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. Significant evidence supporting this long-standing claim was only recently achieved and naturally leads to the question of whether or not this gap is stable. In this talk, we review this and other recent progress on proving gaps for AKLT models and then turn to the question of whether these gaps are stable. One avenue for establishing gap stability pioneered by Bravyi, Hastings and Michalakis is to prove that the finite volume ground states are sufficiently indistinguishable by local observables in the bulk - a property known as Local Topological Quantum Order. We discuss a forthcoming work which uses cluster expansion techniques to prove that the ground states of the AKLT model on the hexagonal lattice and Lieb lattice satisfy LTQO. This talk is based on joint work with Thomas Andrew Jackson and Bruno Nachtergaele.

Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchial lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes.  At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails.  This results in a sequence of random graphs, which can be interpreted as a resistor network.  Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up.  In this talk, I will discuss what happens in the critical case when fair coins are used.  Starting with a new recursive distributional equation (RDE) proposed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance.  In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.

The capillary energy functional is used to model the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 degrees and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 degrees and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.
 

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. Some related, Weyl-sharp estimates of arbitrary eigenvalue ratios can be obtained using similar ideas.  If time permits, I will also describe some optimal results about differences and other combinations of eigenvalues.