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arXiv:cond-mat/9510057 · Physical Review Letters

Universal Short-Time Behavior in Critical Dynamics near Surfaces

Uwe Ritschel, Peter Czerner

Published Nov 20, 1995• 41 citations
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Abstract

We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from $T\ensuremath{\gg}{T}_{c}$ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative ($\ensuremath{\epsilon}$-expansion) and Monte Carlo results.

DOI
10.1103/physrevlett.75.3882

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We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from T\ensuremath≫TcT\ensuremath{\gg}{T}_{c}T\ensuremath≫Tc​ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative (\ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion) and Monte Carlo results.
We use a three-dimensional massive field theory up to the two-loop approximation to study the critical behavior of semi-infinite quenched random Ising-like systems at the special surface transition. Besides, we extend up to the next-to leading order, the previous first-order results of the sqrt[epsilon] expansion obtained by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. The numerical estimates for surface critical exponents in both cases are computed by means of the Padé analysis. Moreover, in the case of the massive field theory we perform Padé-Borel resummation of the resulting two-loop series expansions for surface critical exponents. The most reliable estimates for critical exponents of semi-infinite systems with quenched bulk randomness at the special surface transition, which we can obtain in the frames of the present approximation scheme, are eta(//)=-0.238, Delta(1)=1.098, eta( perpendicular )=-0.104, beta(1)=0.258, gamma(11)=0.839, gamma(1)=1.426, delta(1)=6.521, and delta(11)=4.249. These values are different from critical exponents for pure semi-infinite Ising-like systems and show that in a system with quenched bulk randomness the plane boundary is characterized by a new set of critical exponents at the special surface transition.
We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$>0 called model BA{\mathit{B}}_{\mathit{A}}BA​, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model BB{\mathit{B}}_{\mathit{B}}BB​---and each static surface universality class splits up into two dynamic surface universality classes.The static and dynamic surface critical exponents of these dynamic surface universality classes as well as the crossover exponent associated with the nonconservative surface term are shown to be expressible in terms of static bulk and surface exponents. Model BA{\mathit{B}}_{\mathit{A}}BA​ and its results should apply, e.g., to uniaxial ferromagnets whose rotational spin symmetry is broken in the vicinity of a surface plane. For model A the considered local modification of the dynamics near the surface is irrelevant in the renormalization-group sense and hence does not lead to new dynamic surface universality classes.
We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.
We consider an open system near a quantum critical point that is suddenly moved towards the critical point. The bath-dominated diffusive nonequilibrium dynamics after the quench is shown to follow scaling behavior, governed by a critical exponent that emerges in addition to the known equilibrium critical exponents. We determine this exponent and show that it describes universal prethermalized coarsening dynamics of the order parameter in an intermediate time regime. Implications of this quantum critical prethermalization are: (i) a power law rise of order and correlations after an initial collapse of the equilibrium state and (ii) a crossover to thermalization that occurs arbitrarily late for sufficiently shallow quenches.
We present a method to calculate short-time nonequilibrium universal exponents within the functional-renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial-slip exponent which characterizes the nonequilibrium universal short-time behavior of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.