Real-space renormalization for the finite temperature statics and dynamics of the Dyson Long-Ranged Ferromagnetic and Spin-Glass models

Abstract

The finite temperature dynamics of the Dyson hierarchical classical spins models is studied via real-space renormalization rules concerning the couplings and the relaxation times. For the ferromagnetic model involving Long-Ranged coupling $J(r)\propto r^{-1-\sigma }$ in the region $1/2<\sigma <1$ where there exists a non-mean-field-like thermal Ferromagnetic-Paramagnetic transition, the RG flows are explicitly solved: the characteristic relaxation time $\tau (L)$ follows the critical power-law $\tau (L)\propto L^{z_c(\sigma )}$ at the phase transition and the activated law $\ln \tau (L)\propto L^{\psi }$ with $\psi =1-\sigma$ in the ferromagnetic phase. For the Spin-Glass model involving random Long-Ranged couplings of variance $\overline{J^2(r)}\propto r^{-2\sigma }$ in the region $2/3<\sigma <1$ where there exists a non-mean-field-like thermal SpinGlass-Paramagnetic transition, the coupled RG flows of the couplings and of the relaxation times are studied numerically : the relaxation time $\tau (L)$ follows some power-law $\tau (L)\propto L^{z_c(\sigma )}$ at criticality and the activated law $\ln \tau (L)\propto L^{\psi }$ in the Spin-Glass phase with the dynamical exponent $\psi =1-\sigma =\theta$ coinciding with the droplet exponent governing the flow of the couplings $J(L)\propto L^{\theta }$.

14 pages, 2 figures

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