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

TL;DR

This paper develops a real-space renormalization method to analyze the relaxation dynamics of Ising models near zero temperature, providing explicit solutions for one-dimensional random chains and Cayley trees.

Contribution

It introduces a novel real-space renormalization approach for the quantum Hamiltonian of Ising dynamics near zero temperature, with explicit solutions for specific geometries.

Findings

Relaxation time in 1D random chains expressed via random couplings.

Exponential growth of relaxation time with generations in Cayley trees.

RG results validated against alternative methods in 1D case.

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.