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arXiv:cond-mat/0006448 · Physical review. B, Condensed matter

Surface scaling behavior of isotropic Heisenberg systems: Critical exponents, structure factor, and profiles

M. Krech

Published Sep 1, 2000• 26 citations
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Abstract

The surface scaling behavior of classical isotropic Heisenberg magnets is investigated by Monte Carlo methods in $d=3$ dimensions for various values of the surface-to-bulk coupling ratio ${J}_{1}/J.$ For ${J}_{1}/J<~1.0,$ critical behavior according to the ordinary surface universality class is found. Estimates for magnetic surface exponents are presented and compared to older estimates and their theoretical counterparts. For ${J}_{1}/J>~2.0,$ scaling is still valid with effective exponents which depend on ${J}_{1}/J.$ The surface structure factor ${S}_{1}(p,L)$ is investigated at bulk criticality as a function of the momentum transfer p parallel to the surface and the system size L. For ${J}_{1}/J<~1.0$ and ${J}_{1}/J>~2.0$ the full p dependence of ${S}_{1}(p,L)$ can be captured by generalized shape functions to a remarkable accuracy. Profiles of the magnetization and the energy density also confirm scaling, where for ${J}_{1}/J<~1.0$ the ordinary surface universality class is recovered and for ${J}_{1}/J>~2.0,$ scaling with ${J}_{1}/J$ dependent exponents is found. For ${J}_{1}/J=1.5$ the system displays a striking crossover behavior from spurious long-range surface order to the ordinary surface universality class. For ${J}_{1}/J>~2.0$ the effective scaling laws must be interpreted as nonasymptotic and the value ${J}_{1}/J=1.5$ marks a crossover regime, in which the crossover from the nonasymptotic to the asymptotic (ordinary) surface scaling behavior can be resolved within numerically attainable system sizes.

DOI
10.1103/physrevb.62.6360

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The surface scaling behavior of classical isotropic Heisenberg magnets is investigated by Monte Carlo methods in d=3d=3d=3 dimensions for various values of the surface-to-bulk coupling ratio J1/J.{J}_{1}/J.J1​/J. For J1/J< 1.0,{J}_{1}/J<~1.0,J1​/J< 1.0, critical behavior according to the ordinary surface universality class is found. Estimates for magnetic surface exponents are presented and compared to older estimates and their theoretical counterparts. For J1/J> 2.0,{J}_{1}/J>~2.0,J1​/J> 2.0, scaling is still valid with effective exponents which depend on J1/J.{J}_{1}/J.J1​/J. The surface structure factor S1(p,L){S}_{1}(p,L)S1​(p,L) is investigated at bulk criticality as a function of the momentum transfer p parallel to the surface and the system size L. For J1/J< 1.0{J}_{1}/J<~1.0J1​/J< 1.0 and J1/J> 2.0{J}_{1}/J>~2.0J1​/J> 2.0 the full p dependence of S1(p,L){S}_{1}(p,L)S1​(p,L) can be captured by generalized shape functions to a remarkable accuracy. Profiles of the magnetization and the energy density also confirm scaling, where for J1/J< 1.0{J}_{1}/J<~1.0J1​/J< 1.0 the ordinary surface universality class is recovered and for J1/J> 2.0,{J}_{1}/J>~2.0,J1​/J> 2.0, scaling with J1/J{J}_{1}/JJ1​/J dependent exponents is found. For J1/J=1.5{J}_{1}/J=1.5J1​/J=1.5 the system displays a striking crossover behavior from spurious long-range surface order to the ordinary surface universality class. For J1/J> 2.0{J}_{1}/J>~2.0J1​/J> 2.0 the effective scaling laws must be interpreted as nonasymptotic and the value J1/J=1.5{J}_{1}/J=1.5J1​/J=1.5 marks a crossover regime, in which the crossover from the nonasymptotic to the asymptotic (ordinary) surface scaling behavior can be resolved within numerically attainable system sizes.
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.
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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 $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
Motivated by recent experimental activities on surface critical phenomena, we present a detailed theoretical study of the near-surface behavior of the local order parameter m(z) in Ising-like spin systems. Special attention is paid to the crossover regime between “ordinary” and “normal” transition in the three-dimensional semi-infinite Ising model, where a finite magnetic field H1 is imposed on the surface which itself exhibits a reduced tendency to order spontaneously. As the theoretical foundation, the spatial behavior of m(z) is discussed by means of phenomenological scaling arguments, and a finite-size scaling analysis is performed. Then we present Monte Carlo results for m(z) obtained with the Swendsen-Wang algorithm. In particular the sharp power-law increase of the magnetization, m(z) ≈ H1z1−η∼ord, predicted for a smallH1 by previous work of the authors, is corroborated by the numerical results. The relevance of these findings for experiments on critical adsorption in systems where a small effective surface field occurs is pointed out.