Speaker
Description
In this talk, I will present a mathematical analysis of Dynamical Mean-Field Theory (DMFT), a Green's function method developed in the 1990s, which provides dynamical correlations associated with the Gibbs state of fermionic quantum lattice models (e.g. the Hubbard model). In particular, I will show the existence of a solution to the corresponding equations in the framework of Iterated Perturbation Theory (IPT).
After a reminder of the models considered by this method (the Hubbard and Anderson impurity models), I will introduce the quantum (one-body time-ordered) Green's function, which can be seen as a time-dependent generalization of the one-particle reduced density matrix (1-pdm). In particular, I will introduce the self-energy and the hybridization function, two central objects in DMFT which share the property of being (negatives of) Nevanlinna-Pick functions (analytic maps from the upper half plane to itself).
After introducing these objects, I give the DMFT equations in the IPT approximation and prove the existence of solutions in functional sets modeling Anderson impurity models with infinite-dimensional baths. This is done by reformulating the equations as a fixed point problem in the space of probability measures, a setting in which we apply the Schauder(-Singbal) theorem. Moreover, we established some properties of the solution(s). (joint work with Éric Cancès & Alfred Kirsch, arxiv:2406.03384).
