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. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108…

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. Nevertheless, it admits a a rather simple, full description in terms of a…

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. Perturbing it with a sequence of rank $t$ matrices, with $t$ taking the values $1\le t \le N$, we study the expected difference between the spectra of the  perturbed and the reference matrices as a…

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. The existence of such a spectral gap has far-reaching consequences, e.g…

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. In this talk, I will show that the '63 effective theory…

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.

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). This field is (or should be)  related to arelativistic quantum field theory with one time and d-1 space…

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. I will explain the origins of this conjecture and…

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. In particular, this yields a new short proof of the…

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

Unlike their classical counterparts, quantum antiferromagnetic systems exhibit ground state frustration effects even in one dimension.  A case in point is the SU(2S+1) invariant spin chain, with the interaction between pairs of neighboring S-spins given by…

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.

Using the fractional moment method it is shown that, within the Hartree-Fock…

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…