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Journal of Physics A Mathematical and General

Simulation of domain wall dynamics in the 2D anisotropic Ising model

António Luis Ferreira, Sushil Kumar Mendiratta, E S Lage

Published May 21, 1989• 8 citations
Publisher

Abstract

The authors first show that the movement of a domain wall in a highly anisotropic Ising model of spins in two dimensions can be mapped to a random walk movement of labelled walkers in one dimension. This movement is correlated in the sense that the transition probability of a given walker, labelled i, depends upon the current positions of its labelled neighbours (i-1 and i+1). A Monte Carlo simulation of the walk for various numbers of walkers (up to a maximum of 50) and for quite long times (up to a maximum of 25*10 4 Monte Carlo steps) yields useful and interesting information about the dynamics. They verify that (i) the centre of mass of the walkers executes an unbiased random walk for all times starting from the lowest times; and (ii) the moment of inertia of the walkers asymptotically reaches a constant value which scales with the number of walkers. They also discuss the relation of this work to the dynamics in the 2D Ising model, identifying phi with the dynamical exponent z. The random walk equivalence also provides a geometrical interpretation of the known results for the solid-on-solid model.

DOI
10.1088/0305-4470/22/10/003

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The authors first show that the movement of a domain wall in a highly anisotropic Ising model of spins in two dimensions can be mapped to a random walk movement of labelled walkers in one dimension. This movement is correlated in the sense that the transition probability of a given walker, labelled i, depends upon the current positions of its labelled neighbours (i-1 and i+1). A Monte Carlo simulation of the walk for various numbers of walkers (up to a maximum of 50) and for quite long times (up to a maximum of 25*10 4 Monte Carlo steps) yields useful and interesting information about the dynamics. They verify that (i) the centre of mass of the walkers executes an unbiased random walk for all times starting from the lowest times; and (ii) the moment of inertia of the walkers asymptotically reaches a constant value which scales with the number of walkers. They also discuss the relation of this work to the dynamics in the 2D Ising model, identifying phi with the dynamical exponent z. The random walk equivalence also provides a geometrical interpretation of the known results for the solid-on-solid model.
The dynamics of the spins in the Ising model are analyzed using a virtual walk scenario. The system is quenched from a very high temperature to a lower one using the Glauber scheme in one and two dimensions. A walk is associated with each spin which evolves according to the current state of the spin. The probability distribution of the displacement of the walker is calculated, which shows a distinct change as the temperature is increased. The average displacement as a function of time shows a nonequilibrium region stretched over a much longer time interval compared to the bulk magnetization. Nevertheless, one can still detect a time-dependent estimate of the critical point determined by two different methods. In addition, we introduce a virtual walk generated from the local energy of individual spins. Finite-time scaling of the different quantities estimated in two dimensions shows excellent consistency with the values of the known critical exponents.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d≤2d\le 2d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.