Fluctuations of two-time quantities and time-reparameterization invariance in spin glasses

Abstract

This article is a contribution to the understanding of fluctuations in the out of equilibrium dynamics of glassy systems. By extending theoretical ideas based on the assumption that timereparametrization invariance develops asymptotically we deduce the scaling properties of diverse high-order correlation functions. We examine these predictions with numerical tests in a standard glassy model, the 3d Edwards-Anderson spin-glass, and in a system where time-reparametrization invariance is not expected to hold, the 2d ferromagnetic Ising model, both at low temperatures. Our results enlighten a qualitative difference between the fluctuation properties of the two models and show that scaling properties conform to the time-reparametrization invariance scenario in the former but not in the latter.

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz–Thouless phase

“…This suggests that the quantity φ( x, t)ξ( x ′ , t ′ )/(2T ) may play a special physical role and is worth of being studied. Indeed, special and interesting properties are exhibited by this fluctuating quantity in the case of aging spin-glasses [12]. Similar considerations have led to use this same fluctuating quantity in ferromagnetic systems in [11].…”

“…Let us now generalize what done insofar to a scalar field φ( x, t) defined on a d + 1 dimensional space. Taking into account the space dependence the definitions (11) and (12) are replaced by…”

“…Traditionally these systems have been characterized in terms of the space and time dependence of two-point correlation functions and responses (see, e.g., the review articles [4][5][6]). It has been only relatively recently that interest has moved to the study of fluctuations of these quantities and, in particular, their characterization via higher order correlation and response functions [7][8][9][10][11][12]. These properties should, obviously, give us a more detailed description of the behavior of these highly non-trivial processes.…”

“…In its structure this paper follows the presentation in [14] where aging and the usual fluctuation-dissipation (FD) relation were studied in these same three models with the purpose of highlighting the fact that neither disorder nor glassiness are needed to obtain non-trivial dynamic features. Composite operators formed with the above-mentioned fluctuating quantities are related to higher order correlation and response functions [9,11,12,15], and ratios between such composite operators can then be defined which generalize the largely studied FD ratios. The analysis of these ratios in the asymptotic time limit inform us, in particular, on the relevance of the effective temperature notion [16] (for reviews see [17][18][19]) beyond the linear order.…”

A note on weakly discontinuous dynamical transitions

“…In last instance this is a consequence of the emerging scale invariance of the MCT equation when the critical temperature is approached. We see here a link with the theory of fluctuations during aging dynamics below T d developed by Cugliandolo, Chamon and collaborators [23][24][25][26] where fluctuations are ascribed to the large time emergence of reparametrization invariance With all this in mind, we can write:…”