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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-$1/2$ antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.