This talk is postponed to a later date.
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.
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.
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.
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.
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. Proving the validity of such estimates is, however, not an easy task; up to recently only the linear smooth case was understood. In our work, we provide the first set of examples of states satisfying the SCEs which arise from a nonlinear minimization problem. In particular, two-body interactions up to the repulsive Coulomb potential are included. If time allows it, I will present — as an application — the quantitative convergence of the fermonic N-body ground state, towards the minimizer of the Vlasov functional. Based on joint work with L. Lafleche (ENS Lyon).
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.
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. The model commonly used to study TBG is an effective one, and was derived by Bistritzer and MacDonald. In this talk, I will present recent results on developing a framework to study TBG from first principles. To be more exact, we consider a tight-binding model for the electrons, but make no further approximations. Using a renormalization group technique, we construct a perturbative expansion to study TBG that is convergent when the twisting angle satisfies certain diophantine conditions. This is joint work with V. Mastropietro.
