Suppose we know densities of eigenvalues/energy levels of two Hamiltonians HA and HB. Can we find the eigenvalue distribution of the joint Hamiltonian HA+HB? Free probability theory (FPT) answers this question under certain conditions. My goal is to show that this result is helpful in physical problems, especially finding the energy gap and predicting quantum phase transitions. As an example, I will consider generic artificial topological systems created by a periodic drive, including Floquet Kitaev chain, and show how FPT can be used to predict and characterize disorder- and noise-driven phase transitions.

(pizza and drinks served 10 min. before talk)