The talk is about the dynamics of a tracer particle coupled strongly to a dense non-interacting electron gas in one or two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times, i.e., the electron gas becomes transparent. However, the correct phase factor is non-trivial. To leading order, it is given by mean-field theory, but one also has to include a correction coming from immediate recollision diagrams.
We introduce a notion of beta-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded beta-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schrodinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.
We propose a method, inspired by Free Probability Theory and Random Matrix Theory, that predicts the eigenvalue distribution of quantum many-body systems with generic interactions . At the heart is a 'Slider', which interpolates between two extremes by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them 'free'. By 'free' we mean that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with an arbitrary disorder and with high accuracy . Theory will be illustrated by numerical experiments.
[Joint work with Alan Edelman]
Time permitting we will prove that quantum local Hamiltonians with generic interactions are gapless . In fact, we prove that there is a continuous density of states arbitrary close to the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble.
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 Phys. Rev. Lett. 107, 097205 (2011)
 Phys. Rev. Lett. 109, 036403 (2012)
 R. Movassagh 'Generic Local Hamiltonians are Gapless', (2017)
Scattering amplitudes are the heart of particle physics, forming a bridge between theory and experiment. The last decade has seen the rise of 'amplitudeology', a program for uncovering the hidden mathematical structures of scattering amplitudes. By importing recent results in mathematics, amplitudeology has produced new insights into scattering amplitudes - such as the 'amplitudehedron' - and translated them into practical computational techniques. In this talk, I will discuss a cluster algebra structure of scattering amplitudes in N=4 Super Yang-Mills theory and deep connections with Goncharov polylogarithms. In order to develop a computational framework which exploits this connection, I show how to construct bases of Goncharov polylogarithms that can be used to describe 6-particle scattering at any order in perturbation theory.
Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, described by Lindbladian dynamics, where dissipative processes drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in many-body models.
Machine learning, which draws from a diversity of fields including computer science, mathematics, and physics, has been taking the world by storm due to its flourishing industrial applications. We present a live demo of machine learning in action, where we train a neural network to classify handwritten digits to an appreciable degree in real time. We then proceed to give an introductory overview of the ingredients that go into this task: logistic regression, stochastic gradient descent, and deep learning. (Part II, which connects machine learning to approximation theory and quantum physics, will take place the following week.)
Precision predictions for high energy experiments at the Large Hadron Collider
require the evaluation of Feynman integrals. In this talk I will discuss how
Feynman integrals can be evaluated in terms of multiple polylogarithms using
differential equations and the coproduct.
Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824
Phone: (517) 353-0844
Fax: (517) 432-1562
College of Natural Science