Phase ordering dynamics of a vector order parameter

Direct test of the Gaussian auxiliary field ansatz in nonconserved order parameter phase ordering dynamics

The assumption that the local order parameter is related to an underlying spatially smooth auxiliary field, u(r[over ⃗],t), is a common feature in theoretical approaches to non-conserved order parameter phase separation dynamics. In particular, the ansatz that u(r[over ⃗],t) is a Gaussian random field leads to predictions for the decay of the autocorrelation function which are consistent with observations, but distinct from predictions using alternative theoretical approaches. In this paper, the auxiliary field is obtained directly from simulations of the time-dependent Ginzburg-Landau equation in two and three dimensions. The results show that u(r[over ⃗],t) is equivalent to the distance to the nearest interface. In two dimensions, the probability distribution, P(u), is well approximated as Gaussian except for small values of u/L(t), where L(t) is the characteristic length-scale of the patterns. The behavior of P(u) in three dimensions is more complicated; the non-Gaussian region for small u/L(t) is much larger than that in two dimensions but the tails of P(u) begin to approach a Gaussian form at intermediate times. However, at later times, the tails of the probability distribution appear to decay faster than a Gaussian distribution.

Quench dynamics of the three-dimensional U(1) complex field theory: Geometric and scaling characterizations of the vortex tangle

We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II superconductors, magnetic materials, and aspects of cosmology. We characterize the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls off algebraically and we associate it to a geometric percolation transition that we characterize in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermodynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach toward a state that, within numerical accuracy, shares many features with the one at the percolation threshold; a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops; and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.

Scaling and universality in the aging kinetics of the two-dimensional clock model

We study numerically the aging dynamics of the two-dimensional p -state clock model after a quench from an infinite temperature to the ferromagnetic phase or to the Kosterlitz-Thouless phase. The system exhibits the general scaling behavior characteristic of nondisordered coarsening systems. For quenches to the ferromagnetic phase, the value of the dynamical exponents suggests that the model belongs to the Ising-type universality class. Specifically, for the integrated response function chi(t,s) approximately or = s(-a)chif(t/s), we find a(chi) consistent with the value a(chi)=0.28 found in the two-dimensional Ising model.

Defect statistics in the two-dimensional complex Ginzburg-Landau model

The statistical correlations between defects in the two-dimensional complex Ginzburg-Landau model are studied in the defect-coarsening regime. In particular the defect-velocity probability distribution is determined and has the same high velocity tail found for the purely dissipative time-dependent Ginzburg-Landau (TDGL) model. The spiral arms of the defects lead to a very different behavior for the order parameter correlation function in the scaling regime compared to the results for the TDGL model.

Ordering dynamics of heisenberg spins with torque: crossover, spin waves, and defects

We study the effect of a torque induced by the local molecular field on the phase ordering dynamics of the Heisenberg model when the total magnetization is conserved. The torque drives the zero-temperature ordering dynamics to a new fixed point, characterized by exponents z=2 and lambda approximately 5. This "torque-driven" fixed point is approached at times such that t>>g(2), where g is the strength of the torque. All physical quantities, like the domain size L(t) and the equal and unequal time correlation functions, obey a crossover scaling form over the entire range of g. An attempt to understand this crossover behavior from the approximate Gaussian closure scheme fails completely, implying that the dynamics at late times cannot be understood from the dynamics of defects alone. We provide convincing arguments that the spin configurations can be decomposed in terms of defects and spin waves which interact with each other even at late times. In the absence of the torque term, the spin waves decay faster, but even so we find that the Gaussian closure scheme is inconsistent. In the latter case the inconsistency may be remedied by including corrections to a simple Gaussian distribution. For completeness we include a discussion of the ordering dynamics at T(c), where the torque is shown to be relevant, with exponents z=4-varepsilon/2 and lambda=d (where varepsilon=6-d). We show to all orders in perturbation theory that lambda=d as a consequence of the conservation law.

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)=f0(r/L)+L(-omega)f1(r/L)+., where L is the coarsening length scale. The correction-to-scaling exponent omega and the correction-to-scaling function f1(x) are calculated for both nonconserved and conserved order parameter systems using the approximate Gaussian closure theory of Mazenko. In general omega is a nontrivial 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 one-dimensional XY model, where an exact solution is possible.