## Classical Analogies in the Solution of Quantum Many-Body Problems

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##### Date

2017##### Author

Keser, Aydin Cem

##### Advisor

Galitski, Victor M.

##### DRUM DOI

doi:10.13016/M2PN8XG0W

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Show full item record##### 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.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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