It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved.

Schrödinger operators with Sturmian potentials have been studied extensively, and a central question, whether every gap predicted by gap labeling actually appears in the spectrum, was recently resolved affirmatively by Band, Beckus, and Loewy. We consider two natural generalizations of Sturmian sequences: codings of rotations and quasi-Sturmian sequences. For both classes, we use the Johnson-Schwartzman gap labeling theorem to identify the set of admissible gap labels; in the quasi-Sturmian setting, this gives the first description of the Schwartzman group for these subshifts. For binary codings of rotations, we go further and show that every admissible label is attained by some Schrödinger operator in the associated family. More precisely, for each label predicted by gap labeling, there exists a sampling function for which the corresponding gap is open.

Given a periodic Schrödinger operator with analytic potential and rational frequency α, let S_- denote the intersection of its spectra taken over the phase x in a torus. We show that up to sets of Lebesgue measure zero, S_- associated with α could be obtained asymptotically from S_- associated with rationals approximating α that satisfy certain approximating properties. We will talk more about the proof details. This work is joint with Svetlana Jitomirskaya and Xianzhe Li. This talk is part of the WSMP program.

A key signature of MBL (many-body localization) is the slow rate at which information spreads. In this talk I will describe my recent results with Elgart showing that the infinite random XXZ spin-1/2 chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.

I will not assume that the audience is familiar with random spin chains. I will introduce the infinite random XXZ spin-1/2 chain, state the main result, and describe some important ingredients for the proof.