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

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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 df=((2K))/ 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 teq(L) as a function of the system size L near zero-temperature. For the pure Ising model, we obtain the behavior teq(L) Lα eβ²ʲ ˡᵈˢ where dₛ=df-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 Beq(L) ≡ (teq/β) near zero temperature. For the fractal dimension df=2, we obtain that the dynamical barrier scales as Beq(L)= c L+L¹/² 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 L¹/² 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=dₛ/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≃ 0.299 involved in the statics of the random ferromagnet on the same lattice.