Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to two-dimensional spin systems with point group symmetries by introducing a background flux, revealing new symmetry constraints and their implications.

Contribution

It introduces a method to 'projectify' point group symmetries using background flux, broadening the applicability of the LSM theorem to various symmetry settings.

Findings

LSM theorem holds with background flux in 2D spin systems.

The theorem applies to flux-free systems via the flux insertion argument.

New symmetry constraints are established for time-reversal and site-centered C2 symmetries.

Abstract

We discuss the Lieb-Schultz-Mattis (LSM) theorem in two-dimensional spin systems with on-site ${\mathrm U}(1)\rtimes {\mathbb Z}_2$ spin rotation symmetry and point group $C_{2v}$ symmetry about a site. We ``twist" the point group symmetry by introducing a small uniform U(1) flux to obtain a projective symmetry, similarly to the familiar magnetic translation symmetry. The LSM theorem is proved in presence of the flux and then it is demonstrated that the theorem holds also for the flux-free system. Besides, the uniform flux enables us to show the LSM theorem for the time-reversal symmetry and the site-centered $C_2$-rotation symmetry.