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

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 $J(r) \propto r^{-1-σ}$ in the region $1/2<σ<1$ where there exists a non-mean-field-like thermal Ferromagnetic-Paramagnetic transition, the RG flows are explicitly solved: the characteristic relaxation time $τ(L)$ follows the critical power-law $τ(L)\propto L^{z_c(σ)} $ at the phase transition and the activated law $\ln τ(L)\propto L^ψ $ with $ψ=1-σ$ in the ferromagnetic phase. For the Spin-Glass model involving random Long-Ranged couplings of variance $\overline{J^2(r)} \propto r^{-2σ}$ in the region $2/3<σ<1$ 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 $τ(L)$ follows some power-law $τ(L)\propto L^{z_c(σ)} $ at criticality and the activated law $\ln τ(L)\propto L^ψ $ in the Spin-Glass phase with the dynamical exponent $ψ=1-σ=θ$ coinciding with the droplet exponent governing the flow of the couplings $J(L) \propto L^θ $.