School on Emergent Phenomena in Non-Equilibrium Quantum Many-Body Systems

Sum of powers of principal minors (SPPM) of matrices appears in the calculation of many quantities in physics and applied mathematics. We show that the calculation of the Renyi entropy of fermionic systems, partition function of the Hubbard model and certain problems in machine learning are related to the SPPM problem. Although there is a simple formula to calculate the sum of principal minors of arbitrary matrices, it has been already shown that calculation of other powers is a hard problem (probably no polynomial time algorithm exists). In this talk I will first discuss how the Renyi (Shannon) entropy in quantum chains detects different phases and determines the universality classes. Then I will show how calculating this quantity boils down to a SPPM problem in certain quantum systems such as Ising chain and free fermions. Finally in the main part of the talk I will write a Grassmann representation for a generic matrix SPPM problem. This field theory-like representation shows interesting symmetries such as U^n(1),  symmetric group,  axial U(1), chiral and particle-hole symmetry. Using this representation one can make a mean-field approximation and find an excellent estimate for the Renyi entropy which can detect the phase transition. We show that some of the mentioned symmetries are broken in, for example, the ferromagnetic phase of the Ising chain.