Minimizing couplings in renormalization by preserving short-range mutual information

TL;DR

This paper proposes a novel approach to renormalization in statistical mechanics by focusing on preserving short-range mutual information, which effectively minimizes couplings and enhances the understanding of long-range physics.

Contribution

It introduces a new method for selecting renormalization maps that minimizes short-range mutual information loss, linking it to the reduction of couplings in the renormalized Hamiltonian.

Findings

Minimizing short-range mutual information reduces couplings effectively.

Preserving short-range information relates to long-range physics.

The approach generalizes previous long-range mutual information minimization methods.

Abstract

The connections between renormalization in statistical mechanics and information theory are intuitively evident, but a satisfactory theoretical treatment remains elusive. Recently, Koch-Janusz and Ringel proposed selecting a real-space renormalization map for classical lattice systems by minimizing the loss of long-range mutual information [Nat. Phys. 14, 578 (2018)]. The success of this technique has been related in part to the minimization of long-range couplings in the renormalized Hamiltonian [Lenggenhager et al., Phys. Rev. X 10, 011037 (2020)]. We show that to minimize these couplings the renormalization map should, somewhat counterintuitively, instead be chosen to minimize the loss of short-range mutual information between a block and its boundary. Moreover, the previous minimization is a relaxation of this approach, which indicates that the aims of preserving long-range physics and eliminating short-range couplings are related in a nontrivial way.