A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions

TL;DR

This paper proves that quantum systems with higher gauge-like symmetries in any dimension have exponentially large spectral degeneracies, especially under certain boundary conditions, extending Lieb-Schultz-Mattis type results.

Contribution

It establishes lower bounds on spectral degeneracy for systems with higher symmetries using boundary condition effects, generalizing previous theorems to arbitrary dimensions.

Findings

Systems with non-commuting gauge-like symmetries have exponential spectral degeneracy.

Boundary conditions significantly influence spectral degeneracy in large systems.

Infrared-ultraviolet mixing can lead to large degeneracies despite conventional behaviors.

Abstract

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (Gauge Like) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in approaching the thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels. We briefly discuss why, in spite of the proven large degeneracy associated with infrared-ultraviolet mixing, some systems may still exhibit conventional physical behaviors, i.e., of those of systems with non-extensive degeneracies, due to entropic "order by disorder" type effects.