# Alex Elgart

460 McBryde Hall, Virginia Tech

225 Stanger Street

Blacksburg, VA 24061-1026

I am interested in mathematical physics. Mathematical physics is a branch of pure mathematics with the aim of deriving rigorous results for models suggested by physics.

My research has mainly been concerned with the mathematical study of quantum mechanics, specifically with disordered single- and many-body quantum systems in recent years. The basic phenomenon of localization in the single particle framework is that disorder can cause localization of electron states and thereby manifest itself in properties such as non-spreading of wave packets under time evolution and absence of dc transport. This mechanism for localization was first proposed by the Nobel prize laureate P. W. Anderson in 1958 and is relatively well understood by now, both physically and mathematically.

Many manifestations of single-particle Anderson localization remain valid if one considers a fixed number of interacting particles. The situation is radically different in the many-body setting. Little is known about the thermodynamic limit of an interacting electron gas in a random environment, i.e., an infinite volume limit in which the number of electrons grows proportionally to the volume. Even simplest models where the individual particle Hilbert space is finite dimensional (spin systems) pose considerable analytical and numerical challenges, due to the fact that the number of degrees of freedom involved grows exponentially fast with the size of the system.

The limited evidence from perturbative and numerical approaches supports the persistence of a many-body localized (MBL) phase for one-dimensional spin systems in the presence of weak interactions. The numerics also suggests the existence of transition from a many-body localized (MBL) phase to delocalized phases as the strength of interactions increases.

Mathematically rigorous results on localization in a true many-body system have been until very recently confined to investigations of exactly solvable (quasi-free) models. More recent progress has been achieved primarily in the study of the XXZ spin chain, a system that is not integrable but yet amenable to rigorous analysis.