Talks

While the simplest Feynman diagrams evaluate to multiple polylogarithms, more complicated functions can arise, involving integrals over higher-dimensional manifolds. Surprisingly, all examples of such manifolds in the literature to date are Calabi-Yau. I discuss why this is, and prove that a specific class of "marginal" diagrams give rise to Calabi-Yau manifolds. I demonstrate a bound on the dimensionality of these manifolds with loop order, and present infinite families of diagrams that saturate this bound to all orders.

This seminar will focus on two cases where the interplay of topology and interactions allows for phases that go beyond simple quasi-particle descriptions. Both models are amenable to sign free auxiliary field quantum Monte Carlo simulations.

First, we design a two-dimensional model consisting of four Dirac-fermion layers on the square lattice. The interaction is given by a four-fermion term where each fermion is from a different layer. In the uncorrelated case, the topology is determined by a Z-valued winding number and previous studies, often using the bulk-boundary correspondence and dimensional reduction arguments, predict the reduction to a Z4 classification in the presence of correlations. An adiabatic path between formerly distinct phases has to visit a strongly interacting state that cannot be described on a mean-field level. We study the phase diagram of the full bulk system and find a symmetry broken state separating topological distinct phases. An attempt to frustrate the ordered state introduces a first order phase transition [Fig. 1 (left)].

Second, we consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a Z2 spin liquid and a fer- romagnetic phase. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the Z2 spin liquid phase due to the breakdown of Kondo screening. Since our model has an odd number of spins per unit cell, this phase is a non-Fermi liquid, also called fractionalized Fermi liquid, that violates the conventional Luttinger theorem, which relates the Fermi volume to the particle density in a Fermi liquid. We probe the Kondo breakdown in this non-Fermi liquid phase via conventional observables such as the spectral function, and also by studying the mutual information between the electrons and the spins [Fig. 1 (right)].

Figure 1: Schematic sketch of the phase diagram discussed during the beginning of the seminar (left). Numerical Results consistent with a FL* to magnetic insulator transition as subject of the second half (right).

Mirković-Vilonen cycles are certain algebraic cycles in the affine Grassmannian that give rise to a particular weight basis (the MV basis) under the Geometric Satake equivalence. I will state a conjecture about the Weyl group action on weight-zero MV cycles and equivariant multiplicities. I can prove it for small coweights in type A. Equivalently, I show that the MV basis agrees with the Springer basis. I have similar results for the Ginzburg-Nakajima basis. A primary tool is work of Braverman, Gaitsgory and Vybornov.

We present a consistent theory of classical gravity coupled to quantum field theory. The dynamics is linear in the density matrix, completely positive and trace-preserving, and reduces to Einstein's equations in the classical limit. Several no-go theorems are evaded since the assumption that gravity is classical necessarily modifies the dynamical laws of quantum mechanics -- the theory must be fundamentally stochastic involving finite sized and probabilistic jumps in space-time and in the quantum field. Nonetheless the quantum state of the system can remain pure conditioned on the classical degrees of freedom. The measurement postulate of quantum mechanics is not needed since the interaction of the quantum degrees of freedom with classical space-time necessarily causes collapse of the wave-function. More generally, we derive a form of classical-quantum dynamics using a non-commuting divergence which has as its limit deterministic classical Hamiltonian evolution, and which doesn't suffer from the pathologies of the semi-classical theory. Details at http://arxiv.org/abs/1811.03116