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Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio $K$ with fractal dimension $d_f=(\ln (2K))/\ln 2$. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$ as a function of the system size $L$ near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \sim L^α e^{β2J L^{d_s}} $ where $d_s=d_f-1$ is the interface dimension, and we compute the prefactor exponent $α$. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers $B_{eq}(L) \equiv (\ln t_{eq}/β)$ near zero temperature. For the fractal dimension $d_f=2$, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where $u$ is a Gaussian random variable of non-zero-mean. While the non-random term scaling as $L$ corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent $ψ=1/2$, in agreement with the conjecture $ψ=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent $θ\simeq 0.299$ involved in the statics of the random ferromagnet on the same lattice.