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Series: Math Physics Seminar

Electrons possess both spin and charge. In one dimension, quantum theory predicts that systems of interacting electrons may behave as though their charge and spin are transported at different speeds.We discuss examples of how such many-particle effects may be simulated using neutral atoms and radiation fields. Joint work with Xiao-Feng Shi

Series: Math Physics Seminar

Localization properties of quantum many-body systems have been a very active subject in theoretical physics in the most recent decade. At the same time, finding rigorous approaches to understanding many-body localization remains a wide open challenge. We will report on some recent progress obtained for the case of quantum spin chains, where joint work with A. Elgart and A. Klein has provided a proof of several manifestations of MBL for the droplet spectrum of the disordered XXZ chain.

Series: Math Physics Seminar

TBA

Series: Math Physics Seminar

This talk will be focused on the large deviation theory (LDT) for Schr\"odinger cocycles over a quasi-periodic or skew-shift base. We will also talk about its connection to positivity and regularity of the Lyapunov exponent, as well as localization. We will also discuss some open problems of the skew-shift model.

Series: Math Physics Seminar

Consider a relativistic electron interacting with a nucleus of nuclear charge Z and coupled to its self-generated electromagnetic field. The resulting system of equations describing the time evolution of this electron and its corresponding vector potential are known as the Maxwell-Dirac-Coulomb (MDC) equations. We study the time local well-posedness of the MDC equations, and, under reasonable restrictions on the nuclear charge Z, we prove the existence of a unique local in time solution to these equations.

Series: Math Physics Seminar

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).Pipe, channel and plane flows, however, are flows on infinite spatial domains. We propose to recast the Navier-Stokes equations as a space-time theory, with the unstable invariant solutions now being the space-time tori (and not the 1-dimensional periodic orbits of the classical periodic orbit theory). The symbolic dynamics is likewise higher-dimensional (rather than a single temporal string of symbols). In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.We illustrate the strategy by solving a very simple classical field theory on a lattice modelling many-particle quantum chaos, adiscretized screened Poisson equation, or the ``spatiotemporal cat.'' No actual cats, graduate or undergraduate, have showninterest in, or were harmed during this research.

Series: Math Physics Seminar

The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while the latter is represented by an ellipsoid in three space. They are both experimentally accessible in many physical systems. By considering transport of the spin vector along loops in the unit ball it is shown that the spin fluctuation tensor picks up geometric phase information. For the physically important case of spin one, the geometric phase is formulated in terms of an SO(3) operator. Loops defined in the unit ball fall into two classes: those which do not pass through the origin and those which pass through the origin. The former class of loops subtend a well defined solid angle at the origin while the latter do not and the corresponding geometric phase is non-Abelian. To deal with both classes, a notion of generalized solid angle is introduced, which helps to clarify the interpretation of the geometric phase information. The experimental systems that can be used to observe this geometric phase are also discussed.Link to arxiv: https://arxiv.org/abs/1702.08564

Series: Math Physics Seminar

I'll report on a project, developed in collaboration with Michael Loss, to extend a very simple model of rarefied gas due to Mark Kac and use it to understand some basic issues of Equilibrium and Non-Equilibrium Statistical Mechanics.

Series: Math Physics Seminar

Quantum theory includes many well-developed bounds for wave-functions, which can cast light on where they can be localized and where they are largely excluded by the tunneling effect. These include semiclassical estimates, especially the technique of Agmon, the use of "landscape functions," and some bounds from the theory of ordinary differential equations. With A. Maltsev of Queen Mary University I have been studying how these estimates of wave functions can be adapted to quantum graphs, which are by definition networks of one-dimensional Schrödinger equations joined at vertices.

Series: Math Physics Seminar

In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.