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Journal of Physics A Mathematical and General

Growth of order in vector spin systems; scaling and universality

Alan J. Bray, K. Humayun

Published Dec 21, 1990• 46 citations
Publisher

Abstract

The growth of order in vector spin systems with nonconserved order parameter ('model A') is considered following an instantaneous quench from infinite to zero temperature. The results of numerical simulations in spatial dimension d=2 and spin dimension 2n5 are presented. For n4, a scaling regime (where a characteristic length scale L(t) emerges) is entered for sufficiently long times, with L(t)t 1 2/. The autocorrelation function. A(t) decays with time as A(t)t -(d- lambda ) 2/, and the exponent lambda (n) agrees well with the predictions of the 1/n-expansion. The cases n=2 and 3 are more complicated, due to the non-trivial role played by topological singularities, i.e. vortices (n=2) and Polyakov solitons, (n=3). For n4, universal amplitudes and scaling functions characterizing the energy relaxation and the equal-time correlation function are identified. It is argued that for d3, where an ordered phase exists at low temperature, such universal quantities characterize the entire ordered phase.

DOI
10.1088/0305-4470/23/24/028

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The growth of order in vector spin systems with nonconserved order parameter ('model A') is considered following an instantaneous quench from infinite to zero temperature. The results of numerical simulations in spatial dimension d=2 and spin dimension 2n5 are presented. For n4, a scaling regime (where a characteristic length scale L(t) emerges) is entered for sufficiently long times, with L(t)t 1 2/. The autocorrelation function. A(t) decays with time as A(t)t -(d- lambda ) 2/, and the exponent lambda (n) agrees well with the predictions of the 1/n-expansion. The cases n=2 and 3 are more complicated, due to the non-trivial role played by topological singularities, i.e. vortices (n=2) and Polyakov solitons, (n=3). For n4, universal amplitudes and scaling functions characterizing the energy relaxation and the equal-time correlation function are identified. It is argued that for d3, where an ordered phase exists at low temperature, such universal quantities characterize the entire ordered phase.
The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.
We consider the nonequilibrium behavior of the spin-glass ordered phase within the droplet scaling theory introduced previously. The fundamental long-time nonequilibrium process is assumed to be the thermally activated growth of spin-glass ordered domains. The remanent magnetization, m(t), in zero field is found to decay at long times as m(t)\ensuremath{\sim}${R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$, where Rt{R}_{t}Rt​\ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.
We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.
We study the short-time dynamics of systems that develop quasi-long-range order after a quench to the Kosterlitz-Thouless phase. With the working hypothesis that the universal short-time behavior, previously found in Ising-like systems, also occurs in the Kosterlitz-Thouless phase, we explore the scaling behavior of thermodynamic variables during the relaxational process following the quench. As a concrete example, we investigate the two-dimensional six-state clock model by Monte Carlo simulation. The exponents governing the magnetization, the second moment, and the autocorrelation function are calculated. From them, by means of scaling relations, estimates for the equilibrium exponents zzz and $\ensuremath{\eta}$ are derived. In particular, our estimates for the temperature-dependent anomalous dimension $\ensuremath{\eta}$ that governs the static correlation function are consistent with existing analytical and numerical results, and, thus, confirm our working hypothesis.
The dynamics of spin fluctuations in nonequilibrium and equilibrium states is studied in the one-phase region above Tc{\mathit{T}}_{\mathit{c}}Tc​. The equal-time structure factor is computed after quenching the system from an infinite temperature to T>Tc{\mathit{T}}_{\mathit{c}}Tc​. The result shows growing spatial patterns which can be scaled with temperature-dependent scaling functions. The different-time (dynamic) structure factor is computed by taking time averages while the system is fluctuating in equilibrium states. The equilibrium autocorrelation decays following a diffusive exponential term and a slower but transient term. The relaxation rate due to the diffusion supports the dynamic scaling.