On a bulk gap strategy for quantum lattice models

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys. 63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré 24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.