Abstract:
Impurities (disorder) can dramatically alter the properties of certain quantum systems. For example, a material that normally conducts electrical current will behave like an insulator in the presence of sufficiently strong impurities, i.e., the wave function of the electron is localized in space when it moves in a medium with random impurities. This is the essence of (the one-body) Anderson localization, which is by now well understood both physically and mathematically. Many-body localization (MBL) is the analogue of Anderson localization for large quantum systems when both disorder and interactions are present. Many-body localized systems do not thermalize under their own dynamics, i.e., they suppress transport. Hence they can be used as quantum storage devices.
In this talk, we discuss the current understanding of the MBL phenomena from a physical and mathematical point of view. Then we present recent MBL results for a class of disordered oscillator systems. In particular, we study the non-equilibrium dynamics of a disordered quantum system consisting of coupled harmonic oscillators in a $d$-dimensional lattice. If the system is sufficiently localized, we show that, starting from a broad class of initial product states that are associated with a tiling (decomposition) of the $d$-dimensional lattice, the dynamical evolution of entanglement scales like the surface area of the subsystem (an area law) in all times after averaging the disorder. Such weak evolution of entanglement is considered as a crucial indicator of MBL. Moreover, for dimensions $d\geq 2$, we discuss how our entanglement bound hints a dependency on the geometry of the subsystems.
Biography:
Houssam Abdul-Rahman is a Visiting Assistant Professor of Mathematics at NYU Abu Dhabi. Before that, he was a Postdoctoral Research Associate at the University of Arizona. He is mainly interested in Mathematical Physics models with an emphasis on studying dynamical properties of disordered many-body quantum systems. His work centers on the mathematical approaches to models that are relevant in quantum information theory and condensed matter physics.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai