Fyodorov, Hiary and Keating have conjectured that the maximum of the characteristic polynomial of random matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the size of local maxima of L-function along the critical axis. I will explain the origins of this conjecture and…

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered p-dimensional ground state sector. The index is fractional with the denominator given by p. In particular, this yields a new short proof of the…

PLEASE NOTE THIS TALK FROM 10/8 HAS BEEN RESCHEDLED TO 10/22/2019 -- THANK YOU!

Unlike their classical counterparts, quantum antiferromagnetic systems exhibit ground state frustration effects even in one dimension.  A case in point is the SU(2S+1) invariant spin chain, with the interaction between pairs of neighboring S-spins given by…

We aim at presenting a simple context where known results on Anderson localization for systems of non-interacting particles in a random environment may be extended to systems with weak interactions.

Using the fractional moment method it is shown that, within the Hartree-Fock…

The quantum random energy model serves as a simple cornerstone, and a testing ground, for a number of fields.  It is the simplest of all mean-field spin glass models in which quantum effects due to the presence of a transversal field are studied.  Renewed interest in its spectral properties arose recently in…

The one-dimensional AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki conjectured that the two-dimensional version of their model on the hexagonal lattice also  exhibits a spectral gap. In this talk, we introduce a family of…

In recent years, there has been a surge of activities in proposing "exactly solvable" quantum spin chains with surprising high amount of ground state entanglement--exponentially more than critical systems that have $\log(n)$ von Neumann entropy. We discuss these models from first principles. For a spin chain of length $n$, we prove that the…