Collective non-equilibrium dynamics at surfaces and the spatio-temporal edge

Abstract

Symmetries represent a fundamental constraint for physical systems and relevant new phenomena often emerge as a consequence of their breaking. An important example is provided by spaceand time-translational invariance in statistical systems, which hold at a coarse-grained scale in equilibrium and are broken by spatial and temporal boundaries, the former being implemented by surfaces -unavoidable in real samples -the latter by some initial condition for the dynamics which causes a non-equilibrium evolution. While the separate effects of these two boundaries are well understood, we demonstrate here that additional, unexpected features arise upon approaching the effective edge formed by their intersection. For this purpose, we focus on the classical semi-infinite Ising model with spin-flip dynamics evolving out of equilibrium at its critical point. Considering both subcritical and critical values of the coupling among surface spins, we present numerical evidence of a scaling regime with universal features which emerges upon approaching the spatio-temporal edge and we rationalise these findings within a field-theoretical approach.PACS numbers: 64.60. De, 64.60.Ht, 68.35.Rh Introduction -Thanks to the advances in miniaturisation of the past decades, devices have reached a size at which boundary effects cannot be neglected; in parallel, the physics of surfaces and interfaces has attracted an increasing interest, in particular concerning non-equilibrium dynamical processes, as many applications involve sudden changes in control parameters. Typically, describing these processes requires knowledge of the many microscopic details that vary widely from system to system. However, in suitable circumstances, collective phenomena emerge which make only few coarse-grained, mesoscopic properties relevant for the dynamics: in fact, it is now well-established, both theoretically and experimentally, that the behaviour of a statistical system close to a continuous phase transition can be characterised by few quantities, such as exponents and scaling functions, which depend only on the range and symmetries of the underlying interaction and on the dimensionality of the space. All the microscopically different systems sharing these same gross features form the so-called universality class of the transition. Within each of these classes, the various thermodynamic and structural properties typically show, in the neighbourhood of the critical point, leading algebraic behaviours characterised by common exponents, which constitute the hallmark of the transition. In turn, upon approaching it, the relevant contribution to the various thermodynamic quantities is effectively determined by the fluctuations of the so-called order parameter ϕ (e.g., the local magnetisation for an Ising ferromagnet).

Critical relaxation and the combined effects of spatial and temporal boundaries

“…In particular, we present here the calculations of the first-order corrections to the two-point functions of the theory, generalizing it to a O(n)-symmetric model, i.e., the one in which the order parameter ϕ is a n-vector whose components ϕ i satisfy Langevin equations of the form (2.6). For the case n = 1, the resulting predictions for the emerging scaling behaviour were in fact confirmed by Monte Carlo simulations, as briefly reported in reference [5]. Since we have assumedmotivated by the central limit theoremthe noise η to be Gaussian, we can integrate over it in equation (2.8), thus…”

“…, s) ∼ s 1−θ−θ E (see equation (4.4)) in their presence. (In passing we mention that A turns out to be of the order 1 in the numerical simulations of reference [5].) Noticing that for the ordinary and special cases, the exponent θ E in equations (3.39) and (3.41) takes opposite signs, one can look for differences between the two transitions, which are expected to display faster/slower power laws with respect to the regime s ≪ y z ≪ t , which is dominated instead by the initial-slip physics and invariably leads to observing C (.…”

“…Further confirmation of the presence of edge effects has been sought in reference [5], which reports a numerical Monte Carlo study of the classical Ising model in three spatial dimensions (i.e., n = 1 and d = 3) with Glauber dynamics. The comparison between the theoretical predictions presented here and these numerical data requires some care: in fact, the predictions for θ E reported here are limited to the first-order corrections in a dimensional expansion, which are generally known not to be quantitatively accurate.…”

“…The separate effects on the system of either spatial or temporal boundary [7,8,11,12,21,[26][27][28], including the case in which an initial non-vanishing magnetisation is present [29,30], are well-understood; on the other hand, their interplay has been less extensively studied [5,10,31,32]. Moreover, in reference [31], a power-counting argument has been used for arguing that no new algebraic behaviours arise which are specific to the intersection of the two boundaries (hereafter referred to as the edge) and that all the observed effects are, therefore, a combination of those independently generated by the surface and the quench; almost all the subsequent studies formulated scaling ansatzes based on this assumption.…”

Short-time universal scaling in an isolated quantum system after a quench

“…This is known to occur for quenches in * These authors contributed equally. classical systems in the presence of a thermal bath [49][50][51][52] and, more recently, for quantum impurities [53,54] or open quantum systems [55,56]. A quench introduces a "temporal boundary" by breaking the time-translational invariance (TTI) that characterizes equilibrium dynamics, causing the emergence of short-time universal scaling, analogous to universal short-distance scaling in the presence of spatial boundaries in equilibrium [57][58][59].…”