Translation constraints on quantum phases with twisted boundary conditions

TL;DR

This paper explores how boundary conditions, especially twisted ones on nonorientable spaces like the Klein bottle, impose constraints on quantum phases and ground state degeneracy in two-dimensional quantum magnets.

Contribution

It introduces a novel analysis of Lieb-Schultz-Mattis-type constraints using twisted boundary conditions and quantum anomalies on nonorientable surfaces.

Findings

Translation symmetry on the Klein bottle forbids a unique gapped ground state.

Flux insertion reveals ground-state degeneracy on magnetization plateaus.

Boundary conditions can encode quantum anomalies affecting phase properties.

Abstract

Bulk properties of quantum phases should be independent of a specific choice of boundary conditions as long as the boundary respects the symmetries. Based on this physically reasonable requirement, we discuss the Lieb-Schultz-Mattis-type ingappability in two-dimensional quantum magnets under a boundary condition that makes evident a quantum anomaly underlying the lattice system. In particular, we direct our attention to those on the checkerboard lattice which are closely related to frustrated quantum magnets on the square lattice and on the Shastry-Sutherland lattice. Our discussion is focused on the adiabatic U(1) flux insertion through a closed path in a boundary condition twisted by a spatial rotation and a reflection. Two-dimensional systems in this boundary condition are effectively put on a nonorientable space, namely the Klein bottle. We show that the translation symmetry on the Klein-bottle space excludes the possibility of the unique and gapped ground state. Taking advantage of the flux insertion argument, we also discuss the ground-state degeneracy on magnetization plateaus of the Heisenberg antiferromagnet on the checkerboard lattice.