Enforced Gaplessness by Combining Non-Anomalous Symmetries

Symmetry-enforced gaplessness is a phenomenon in which a quantum system is forced to be gapless due to its global symmetry. Previous works have mainly focused on anomalous symmetries. Another approach, using symmetry spans, which specify two embeddings of a common symmetry into two larger symmetries, was proposed by the authors. In this work, we show that a symmetry span can enforce gaplessness even when both larger symmetries are invertible and non-anomalous: each alone admits a trivially gapped phase, but because the embeddings differ by a twist associated with a symmetry-protected topological phase, the symmetry they generate admits no gapped phase with finitely many ground states. We present such a span of a finite group and $\mathrm{U}(1)^2$ in 1+1 dimensions, with continuum realizations and a lattice Hamiltonian, with a finite-dimensional local Hilbert space and finitely many parameters, that is gapless in any region of its parameter space. We also discuss a 2+1-dimensional example and spans of only finite symmetries.