Classical Analogies in the Solution of Quantum Many-Body Problems

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## 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.