arXiv:cond-mat/9902362 · Physical Review Letters

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

Alan J. Bray, A. J. Briant, D. K. Jervis

Published Feb 14, 2000• 96 citations

Abstract

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, xi(t) approximately 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, xi(t) approximately t(1/2) if no free vortices are present in the initial state, while xi(t) approximately (t/lnt)(1/2) if free vortices are present.

DOI: 10.1103/physrevlett.84.1503

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arXiv:cond-mat/9902362 · Physical Review Letters

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

Alan J. Bray, A. J. Briant, D. K. Jervis

Published Feb 14, 2000• 96 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.84.1503

Related papers

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arXiv:cond-mat/0003447Physical review. B, Condensed matterJun 1, 200044 citations

ψ

, is introduced through the topological current theory. By solving the scalar field, the interaction energy of vortices in a 2D XY model is revisited. We study the dynamical evolution of vortices and present the branch conditions for generating, annihilating, crossing, splitting and merging of vortices. During the growth or annihilation of vortices, the dynamical scaling law of relevant length in a 2D XY model,

ξ(t)\propto(t-t^*)^{1/z}

, is obtained in the neighborhood of the limit point, given the dynamic exponent

z=2

. This dynamical scaling behavior is consistent with renormalization group theory, numerical simulations, and experimental results. Furthermore, it is found that during the crossing, splitting and merging of vortices, the dynamical scaling law of relevant length is

ξ(t)\propto(t-t^*)

. However, if vortices are at rest during splitting or merging, the dynamical scaling law of relevant length is a constat.

We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to

{\mathit{T}}_{\mathit{c}}

from random initial states. As in the case of subcritical quenches, we observe dynamical scaling of the equal time order parameter correlation function G(r,t,t), but here the nonequilibrium correlation length grows as L(t)\ensuremath{\sim}

{\mathit{t}}^{1/\mathit{z}}

, where z=4/15 is the usual dynamic critical exponent for this system. Moreover, we find dynamical scaling for correlations with the initial condition G(r,0,t). The temporal decay of autocorrelations at

{\mathit{T}}_{\mathit{c}}

yields a critical exponent ${\ensuremath{\lambda}}_{\mathit{c}}$=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}${\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$(t)G\ifmmode \tilde{}\else \~{}\fi{}(r/L(t)).

The dynamic critical exponent z is determined numerically for the d-dimensional

\mathrm{XY}

model

(d=2,

3, and 4) subject to relaxational dynamics and resistively shunted junction dynamics. We investigate both the equilibrium fluctuation and the relaxation behavior from nonequilibrium towards equilibrium, using the finite-size scaling method. The resulting values of z are shown to depend on the boundary conditions used, the periodic boundary condition, and fluctuating twist boundary condition (FTBC), which implies that the different treatments of the boundary in some cases give rise to different critical dynamics. It is also found that the equilibrium scaling and the approach to equilibrium scaling for the same boundary condition do not always give the same value of z. The FTBC in conjunction with the finite-size scaling of the linear resistance for both type of dynamics yields values of z consistent with expectations for superfluids and superconductors:

z=2,

3/2, and 2 for

d=2,

3, and 4, respectively.