Insulator, conductor and commensurability: a topological approach

TL;DR

This paper explores how the conduction properties of many-particle systems on periodic lattices are topologically linked to their energy spectra, revealing specific spectral features associated with insulating states.

Contribution

It introduces a topological framework connecting particle density fractions to low-energy spectral states in insulators across one and two dimensions.

Findings

Insulators with fractional particle densities have characteristic low-energy states.

The spectral properties depend on the irreducible fraction of particles per unit cell.

The approach applies to systems at zero temperature.

Abstract

I discuss a topological relation of the conduction property of a many-particle system on a periodic lattice at zero temperature to the energy spectrum. When the particle number per unit cell is an irreducible fraction $p/q$, an insulator must have $q$ low-lying states of energy $o(1/L)$ in one dimension and of energy $o(1)$ in two dimensions, where $L$ is the linear system size.