Research Interests

I apply analytical techniques to study the existence and stability of spectral gaps in quantum lattice systems. Many important properties of these systems are determined by the low-lying energy spectrum and whether there is a spectral gap above the ground state energy. A non-vanishing spectral gap is a key assumption in many results in condensed matter physics and quantum information theory. While the importance of the gap is known, there are only a handful methods for establishing its existence and few rigorously proven results. My research focuses on developing techniques for determining spectral gaps, and the study of gapped ground state phases. For a thorough discussion of my past and current research, please see my research statement.

Non-Vanishing Spectral Gaps

One pillar of my research program focuses on developing methods for estimating gaps by rigorously answering the gap question for fundamental models. The spectral gap is known to be generically undecidable, making it necessary to develop new techniques for estimating the spectral gap. The spectral gap question is often more difficult for multi-dimensional models (as the degeneracy of ground state space if often related to boundary conditions), and so there is a particular interest in developing methods suitable for these systems. Models I've worked on include:

  • Haldane pseudopotential: These Hamiltonian are expected to faithfully describe all important features of fractional quantum Hall systems and rotating Bose gases. The fractional quantum Hall effect (FQHE) is a phenomenon that results from interacting charge degrees of freedom on a two-dimensional manifold with a perpendicular magnetic field. It was later realized that rapidly rotating Bose gases also behave like fractional quantum Hall systems. Two properties characterize this state of matter: the incompressibility of the liquid into which the electrons condense, and the existence of a spectral gap above which are excitations with fractional charge. While numerical evidence supports that the Haldane pseudopotentials satisfy these conditions, it is still a major open problem to prove them rigorously. We are currently working towards this goal, and have already proved these properties for truncated versions of the one third and half filled models in a cylinder geometry. With open boundary conditions, our estimates reflect the presence of edge states whose energy tends to zero with the radius of the cylinder. In our work for the 1/2-filled model, we were also able to prove a bulk gap for the model with periodic boundary conditions. Our method is robust and also applies to the 1/3-filled model.

    (Collaborators: B. Nachteragele and S. Warzel)

  • Multi-Dimensional AKLT Models: The one-dimensional AKLT model was the first quantum spin chain proved to be in the Haldane phase, which includes a spectral gap. Generalizations of the AKLT model have also been introduced, and conjectured to be gapped. These models are also of significant interest, e.g., as they are the canonical example tensor network state (TNS) models, and constitute universal quantum computation resources. My goal is to produce analytical methods for proving spectral gaps of more general TNS models by analyzing key AKLT models.

    (Collaborators: H. Abdul-Rahman, M. Lemm, A. Lucia, and B. Nachtergaele)

Stability of Gapped Ground State Phases

A second central topic of my research is the study of quantum phases of matter and the stability of the spectral gap. A quantum phase of matter is a collection of models that can be connected along a smooth path of gapped Hamiltonians. Many properties are preserved for models belonging to the same phase. A natural next question for a gapped model is whether the gap remains open in the presence of perturbations. Such stability is of particular interest, e.g., for models in topological phases due to their potential for producing robust quantum memory. We have been exploring how to adapt a technique for stability in topological systems to more general gapped phases such as, e.g., models with broken discrete symmetries and topological insulators.

(Collaborators: B. Nachtergaele and R. Sims)

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