arXiv:cond-mat/9806175 · Physical review. B, Condensed matter

Edge critical behavior at the surface transition of Ising magnets

Michel Pleimling, Walter Selke

Published Jan 1, 1999• 9 citations

Abstract

Using Monte Carlo techniques, Ising models with ferromagnetic nearest-neighbor interactions on a simple cubic lattice are studied. At the surface transition, the critical exponent ${\ensuremath{\beta}}_{2}$ of the edge magnetization is found to be nonuniversal, depending on the edge and edge-surface couplings, in contrast to the situation at the ordinary transition. Results are compared to those for two-dimensional Ising magnets with chain and ladder defects.

DOI: 10.1103/physrevb.59.65

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arXiv:cond-mat/9806175 · Physical review. B, Condensed matter

Edge critical behavior at the surface transition of Ising magnets

Michel Pleimling, Walter Selke

Published Jan 1, 1999• 9 citations

arXiv HTML PDF

Abstract

Using Monte Carlo techniques, Ising models with ferromagnetic nearest-neighbor interactions on a simple cubic lattice are studied. At the surface transition, the critical exponent

{\ensuremath{\beta}}_{2}

of the edge magnetization is found to be nonuniversal, depending on the edge and edge-surface couplings, in contrast to the situation at the ordinary transition. Results are compared to those for two-dimensional Ising magnets with chain and ladder defects.

DOI: 10.1103/physrevb.59.65

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Michel Pleimling, Walter Selke

arXiv:cond-mat/9908291Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJan 1, 200011 citations

Edge critical exponent for the three-dimensional Ising model by Monte Carlo simulations

K. K. Mon, J. L. Vallés

We present the first Monte Carlo estimate of the edge critical exponent ${\ensuremath{\Delta}}_{2}$ for the three-dimensional nearest-neighbor ferromagnetic Ising model at the ordinary edge transition. The finite-size dependence of the spin-spin correlation function is used to obtain the edge-bulk exponent ${\ensuremath{\eta}}_{0}$,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives ${\ensuremath{\Delta}}_{2}$/${\ensuremath{\nu}}_{b}$=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .We present the first Monte Carlo estimate of the edge critical exponent ${\ensuremath{\Delta}}_{2}$ for the three-dimensional nearest-neighbor ferromagnetic Ising model at the ordinary edge transition. The finite-size dependence of the spin-spin correlation function is used to obtain the edge-bulk exponent ${\ensuremath{\eta}}_{0}$,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives ${\ensuremath{\Delta}}_{2}$/${\ensuremath{\nu}}_{b}$=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .

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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.

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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.

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

Surface critical behavior of the smoothly inhomogeneous Ising model

Theodore W. Burkhardt, Ihnsouk Guim

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large $m$ , $m$ being the distance from the edge. The case $A<0$ of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when $A>0$ . At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large $m$, $m$ being the distance from the edge. The case $A<0$ of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when $A>0$. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.

Physical review. B, Condensed matterJan 1, 198430 citations

Using Monte Carlo techniques, Ising models with ferromagnetic nearest-neighbor interactions on a simple cubic lattice are studied. At the surface transition, the critical exponent

{\ensuremath{\beta}}_{2}

{\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.

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large

m

m

being the distance from the edge. The case

A<0

of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when

A>0

. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.