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Physical review. B, Condensed matter

Critical dynamics of unstable states

Yannis Drossinos, David Ronis

Published Sep 1, 1990• 0 citations
Publisher

Abstract

A one-loop, renormalized, non-Markovian equation of motion for the average order parameter of the time-dependent, scalar, nonconserved Landau-Ginzburg model in the critical region is presented. A new time-dependent length scale, a coherence length of critical-like fluctuations, is defined via a memory-dependent dynamic match condition. Two coupled equations for the new length scale and the average order parameter are numerically solved for instantaneous quenches into the unstable region of the phase diagram. The instantaneous free energy that governs the dynamics is not equal to the adiabatic free energy, and the new length scale increases rapidly in regions of negative instantaneous susceptibility. For quenches at zero external field through the critical temperature the structure factor decomposes into two independent parts at intermediate and late times. The late-time contribution is approximately given by a mean-field, critical structure factor. Both contributions admit scaling forms, albeit with different scaling lengths.

DOI
10.1103/physrevb.42.4694

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A one-loop, renormalized, non-Markovian equation of motion for the average order parameter of the time-dependent, scalar, nonconserved Landau-Ginzburg model in the critical region is presented. A new time-dependent length scale, a coherence length of critical-like fluctuations, is defined via a memory-dependent dynamic match condition. Two coupled equations for the new length scale and the average order parameter are numerically solved for instantaneous quenches into the unstable region of the phase diagram. The instantaneous free energy that governs the dynamics is not equal to the adiabatic free energy, and the new length scale increases rapidly in regions of negative instantaneous susceptibility. For quenches at zero external field through the critical temperature the structure factor decomposes into two independent parts at intermediate and late times. The late-time contribution is approximately given by a mean-field, critical structure factor. Both contributions admit scaling forms, albeit with different scaling lengths.
The critical dynamics of superconductors in the charged regime is reconsidered within field-theory. For the dynamics the Ginzburg-Landau model with complex order parameter coupled to the gauge field suggested earlier [Lannert et al. Phys. Rev. Lett. 92, 097004 (2004)] is used. Assuming relaxational dynamics for both quantities the renormalization group functions within one loop approximation are recalculated for different choices of the gauge. A gauge independent result for the divergence of the measurable electric conductivity is obtained only at the weak scaling fixed point unstable in one loop order where the time scales of the order parameter and the gauge field are different.
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.
We extend standard finite-size scaling methods to study the dynamical evolution of an unstable state far from equilibrium as the system undergoes a first order phase transition. We suggest that the nonequilibrium structure factor S(q, t, L), at late times and for large enough lattice sizes, scales as S(q, t, L)=LdF(qL, t1/x/L). L is the linear dimension of the system and 1/x is the domain growth exponent. We obtain x=2 in the case of the kinetic Ising model with a nonconserved order parameter. For a critical quench in a system with conserved order parameter, scaling of the peak of the structure factor gives 1/x≈0.27. Higher wavenumbers, however, are more consistent with x=3.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
The critical dynamics of superconductors in the charged regime is reconsidered within field-theory. For the dynamics, the Ginzburg-Landau model with complex order parameter coupled to the gauge field suggested earlier [Lannert et al. Phys. Rev. Lett. 92, 097004 (2004)] is used. Assuming relaxational dynamics for both quantities, the renormalization group functions within one loop approximation are recalculated for different choices of the gauge. A gauge independent result for the divergence of the melectric conductivity is obtained only at the weak scaling fixed point unstable in one loop order where the timescales of the order parameter and the gauge field are different.
We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with nonconserved n-component order parameter (model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations $g(|r\ensuremath{-}{r}^{\ensuremath{'}}|)\ensuremath{\sim}|r\ensuremath{-}{r}^{\ensuremath{'}}{|}^{\ensuremath{-}a},$ $a=4\ensuremath{-}\ensuremath{\delta},$ previously introduced by Weinrib and Halperin. The parameter $\ensuremath{\delta}\ensuremath{\sim}\ensuremath{\epsilon},$ where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the usual small parameter within the Wilson-Fisher renormalization group theory. Using a double expansion in $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ we calculate the critical exponent zzz up to second order on the small parameters. We show that quenched impurities of this kind affect the critical dynamics already in first order of $\ensuremath{\epsilon}$ and $\ensuremath{\delta},$ leading to a relevant correction for the mean field value of the exponent z.z.z.