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 [1]. 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 [2]. 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 [3]. 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.

References:
[1] Phys. Rev. Lett. 107, 097205 (2011)
[2] Phys. Rev. Lett. 109, 036403 (2012)
[3] R. Movassagh “Generic Local Hamiltonians are Gapless”, (2017)
arXiv:1606.09313v2 [quant-ph]