Hydrodynamics of quantum matter

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Imagine preparing a set of qubits in a simple product state, and time evolving it under a given Hamiltonian which couples them. The state of the system is going to become more and more complex as entanglement is going to build up. If one wanted to store information about the system on a classical computer, the number of bits required would increase exponentially with time. Yet, after a while one expects the system to reach a thermal equilibrium, that should be describable in terms of a few macroscopic quantities, like temperature. In the field of quantum dynamics, the goal is to understand this extremely rich process, from the microsopic quantum chaos at short times, to the emergent hydrodynamics at long times. Hydrodynamics is a universal description of fluids which emerges at length and time scales much larger than those that govern microscopic, atom-level processes. While it has been useful in almost all fields of physics, its applications in solid state physics have been fairly restricted. Recently, however, important strongly correlated systems have been found which exhibit universal transport properties which cannot be explained by a single particle picture, and for which a non-perturbative approach like hydrodynamics is warranted. The opportunity provided by these systems is twofold: it enables the study of novel regimes of transport with unique properties, and it also generates exotic types of hydrodynamic theories which would not occur otherwise ``in vacuum''. My first direction is concerned with a new regime of electronic transport in which electrons behave like a viscous fluid, and for which the ubiquitous Ohm's law is replaced by the much richer Navier-Stokes equation. Interest in this regime was recently amplified by a series of experiments in 2D materials (e.g. graphene, PdCoO2). As a new Assistant Professor at the University of Toronto, I propose to make use of my previous innovations, and to further develop the theory, which will allow me to continue my deep interaction with top experimental groups in this field (Andrew Mackenzie's, Shahal Ilani's, Nobel laureate Andre Geim's), and to continue to produce new insights into this novel regime of quantum matter. Moving beyond the semiclassical regime, I will also study the dynamics of systems deep in the quantum regime by using a new formalism my collaborators and I have recently proposed. This formalism utilizes a recursion method in operator space to reveal the exponentially growing complexity of observables. It can also be used to study many-body quantum chaos at short times, and hydrodynamics at long times. This is a very exciting time. Experimental studies of hydrodynamic behavior of electrons in solids are producing new and unexpected advances every few months. With the help of NSERC funding, I look forward to building a new theory group at the University of Toronto that will explore this new frontier of condensed matter physics.