We investigate the crossover of critical behavior for the dynamic phase transition (DPT) in ferromagnetic thin films using Monte Carlo simulations of the kinetic Ising model, focusing on the scaling behavior of the dynamic order parameter under a time-dependent external magnetic field. Specifically, we study the transition of the critical behavior of such multilayer film systems from two-dimensional (2D) to three-dimensional (3D) as a function of the film thickness and the distance to the critical point, which enables dimensional crossover observations. Our results indicate that the effective critical exponents exhibit a clear transition in their scaling behavior, with thinner films showing 2D-like characteristics and thicker films displaying 3D-like behavior, for both the DPT and the thermodynamic phase transitions (TPT). Quantitatively, the crossover from 2D to 3D behavior occurs at larger film thicknesses for the DPT compared to the TPT, suggesting that DPT and TPT are governed by distinctly different length scales and underlying surface effects. These findings are in agreement with experimental observations in ultrathin Co films, where dynamic and thermodynamic critical exponents were found to differ. Therefore, our study provides an in-depth explanation for critical phenomena in thin-film ferromagnets driven by a time-dependent magnetic field. By comparing the dimensional crossover properties of both TPT and DPT, we present a comprehensive understanding of how thin-film geometry and surface effects influence the scaling laws and critical behavior in nonequilibrium systems.
We have used Monte Carlo simulations and mean-field analysis to observe the magnetic behavior of Ising thin films with cubic lattice structures as a function of temperature and thickness, especially in the critical region. Magnetization and magnetic susceptibility, including layer variation, are investigated. We find that the magnetic behavior changes from two-dimensional to three-dimensional character with increasing film thickness. Both the crossover of the critical temperature from a two-dimensional to a bulk value and the shift exponent are observed. Nevertheless, with support from a scaling function, the simulations show that the effective critical exponents for films with large enough layer extents only vary a little from their two-dimensional values. This, in particular, provides an indication of two-dimensional universality in the thin films.
Abstract By the use of both the standard mean field theory and the effective field theory with correlations the phase transition of the transverse Ising thin films is investigated with special emphasis to the behavior of the susceptibility χ(T) outside the critical regime. The exchange interaction and transverse field at the surfaces of the thin film take values different from those in the bulk. Various non‐monotonic temperature dependences of the effective exponent γ(T) introduced by Kouvel and Fisher are found, depending on the ratios of these properties between the surfaces and the bulk as well as the value of film thickness L. They are clearly different from the behavior in the bulk. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
In this review, we show our results with new interpretation on the critical exponents of thin films obtained by high-performance multi-histogram Monte Carlo simulations. The film thickness
Nz consists of a few layers up to a dozen of layers in the
z direction. The free boundary condition is applied in this direction while in the
xy plane periodic boundary conditions are used. Large
xy plane sizes are used for finite-size scaling. The Ising model is studied with nearest-neighbor (NN) interaction. When
Nz=1, namely the two-dimensional (2D) system, we find the critical exponents given by the renormalization group. While, for
Nz>1, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate slightly but systematically from the 2D values. If we use these values of critical exponents in the hyperscaling relation with
d=2, then the hyperscaling relation is violated. However, if we use the hyperscaling relation and the critical exponents obtained for
Nz>1 to calculate the dimension of the system, we find the system dimension slightly larger than 2. This can be viewed as an "effective" dimension. More discussion is given in the paper. We also show the cross-over between the first- and second-order transition while varying the film thickness in an antiferromagnetic FCC Ising frustrated thin film. In addition, we will show evidence that when a 2D system has two order parameters of different symmetries with a single transition, the critical exponents are new, suggesting a universality class of coupled two-symmetry breakings. In this case, the 2D hyperscaling does not hold. Another case is the 3D Ising model coupled to the lattice vibration: the critical exponents deviate from the 3D Ising ones, the results suggest the violation of the hyperscaling.
In this review, we show our results with new interpretation on the critical exponents of thin films obtained by high-performance multi-histogram Monte Carlo simulations. The film thickness Nz consists of a few layers up to a dozen of layers in the z direction. The free boundary condition is applied in this direction while in the xy plane periodic boundary conditions are used. The large xy plane size, from 702 to 1602, is used for finite-size scaling. The Ising model is used with nearest-neighbor (NN) interaction. When Nz=1, namely the two-dimensional (2D) system, we find strictly the critical exponents given by the renormalization group. While, for Nz>1, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate systematically from the 2D values and get closer to the 3D critical exponents for Nz>13 . If we argue that as long as the thickness Nz is small enough, the correlation length in the z direction is finite, it does not affect the nature of the phase transition, namely it remains in the 2D universality class. This argument is in contradiction with our numerical results which show a systematic deviation from 2D values: If we use these values of critical exponents in the hyperscaling relation with d=2, then the hyperscaling relation is violated. However, if we use the hyperscaling relation and the critical exponents obtained for Nz>1 to calculate the dimension of the system, we find the system dimension between 2 and 3. This can be viewed as an "effective" dimension. More discussion is given in the paper. We also show the cross-over between the first- and second-order transition while varying the film thickness. In addition, we will show evidence that when a 2D system has two order-parameters of different symmetries with a single transition, the critical exponents are new, suggesting a universality class of coupled two-symmetry breakings. In this case, the 2D hyperscaling does not hold. Another case is the 3D Ising model coupled to the lattice vibration: the critical exponents deviate from the 3D Ising ones, the results suggest the violation of the hyperscaling.
We have numerically determined the behavior of the magnetic susceptibility upon approach of the critical point in two-dimensional spin systems with an interaction range that was varied over nearly 2 orders of magnitude. The full crossover from classical to Ising-like critical behavior, spanning several decades in the reduced temperature, could be observed. Our results convincingly show that the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}$ changes nonmonotonically from its classical to its Ising value when approaching the critical point in the ordered phase. In the disordered phase the behavior is monotonic. Furthermore, the hypothesis that the crossover function is universal is supported.