Gappability Index for Quantum Many-Body Systems

We propose an index ${\mathcal{I}}_{G}$ which characterizes the degree of gappability, 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 theories 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-$1/2$ antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.