Linked Cluster Expansions via Hypergraph Decompositions

TL;DR

This paper introduces a hypergraph-based method for linked cluster expansions that efficiently handles complex many-site interactions in quantum spin models, enabling more accurate perturbative calculations.

Contribution

It presents a novel hypergraph decomposition approach using hypergraph isomorphism and reduced K"onig representations for linked cluster expansions in quantum models.

Findings

Successfully applied to the plaquette Ising model in a transverse field

Calculated ground-state energy and excitation gap with improved efficiency

Demonstrated the method's applicability to three-dimensional lattice models

Abstract

We propose a hypergraph expansion which facilitates the direct treatment of quantum spin models with many-site interactions via perturbative linked cluster expansions. The main idea is to generate all relevant subclusters and sort them into equivalence classes essentially governed by hypergraph isomorphism. Concretely, a reduced K\"onig representation of the hypergraphs is used to make the equivalence relation accessible by graph isomorphism. During this procedure we determine the embedding factor for each equivalence class, which is used in the final resummation in order to obtain the final result. As an instructive example we calculate the ground-state energy and a particular excitation gap of the plaquette Ising model in a transverse field on the three-dimensional cubic lattice.