Classical Analogies in the Solution of Quantum Many-Body Problems

Abstract

We consider three quantum many-body systems motivated by recent developments in condensed matter physics, namely topological superconductivity, strongly interacting Bose-Einstein condensates and many-body localization with periodically driven systems. In each of the three problems, an analogy with classical mechanics is employed in the solution of the problem and the interpretation of results. These analogies, in addition to facilitating the solution, illustrate how unique features of classical mechanics or macroscopic phenomena such as macroscopic order parameter and observables, hydrodynamics, spacetime curvature, noise and dissipation, chaos and delocalization emerge out of quantum mechanics. The three problems we study are as follows. In the 1st problem, we use quasiclassical methods of superconductivity to study the superconducting proximity effect from a topological p-wave superconductor into a disordered quasi-one-dimensional metallic wire. We demonstrate that the corresponding Eilenberger equations with disorder reduce to a closed nonlinear equation for the superconducting component of the matrix Green's function. Remarkably, this equation is formally equivalent to a classical mechanical system (i.e., Newton's equations), with the Green's function corresponding to a coordinate of a fictitious particle and the coordinate along the wire corresponding to time. This mapping allows us to obtain exact solutions in the disordered nanowire in terms of elliptic functions. A surprising result that comes out of this solution is that the p-wave superconductivity proximity induced into the disordered metal remains long range, decaying as slowly as the conventional s-wave superconductivity. It is also shown that impurity scattering leads to the appearance of a zero-energy peak. In the second problem, we consider a system of bosons in the superfluid phase. Collective modes propagating in a moving superfluid are known to satisfy wave equations in a curved spacetime, with a metric determined by the underlying superflow. We use the Keldysh technique in a curved spacetime to develop a quantum geometric theory of fluctuations in superfluid hydrodynamics. This theory relies on a ``quantized" generalization of the two-fluid description of Landau and Khalatnikov, where the superfluid component is viewed as a quasi-classical field coupled to a normal component { the collective modes/phonons representing a quantum bath. This relates the problem in the hydrodynamic limit to the \quantum friction" problem of Caldeira-Leggett type. By integrating out the phonons, we derive stochastic Langevin equations describing a coupling between the superfluid component and phonons. These equations have the form of Euler equations with additional source terms expressed through a fluctuating stress-energy tensor of phonons. Conceptually, this result is similar to stochastic Einstein equations that arise in the theory of stochastic gravity. We formulate the fluctuation-dissipation theorem in this geometric language and discuss possible physical consequences of this theory. In the third problem, we investigate dynamical many-body localization and delocalization in an integrable system of periodically-kicked, interacting linear rotors. The linear-in-momentum Hamiltonian makes the Floquet evolution operator analytically tractable for arbitrary interactions. One of the hallmarks of this model is that depending on certain parameters, it manifests both localization and delocalization in momentum space. We present a set of \emergent" integrals of motion, which can serve as a fundamental diagnostic of dynamical localization in the interacting case. We also propose an experimental scheme, involving voltage-biased Josephson junctions, to realize such many-body kicked models.