Applied Math Seminar
The Green's function view of interacting quantum physics
Speaker: Jason Kaye, Flatiron Institute
Location: Warren Weaver Hall 1302
Date: Friday, December 5, 2025, 2:30 p.m.
Synopsis:
The central challenge in simulating the physics of many interacting quantum particles is the exponential growth of the many-body state space dimension with respect to the number of possible states of a single particle. In the special case of non-interacting particles, the many-body space factorizes and this scaling disappears, but for the strongly interacting systems of contemporary interest in condensed matter theory and experiment, correctly treating interaction effects is critical. Green's function methods offer an alternative to the standard wavefunction-based picture of quantum physics, suggesting systematic and computationally tractable strategies to approximate the effects of particle interactions.
This talk will give a pedagogical introduction to some basic characters of quantum many-body theory, including creation and annihilation operators, the Hubbard model, the single-particle Green's function, the self-energy, and quantum embedding via dynamical mean-field theory (DMFT). We will see that the single-particle Green's function of many-body theory naturally generalizes its more familiar classical counterpart, and that it provides a practical link with experiment via the spectral function. The Green's function framework leads to many interesting applied mathematics problems, and we will focus in particular on the solution of the quantum impurity problem, the bottleneck of DMFT. I will describe a new solver which uses a simple and well-known trick from computational mathematics, sum-of-exponentials expansion and separation of variables, to evaluate the Feynman diagrams arising from a perturbative expansion of the impurity problem.