Physical review. B, Condensed matter

Critical behavior of the surface-layer magnetization at the extraordinary transition in the three-dimensional Ising model

David Paul Landau, Kurt Binder

Published Mar 1, 1990• 52 citations

Abstract

We have used a vectorized multispin-coding Monte Carlo method to determine the behavior of the surface-layer magnetization ${\mathit{m}}_{1}$ at the bulk transition in a simple-cubic Ising film with strongly enhanced surface coupling, i.e., at the extraordinary transition. In contrast to recent renormalization-group calculations we find no evidence for a discontinuous slope in the temperature dependence of ${\mathit{m}}_{1}$; the data are consistent with a free-energy-like (T-${\mathit{T}}_{\mathit{c}}$${)}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ behavior plus background terms.

DOI: 10.1103/physrevb.41.4786

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

David Paul Landau, Kurt Binder

Physical review. B, Condensed matterMar 1, 1990146 citations

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

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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.

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

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

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Martin Hasenbusch B

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Theodore W. Burkhardt, Erich Eisenriegler

Physical review. B, Solid stateOct 1, 1977124 citations

Physical review. B, Condensed matter

Critical behavior of the surface-layer magnetization at the extraordinary transition in the three-dimensional Ising model

David Paul Landau, Kurt Binder

Published Mar 1, 1990• 52 citations

Publisher

Abstract

We have used a vectorized multispin-coding Monte Carlo method to determine the behavior of the surface-layer magnetization

{\mathit{m}}_{1}

at the bulk transition in a simple-cubic Ising film with strongly enhanced surface coupling, i.e., at the extraordinary transition. In contrast to recent renormalization-group calculations we find no evidence for a discontinuous slope in the temperature dependence of

{\mathit{m}}_{1}

; the data are consistent with a free-energy-like (T-

{\mathit{T}}_{\mathit{c}}$${)}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}

behavior plus background terms.

DOI: 10.1103/physrevb.41.4786

Related papers

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.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 matterMar 1, 1990146 citations

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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 $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.

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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.

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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

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.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 stateOct 1, 1977124 citations

We have used a vectorized multispin-coding Monte Carlo method to determine the behavior of the surface-layer magnetization

{\mathit{m}}_{1}

{\mathit{m}}_{1}

; the data are consistent with a free-energy-like (T-

{\mathit{T}}_{\mathit{c}}$${)}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}

behavior plus background terms.

{\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.