Dynamical Barriers in the Dyson Hierarchical model via Real Space Renormalization

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: $\ln t_{eq}(L)\simeq \beta (\frac{4J_0}{3(2^{1-\sigma }-1)})L^{1-\sigma }$. For $\sigma =1$ where the static critical temperature vanishes $T_c=0$, we obtain that dynamical barriers grow logarithmically as : $\ln t_{eq}(L)\simeq [\beta (\frac{4J_0}{3\ln 2})-1]\ln L$. We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.

13 pages

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• Quantum mechanical and information theoretic view on classical glass transitions
• Poisson statistics of eigenvalues in the hierarchical Anderson model
• Overlap Interfaces in Hierarchical Spin-Glass models