A Markov chain approach to renormalization group transformations

TL;DR

This paper explicitly characterizes the renormalized Hamiltonian of a 1D Ising model after decimation, using Markov chains to analyze the renormalization flow and eigenvalues related to critical exponents, illustrating universality.

Contribution

It introduces a Markov chain approach to analyze renormalization group transformations for the 1D Ising model, providing new methods to verify critical exponent relations.

Findings

Eigenvalues of the linearized transformation encode critical exponents.

The Markov-Gibbs equivalence facilitates understanding of decimation effects.

The universality of the renormalization map is demonstrated.

Abstract

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov-Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear important information about all six of the critical exponents. This illustrates the universality property of the renormalization group map in this case.