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Europhysics Letters (EPL)

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

Michele Vendruscolo, Mauro Rovere, Annalisa Fasolino

Published Nov 15, 1992• 14 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/20/6/013

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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.
The phase diagram of the three-dimensional transverse Ising system with antiferromagnetic intrasurface J S and surface–bulk J ⊥ interactions modified with respect to the ferromagnetic bulk exchange interaction J B is studied by the use of an effective field method in the framework of a single-site cluster theory. In this investigation, the state equations are derived by means of differential operator technique. Using these coupled equations within the four-layer approximation, it has been shown that the new parameter J ⊥ strongly affects the phase diagram obtained recently in the case J ⊥ = J B. Indeed, our calculations reveal some qualitatively interesting features both in the absence and in the presence of the surface and bulk transverse fields.
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.
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.
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 effect of edge on wetting and layering transitions of a three-dimensional spin-1/2 Ising model is investigated, in the presence of longitudinal and surface magnetic fields, using mean field theory and Monte Carlo simulations. For T=0, the ground state phase diagram shows that there exist only three allowed transitions, namely, surface and bulk transition, surface transition, and bulk transition. However, there exist a surface intralayering temperature T(s)(L), above which the surface and the intralayering surface transitions occur. While the bulk layering and intralayering transitions appear above another finite temperature T(b)(L)(>or=T(s)(L)). These surface and bulk intralayering transitions are not seen in the perfect surfaces case. Numerical values of T(s)(L) and T(b)(L), computed by Monte Carlo method are found to be smaller than those obtained using mean field theory. However, the results predicted by the two methods become similar, and are exactly those given by the ground state phase diagram, for very low temperatures. On the other hand, the behavior of the local magnetizations as a function of the external magnetic field, shows that the transitions are of the first order type. T(s)(L) and T(b)(L) decrease when increasing the system size and/or the surface magnetic field. In particular, T(b)(L) reaches the wetting temperature T(w) for sufficiently large system sizes.
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 yt1=0.718(2){y}_{{t}_{1}}=0.718(2)yt1​​=0.718(2) and yh1=1.6465(6){y}_{{h}_{1}}=1.6465(6)yh1​​=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.