Mathematical Physics Seminar

[* This talk presents the core of 2 disjoint work *]

 

Abstract:

1. Eigenvalue Attraction-- Much work has been devoted to the understanding of the motion of eigenvalues in response to randomness. The folklore of random matrix analysis, especially in the case of Hermitian matrices, suggests that the eigenvalues of a perturbed matrix repel. We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. We apply the results to the Hatano-Nelson model, random perturbations of a fixed matrix, real stochastic processes with zero-mean and independent intervals and discuss open problems.

 

Reference : 

Journal of Statistical Physics, Feb. 2016, Volume 162, Issue 3, pp 615-643

http://link.springer.com/article/10.1007/s10955-015-1424-5

 

2. Disordered Hamiltonian Density of States-- The method of "Isotropic Entanglement" (IE), inspired by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue distribution of quantum many-body systems with generic interactions. At the heart is a "Slider", which interpolates between two extrema by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them isotropically. Isotropic means 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 arbitrary disorder and with high accuracy. Theory will be illustrated by numerical experiments.

[Joint work with Alan Edelman]

References:
Phys. Rev. Lett. 107, 097205 (2011)
Phys. Rev. Lett. 109, 036403 (2012)