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Phase Transitions

Surface phase transitions of the transverse Ising model with modified surface–bulk coupling

T. Lahcini, Noureddine Benayad

Published Mar 1, 2009• 9 citations
Publisher

Abstract

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.

DOI
10.1080/01411590802575192

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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.
The transverse Ising system with a diluted surface and a modified surface-bulk interaction J(subscript ┴) is studied by the use of an effective field method within the framework of a single-site cluster theory. The surface antiferromagnetic exchange interaction JS is distributed randomly and has a concentration p. The new parameter p strongly affects the phase diagram obtained in the pure case (p=1). Indeed, some qualitatively interesting features have been found, both in the absence and in the presence of the surface transverse field. In particular, we defined a critical value p(superscript *) of p, separating two qualitatively different behaviors of the system.
The effect of surface and bulk transverse fields on the phase transitions of a semi-infinite spin-1 ferromagnetic Ising model is investigated within a finite cluster approximation with an expansion technique for cluster identities of spin-1 localized spin systems. The phase diagrams are determined as a function of the ratio of bulk and surface transverse fields, the temperature and the transverse field of the surface. They show that the systems exhibit a variety of phase transitions. A dead layer is also obtained in a special region of space parameters.
Abstract The phase diagram of the three‐dimensional transverse Ising system with competing surface and bulk exchange interactions is studied by the use of an effective‐field method within the framework of a single‐site cluster theory. In addition to this pure case, which shows qualitatively interesting features, we investigated the effects of the bond dilution on the obtained phase diagram. In particular, we studied the influence of the surface dilution parameter on the surface transitions. We find a number of characteristic behaviours of the surface both in the absence and in the presence of the surface transverse field. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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 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.