Dynamical Barriers in the Dyson Hierarchical model via Real Space Renormalization

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.