Georgia Institute of Technology College of Sciences

The classification of quantum phases of matter is a fundamental topic in the study of quantum many-body systems. One main question in this study is whether or not there is a nonvanishing spectral gap above the ground state energy and, if so, whether or not this gap is stable under sufficiently short-range perturbations. In their seminal work, Affleck, Kennedy Lieb and Tasaki (AKLT) conjectured the existence of a spectral gap for the model they defined on the hexagonal lattice.

I will give a brief overview of spectral theory of discrete Schrodinger-type operators on periodic graphs and discuss the recent result (joint with Matthew Faust) on the absence of flat bands for generic potentials.

In this talk we consider the Hartree energy functional for an ensemble of identical fermions. We prove that the minimizers satisfy an important set of semi-classical commutator estimates (SCEs), which encode the uniform regularity of the states in the semi-classical parameter. In the recent years, the SCEs have been shown to play a key role in the quantitative derivation of Hartree-Fock and Vlasov dynamics from large systems of fermions, and are typically implemented as assumptions in the initial data.

I'll review a few known results about quantum graphs that maximize or minimize eigenvalues or combinations of eigenvalues, and will then concentrate on ratios of eigevalues under topological constraints on the graph.  In particular a new discovery with James Kennedy and Gabriel Ramos is that the largest ratio of the first two eigenvalues of the Laplacian on a finite tree graph with Dirichlet conditions at the ends is achieved by equilateral stars.

The study of the electronic properties of twisted bilayer graphene (TBG) has garnered much attention from the condensed matter community recently. TBG is obtained by stacking two graphene monolayers on top of each other, and rotating one of them with respect to the other. Theoretical and experimental analyses have found that the electronic properties of TBG depend very strongly on the angle between the layers. In fact, a handful of “magic” angles have been predicted at which TBG becomes a superconductor, and this has even been verified experimentally.

We present joint work with Artur Avila on delocalizing Schr\"odinger operators in arbitrary dimensions via arbitrarily small perturbations of the potential. As a consequence we obtain an analog of Simon's Wonderland Theorem for the case of dynamically defined potentials. We will discuss a mechanism based on the Feynman-Hellmann Theorem, whose infinite volume limit is instrumental in establishing delocalization in infinite volume. Furstenberg's correspondence principle then yields the desired delocalization statement in finite volume.

The local spacing of zeros of orthogonal polynomials is studied using scaling limits of Christoffel--Darboux kernels. Different limit kernels are associated with different universality classes, e.g. sine kernel with bulk universality and locally asymptotically uniform zero spacing. In recent years, new results have been obtained by using the de Branges theory of canonical systems.

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed.

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors D\ge2 except for D=4, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the distribution of the matrix element fluctuations for a randomly chosen eigenbasis looks Gaussian in the semiclassical limit N\to\infty, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases.