Dynamics of Ising models near zero temperature : Real Space Renormalization Approach

We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time $t_{eq}$ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue $E_1=1/t_{eq}$ of the quantum Hamiltonian $H$. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time $t_{eq}$ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in $d=1$ is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio $K$, we compute the exponential growth of $t_{eq}(N)$ with the number $N$ of generations.