Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice

TL;DR

This paper extends Lieb-Schultz-Mattis arguments to quantum many-particle systems on periodic lattices, establishing that a finite excitation gap requires an integer particle number per unit cell, regardless of dimension, interaction, or statistics.

Contribution

It generalizes the Lieb-Schultz-Mattis theorem to include systems with conserved particle number in arbitrary dimensions, linking excitation gaps to particle filling.

Findings

Finite excitation gap only when particle number per unit cell is integer.

Extension of Lieb-Schultz-Mattis argument using Laughlin's quantized Hall conductivity.

Applicable to quantum spin systems and various particle statistics.

Abstract

Combined with Laughlin's argument on the quantized Hall conductivity, Lieb-Schultz-Mattis argument is extended to quantum many-particle systems (including quantum spin systems) with a conserved particle number, on a periodic lattice in arbitrary dimensions. Regardless of dimensionality, interaction strength and particle statistics (bose/fermi), a finite excitation gap is possible only when the particle number per unit cell of the groundstate is an integer.