Physics

Describing the emergence of universality in far-from-equilibrium regimes of many-body quantum systems is a current frontier in physics. One key challenge in non-equilibrium regimes is that the system can explore a large fraction of phase space, therefore identifying relevant degrees of freedom across many length and energy scales is challenging. In these lectures I will focus on isolated manybody systems that are initially quenched from an initial excited states and describe how universality can emerge in an intermediate-time but long-lived prethermal regime. Our main goal will be to identify the relevant degrees of freedom and compute the universal scaling exponents in the prethermal regime, with a special focus on magnetic spin systems and `fractonic' fluids.In the first lecture, I will set the stage by reviewing the universal non-equilibrium dynamics of weaklyinteracting bose gases, which have been extensively studied both theoretically and experimentally. We will begin the discussion by identifying all the possible excitations that may exist at different energy densities. Using kinetic theory for weakly-interacting particles, we will discuss the non-thermal solution associated to wave-turbulence and compute the universal scaling exponents. We will end the lecture by providing a holistic picture of thermalization in bosonic gases.In the second lecture, we will add an additional layer of complexity by considering spin models with a global SU(2) symmetry. We will find the effective theory describing excitations at intermediate energy scales and derive a kinetic model for such excitations. I will show that, in the special case of d=2, the systems exhibits unusually slow thermalization due to the existence of slow modes with quasi-longrange character, giving rise to rise to a universal self-similar regime with anomalous scaling exponents.In the third lecture, we will discuss `fractonic' models which contain kinematic constraints, such as dipole moment conservation. We will use a hydrodynamic approach to study dynamics close to the thermal state, and consider cases with and without momentum conservation. For momentumconserving fracton fluids, we will find that the hydrodynamic description is intrinsically unstable and leads to a higher-dimensional analogue of the Kardar-Parisi-Zhang (KPZ) universality class.

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.