arXiv subjects

A. J. Bray

Publications and source records attributed to A. J. Bray.

At least 37 records · Page 2Linked to original sources

arXiv1999

The Link Overlap and Finite Size Effects for the 3D Ising Spin Glass

We study the link overlap between two replicas of an Ising spin glass in three dimensions using the Migdal-Kadanoff approximation and scaling arguments based on the droplet picture. For moderate system sizes, the distribution of the link overlap shows the asymmetric shape and large sample-to-sample variations found in Monte Carlo simulations and usually attributed to replica symmetry breaking. However, the scaling of the width of the distribution, and the link overlap in the presence of a weak coupling between the two replicas are in agreement with the droplet picture. We also discuss why it is impossible to see the asymptotic droplet-like behaviour for moderate system sizes and temperatures not too far below the critical temperature.

cond-mat.dis-nn↗

arXiv1999

Dynamics of Phase Separation under Shear: A Soluble Model

The dynamics of phase separation for a binary fluid subjected to a uniform shear are solved exactly for a model in which the order parameter is generalized to an n-component vector and the large-n limit taken. Characteristic length scales in directions parallel and perpendicular to the flow increase as (t^5/\ln t)^{1/4} and (t/\ln t)^{1/4} respectively. The structure factor in the shear-flow plane exhibits two parallel ridges as observed in experiment.

cond-mat.stat-mech↗

arXiv1999

Corrections to Scaling in the Phase-Ordering Dynamics of a Vector Order Parameter

Corrections to scaling, associated with deviations of the order parameter from the scaling morphology in the initial state, are studied for systems with O(n) symmetry at zero temperature in phase-ordering kinetics. Including corrections to scaling, the equal-time pair correlation function has the form C(r,t) = f_0(r/L) + L^{-omega} f_1(r/L) + ..., where L is the coarsening length scale. The correction-to-scaling exponent, omega, and the correction-to-scaling function, f_1(x), are calculated for both nonconserved and conserved order parameter systems using the approximate Gaussian closure theory of Mazenko. In general, omega is a non-trivial exponent which depends on both the dimensionality, d, of the system and the number of components, n, of the order parameter. Corrections to scaling are also calculated for the nonconserved 1-d XY model, where an exact solution is possible.

cond-mat.stat-mech↗

arXiv1999

Breakdown of Scaling in the Nonequilibrium Critical Dynamics of the Two-Dimensional XY Model

The approach to equilibrium, from a nonequilibrium initial state, in a system at its critical point is usually described by a scaling theory with a single growing length scale, $ξ(t) \sim t^{1/z}$, where z is the dynamic exponent that governs the equilibrium dynamics. We show that, for the 2D XY model, the rate of approach to equilibrium depends on the initial condition. In particular, $ξ(t) \sim t^{1/2}$ if no free vortices are present in the initial state, while $ξ(t) \sim (t/\ln t)^{1/2}$ if free vortices are present.

cond-mat.stat-mech↗

arXiv1999

Reply to "Comment on Evidence for the droplet picture of spin glasses"

Using Monte Carlo simulations (MCS) and the Migdal-Kadanoff approximation (MKA), Marinari et al. study in their comment on our paper the link overlap between two replicas of a three-dimensional Ising spin glass in the presence of a coupling between the replicas. They claim that the results of the MCS indicate replica symmetry breaking (RSB), while those of the MKA are trivial, and that moderate size lattices display the true low temperature behavior. Here we show that these claims are incorrect, and that the results of MCS and MKA both can be explained within the droplet picture.

cond-mat.dis-nn↗

arXiv1998

Comment on "General Method to Determine Replica Symmetry Breaking Transitions"

In a recent letter Marinari et al [Phys. Rev. Lett. 81, 1698 (1998)] introduced a new method to study spin glass transitions and argued that by probing replica symmetry (RS) as opposed to time reversal symmetry (TRS), their method unambiguously shows that replica symmetry breaking (RSB) occurs in short-range spin glasses. In this comment we show that while the new method is indeed useful for studying transitions in systems where TRS is absent (such as the p-spin model studied by them), the conclusion that it shows the existence of RSB in short-range spin glasses is wrong.

cond-mat.dis-nn↗

arXiv1998

Phase-Ordering Dynamics with an Order-Parameter-Dependent Mobility: The Large-n Limit

The effect of an order-parameter dependent mobility (or kinetic coefficient), on the phase-ordering dynamics of a system described by an n-component vector order parameter is addressed at zero temperature in the large-n limit. We consider cases in which the mobility or kinetic coefficient vanishes when the magnitude of the order parameter takes its equilibrium value. In the large-n limit, the system is exactly soluble for both conserved and non-conserved order parameter. In the non-conserved case, the scaling form for the correlation function and it's Fourier transform, the structure factor, is established, with the characteristic length growing as a power of time. In the conserved case, the structure factor is evaluated and found to exhibit a multi-scaling behaviour, with two growing length scales differing by a logarithmic factor. In both cases, the rate of growth of the length scales depends on the manner in which the mobility or kinetic coefficient vanishes as the magnitude of the order parameter approaches its equilibrium value.

cond-mat.stat-mech↗

arXiv1998

Defect Relaxation and Coarsening Exponents

The coarsening exponents describing the growth of long-range order in systems quenched from a disordered to an ordered phase are discussed in terms of the decay rate, omega(k), for the relaxation of a distortion of wavevector k applied to a topological defect. For systems described by order parameters with Z(2) (`Ising'), and O(2) (`XY') symmetry, the appropriate defects are domain walls and vortex lines respectively. From omega(k) ~ k^z, we infer L(t) ~ t^(1/z) for the coarsening scale, with the assumption that defect relaxation provides the dominant coarsening mechanism. The O(2) case requires careful discussion due to infrared divergences associated with the far field of a vortex line. Conserved, non-conserved, and `intermediate' dynamics are considered, with either short-range or long-range interactions. In all cases the results agree with an earlier `energy scaling' analysis.

cond-mat.stat-mech↗

arXiv1997

Corrections to Scaling in Phase-Ordering Kinetics

The leading correction to scaling associated with departures of the initial condition from the scaling morphology is determined for some soluble models of phase-ordering kinetics. The result for the pair correlation function has the form C(r,t) = f_0(r/L) + L^{-ω} f_1(r/L) + ..., where L is a characteristic length scale extracted from the energy. The correction-to-scaling exponent ωhas the value ω=4 for the d=1 Glauber model, the n-vector model with n=\infty, and the approximate theory of Ohta, Jasnow and Kawasaki. For the approximate Mazenko theory, however, ωhas a non-trivial value: omega = 3.8836... for d=2, and ω= 3.9030... for d=3. The correction-to-scaling functions f_1(x) are also calculated.

cond-mat.stat-mech↗

arXiv1997

Mixed phases in U(N) superconductivity

We consider a general class of type-II superconductors which are described by N complex order parameters (OP), with an overall U(N) symmetry for the energy functional. In the lowest energy state, each OP adopts its own particular lattice structure, making the spatial variation of the overall condensate very complex. We present detailed results for the case N=2. We also treat the limit N -> infinity, which allows us to resolve a recent controversy, by showing that all previous studies are fundamentally flawed.

cond-mat.supr-con↗

arXiv1997

Persistence exponents for fluctuating interfaces

Numerical and analytic results for the exponent θdescribing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β< 1; for β= 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θ_S = 1 - β. The exponent θ_0 for the flat initial condition appears to be nontrivial. We prove that θ_0 \to \infty for β\to 0, θ_0 \geq θ_S for β< 1/2 and θ_0 \leq θ_S for β> 1/2, and calculate θ_{0,S} perturbatively to first order in an expansion around the Markovian case β= 1/2. Using the exact result θ_S = 1 - β, accurate upper and lower bounds on θ_0 can be derived which show, in particular, that θ_0 \geq (1 - β)^2/βfor small β.

cond-mat.stat-mech↗

arXiv1997

Velocity Distribution of Topological Defects in Phase-Ordering Systems

The distribution of interface (domain-wall) velocities ${\bf v}$ in a phase-ordering system is considered. Heuristic scaling arguments based on the disappearance of small domains lead to a power-law tail, $P_v(v) \sim v^{-p}$ for large v, in the distribution of $v \equiv |{\bf v}|$. The exponent p is given by $p = 2+d/(z-1)$, where d is the space dimension and 1/z is the growth exponent, i.e. z=2 for nonconserved (model A) dynamics and z=3 for the conserved case (model B). The nonconserved result is exemplified by an approximate calculation of the full distribution using a gaussian closure scheme. The heuristic arguments are readily generalized to conserved case (model B). The nonconserved result is exemplified by an approximate calculation of the full distribution using a gaussian closure scheme. The heuristic arguments are readily generalized to systems described by a vector order parameter.

cond-mat.stat-mech↗

arXiv1997

Non-Markovian Persistence and Nonequilibrium Critical Dynamics

The persistence exponent θfor the global order parameter, M(t), of a system quenched from the disordered phase to its critical point describes the probability, p(t) \sim t^{-θ}, that M(t) does not change sign in the time interval t following the quench. We calculate θto O(ε^2) for model A of critical dynamics (and to order εfor model C) and show that at this order M(t) is a non-Markov process. Consequently, θis a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Majumdar and Sire [Phys. Rev. Lett. _77_, 1420 (1996); cond-mat/9604151].

cond-mat.stat-mech↗

arXiv1996

Global Persistence Exponent for Critical Dynamics

A `persistence exponent' $θ$ is defined for nonequilibrium critical phenomena. It describes the probability, $p(t) \sim t^{-θ}$, that the global order parameter has not changed sign in the time interval $t$ following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the $n=\infty$ limit of the $O(n)$ model, to first order in $ε= 4-d$, and for the 1-d Ising model. Numerical results are obtained for the 2-d Ising model. We argue that $θ$ is a new independent exponent.

cond-mat↗

arXiv1996

Coarsening Dynamics of a One-Dimensional Driven Cahn-Hilliard System

We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field $E$ has an asymmetric effect on the solution for a single stationary domain wall (or `kink'), the direction of the field determining whether the analytic solutions found by Leung [J.Stat.Phys.{\bf 61}, 345 (1990)] are unique. The dynamics of a kink-antikink pair (`bubble') is then studied. The behaviour of a bubble is dependent on the relative sizes of a characteristic length scale $E^{-1}$, where $E$ is the driving field, and the separation, $L$, of the interfaces. For $EL \gg 1$ the velocities of the interfaces are negligible, while in the opposite limit a travelling-wave solution is found with a velocity $v \propto E/L$. For this latter case ($EL \ll 1$) a set of reduced equations, describing the evolution of the domain lengths, is obtained for a system with a large number of interfaces, and implies a characteristic length scale growing as $(Et)^{1/2}$. Numerical results for the domain-size distribution and structure factor confirm this behavior, and show that the system exhibits dynamical scaling from very early times.

cond-mat↗

arXiv1996

Strong-coupling behaviour in discrete Kardar-Parisi-Zhang equations

We present a systematic discretization scheme for the Kardar-Parisi-Zhang (KPZ) equation, which correctly captures the strong-coupling properties of the continuum model. In particular we show that the scheme contains no finite-time singularities in contrast to conventional schemes. The implications of these results to i) previous numerical integration of the KPZ equation, and ii) the non-trivial diversity of universality classes for discrete models of `KPZ-type' are examined. The new scheme makes the strong-coupling physics of the KPZ equation more transparent than the original continuum version and allows the possibility of building new continuum models which may be easier to analyse in the strong-coupling regime.

cond-mat↗

arXiv1996

Soluble Infinite-Range Model of Kinetic Roughening

A modified Kardar-Parisi-Zhang (KPZ) equation is introduced, and solved exactly in the infinite-range limit. In the low-noise limit the system exhibits a weak-to-strong coupling transition, rounded for non-zero noise, as a function of the KPZ non-linearity. The strong-coupling regime is characterised by a double-peaked height distribution in the stationary state. The nonstationary dynamics is quite different from that of the stationary state.

cond-mat↗

arXiv1996

Dimensional Crossover in the Large N Limit

We consider dimensional crossover for an $O(N)$ Landau-Ginzburg-Wilson model on a $d$-dimensional film geometry of thickness $L$ in the large $N$-limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using ``environmentally friendly'' renormalization with those found using a direct, non-renormalization group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding non-trivial relations between the various thermodynamic scaling functions.

cond-mat↗