0.715$ (1) and ${y}_{h1}^{(\mathrm{s})}=1.636$ (1) for the Ising model, and ${y}_{t1}^{(\mathrm{s})}=0.608$ (4) and ${y}_{h1}^{(\mathrm{s})}=1.675$ (1) for the $XY$ model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.

arXiv:cond-mat/0504173Physical Review EJul 27, 200586 citations

Critical parameters for thed=3 Ising model in a film geometry

C. Ruge, Friedrich Wagner

We simulated the critical behavior of the d=3 Ising model with surfaces with a cluster Monte Carlo method at the multicritical point and at the ordinary transition. The surface exponents ${\mathrm{\ensuremath{\beta}}}_{1}$ have been determined using finite-size scaling. The errors of the resulting values ${\mathrm{\ensuremath{\beta}}}_{1}^{\mathit{m}}$=0.2375(15) and ${\mathrm{\ensuremath{\beta}}}_{1}^{0}$=0.807(4) are an order of magnitude smaller than in previous determinations. Whereas the values of the exponents agree with the prediction of the \ensuremath{\epsilon} expansion, the universal ratios of susceptibility and magnetization amplitudes at the special transition are at variance with those predictions.We simulated the critical behavior of the d=3 Ising model with surfaces with a cluster Monte Carlo method at the multicritical point and at the ordinary transition. The surface exponents ${\mathrm{\ensuremath{\beta}}}_{1}$ have been determined using finite-size scaling. The errors of the resulting values ${\mathrm{\ensuremath{\beta}}}_{1}^{\mathit{m}}$=0.2375(15) and ${\mathrm{\ensuremath{\beta}}}_{1}^{0}$=0.807(4) are an order of magnitude smaller than in previous determinations. Whereas the values of the exponents agree with the prediction of the \ensuremath{\epsilon} expansion, the universal ratios of susceptibility and magnetization amplitudes at the special transition are at variance with those predictions.

Physical review. B, Condensed matterAug 1, 199529 citations

Monte Carlo study of surface critical phenomena: The special point

Martin Hasenbusch B

We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find ${y}_{{t}_{1}}=0.718(2)$ and ${y}_{{h}_{1}}=1.6465(6)$ for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find ${y}_{{t}_{1}}=0.718(2)$ and ${y}_{{h}_{1}}=1.6465(6)$ for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.

arXiv:1108.2425Physical Review BOct 7, 201131 citations

{\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.

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the

\mathrm{O}(n)

models on the simple-cubic lattice with

n=1

, 2, and 3, i.e., the Ising,

XY

, and Heisenberg models. For the critical couplings we find

{K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659

(10) and

{K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003

(2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be

{y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)

, 0.781 (2), and 0.813 (2) for

n=1

, 2, and 3, respectively. Then we vary the surface coupling

{K}_{1}

and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are

{y}_{t1}^{(\mathrm{s})}=0.715

(1) and

{y}_{h1}^{(\mathrm{s})}=1.636

(1) for the Ising model, and

{y}_{t1}^{(\mathrm{s})}=0.608

(4) and

{y}_{h1}^{(\mathrm{s})}=1.675

(1) for the

XY

model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.

{y}_{{t}_{1}}=0.718(2)

and

{y}_{{h}_{1}}=1.6465(6)

for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.