Abstract
We revisit here the problem of the collective non-equilibrium dynamics of a classical statistical system at a critical point and in the presence of surfaces. The effects of breaking separately space- and time-translational invariance are well understood, hence we focus here on the emergence of a non-trivial interplay between them. For this purpose, we consider a semi-infinite model with $O(n)$-symmetry and purely dissipative dynamics which is prepared in a disordered state and then suddenly quenched to its critical temperature. We determine the short-distance behaviour of its response function within a perturbative approach which does not rely on any a priori assumption on the scaling form of this quantity.
Highlights
It is a trivial observation that any physical system has a finite extent; as a consequence, descriptions which assume translational invariance can at most capture its bulk features, with surface effects representing subleading corrections which decay upon moving away from the boundaries
In order to highlight the effect of the boundaries on the collective dynamics, hereafter we focus on the case in which the system is quenched right at its critical point, i.e., we fix r = rc for t > 0, where rc is the critical value of the parameter r ; beyond the Gaussian approximation rc still vanishes if the analysis is done by using dimensional regularization to calculate the relevant integrals
We investigated the universal properties of a Landau-Ginzburg model with n-component vector order parameter, O(n) symmetry, and a purely dissipative dynamics [25] in which the translational symmetries in space and time are broken by the presence of a surface and by suddenly quenching the temperature, respectively
Summary
It is a trivial observation that any physical system has a finite extent; as a consequence, descriptions which assume translational invariance can at most capture its bulk features, with surface effects representing subleading corrections which decay upon moving away from the boundaries This decay is controlled by the presence of an inherent length scale, which sets the “range” of these surface effects. When the correlation length ξ becomes large with respect to microscopic scales, collective behaviours emerge which are not determined by the underlying microscopic structure, but by those coarse-grained properties which do not really depend on the considered scale, such as symmetries, range of interactions and the (effective) spatial dimensionality These circumstances, which represent a hallmark of systems undergoing a continuous phase transition, lead to universality. Equilibrium transitions at surfaces and non-equilibrium critical dynamics after a quench
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