Low-temperature dynamics of long-ranged spin-glasses: full hierarchy of relaxation times via real-space renormalization

Abstract

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via realspace renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.

Strong disorder RG approach – a short review of recent developments

“…As a final remark, it is important to stress that for many-body classical models, the fact that the above RG rules are defined in configuration space clearly limits the numerical implementation to small sizes. As a consequence, various other types of real-space RG procedures have been developed for the dynamics of classical spin models, in particular boundary-RG for the dynamics in d = 1 [263] and on the Cayley tree [264] or block-RG for for the dynamics of Long-Ranged ferromagnetic [265] or Spin-Glass models [266,267].…”

Numerical study of the dynamics of some long range spin glass models

“…In a recent paper, Monthus and Garel [11] give numerical evidence that the equilibrium relaxation time τ of the 1dLR model with Gaussian distributed quenched couplings scales like E(ln(τ )) ∝ N ψ with ψ ≈ 1/3, like in the SK model (see [12,13,14,15] and references therein), for all values of σ ∈ [0, 1]. This is a surprising result since the static properties of the model do depend on σ for σ > 2/3, and it contradicts the later analytical prediction that [16] ψ = 1 − σ in this region. The results of [11] have been obtained for the temperature T = T c /2, but it is natural to expect that ψ is temperature independent below T c .…”

Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for Long-Ranged Quantum Spin Chains with or without disorder

“…More generally, the Dyson hierarchical versions of many long ranged models have been considered since the original Dyson hierarchical classical ferromagnetic Ising model [45] that has been much studied by both mathematicians [46][47][48][49] and physicists [50][51][52][53], including the properties of the dynamics [54,55]. In the field of classical disordered systems, equilibrium properties have been analyzed for random fields Ising models [56,57] and for spin-glasses [58][59][60][61][62][63], while the dynamical properties are discussed in Refs [55,64]. Finally, let us mention that Dyson hierarchical models have been also considered for Anderson localization models [65][66][67][68][69][70][71][72] and for Many-Body-Localization [73].…”