Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

Abstract

The random-field Ising model shows extreme critical slowdown that has been described by activated dynamic scaling: the characteristic time for the relaxation to equilibrium diverges exponentially with the correlation length, ln τ ∼ ξ ψ /T , with ψ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions d less than a critical value dDR ≃ 5.1, also associated with dimensional-reduction breakdown, ψ = θ with θ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d > dDR on the other hand, ψ = θ − 2λ where θ = 2 and λ > 0 a new exponent. At the upper critical dimension d = 6, λ = 1 so that ψ = 0, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches and droplets. We also propose a way to check the two regimes by computer simulations of long-range 1-d systems.

Does the Adam-Gibbs relation hold in simulated supercooled liquids?

“…We obtain θ = 1.35± 0.06 for GFL and θ ≃ 1.84±0.09 for FP. These values are compatible with either the theoretical prediction θ = d/2 by Kirkpatrick et al [3], or with that of Franz θ = d − 1 [86].…”

Random-field Ising and O(N) models: theoretical description through the functional renormalization group

“…The renormalized temperature flows to zero but the limit is highly nonuniform in the field-dependent cumulants and proceeds via a "thermal boundary layer", as first found in the case of the random elastic manifold model [69][70][71] . This manifestation of the dangerous irrelevance of the temperature leads to anomalous thermal fluctuations and activated dynamic scaling in the RFIM that can both be described by the nonperturbative FRG 77,96 .…”

“…A crucial point is that the exact FRG equations derived within the different formalisms of course coincide when applied to the same situation 75,77,96,108 . In the following presentation we will consider the equilibrium behavior of in the presence of a random field and present the FRG in the context of the Boltzmann-Gibbs formalism which is conceptually simpler and requires lighter notations.…”

Phase diagram and criticality of the random anisotropy model in the large- N limit

“…Such a thermal rounding is however found in the perturbative FRG near d = 4 for any finite N and is furthermore related to the physical picture of finite-T droplet excitations. 21,25,27,29 We therefore take as the most plausible hypothesis that nonanalyticities in the disorder cumulants are rounded by a finite temperature at finite N , unless the long-distance behavior is controlled by a zero-temperature fixed point. In the latter case, when starting the 1-PI FRG flow from the region of parameter space where a cusp is encountered in, say, the fully transverse renormalized disorder function ∆ k (ρ, z), at T = 0, one runs for a small enough bare temperature T into a thermal boundary layer, 21,25,27,29 …”

“…We apply and compare two formalisms: on the one hand, the 2-particle irreducible (2-PI) formalism, and the associated Schwinger-Dysonlike self-consistent equations for the pair correlation functions, which allows one to make contact with the putative replica-symmetry breaking; on the other hand, the 1-particle irreducible (1-PI) functional renormalization group (FRG), which has proven a powerful tool to study the emergence of nonanalytic renormalized disorder cumulants near zero-temperature fixed points and to unveil the associated physics. 7,9,13,[18][19][20][21][22][23][24][25][26][27] Both formalisms are a priori exact in the large N limit but, as shown in the case of a manifold pinned in a random environment, 16,17 studying the 1-PI FRG flow provides a systematic way to find solutions in the regions of parameter space where the self-consistent Schwinger-Dyson-like equations apparently cease to have stable solutions. Quite importantly, it is also generalizable to finite N cases.…”