In this talk I will give an overview of recent progress regarding mathematical questions and proofs of the spectral properties of Haldane pseudo-potentials.

# Events Archive

## Mathematical Physics Seminar

### Mathematical Physics Seminar, Tues, Nov 30, 4:30 pm, Jadwin A-06, Simone Warzel, TU-Munich, “The Spectral Gap in Fractional Hall Systems”

### Mathematical Physics Seminar, Tue Nov. 9, 4:30 p.m. Jadwin A-06 - Simon Becker, Courant Institute, "Mathematical Properties of Twisted Bilayer Graphene"

Twistronics is the study of how the angle (the twist) between layers of two-dimensional materials can change their electronic structure.

### Mathematical Physics Seminar, Tues, Nov 2, 4:30pm, Jadwin A06, Jacob Shapiro, Princeton University, “Depinning in the integer-valued Gaussian field and the BKT phase of the 2D Villain model”

It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning.

### The Universe from a single Particle - Michael Freedman - Microsoft Q

I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) - can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates.

### Mathematical Physics Seminar, Tues, Oct 26, 4:30pm, Jadwin A06, Terry Loring, University of New Mexico, “EMERGENT TOPOLOGY FROM FINITE VOLUME TOPOLOGICAL INSULATORS”

Finite volume (or area) models for topological insulators are closer to experiment than in_nite volume models. However, they are only indirectly connected to Brillouin zone and so we need to look elsewhere for a topological space.

### Mathematical Physics Seminar, Tue, Oct 12, 4:30pm Jadwin A-06, Giorgio Cipolloni, PCTS, "Extension of Wigner's Vision to Quantum Chaos"

After reviewing recent progress in random Hermitian and non-Hermitian matrix theory, we prove that eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH), which is a strong form of Quantum Unique ergodicity (QUE) with optimal speed of convergence.

### Mathematical Physics Seminar: Simone Warzel,TU-Munich, "Spectral Gaps, Incompressibility and Fragmented Matrix-Product State in a Fractional Quantum Hall System"

In the thin cylinder regime Haldane’s pseudo-potential corresponding to one-third filling results in a frustration-free fermionic lattice Hamiltonian which is dipole-conserving with an added electrostatic interaction. Its zero-energy eigenspace is exponentially large.

### Mathematical Physics Seminar: Uzy Smilansky, Weizmann Institute, "Systematics of spectral shifts in random matrix ensembles"

We chose a $N\times N$ Hermitian matrix randomly picked from one of the random Gaussian matrix ensembles $(\beta =1,2,4)$ - the reference matrix.

### Mathematical Physics Seminar: Marius Lemm, Harvard University, "Spectral Gaps in Quantum Spin Systems"

Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state.

### Mathematical Physics Seminar: Ian Jauslin, Princeton University, "A Simplified Approach to Interacting Bose Gases"

I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation.

### Mathematical Physics Seminar: Michael Kiessling, Rutgers University, "Quantization of the electromagnetic field as a consequence of old-fashioned semi-relativistic quantum mechanics of radiating atoms''

Born's probabilistic interpretation of Schroedinger's wave function is shown to lead to a semi-relativistic quantum mechanics of atoms, molecules, etc., coupled with electromagnetic radiation. No second quantization is invoked, yet the photon naturally shows up in this formulation.

### Mathematical Physics Seminar: Charles Newman, Courant Institute, "A Gaussian Process Related to the Mass Spectrum of the Near-Critical Ising Model"

The continuum scaling limit of the Ising model in d dimensions at the critical temperature whose magnetic field properly scales to zero with lattice spacing is (or should be) a non-Gaussian generalized random field Phi for d = 2 (and d = 3).

### Math Physics Seminar | Paul Bourgade, NYU | "Branching processes in random matrix theory and analytic number theory" | Jadwin A06

Fyodorov, Hiary and Keating have conjectured that the maximum of the characteristic polynomial of random matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the size of local maxima of L-function along the critical axis.

### Math Physics | Martin Fraas, Virginia Tech | "A many-body Fredholm index for ground state spaces and Abelian anyons" | Jadwin A06

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered p-dimensional ground state sector. The index is fractional with the denominator given by p.

### Math Physics Seminar | Michael Aizenman, Princeton | ``Geometric analysis of frustration in a class of SU(2S+1) invariant quantum spin chains’’ | Jadwin A06

PLEASE NOTE THIS TALK FROM 10/8 HAS BEEN RESCHEDLED TO 10/22/2019 -- THANK YOU!

### Math Physics Seminar | Rodrigo Matos, Michigan State | "Localization in the Disordered Hubbard Model within Hartree-Fock Theory" | Jadwin A06

We aim at presenting a simple context where known results on Anderson localization for systems of non-interacting particles in a random environment may be extended to systems with weak interactions.

### Math Physics Seminar, | Simone Warzel, TU, Munich | "The Quantum Random Energy Model" | Jadwin A06

The quantum random energy model serves as a simple cornerstone, and a testing ground, for a number of fields. It is the simplest of all mean-field spin glass models in which quantum effects due to the presence of a transversal field are studied. Renewed interest in its spectral properties arose recently in connection with many-body...

### Math Physics Seminar | Amanda Young, U. of Arizona, Tucson | "A gapped family of two-dimensional AKLT models" | Jadwin 343

The one-dimensional AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki conjectured that the two-dimensional version of their model on the hexagonal lattice also exhibits a spectral gap.

### Math Physics Seminar | Ramis Movassagh, IBM Research | "Highly entangled spin chains: Exactly solvable counter-examples to the area law" | Jadwin 343

In recent years, there has been a surge of activities in proposing "exactly solvable" quantum spin chains with surprising high amount of ground state entanglement--exponentially more than critical systems that have $\log(n)$ von Neumann entropy. We discuss these models from first principles.

### Math Physics Seminar | Jacob Shapiro, Columbia U. | "The topology of strongly-disordered chiral chains" | Jadwin Hall 343

Abstract: We show how in the strongly-disordered, gapless regime, the topological properties of chiral chains (class AIII 1D in the Kitaev table) can be read off from the Lyapunov spectrum of the system at zero energy, and prove that these objects are topologically stable in the strongly-disordered gapless regime.

(Based on joint...