Hilbert band complexes and their applications

TL;DR

This paper introduces a novel mathematical framework using Hilbert basis to analyze band connectivity in condensed matter physics, providing a complete solution to band reducibility and decomposition.

Contribution

It develops a new method translating symmetry conditions into band balance equations, revealing that band structures can be described by a positive affine monoid and introducing Hilbert band complexes as fundamental building blocks.

Findings

Hilbert basis determines band complex reducibility and decomposition

Algorithms for constructing and merging Hilbert band complexes are developed

Examples include complexes related to bipartite graphs and unbounded growth scenarios

Abstract

The study of band connectivity is a fundamental problem in condensed matter physics. Here, we develop a new method for analyzing band connectivity, which completely solves the outstanding questions of the reducibility and decomposition of band complexes. By translating the symmetry conditions into a set of band balance equations, we show that all possible band structure solutions can be described by a positive affine monoid structure, which has a unique minimal set of generators, called Hilbert basis. We show that Hilbert basis completely determine whether a band complex is reducible and how it can be decomposed. The band complexes corresponding to Hilbert basis vectors, termed as Hilbert band complexes (HBCs), can be regarded as elementary building blocks of band structures. We develop algorithms to construct HBCs, analyze their graph features, and merge them into large complexes. We find some interesting examples, such as HBCs corresponding to complete bipartite graphs, and complexes which can grow without bound by successively merging a HBC.