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Physical Review Letters

Correlations and Coarsening in the q -State Potts Model

Clément Sire, Satya N. Majumdar

Published May 22, 1995• 49 citations
Publisher

Abstract

We study the nonequilibrium dynamics of the $q$-state Potts model following a quench from the high-temperature disordered phase to zero temperature. We calculate, within a Gaussian closure approximation, the time-dependent two-point correlation functions for general $q$. These correlation functions obey dynamic scaling with a length scale $L(t)\ensuremath{\sim}{t}^{1/2}$, while the autocorrelation function decays as $L(t{)}^{\ensuremath{-}\ensuremath{\lambda}(q)}$. We also establish a correspondence of this model to the Ising model evolving with a fixed magnetization $(2/q\ensuremath{-}1)$. Extensive numerical simulations of the Potts model in two dimensions show good agreement with our theory.

DOI
10.1103/physrevlett.74.4321

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We study the nonequilibrium dynamics of the qqq-state Potts model following a quench from the high-temperature disordered phase to zero temperature. We calculate, within a Gaussian closure approximation, the time-dependent two-point correlation functions for general qqq. These correlation functions obey dynamic scaling with a length scale L(t)\ensuremath∼t1/2L(t)\ensuremath{\sim}{t}^{1/2}L(t)\ensuremath∼t1/2, while the autocorrelation function decays as L(t)\ensuremath−\ensuremathλ(q)L(t{)}^{\ensuremath{-}\ensuremath{\lambda}(q)}L(t)\ensuremath−\ensuremathλ(q). We also establish a correspondence of this model to the Ising model evolving with a fixed magnetization (2/q\ensuremath−1)(2/q\ensuremath{-}1)(2/q\ensuremath−1). Extensive numerical simulations of the Potts model in two dimensions show good agreement with our theory.
We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}t1/2{\mathit{t}}^{1/2}t1/2. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.
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The nonequilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with O(N) symmetry, starting from a ground state in the disordered phase. In the limit of infinite N, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.
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