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

Abstract

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.

30 pages, v2=final version

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Cited by in corpus (7)

• Strong Disorder RG approach - a short review of recent developments
• Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment
• Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
• Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices
• Dynamical barriers for the random ferromagnetic Ising model on the Cayley tree : traveling-wave solution of the real space renormalization flow
• Phase ordering dynamics of reconstituting particles
• Exact solution of the Glauber-Ising model on the finite-length semi-open chain