Generic Local Hamiltonians are Gapless
June 26th, 2018 RAMIS MOVASSAGH IBM t. J. Watson Research Center

2018/06/05-12, Lecture 1&2: ‘Supercritical Entanglement in Exactly Solvable Models: Counter-Examples to the Area Law for Quantum Matter’

In recent years, there has been a surge of activities in proposing exactly solvable quantum spin chains with the surprisingly high amount of ground state entanglement entropies--beyond what one expects from critical systems described by conformal field theories (i.e., super-logarithmic violations of the area law). We will introduce entanglement and discuss these models. We prove that the ground state entanglement entropy is \sqrt(n) and in some cases even extensive (i.e., ~n). These models have rich connections with combinatorics, random walks, and universality of Brownian excursions. Lastly, we develop techniques that enable proving the gap of these models. As a consequence, the gap scaling rules out the possibility of these models having a relativistic conformal field theory description.

In the last lecture, we will detail the mathematical physics of these models including von Neumann and Renyientropies, correlation function calculations, and underlying Markov chains for quantifying the gap.

2018/06/19, Lecture 3: ‘’Gap of generic local Hamiltonians, and Hamiltonian Density of States’

We prove that quantum local Hamiltonians with generic interactions are gapless. In fact, we prove that there is a continuous density of states arbitrary above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for herein may include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. We calculate the scaling of the gap with the system’s size in the case that the local terms are distributed according to gaussianβ−orthogonal random matrix ensemble.

We then will present a very new result that predicts the density of states (DOS) or the eigenvalue distribution of sums of matrices from the knowledge of the summands. This theory and techniques utilize modern free probability theory and other ideas from random matrices. Applications include the DOS of generic local Hamiltonians, Anderson model, and other ideas from random matrices. Applications include the DOS of generic local Hamiltonians, Anderson model, and disordered Floquetsystems.

2018/06/26, Lecture 4: ‘’Eigenvalue Attraction.Optical Bernoulli Forces’’

Optical Bernoulli Forces
: By Bernoulli’s law, an increase in the relative speed of a fluid around a body is accompanied by a decrease in the pressure. Therefore, a rotating body in a fluid stream experiences a force perpendicular to the motion of the fluid because of the unequal relative speed of the fluid across its surface. It is well known that light has a constant speed irrespective of the relative motion. Does a rotating body immersed in a stream of photons experience a Bernoulli-like force? We show that, indeed, a rotating dielectric cylinder experiences such a lateral force from an electromagnetic wave. In fact, the sign of the lateral force is the same as that of the fluid-mechanical analogas long as the electric susceptibility is positive (\epsilon>\epsilon_{0}), but for negative-susceptibility materials (e.g. metals) we show that the lateral force is in the opposite direction. Because these results are derived from a classical electromagnetic scattering problem, Mie-resonance enhancements that occur in other scattering phenomena also enhance the lateral force.

Tuesday, June 5-12-19-26, 2018, 10:15. ICFO’s Blue Lecture Room