Global symmetry is often useful in specifying ground states of quantum many-body systems; some symmetries forbid unique gapped ground states via anomaly-matching arguments, also known as Lieb–Schultz–Mattis type constraints. However, ruling out any gapped ground states, including degenerate ones, is significantly harder. Although anomalies of continuous symmetries can exclude such phases, realizing these anomalous symmetries exactly on lattices with finite-dimensional on-site Hilbert spaces is presumably impossible.

In this talk, we propose a different (but likely related) approach to exclude all gapped ground states in one spatial dimension. Specifically, we study situations where two different symmetries share a common symmetry—for example, where the two symmetries are non-invertible or continuous, and the common one is a non-anomalous group symmetry. We argue that such configurations, which we call symmetry spans, can forbid gapping. We clarify the argument for symmetry spans and gaplessness and illustrate it through concrete examples.

This talk is based on ongoing work with Kantaro Ohmori (RIKEN iTHEMS).