Dynamical Barriers in the Dyson Hierarchical model via Real Space Renormalization

TL;DR

This paper analyzes the dynamical barriers in the Dyson hierarchical Ising model near zero temperature, deriving explicit growth laws for equilibrium times as a function of system size for different parameter regimes.

Contribution

It provides explicit calculations of dynamical barriers and equilibrium times in the Dyson hierarchical model using Real Space Renormalization, extending previous work.

Findings

For σ<1, barriers grow as a power-law with system size.

For σ=1, barriers grow logarithmically with system size.

Finite contributions to barriers depend on transition rate choices.

Abstract

The Dyson hierarchical one-dimensional Ising model of parameter $\sigma>0$ contains long-ranged ferromagnetic couplings decaying as $1/r^{1+\sigma}$ in terms of the distance $r$. We study the stochastic dynamics near zero-temperature via the Real Space Renormalization introduced in our previous work (C. Monthus and T. Garel, arxiv:1212.0643) in order to compute explicitly the equilibrium time $t_{eq}(L)$ as a function of the system size $L$. For $\sigma<1$ where the static critical temperature for the ferromagnetic transition is finite $T_c>0$, we obtain that dynamical barriers grow as the power-law: $\ln t_{eq}(L) \simeq \beta (\frac{4 J_0}{3(2^{1-\sigma}-1)}) L^{1-\sigma}$. For $\sigma=1$ where the static critical temperature vanishes $T_c=0$, we obtain that dynamical barriers grow logarithmically as : $\ln t_{eq}(L) \simeq [\beta (\frac{4 J_0}{3 \ln 2}) -1] \ln L $. We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.