A central question in the study of quantum many body systems is to classify models by their phase of matter, and a fundamental quantity in this classification is the existence or non-existence of a spectral gap above the ground state energy. While the importance of the spectral gap is well known, it is a notoriously difficult to rigorously determine the existence of a nonvanishing gap. For frustration-free quantum lattice models, there are only two classes of general methods for proving spectral gaps: methods based on localizing excitations via ground state projections, and finite size criteria.
In this talk, we will review methods from each of these classes and apply them to two different models: the AKLT model on the decorated honeycomb lattice, and a truncated Haldane pseudopotential for the fractional quantum Hall effect. While the approaches used to prove these gaps have some commonalities, the different properties of the ground states play a key role in how one approaches proving a spectral gap. In the former, the tensor network state description of the ground states is at the heart of the proof. For the latter, the ground states are exponentially degenerate in the system size, and to overcome this combinatorial issue one employs a strategy based on identifying invariant subspaces. For both models we will describe the ground states and their properties, and discuss the strategies for proving the gap.
This talk is based off joint two joint works. The first with H. Abdul-Rahman, M. Lemm, A. Lucia and B.Nachtergaele, and the second with S. Warzel.