Gappability Index for Quantum Many-Body Systems

We propose an index ${\mathcal{I}}_G$ which characterizes the degree of ingappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry $G$. ${\mathcal{I}}_G$ represents the dimension of the subspace of ambient uniquely-gapped in the entire $G$-invariant “theory space”. The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case ${\mathcal{I}}_G=0$ (completely ingappable) for the symmetry $G$ including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-$1\mathrm{/}2$ antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.