ABSTRACT
Title of dissertation: CLASSICAL ANALOGIES
IN THE SOLUTION OF
QUANTUM MANY-BODY PROBLEMS
Aydin Cem Keser, Doctor of Philosophy, 2017
Dissertation directed by: Professor Victor M. Galitski
Department of Physics
We consider three quantum many-body systems motivated by recent develop-
ments 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 first problem, we use quasiclassical methods of superconductivity to
study the superconducting proximity effect from a topological p-wave superconduc-
tor 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. Re-
markably, 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 equa-
tions 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 geo-
metric 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” prob-
lem 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. Conceptu-
ally, this result is similar to stochastic Einstein equations that arise in the theory
of stochastic gravity. We formulate the fluctuation-dissipation theorem in this geo-
metric 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 ro-
tors. 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 delocal-
ization in momentum space. We present a set of “emergent” integrals of motion,
which can serve as a fundamental diagnostic of dynamical localization in the in-
teracting case. We also propose an experimental scheme, involving voltage-biased
Josephson junctions, to realize such many-body kicked models.
CLASSICAL ANALOGIES IN THE SOLUTION OF QUANTUM
MANY-BODY PROBLEMS
by
Aydin Cem Keser
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor Victor M. Galitski, Chair/Advisor
Professor Theodore L. Einstein
Professor Theodore Jacobson
Professor Dionisios Margetis, Dean’s Representative
Professor Ian Spielman
c© Copyright by
Aydin Cem Keser
2017
Dedication
to my father
ii
Acknowledgments
This thesis would not have been possible without people who contributed with
their ideas to my research projects, and people who accompanied me through out
my graduate school years.
First and foremost, I want to thank my advisor Victor Galitski. He gave me a
chance to join his group and to work on a set of diverse, interesting and challenging
problems. He taught me that being a scientist is about forging a character that ex-
hibits courage in the face of the unknown, candid curiosity and integrity, in addition
to acquiring knowledge and skills.
I would also like to thank the thesis defense committee members for their
efforts to carefully read and comment on this thesis.
I would like to thank my collaborators, Valentin Stanev, Sriram Ganeshan and
Gil Refael. Their ideas, guidance and support were essential for the research I am
presenting in this thesis. I am also grateful to the Condensed Matter Theory Center,
its director Sankar Das Sarma, and the Joint Quantum Institute for providing great
environment for doing physics. The people who work here, and frequent seminars
and talks were very inspiring and helpful. I also want to mention my officemates
(chronologically): Sergey Pershoguba, Juraj Radic, Hilary Hurst, Yahya Alavirad.
I enjoyed our time together and all the conversations and discussions about physics
or life in general. I am also grateful to all the other Galitski group members with
whom I spent my time at UMD.
In addition to this I would like to thank my mother and father for their endless
iii
support. I would like to commemorate my late uncle and late grandfather, both
immigrant workers at some point in their lives. Your legacy will live on.
I would like to thank Natalia Budyldina for all the years we spent together,
and for carrying the emotional burden of graduate school with me.
Likewise, I would like to thank my roommates. It has been great sharing the
same living space with them.
I would also like to thank my friends in the area: Ali and Helen and of course
Benjamin Yapici for making me a member of their family, Berk Gurakan and Maya
Kabkab, Caner Celik and Sinem Kilic, Burak Cesme, Ugur Cetiner, Bahrom Oripov,
Lori Sen, Gokce Basar and Tania Marie, Murat Mercan and all the members of the
METU alumni association in DC and the Pir Sultan Abdal Cultural Association of
DC.
I would like to thank Monash University and Mannix College for their generous
hospitality and Dr. Simon Caterson, whom I had the chance to meet during the
amazing time I spent in Australia.
Finally, I would like to mention my endless gratitude to Sarah Black for trans-
forming me from a hesitant and risk averse young dreamer towards a resolutely
persistent man.
iv
Table of Contents
Dedication ii
Acknowledgments iii
Table of Contents v
List of Figures vii
List of Abbreviations ix
List of Publications x
1 Introduction 1
1.1 Classical and Quantum mechanics . . . . . . . . . . . . . . . . . . . 1
1.1.1 Macroscopic wave functions and order parameter fields . . . . 5
1.1.2 Chaos and localization . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Keldysh contour and nonequilibrium quantum theory . . . . . . . . . 10
1.2.1 Keldysh field integral for bosons . . . . . . . . . . . . . . . . . 11
1.2.2 Quantum particle in contact with an environment . . . . . . . 14
1.2.3 Keldysh field integral for fermions . . . . . . . . . . . . . . . . 17
1.2.4 BCS theory of superconductivity in the Keldysh formalism . . 20
2 Long range p-wave proximity effect into a disordered metal 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 The s-wave case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 The p-wave case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Classical particle analogy . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 The p-wave wire with normal segment . . . . . . . . . . . . . . . . . 34
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Analogue stochastic gravity in strongly interacting Bose-Einstein condensates 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 The model and energy scales . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Background field formalism on the Keldysh contour for Bogoliubov
quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Covariant phonon action . . . . . . . . . . . . . . . . . . . . . 57
3.4 Analogue Einstein equations and two-fluid hydrodynamics . . . . . . 61
v
3.4.1 Hilbert stress-energy operator of the covariant phonon field . . 63
3.4.2 Semiclassical analogue Einstein equations and the “phonon
matter” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.3 Canonical versus covariant conserved currents . . . . . . . . . 65
3.4.4 Mass-energy-momentum balance and the covariant conserva-
tion of stress-energy operator . . . . . . . . . . . . . . . . . . 69
3.5 Stochastic analogue Einstein equations . . . . . . . . . . . . . . . . . 74
3.5.1 Linear response and the covariant stress-energy correlator . . . 74
3.5.2 Global thermal equilibrium and fluctuation dissipation relation 76
3.5.3 Hydrodynamic fluctuations around a deterministic solution . 80
3.5.4 Stochastic analogue Einstein equation . . . . . . . . . . . . . . 82
3.5.5 Procedure to compute background metric fluctuations from
the analogue Einstein equation . . . . . . . . . . . . . . . . . 82
3.6 Equilibrium ‘Minkowski’ fluctuations . . . . . . . . . . . . . . . . . . 84
3.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Dynamical many-body localization in an integrable model 92
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 Energy dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.2 Momentum dynamics . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Dynamical many-body localization and the structure of ~α . . . . . . . 100
4.4.1 Case I: α1 . . . αd are distinct generic irrationals . . . . . . . . . 100
4.4.2 Case II: α1 = α2 and α2 . . . αd are distinct generic irrationals . 101
4.4.3 Case III: α1 = 1/2 and α2 . . . αd are distinct generic irrationals102
4.5 Derivation of main results . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 Derivation of the evolution of energy and momentum averages 104
4.5.2 Construction of integrals of motion . . . . . . . . . . . . . . . 106
4.5.3 Floquet Hamiltonian and quasienergy eigenstates . . . . . . . 107
4.6 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Conclusions 112
A Mapping between Quantum Kicked Rotor and the tight-binding model 116
A.1 Lattice model of single quantum kicked rotor . . . . . . . . . . . . . . 116
A.2 d-dimensional lattice model . . . . . . . . . . . . . . . . . . . . . . . 117
B Derivation of the canonical conservation law 120
C Noise and response kernels 122
Bibliography 125
vi
List of Figures
2.1 The potential landscape of a classical particle with motion describing
the Green’s function, for a normal metallic segment, in contact with
a superconductor. Depending on the superconductor (s or p-wave)
potential is either Vs or Vp. In the clean limit both converge to V
clean. 33
2.2 Contour plot of the DOS of an infinite wire. There is moderate dis-
order (β = 1) in the normal segment (x > 0). The solid yellow marks
the regions that are beyond the plot range (where N/N0 > 3.5). No-
tice the zero-energy peak in the normal segment. . . . . . . . . . . . . 37
2.3 DOS at the junction of infinite normal and superconducting segments.
Three cases for the disorder in the normal segment are plotted: weak
(β = 0.1, blue), moderate (β = 1, purple), and strong (β = 10, red).
Notice the suppression of DOS with the increase of disorder strength. 38
2.4 Components of gˆ,(g1: blue, g2: purple, g3: red) for a wire with infinite
p-wave section and finite disordered section of length L = 5ξ0. Top
panel: weak disorder(β = 1/(2τimp∆0) = 0.1). Middle panel: moder-
ate disorder (β = 1). And bottom panel: strong disorder (β = 10).
The Matsubara frequency is set to ω = ∆0/2. . . . . . . . . . . . . . 40
2.5 The weight of the zero-energy mode M(x) in a normal section with
length L = 5ξ0 for three disorder strengths (blue: β = 0.1, purple:
β = 1, red: β = 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 The Keldysh contour and its forward/backward branches labeled by
the path index s = ±: Each degree of freedom and the associated
sources are defined twice, one on each branch. The values are matched
at tf , for example ρ
+
f = ρ
−
f . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 The Keldysh contour for the phonon field. The metric tensor depends
on the background variables ρ±0 and θ
±
0 and therefore is also defined
twice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
4.1 (Top panel) Plots showing evolution of the root mean square deviation
of energy, σ(E) =
√〈H20 〉 − 〈H0〉2, as a function of time, labeled by
the number of kicks, N . (Bottom Panel) Plots showing evolution of
the root mean square deviation of individual momenta in an ensemble
of 10 particles σ(pi) =
√〈p2i 〉 − 〈pi〉2 as a function of time, labeled by
the number of kicks, N . The initial states of the rotors are assumed
to be definite momentum states. The total number of particles in
this interacting ensemble is d = 10 and circular boundary conditions
apply. We consider two body interaction term to be Jij = cos(θi−θj).
The periodic kicking potential is given by K(θ) =
∑∞
m=1 km cos(mθ)
for all particles. The m-th Fourier coefficient is fixed by km = z˜/m
2,
where z˜ is a random complex number with real and imaginary parts
uniformly distributed in the interval [0,1]. We make sure that K(θ) is
real. We choose ~α corresponding to three scenarios: Figs. (AI, AII):
If ϕ denotes the golden ratio, αj = (j/d)ϕ− (1/2)ϕ are all irrationals
for j = 1 . . . 10. Figs. (BI, BII): We consider αj distinct irrationals as
in Case A, but set α1 = α2, which results in the resonant growth of
σ(p1) and σ(p2), while σ(E) is bounded. Figs. (CI, CII): We consider
αj as in Case A and set α1 = 1/2, which results in the resonant
growth of σ(E) and σ(pi). . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Schematic of chain of voltage-biased (Vi) superconducting grains cou-
pled to each other (Ji) using Josephson junctions. Each grain is
connected (Ki) to a grounded common macroscopic superconductor
stroboscopically through a strong Josephson coupling. . . . . . . . . . 109
viii
List of Abbreviations
BCS Bardeen-Cooper-Schrieffer
BEC Bose-Einstein Condensate
BBGKY Bogoliubov-Born-Green-Kirkwood-Yvon
BdG Bogoliubov-de Gennes
IOM Integral of Motion
MBL Many Body Localization
ODE Ordinary Differential Equation
PDE Partial Differential Equation
QFT Quantum Field Theory
RMS Root Mean Square
ix
List of Publications
The dissertation is based on the following publications:
1. Aydin Cem Keser, Valentin Stanev, and Victor Galitski. Long range p-wave
proximity effect into a disordered metal. Phys. Rev. B 91, 094518 (2015) .
2. Aydin Cem Keser, Sriram Ganeshan, Gil Refael, and Victor Galitski. Dynam-
ical many-body localization in an integrable model. Phys. Rev. B 94, 085120
(2016).
3. Aydin Cem Keser, and Victor Galitski. Analogue Stochastic Gravity in Strongly-
Interacting Bose-Einstein Condensates. arXiv:1612.08980 (2017).
x
Chapter 1: Introduction
1.1 Classical and Quantum mechanics
In this thesis, we will solve or extend the solution of three quantum many-
body problems. These problems in addition to their practical value in the context
of recent developments in condensed matter physics, such as zero-energy modes in
topological systems, strongly interacting Bose condensates and many-body localized
systems, serve an auxiliary purpose as well. Each model is connected to a seemingly
unrelated analogous classical system that elucidates the underlying physics or helps
us address a broader issue in modern physics, such as the emergence of classical
degrees of freedom, spacetime and quantum chaos. Before introducing these three
problems, we will give a very brief qualitative review of the relation between classical
and quantum mechanics.
Classical mechanics is a theory of nature that describes the motion of an N -
body system in the 6N dimensional phase space. All observables are functions
of 3N configuration space variables and their 3N conjugate momenta. [1, 2] The
N →∞ limit leads to an idealization called the classical field and is the subject of
continuum mechanics. The geometry of spacetime on which physical systems live
is thought of as a continuum itself whose dynamics is determined by the Einstein
1
equation. [3] When the gravitational field is weak, the solution to this equation ap-
proaches the Minkowski geometry. [4] In this geometry, phenomena are described
by the Lorentz covariant equations of motion. For example, electrodynamics is de-
scribed by Maxwell’s equations. Finally, when the typical speeds are slow compared
to the speed of light, Galilean relativity, Newton’s laws of motion and universal grav-
itation hold to a good approximation. In situations where it is impossible to exactly
locate the system in phase space, the equations of motion for a probability distri-
bution is written [5]. Finally where the number of constituents are too numerous
and undergo random collisions, aggregate quantities are described by thermodynam-
ics. [6]
Ever since the conception of quantum theory, the problem of how the classical
description of nature emerges out of quantum mechanics has attracted a great deal
of interest, and it is still at the heart of many interesting physical problems. [7, 8]
Roughly speaking, classical mechanics is thought of as the ~→ 0 limit of quantum
theory. (The numerical value of the Planck constant measured in terms of the
typical scales of the system is small ). Depending on the mathematical formalism
that describes quantum mechanics, this limit manifests itself in myriad ways. In
the path integral quantization, the ratio of the action functional to Planck constant
appears in a complex exponential that represents the amplitude of a matter wave.
When this quantity grows, the path integral tends to zero due to a rapidly oscillating
integrand, except for classical trajectories, for which the action is extremized. [9]
In the canonical quantization, the fundamental commutation relation tends to zero,
which means that the expectation value of a product of position and momentum
2
operators reduces to the product of their expectation values, each of which obeys
the classical equation of motion by Ehrenfest’s principle. [10, 11] In phase space
formulation, the Moyal bracket reduces to the Poisson bracket, and the equations
of motion for observables obey Hamilton’s equation of motion. [12]
To get a grasp of the situation, consider a harmonic oscillator made up of a
mass of 1 kg and spring constant of 1 kg s−2. Assuming that a sinusoidal mode
with an amplitude of 1 m is excited, the energy is 0.5 J and the time scale is 1
s. Compare the product of energy and time scales to the metric value of reduced
Planck constant which is 34 orders of magnitude smaller. Alternatively, we can
compute the quantum of energy ~ω in the system to find the occupation number
to be on the order of 1033 particles (phonons). In other words, the quanta are too
small to be perceived.
An important issue worth noting is the measurement problem. So long as
one stays in the classical realm where ~ is small, as everybody did till the 20th
century, the above considerations roughly (we will explain this further 1.1.2) explain
the emergence of classical mechanics. However, the problem deepens when one
considers a quantum system, typically surrounded by an experimental physicist and
her apparatus which are both in the classical realm. As in Schroedinger’s thought
experiment [13], the result of a quantum phenomenon might trigger a macroscopic
classical event, for example the death of a cat. How the quantum state collapses
to a definite classical outcome is the so-called measurement problem that is at the
heart of many scientific and philosophical issues in quantum theory. [8, 14]
In the many-body context, quantum mechanics is responsible for many macro-
3
scopic electromagnetic and thermal phenomena, and the problem of how macro-
scopic observables follow from the many-body wave function encompasses an entire
branch of physics called condensed-matter physics.
Therefore, the transition to classical mechanics, or emergence of classical de-
grees of freedom, is complicated through interactions that entangle the underlying
quantum degrees of freedom. This seems to indicate that the measurement prob-
lem is related to how a quantum system is interacting with its environment, which
can be a macroscopic measurement device or simply the rest of the particles in the
system. This, we believe, should convince the reader that the quantum to classical
transition is essentially a many-body phenomenon.
The process of classicalization of a system due to interactions with an envi-
ronment or a ‘bath’ is called decoherence. One interpretation is that only classical
looking degrees of freedom survive the decoherence process. [8] A simple toy model
is a quantum particle that is coupled to a thermal bath i.e. a family of oscillators.
The so called Caldeira-Leggett model, that we will solve in the Keldysh field integral
method, describes how classical equations of motion, dissipation and noise can be
calculated from quantum mechanics, assuming that decoherence takes place. [15]
The complete resolution of the measurement problem and transition to classical
mechanics is beyond the scope and aims of this thesis, therefore we refer the reader
to the above references. However, as in the Calderra-Leggett model, the models we
consider in this thesis will help us understand how various unique features of clas-
sical mechanics can come out of the many-body quantum mechanics at least from
an operational point of view.
4
1.1.1 Macroscopic wave functions and order parameter fields
At low temperatures a many-body quantum system, due to the vast number
of its constituents, is approximated by a quantum field. [15] (This ‘low’ tempera-
ture can be as large as thousands of Kelvins in metals due to the vast number of
electrons that occupy high energy states as a result of the exclusion principle). The
quantum field is described as modes (quasiparticles) excited over a vacuum state.
The concept of a vacuum state of a many-body system is intimately connected to
the phases of matter. In metals it is an incompressible sea of electrons; in a BEC,
it is a macroscopic wave function, that behaves like a fluid through Madelung’s de-
scription. In any case, the typical procedure is to start from a phase (that forms by
spontaneously breaking a symmetry) and consider the excitations above the vacuum
defined by the phase. [16]
One of the holy grails of the condensed matter theory is the phases of matter
that has macroscopic quantum properties, such as superfluids and superconductors.
In these systems, a macroscopic number of fermions or bosons get locked and form
a single giant wave function ( a classical matter wave similar to a classical light field
). This classical wave is called the order parameter field of the phase. The many-
body wave function is, by the above mentioned general picture, described by the
quantum field of quasiparticles that are excited over the vacuum defined by the order
parameter. The order parameter field, together with correlation functions of the
quasiparticle field, obeys a set of partial differential equations [17,18]. The resulting
description predicts how macroscopic observables arise from quantum mechanics;
5
hence, it answers the quantum to classical transition question for the system at
hand. Furthermore, once a PDE is obtained, one can conjure up an unrelated
classical system that mimics its behavior.
In Part 2, we consider a superconductor, where the order parameter (Cooper
field), together with the matrix of two-point correlation functions, satisfies the PDE
called the Gorkov equation supplemented by a self-consistency condition. Further
simplification is obtained by considering the slow varying dynamics of the correlation
functions. In thermal equilibrium, the PDE reduces to a set of three ODE’s with
non-linear terms due to the disorder in the superconductor. By identifying a hidden
constant of motion, we reduce the ODE system to a single second-order ODE, that
can be thought of as the equation of motion of a classical particle moving in a
potential. Using this simple and intuitive system, we compute the Green’s function
of a topological superconducting wire in contact with a dirty lead, a problem of great
practical interest in the field of topological quantum computing. Our calculation
reveals the smoking gun for the long-sought Majorana zero mode in this system.
In Part 3, we start with a bosonic system in the superfluid phase. Writing the
superfluid wave function in hydrodynamic variables, we write the fluid equations
of motion that describe the interplay between the superfluid ‘classical’ degrees of
freedom and the quantum field of excitations, that is phonons. As in the Caldeirra-
Leggett model, the bath of quasaiparticles create dissipative and noisy dynamics
on the background field. Another interesting layer of analogy appears due to the
effective covariance of the phonon field. Phonons in this theory appear as though
they are propagating in a pseudo-Riemannian metric defined by the background flow.
6
In this respect, the momentum, mass and energy conservation laws of the superfluid
are analogous to Einstein’s equations and the covariant conservation of the stress
energy tensor for matter. We use this analogy to compute the noise correlator and
transport coefficients in the system in a manifestly covariant form. This analogy
provides a hint for the emergence of the classical spacetime from the yet-to-be-
found quantum theory of everything. Moreover, one can discuss the vacuum energy,
cosmological constant and other quantum gravitational phenomena such as black
holes, Hawking radiation and covariance anomalies in this context.
1.1.2 Chaos and localization
Even for macroscopic objects, the ~ → 0 limit does not straightforwardly ex-
plain classical behavior all together, because the limit is singular. [19] When ~ = 0
strictly, classical mechanics predicts chaos. Differential equations that describe clas-
sical systems with positive Lyapunov exponents is ubiquitous. A positive Lyapunov
exponent means that the distance between two particles in the system or in the
ensemble grows exponentially with time, despite being infinitesimally small at the
beginning. The linearity and the unitarity of quantum mechanics and the ensuing
integrability prohibits this behavior for isolated systems. [20] Although, at first the
phase space distribution of a quantum system explores the phase space in a way
similar to a classical ensemble at a certain time scale quantum effects take over
and the phase space diffusion stops. [21, 22] This time scale, the Ehrenfest time, is
perceivably short even for astronomical bodies, despite the fact that ~ is extremely
7
small yet non-zero. For example, the inverse Lyapunov exponent i.e. time scale
for chaos to develop for the rotation of Hyperion, the non-round moon of Saturn
is approximately 30 days. The Ehrenfest time for Hyperion is 37 years. However,
chaotic motions of macroscopic bodies persist. [23]
The resolution is again hidden in the decoherence process. The rotation angle
of Hyperion is not a quantum variable. Hyperion is showered by a variety of particles
that causes decoherence on a time scale of order 10−53 s. [24] The environment
restores chaos.
Prohibition of chaos in quantum mechanics has an important consequence in
many-body physics: Anderson localization. In a disordered lattice, the diffusion of
a wave packet stops due to quantum destructive self-interference. The Ehrenfest
time is replaced by the localization length. [25,26]
An illustrative model, dubbed the Maryland Model, that connects the pre-
vention of dynamical chaos in quantum mechanics and localization was put forward
by Fishman, Grempel and Prange. [25] The quantized version of a quintessential
chaotic system, the standard map that describes the motion of a rotor kicked with
a constant period, is mapped onto a tight binding model. The sites of the lattice
are angular momentum quantum numbers, so that dynamical chaos maps to delo-
calization. It is shown that, unless certain resonances are avoided, the phase space
diffusion comes to a halt, in other words wave functions of the lattice are localized.
An integrable version of the Maryland model was also proposed, where localiza-
tion is controlled by the rotation number of the rotor, i.e. the number of turns it
completes between each kick. [27, 28]
8
For a given subsystem located deep inside a many-body system, the environ-
ment is the rest of the system itself. So far, it has been believed that localization
would be destroyed due to the interacting system acting on its components as a bath.
As a result electron transport has been expected to be restored, just as the chaotic
orbit of Hyperion persists due to decoherence enforced by the showering light and
cosmic particles. This process of an interacting system acting as a bath to its sub-
systems is called thermalization and it is what makes the predictions of equilibrium
statistical mechanics applicable to a quantum many-body system. [29–31]
However, recently a class of interacting models have been proposed where
thermalization does not occur, causing a phenomenon called many-body localiza-
tion. [32–39] A many-body localized system can not be described by equilibrium
statistical mechanics because it does not thermalize; instead, it is described by a set
of local conserved quantities that remember their initial values. [40]
In Part 4, inspired by the Maryland model, we consider a set of interacting
integrable rotors and discuss localization in the context of integrals of motion and
resonances in the energy-momentum transfer due interactions and kicking.
Quite surprisingly, the quantum and classical behavior of the integrable Mary-
land model are very similar. [41] For this reason, as a side note, a similar Hamiltonian
is proposed as a building block for a deterministic quantum theory. [42]
9
1.2 Keldysh contour and nonequilibrium quantum theory
The goals we laid out in the introduction requires formulating quantum theory
of fields and particles and, extracting classical/macroscopic observables. The path
integral formulation is an ideal candidate for this. Moreover, choosing a closed
time integration contour provides great utility in non-equilibrium and disordered
systems, as well as noisy and dissipative scenarios. In this section we will provide a
brief review of Keldysh field integral.
Nonequilibrium quantum field theory is developed by the pioneers of the quan-
tum theory such as Schwinger, Kadanoff, Baym, Konstantinov, Perel and Keldysh
around the same time that Matsubara developed the thermal field theory tech-
nique. [43–46] The method is applied to a variety of problems in non-equilibrium
statistical mechanics, quantum Brownian motion, transport problems and disor-
dered systems and it forms the backbone of nonequilibrium kinetic theory. [15, 47]
The technique we refer to as Keldysh formalism is also referred to as Closed Time
Path formalism due to the choice of the integration contour of the path integral.
Calzetta and Hu applied this formalism to fields on curved spacetime [48] and the
marriage between non-equilibrium fields and gravity gave rise to the stochastic grav-
ity paradigm. [49].
In the Keldysh formalism one replaces the vacuum persistence amplitude of
the “in-out” formalism with the following partition function,
Z[V ] = Tr
{
UˆC(Vˆ )%ˆ(−∞)
}
(1.1)
10
In this equation %ˆ(−∞) is the (normalized ) density matrix of the system
at an early time, and Tr%ˆ(−∞) = 1. The operator UˆC is the evolution operator
path ordered on a closed time contour.The closed time contour starts at an early
time t → −∞. extends to t → ∞, forming the forward branch of the contour,
and reverses back to t → −∞ ,forming the backward branch of the contour. The
Hamiltonian for the evolution is Hˆ = Hˆ0 + Vˆ
±(t) where the external time dependent
potential is allowed to be different on the forward and backward branches of the
contour. If Vˆ ±(t) denotes the forward and backward values of the external potential,
V +(t) = V −(t) renders UC = 1 and the partition function becomes Z = 1. This has
a technical advantage over the usual in/out formalism, where Z = 〈in|out〉 is not
necessarily unity and should be accounted for while computing amplitudes.
1.2.1 Keldysh field integral for bosons
Specifically, suppose we have a scalar real massless free field φ that obeys the
scalar wave equation. The partition function is then written as a functional integral
over the closed time path C, and it reads
Z =
∫
D [φ]ei
∫
C d
4x d4x′ φ¯(x)Dˆ−1(x,x′)φ(x′) (1.2)
where D is the Green’s function. The objective is to determine the Green’s function,
given the initial density matrix. Let φ± denote the values of the forward/backward
values of the field , thereby effectively doubling the number of degrees of freedom
in the problem. Let S denote the classical action of the field φ. Labeling the initial
11
and final values of the fields as φ±i = φ
±(−∞) and φ±f = φ±(+∞), respectively, we
can write the Feynman path integral representation of Z in Eq. (1.1) as
Z =
∫
D [φ±]%[φ+i , φ
−
i ]δ(φ
+
f − φ−f )ei(S[φ
+]−S[φ−]) (1.3)
Here S, the action integral, is taken from t → −∞ to t → +∞. Notice that we
explicitly imposed the initial and final conditions on the fields. Keldysh discovered
that the initial and final value constraints can be handled by constructing the fields
φc(q) =
1
2
(
φ+ ± φ−) (1.4)
φc(q) are classical and quantum components of the field. Defining ~φ = (φc φq)T and
absorbing the density matrix into the action and the Jacobian of the transformation
in Eq. (1.4) in the integration measure we get the following path integral for the
field vector:
Z =
∫
D [~φ] exp
[
i
2
∫
d4xd4x′ ~φT (x)Dˆ−1(x, x′)~φ(x′)
]
(1.5)
where the inverse Green’s function matrix reads
Dˆ−1(x, x′) =
 0 [DA]−1
[DR]−1 [D−1]K

x,x′
(1.6)
The retarded (advanced) Green’s functions DR(A) are lower(upper) triangular
matrices in the time domain that are independent of the initial density matrix. They
are formally written as:
− i 〈φc(q)(x)φq(c)(x′)〉 = DR(A)(x, x′) = 1
2
{
(i∂t ± i0)2 +∇2~r
}−1
δ(x− x′) (1.7)
12
So we can write them as a Fourier integral over 4-momentum
DR(A)(x, x′) =
1
2
∫
d4k
(2pi)4
e−ik(x−x
′)
(k0 ± i0)2 − |~k|2 (1.8)
and therefore
DR(x− x′) = DR(y) = − 1
4pi
θ(y0)δ (yµyµ) (1.9)
They satisfy the properties
[DA]T = DR (1.10a)
DA(k0, ~k) = DR(−k0, ~k) = [DR]∗(k0, ~k) (1.10b)
DA(R)(y0 = 0, ~y) = 0 (1.10c)
The last one means, in the continuum limit the step function in Eq. (1.9) is regu-
larized as θ(0) = 0.
The Keldysh Green’s function contains the information about the initial cor-
relations, and it is the correlator
DK(x, x′) = −i 〈φc(x)φc(x′)〉 (1.11)
It is parametrized by using an antisymmetric kernel F
DK(x− x′) = DR ◦ F − F ◦DA (1.12a)
[D−1]K = [DR]−1 ◦ F − F ◦ [DA]−1 (1.12b)
F T = −F (1.12c)
Here F (x, x′) contains the initial correlations hence it depends on %ˆ(−∞) and the
symbol ◦means convolution. The Wigner transform of F is the distribution function
13
and in equilibrium it is F () = coth(− µ)/2T , a manifestation of the Fluctuation-
Dissipation Relation (FDR). For real bosons, F () is real and odd; therefore, the
chemical potential vanishes.
Note that the action in Eq. (1.5) vanishes when quantum components of fields
are put to zero,
S0|φq = 0 and δS0
δφq
∣∣∣∣
φq=0
= 0 (1.13)
gives the classical equation of motion for the field φc. This justifies the names chosen
for the field components φc, φq. The above properties of the Keldysh action and its
Green’s functions are collectively referred as the ‘causality structure’. The causality
structure is intact when external potentials or interactions are switched on.
1.2.2 Quantum particle in contact with an environment
Here we will derive the Langevin equation for the Brownian motion of a quan-
tum particle in contact with a bath. The particle has the following action in the
Keldysh formalism:
Sp[X] =
∫
C
dt
1
2
X˙2 − V (X) (1.14)
Denoting the coordinate over the forward/backward branches as X± and per-
forming the rotation to classical and quantum components
Xcl =
1
2
(X+ +X−); Xq =
1
2
(X+ −X−) (1.15)
14
the action reads
Sp[X] =
∫ ∞
−∞
dt 2X˙clX˙q − V (Xcl +Xq)+ V (Xcl −Xq) (1.16)
Note that variational derivative with respect to Xq at the limit Xq → 0 gives
the classical equation of motion
X¨cl + V ′(Xcl) = 0 (1.17)
In addition to the particle, we have the heat bath, commonly assumed to be a
collection of bosonic degrees of freedom with a wide spectrum, i.e. a collection of
harmonic oscillators with characteristic frequencies indexed by s. The action for the
bath and bath-system coupling will be taken as
Sbath[~φs] =
1
2
∑
s
∫ ∞
−∞
dt~φTsD
−1
s
~φ (1.18a)
Scoupling =
∑
s
gs
∫
C
dtXφs =
∑
s
gs
∫ ∞
−∞
dt ~XTσ1~φs (1.18b)
The choice for the form of the couplings make the partition function amenable
to Gaussian integration so that
Z =
∫
D [~φs]D ~Xe
i[Sp+Sbath+Scoupling ] =
∫
D [ ~X]ei[Sb+Sdiss] (1.19)
where the resulting dissipative term reads
Sdiss =
1
2
∫ ∫
dtdt′ ~XT (t)D−1(t− t′) ~X(t′) (1.20)
with the dissipative kernel
15
D−1(t− t′) = −σ1
(∑
s
g2sDs(t− t′)
)
σ1 (1.21)
The kernel D inherits is causality structure from the bosonic Green’s function
D. The retarded and advanced components in Fourier space reads
[D−1(ω)]R(A) = −1
2
∑
s
g2s
(ω ± i0)2 − ω2s
(1.22)
The spectral density of the path is defined as
J(ω) = pi
∑
s
(g2s/ωs)δ(ω − ωs) (1.23)
and it is assumed to behave as J(ω) = 4γω at low frequencies. With this definition
the dissipative kernel becomes
[D−1(ω)]R(A) = 4γ
∫
dω′
2pi
ω′2
ω′2 − (ω ± i0)2 = const± 2iγω (1.24)
The constant contributes to the definition of the classical potential in which
the particle is moving. In thermal equilibrium the Keldysh component follows from
these as
[D−1(ω)]K = ([D−1(ω)]R − [D−1(ω)]A) coth ω
2T
= 4iωγ coth
ω
2T
≈ 8iTγ (1.25)
where the approximation holds in the classical limit where ~ω/kT  1.
All in all the total action in real space after integrating out the bath reads:
Stot[ ~X] =
∫ ∞
−∞
dt 2X˙clX˙q − V (Xcl +Xq)+ V (Xcl −Xq)− 2XqγX˙cl + 8iγT (Xq)2
(1.26)
16
The variational derivative around X1 = 0 will produce a damped equation of
motion. The full Langevin equation can be obtained by decoupling the (Xq)2 by
using an auxiliary noise variable.
Z =
∫
D [ ~X]eiStot =
∫
D [ ~X, ξ]e
i
4γT
∫
dt ξ2e−i
∫
dt 2Xq(X¨cl+V ′(Xcl)+γX˙cl−ξ) (1.27)
where, the higher terms in the external potential are dropped, as they don’t con-
tribute to the equations of motion. The Langevin equation immediately follows after
applying Eq. (1.13) to Eq. (1.27).
X¨ + V ′(X) + γX˙ = ξ (1.28)
The correlation function of noise can be computed from Eq. (1.27) as
〈ξ(t)ξ(t′)〉 = 2γTδ(t− t′) (1.29)
The fluctuation dissipation relation manifests itself in this equation, as the
noise correlator is proportional to the friction coefficient γ.
1.2.3 Keldysh field integral for fermions
The path integral for a fermionic field is written in terms of two independent
Grassmann fields ψ and ψ¯. On a closed time path contour C, the functional integral
for a free fermion field with Green’s function G reads
Z =
∫
D [ψ¯ψ]ei
∫
C d
4xd4x′ ψ¯(x)Gˆ−1(x,x′)ψ(x′) (1.30)
17
Similar to the boson case, let ψ± and ψ¯± denote the forward/backward values of
the Grassmann fields. The following Keldysh parametrization
ψ1 =
1√
2
(
ψ+ + ψ−
)
, ψ2 =
1√
2
(
ψ+ − ψ−) (1.31a)
ψ¯1 =
1√
2
(
ψ¯+ − ψ¯−) , ψ¯2 = 1√
2
(
ψ¯+ + ψ¯−
)
(1.31b)
proposed by Larkin and Ovchinnikov and defining
~Ψ =
ψ1
ψ2
 , ~¯Ψ =
ψ¯1
ψ¯2
 (1.32)
brings the action into the following
Z =
∫
D [~¯Ψ~Ψ] exp
[
i
∫
d4xd4x′ ~¯ΨT (x)Gˆ−1(x, x′)~Ψ(x′)
]
(1.33)
where the inverse Green’s function matrix reads
Gˆ−1(x, x′) =
[GR]−1 [G−1]K
0 [GA]−1

x,x′
(1.34)
The components of the Green’s function satisfy the following symmetry prop-
erties
GA =
[
GR
]†
, GK = − [GK]† (1.35)
Therefore, the Keldysh component can be parametrized by using a hermitian matrix
F = F † as
GK = GR ◦ F − F ◦GA (1.36)
18
where the symbol ◦ means convolution. In thermal equilibrium the Green’s function
depends only on the difference between time components, due to the time transla-
tional symmetry. Fourier transform in this time coordinate produces the equilibrium
distribution function F as a function of energy  as
F () = 1− 2nF () = tanh − µ
2T
(1.37)
To be more specific, consider the free gas of electrons under a uniform Zeeman
field HZ that has the Hamiltonian
Hˆ =
pˆ2
2m
+ ~HZ · ~ˆs (1.38)
With the spin degrees of freedom σ, σ′ = ±, the Keldysh action of the free
Fermi gas reads
S0 =
∑
σ,σ′
∫
d4xd4x′ ~¯ΨTσ (x)Gˆ
−1
σσ′(x, x
′)~Ψσ′(x′) (1.39)
Choosing the direction of HZ as the z axis and using the fact that energy, momentum
and the z-component of spin are good quantum numbers, and defining the Fourier
decomposition of the fermion fields as
ψ(~r, t) =
∑
~k
ψ(~k, t)ei
~k·~r, ψ¯(~r, t) =
∑
~k
ψ¯(~k, t)e−i
~k·~r (1.40)
the bare Green’s functions can be immediately written down as
G
R(A)
σσ′ (
~k, ) = δσσ′
(
− ~k,σ ± i0
)−1
(1.41a)
GKσσ′(
~k, ) = −2piiδσσ′F ()δ(− ~k,σ) (1.41b)
where
~k,σ =
|~k|2
2m
+ σHZ (1.42)
19
1.2.4 BCS theory of superconductivity in the Keldysh formalism
The free electron gas is just an idealization, as collisions and momentum trans-
fer between electrons occur due to interactions. The two-body interaction term can
be captured by including the interaction term in the action S = S0 + Sint where in
reciprocal space we have
Sint = −1
2
∫
C
dt
∑
~q.~k,~k′
∑
σσ′
U(~q)ψ¯~k,σψ¯~k′,σ′ψ~k′+~q.σ′ψ~k−~q,σ (1.43)
Depending on the magnitudes of the momenta of the incoming electrons, we
can have three classes of scattering events. First, the incoming electrons have
momenta comparable to the Fermi momentum so that k, k′ ∼ kF and the trans-
ferred momentum q  kF is small. Second, the sum of the incoming momenta
|~k + ~k′| ∼ 2kF is close to twice Fermi momentum; and third, the difference of the
momenta |~k−~k′| ∼ 2kF is nearly twice Fermi momentum. In the last two cases, the
momentum transfer is on the order of kF .
The second type, where |~k − ~k′| ∼ 2kF is the most important, as it produces
Cooper pairs in the following way. Relabeling the momenta so that the transferred
momentum is exactly |~k + ~k′|, we get
SBCS = −1
2
∫
C
dt
∑
~q,~k,~k′
∑
σσ′
U(~k + ~k′)ψ¯~k,σψ¯−~k+~q,σ′ψ~k′+~q,σ′ψ−~k′,σ (1.44)
where the subscript BCS stands for Bardeen-Cooper-Schrieffer theory that is based
on the idea that interactions that flip the momenta of the incoming pair binds them
20
into bosonic pairs. In BCS theory, the interaction between electrons are mediated
by phonons with momentum ∼ kF is frequency independent and attractive and
is −λ/ν where ν is the density of states at Fermi energy and λ > 0. Indeed, in
this case, the interaction is of pairing type because the interaction vanishes due to
fermionic commutation relations when the spins of the interacting electrons are the
same. Defining the Cooper pair fields as:
Φ(~q, t) =
∑
ψ~k+~q↓(t)ψ−~k↑(t), Φ¯(~q, t) =
∑
ψ¯−~k↑(t)ψ¯~k+~q↓(t) (1.45)
The interaction in real space reads
SBCS =
λ
ν
∫
C
d4x Φ¯(x)Φ(x) (1.46)
A bosonic degree of freedom ∆ is introduced so that the total action becomes
quadratic in the fermionic fields. This effectively decouples the Cooper fields at
expense of creating a bosonic condensate ∆. To do this observe that, BCS action
can be written as
exp(iSBCS) =
∫
D [∆]ei
∫
C d
4x[− νλ |∆|2+∆Φ¯+∆∗Φ] (1.47)
Now, the partition function for the free and interacting parts S0 + SBCS can
be expressed in terms of a single quadratic action, the Bogoluibov de Gennes action
SBdG
∫
D [ψ¯ψ]e(iS0+iSBCS) =
∫
D [∆]e−
ν
λ
i
∫
C d
4x|∆|2
∫
D [ψ¯ψ]eiSBdG (1.48)
21
where
SBdG =
∫
C
d4x
(
ψ¯↑ −ψ↓
)i∂t + 12m(∇r + i ~A)2 − Vdis ∆
−∆∗ −i∂t + 12m(∇r − i ~A)2 − Vdis

ψ↑
ψ¯↓

(1.49)
The crystal dislocations and non-magnetic impurities impose a disordered elec-
tric potential on the system, which we call Vdis. In addition, we assumed the Zeeman
coupling is small and ignored it.
This 2×2 kernel is defined in the particle-hole or the so-called Nambu space. In
a general situation coupling terms exist between the particle hole spinor (ψ↑, ψ¯↓) and
the spinor with opposite spin structure (ψ↓,−ψ¯↑). So the BdG kernel is 4×4 acting
on bi-spinors that live in the direct product of Nambu and spin spaces. Here, due
to the absence magnetic impurities and spin orbit coupling, the spin space becomes
trivial and we get two copies of the Nambu space.
However, we still need to enlarge the kernel if we want to double the degrees
of freedom to account for the closed-time contour. So we are working in the product
of Nambu and Keldysh spaces. In this space, we can write
~Ψ =
(
ψ1↑, ψ¯1↓, ψ2↑, ψ¯2↓
)T
, ~¯Ψ =
(
ψ¯1↑, −ψ1↓, ψ¯2↑, −ψ2↓
)
(1.50)
SBdG =
∫ ∫
d4xd4x′ ~¯Ψ(x)
[
Gˆ−10 (x, x
′)− Vˆdis(x, x′)
]
~Ψ(x′) (1.51)
where G0 is the Green’s function of the clean superconducting system and the dis-
22
order potential Vˆdis(x, x
′) = V (x)δ(x − x′)1 i.e. it is diagonal in spacetime as well
as Keldysh and Nambu spaces. The derivation of Gorkov equations for the Green’s
function and quasiclassical Green’s functions for a disordered superconductor follows
from the Keldysh BdG action in Eq. (1.51). [50]
23
Chapter 2: Long range p-wave proximity effect into a disordered metal
2.1 Introduction
Superconducting heterostructures have attracted a lot of attention recently
as possible hosts of Majorana fermions [51–59]. One of the important outstanding
questions in the studies of these heterostructures is the interplay between topologi-
cal superconductivity and disorder [60–63]. Here we explore this issue, focusing on
the leakage of p-wave superconductivity into a disordered metal. Na¨ıvely, it may
not appear to be a particularly meaningful question, because unconventional super-
conductivity is known to be suppressed by disorder per Anderson’s theorem [64].
However, Anderson’s theorem is only relevant to an intrinsic superconductor and
has little to do with a leakage of superconductivity.
The linearized Usadel equations are standard tools in studies of proximity
effects [65, 66]. Their derivation, however, assumes that an anisotropic component
of the superconducting condensate’s wave function is small compared to the isotropic
one, which is not the case in the systems we are interested in. Here, we focus on
the more general Eilenberger equations [18,67], which allow us to straightforwardly
model systems with complicated geometries and varying degree of disorder. (In
the context of topological superconductivity, similar approach has been used in
24
Refs. [68–71].)
In this work, we consider an infinite quasi-one-dimensional system (nanowire),
at least part of which is superconducting. To describe the electronic correlations
in the system we utilize the quasiclassical Green’s function gˆ – a matrix in Nambu
and spin space [18]. It is obtained from the full microscopic Green’s function and
faithfully captures the long length scale features of the system. For a recent compar-
ison between quasiclassical and fully microscopic calculation of the same structure,
see, for example [72]. In a one-dimensional model, gˆ depends on the Matsubara fre-
quency (ω), the center-of-mass coordinate of the pair (x), and the direction of the
momentum at the Fermi points (ζ ≡ px/pF is +1/−1 for right/left going particles).
It obeys the Eilenberger equation [18,66,67]
ζvF∂xgˆ = −[ωτˆ3, gˆ] + i[∆ˆ, gˆ]− 1
2τimp
[〈gˆ〉, gˆ]. (2.1)
Here τˆ are the Pauli matrices in Nambu space. The effects of impurities enter the
equation through the last term, in which τimp is the mean time between collisions,
and 〈...〉 denotes an average over the Fermi surface (actually, two disconnected points
in the one-dimensional case). Since we are interested in wires in which supercon-
ductivity is induced by a proximity with a bulk superconductor, we treat ∆ˆ as an
external parameter, which, furthermore, we assume constant throughout the wire.
This allows us to ignore self-consistency, and simplifies the calculations.
The outline of this chapter is as follows. In Sec. 2.2 and Sec. 2.3, we obtain
exact solutions of Eq. (2.1) for s-wave and p-wave order parameters respectively. In
Sec. 2.4 we introduce an intuitive picture for the behavior of the solution by mapping
25
the equations to the equation of motion of a classical particle in an external force
field. In Sec. 2.5 we study superconducting correlations induced by proximity in
a metallic wire. In particular, we demonstrate that the p-wave correlations can
be surprisingly long-ranged, even in the presence of disorder. We also show that
impurity scattering produces a zero-energy peak in the density of states (DOS).
We summarize the results of our work in Sec 2.6. Although self-consistency is not
relevant for the experimental setup we are considering, in the Appendix we provide
for completeness the solutions of the fully self-consistent Eilenberger equations for
both s-wave and p-wave quasi-one-dimensional superconductors.
2.2 The s-wave case
We decompose gˆ in Nambu space using the Pauli matrices τi: gˆ = −ig1τˆ1 +
g2τˆ2 + g3τˆ3. The off-diagonal scalar functions g1 and g2 describe the superconduct-
ing particle-particle correlations, whereas the diagonal component g3 contains the
particle-hole correlations. These functions have to satisfy the normalization condi-
tion −g21 + g22 + g23 = 1.
In the case of an s-wave superconductor, ∆ˆ is a spin-singlet and, ignoring
the spin indices, it can be written as ∆0iτ2 – this is equivalent to choosing the
order parameter to be purely real. We ignore self consistency and assume that ∆ is
constant in space. The function g2 is in the same channel as ∆ˆ, and thus encodes the
s-wave pairing correlations. The g1 function has more interesting origin – it describes
the p-wave, odd-frequency superconducting correlations, induced by boundaries or
26
other inhomogeneities, and disappearing in bulk uniform superconductors [73–76].
The component g1 is odd in momentum; therefore, its Fermi surface average vanishes
identically: 〈g1〉 = 0. The components g2 and g3 are even in momentum; therefore,
〈g2〉 = g2 and 〈g3〉 = g3 are satisfied. With these considerations, Eq. 2.1 can be
now written as a set of three coupled first order differential equations for the scalar
functions gi:
ζvF∂xg1 = −2ωg2 + 2∆g3, (2.2a)
ζvF∂xg2 = −2ωg1 − 1
τimp
g1g3, (2.2b)
ζvF∂xg3 = 2∆g1 +
1
τimp
g1g2. (2.2c)
To be integrable this system of equations has to have two constants of integra-
tion. The norm of gˆ, which is −g21 + g22 + g3 = 1 is one of them. We have identified
another constant Cs, given by
Cs = g
2
1/(2τimp) + 2∆g2 + 2ωg3, (2.3)
and obeying ∂xCs = 0. Note that the value of Cs can be fixed using the appropriate
boundary conditions. Using it we can reduce the system given by Eq. (2.2) to a
single second-order differential equation for g1:
v2F∂
2
xg1 =
(
4ω2 + 4∆2 +
Cs
τimp
)
g1 − 1
2τimp
g31. (2.4)
Let us note several interesting limits for this equation. The τimp → ∞ is the clean
superconductor limit, which was considered in Refs. [73, 74]. The τimp → 0 is the
27
strong disorder limit, in which Eq. (2.4) leads to results equivalent to those obtained
by the Usadel equations [66]. Finally, ∆ → 0 is the normal metal limit, where non
trivial (proximity) solutions follow from superconducting boundary conditions.
The bulk superconducting energy ∆0 is a convenient energy scale for the sys-
tem. Even in a normal metal, where ∆ = 0, non-trivial solutions can appear be-
cause of proximity with a superconductor. In this case we can still use ∆0, the
bulk gap parameter value in the neighboring superconductor, as a convenient en-
ergy scale. To streamline the notation we denote the coefficient from Eq. (2.4) by
αs = ω
2 + ∆2 + Cs/(4τimp), and normalize Cs and αs by the superconducting en-
ergy ∆0, to get C˜s = Cs/∆0 and α˜s = αs/∆
2
0. We also introduce the dimensionless
coordinate x˜ = x/ξ0 (where ξ0 ≡ vF/∆0 is the superconducting coherence length)
and the dimensionless disorder strength β = 1/(2∆0τimp). With these substitutions
we can write Eq. (2.4) as
∂2x˜g1 = 4α˜sg1 − 2β2g31. (2.5)
Once the boundary values of the Green’s function components are given, the
value of Cs and therefore αs is fixed. Then we can solve Eq. (2.5) without explicit
reference to the other components of gˆ. Multiplying both sides of Eq. (2.5) with
∂x˜g1, the solution can be obtained implicitly as an integral:∫ g1(x˜2)
g1(x˜1)
dg1
±
[
4α˜sg21 − β2g41 + 2E˜s
]1/2 = ζx˜2 − ζx˜1. (2.6)
Here, E˜s is a constant of integration (in itself a function of Cs), about which we will
have more to say in Sec. 2.4. Note that the Fermi index ζ = ±1 appears on the
right side of Eq. (2.6), therefore the function g1 is manifestly odd in momentum. We
28
determine the plus/minus sign in front of the integrand by demanding consistency
with the boundary conditions and the integration path so that the signs of both sides
in Eq. (2.6) match. We will present an intuitive way to understand this solution in
Sec. 2.4 and show that only monotonic solutions are physically acceptable. For a
solution that starts from x˜ = 0, it is convenient to recast the integral in Eq. (2.6)
in terms of the inverse elliptic function sn−1, which leads to
sn−1
(
g1(x˜
′)
(ρ+s )
1/2
∣∣∣∣ρ+sρ−s
) ∣∣∣∣x˜
0
= ±ζβx˜[−ρ−s ]1/2, (2.7a)
ρ±s =
1
β2
(
2αs ±
[
4α2s + 2E˜sβ
2
]1/2)
. (2.7b)
where x˜′ is a dummy variable.
Once we have g1(x), we can obtain the other components by using the constant
of integration Cs and the system in Eq. (2.2), to get
g2(x) =
∆˜
(
C˜s − βg21(x)
)
− ω˜ζ∂x˜g1(x)
2
(
ω˜2 + ∆˜2
) , (2.8a)
g3(x) =
ω˜
(
C˜s − βg21(x)
)
+ ζ∆˜∂x˜g1(x)
2
(
ω˜2 + ∆˜2
) . (2.8b)
2.3 The p-wave case
In the case of a p-wave wire we consider spinless fermions. As in the previous
section, we decompose gˆ using the Pauli matrices τˆi. The order parameter can be
written as ζ∆0iτ2, and we again assume that ∆ is a constant in space. The difference
from the s-wave case arises from the fact that now g2 is p-wave, and g1 contains the
secondary s-wave (odd-frequency) correlations [76,77], and references therein.
29
The components of gˆ again obey three coupled differential equations, which
differ from the s-wave case, due to the Fermi surface averaging in the last term of
Eq. (2.1). In the p-wave case the order parameter has p-wave symmetry; therefore,
we get 〈g1〉 = g1, 〈g2〉 = 0. Note that 〈g3〉 = g3 applies (particle-hole correlations
are s-wave-like). With these we have
ζvF∂xg1 = −2ωg2 + 2ζ∆g3 − 1
τimp
g2g3, (2.9a)
ζvF∂xg2 = −2ωg1, (2.9b)
ζvF∂xg3 = 2ζ∆g1 − 1
τimp
g1g2. (2.9c)
In the clean case, these equations are linear and easily solved [70,73,74]. Impurities
introduce nonlinear coupling, proportional to 1/τimp. Nevertheless, as we demon-
strate, these equations can be treated analytically. The next several steps closely
follow the discussion in the preceding section. We have again identified a constant
Cp obeying ∂xCp = 0. It is given by
Cp = −g22/(2τimp) + 2ζ∆g2 + 2ωg3. (2.10)
Using it we can derive from the system in Eqs. (2.9) a single second-order equation
for g2 :
v2F∂
2
xg2 = −2ζ∆Cp + 4αpg2 −
3ζ∆
τimp
g22 +
g32
2τ 2imp
, (2.11)
where for convenience we have introduced αp = ω
2 + ∆2 + Cp/(4τimp). Notice the
difference with Eq. (2.4), which is for g1.
Again, we use the energy scale ∆0 (see Sec. 2.2 for more explanation) to intro-
duce normalized coordinate x˜ = x/ξ0 (where ξ0 ≡ vF/∆0), dimensionless disorder
30
strength β = 1/(2∆0τimp), and normalize Cp and αp as C˜p = Cp/∆0 and α˜p = αp/∆
2
0.
With these substitutions Eq. (2.11) becomes
∂2x˜g2 = −2ζ∆˜C˜p + 4α˜pg2 − 6ζβ∆˜g22 + 2β2g32. (2.12)
Now we integrate the equation Eq. (2.12), which can be done without any
explicit reference to the other two components:
∫ g2(x˜2)
g2(x˜1)
dg2[
−4ζ∆˜C˜pg2 + 4α˜pg22 − 4ζ∆˜g32 + β2g42 + 2E˜p
]1/2 = ±ζ(x˜2 − x˜1) (2.13)
The Fermi momentum appears on the right hand side of equation Eq. (2.13)
and g2 is odd in momentum, as it should. We determine the plus/minus sign of the
integral in Eq. (2.13) in order to be consistent with the boundary conditions and
the integration path.
In this chapter, we are interested in the behavior of the solution in a disordered
normal metal next to a superconductor, so we consider the special case ∆ = 0, and
recast the integral in Eq. (2.13) in terms of the inverse elliptic function sn−1 to get
Eq. (2.14):
sn−1
(
g2(x˜
′)
(ρ+p )
1/2
∣∣∣∣ρ+pρ−p
) ∣∣∣∣x˜
0
= ±ζβx˜
ξ0
[ρ−p ]
1/2, (2.14a)
ρ±p =
1
β2
(
−2αp ±
[
4α2p − 2E˜pβ2
]1/2)
. (2.14b)
Once we have g2(x), we can obtain g1(x) and g3(x) by using the constant of
31
integration Cp and the system in Eq. (2.9), to get
g1(x) =
−ζ∂x˜g2(x)
2ω˜
, (2.15a)
g3(x) =
C˜p + βg2(x)
2 − 2ζ∆˜g2(x)
2ω˜
. (2.15b)
2.4 Classical particle analogy
There is a surprising but intuitive way to understand the results from the
previous two sections. We can think of Eq. 2.6 as an equation of motion of a
classical particle with coordinate g1, in an external force field. In this interpretation
the normalized position ζx˜ takes the role of the dynamical time of this classical
particle. The Hamiltonian of the classical particle is given by the following equation:
Hs[g1] =
1
2
(∂x˜g1)
2 − 2α˜sg21 +
β2
2
g41. (2.16)
This Hamiltonian describes a particle in a double-well potential Vs[g1] =
−2α˜sg21 + β2g41/2, as shown in Fig. 2.1. We denote the conserved energy of this
Hamiltonian as E˜s. We can furnish the solution to the “equation of motion” by in-
verting the integral in equation Eq.(2.6) that sums up to the elapsed time ζx˜2− ζx˜1
between initial and final coordinates g1(x˜1) and g2(x˜2), respectively.
The double well potential Vs[g1], allows non-monotonic solutions. However,
the classical turning points of this potential scale as ±(ωτimp) at high frequency,
and for both ω → ∞ or τimp → ∞ the non-monotonic motion has unbounded
amplitude; hence, these solutions are not physical. Therefore, we will only consider
the monotonic solutions. If we denote the poles of the integrand as ρ±s then we can
32
Figure 2.1: The potential landscape of a classical particle with motion
describing the Green’s function, for a normal metallic segment, in contact
with a superconductor. Depending on the superconductor (s or p-wave)
potential is either Vs or Vp. In the clean limit both converge to V
clean.
conveniently write the integral in Eq. (2.6) with monotonic integration path that
starts from the point g1(x˜ = 0), in terms of the inverse Jacobi elliptic function sn
−1
as in equation Eq. (2.7).
In a similar manner, we can interpret Eq. (2.12) as the equation of motion
of a classical particle with coordinate g2, moving in an external potential. The
Hamiltonian of this classical particle is given in Eq. (2.17)
Hp[g2] =
1
2
(∂x˜g2)
2 + 2ζ∆˜C˜pg2 + 2α˜pg
2
2 − 2ζ∆˜g32 +
β2
2
g42. (2.17)
The Hamiltonian given by equation Eq. (2.17) describes the particle in the
external potential potential Vp[g2] = 2ζ∆˜C˜pg2 +2α˜pg
2
2−2ζ∆˜g32 +β2g42/2. We denote
the conserved energy of this Hamiltonian as E˜p. We can again find the solution to
the “equation of motion” by inverting the integral in equation Eq.(2.6) that sums
33
up to the elapsed time ζx˜2 − ζx˜1 between initial and final coordinates g2(x˜1) and
g2(x˜2), respectively.
In the special case of interest, where we are solving the equations in a disor-
dered metal, ∆ = 0, therefore the potential Vp[g2], shown in Fig. 2.1, permits only
monotonic solutions (given in Eq. (2.14)).
2.5 The p-wave wire with normal segment
We now proceed to study the leakage of superconductivity in a metallic wire.
We consider an infinite wire extending along the x-axis with two segments that
meet at x = 0. The infinite segment on the left (x < 0) is made of clean p-wave
superconductor with order parameter ζ∆0. The segment on the right (x > 0)
is made of a diffusive normal metal (the order parameter is zero). We are, in
fact, considering a quasi-one-dimensional system, with several conducting channels,
rather than a truly one-dimensional wire, which would be in an insulating state even
for infinitesimal disorder strength.
We want a solution that for x → −∞ reproduces the mean field result for a
uniform clean p-wave superconductor. Introducing the parameter B and the dimen-
sionless variables Ω˜ = Ω/∆0, ω˜ = ω/∆0, we can write such a solution [70, 78, 79]:
g1(x) = (1/ω˜)[1− Ω˜B] exp(2Ω˜x/ξ0), (2.18a)
g2(x) = ζ(1/Ω˜)
(
1− [1− Ω˜B] exp(2Ω˜x/ξ0)
)
, (2.18b)
g3(x) =
{
[1− Ω˜B]/(Ω˜ω˜)
}
exp(2Ω˜x/ξ0) + ω˜/Ω˜. (2.18c)
34
B has to be determined from the boundary conditions at the junction (x = 0). For
simplicity, we consider the case of perfectly transparent boundary, which guarantees
the continuity of the Green’s functions at the junction (More realistic modeling of
the boundary requires more complicated boundary conditions) [80,81]. (Note that
Eqs. (2.18) were derived under the assumption that the order parameter is constant
in the clean p-wave superconductor.)
Now we consider the diffuse normal segment with infinite length. Then, for
x → ∞ we have g1 → 0, g2 → 0 and g3 → sgn(ω), and using the constant of
integration C˜p(x = 0) = C˜p(x → ∞) = 2|ω˜| we get a quadratic equation, with one
of the two solutions given as:
B =
1
β
[
−1 +
√
1 + 2β(Ω˜− ω˜)
]
, (2.19)
and we discard the other solution because it leads to a non-monotonic solution.
We can understand intuitively the behavior of g2 by invoking the classical anal-
ogy. The particle in potential Vp starts at “position” g2(0) = ζB, with velocity
∂x˜g2(0) = −2ω˜ζg1(0) = −2ζ(1 − Ω˜B), and moves towards its unstable equilibrium
point g1(+∞) = 0, gradually slowing down until ∂x˜g2(+∞) = 0, from which we
deduce E˜p = 0. Indeed, it takes infinite amount of time for the particle to reach the
point g2 = 0, a fact we see from the diverging integral in Eq. (2.13) for E˜p = 0, as
g2 → 0.
For E˜p = 0 the elliptic integral leads to inverse hyperbolic functions, and defin-
ing the dimensionless constant κ = [1 + β2B2/(4α˜p)]
1/2, we can write the solution
35
for g2:
g2(x) =
ζB
cosh(x/ξ′) + κ sinh(x/ξ′)
. (2.20)
Here ξ′ = ξ0/(2α˜
1/2
p ) gives the effective decay length of the solution (at T = 0). In
physical units it is
ξ′ =
vF√
4ω2 + 2|ω/τimp|
. (2.21)
In the dirty limit we have ξ′ =
√
D/|ω|, where D is the diffusion coefficient. Finally,
in the clean limit g2 converges to ζB exp(−2|ω˜|x/ξ0), as expected [70].
The other two components of the Green’s function can be derived from g2 using
C˜p and the Eilenberger equations: g1 = −ζξ0∂xg2/(2ω˜) and g3 = sgn(ω˜)+βg22/(2ω˜).
As expected, impurities suppress g2 relative to g1. However, they both decay in the
normal segment over the same length scale, given by Eq. (2.21). This decay is long-
range, and furthermore, with exactly the same length scale obtained for the case of
s-wave order parameter [73, 74]. Thus, the na¨ıve expectation of strong suppression
of the p-wave correlations is misleading in this case. This is one of the main points
of this chapter ( Also, note that the order parameter amplitude ∆0 appears only in
the boundary conditions at x = 0. Thus, even a complete self-consistent treatment
of the superconducting segment would not change the decay length, only the overall
prefactor), and it is also supported by the fact that the same decay length scale
appears also in the case of an s-wave superconductor with magnetic disorder [75],
which is known to be analogous in some ways to a p-wave superconductor with
potential disorder.
We can now obtain the DOS of the system from Re[g3(ω → −i + δ)] [70].
36
Figure 2.2: Contour plot of the DOS of an infinite wire. There is mod-
erate disorder (β = 1) in the normal segment (x > 0). The solid yellow
marks the regions that are beyond the plot range (where N/N0 > 3.5).
Notice the zero-energy peak in the normal segment.
On Fig. 2.2 we show it for a system with moderate disorder. Several things are
apparent from this plot. First, for low energies there is a significant decrease in the
DOS of the normal segment, caused by the proximity effect; however, it is not a
real gap, since the DOS stays finite. This decrease is entirely due to the impurities,
which “trap” the superconducting correlations close to the boundary (in the clean
case the DOS is constant for x > 0 [70]). The impurity-induced term in g3 also has
a divergence in the limit of small frequencies (g3 ∼ 1/ω), which leads to an infinite
peak in the DOS (see Fig. 2.3). This zero-energy peak has the same origin as the
Majorana edge state (namely, the sign change in the order parameter [78, 79, 82]).
37
From Eq. (2.20), we can see that the weight of the zero-energy peak has a power-law
decay ∼ (1 + βBx/ξ0)−2 into the metalic segment.
Figure 2.3: DOS at the junction of infinite normal and superconduct-
ing segments. Three cases for the disorder in the normal segment are
plotted: weak (β = 0.1, blue), moderate (β = 1, purple), and strong
(β = 10, red). Notice the suppression of DOS with the increase of dis-
order strength.
As a side note, in the case of an s-wave superconductor, the solution of Eq. (2.4)
is g1 = ζA[cosh(x/ξ
′)+κs sinh(x/ξ′)]−1, with A and κs that can be derived by match-
ing the solutions at the junction. However, unlike the p-wave case, the g1 component
at the boundary is proportional to ω˜. This dependence on ω˜ changes the behavior
of the DOS. From g3 = sgn(ω˜)−β/(2ω˜)g21, we can see that the low frequency limit is
finite, and there is no zero-energy peak in the s-wave case (Analogous calculation in
the s-wave case leads not to a peak, but linear suppression of DOS at low energies.
The overall DOS profiles are very similar to those obtained earlier numerically ) [83].
38
If the normal segment has finite length L, we impose the condition g2(L) = 0
(the p-wave component is suppressed by boundary reflection). Then the solution
follows immediately from Eq. 2.14 as g2(x) = ζ(ρ
+
p )
1/2sn[β(ρ−p )
1/2(x − L)/ξ0], with
elliptic parameter m = ρ+p /ρ
−
p . However, this expression has limited value, since BL,
which should be obtained from matching the two solutions for g2 at x = 0, enters
the expression through the parameters ρ±p , and is difficult to find. Fortunately, an
approximate analytic form for BL can be obtained. Numerical investigation suggests
that BL can be approximated by B[1− exp(−2L/λB)], with λB = Bξ0 (it controls
how quickly BL approaches to the infinite wire limit). Once we have BL, we can
write g2 in a form that manifestly converges to that of the L = ∞ case [84]. The
common elliptic parameter of the elliptic functions is (ρ−p − ρ+p )/ρ−p , and it lies in
the interval [0, 1]. With these definition we get:
g2(x) = ζ
BLdn
(
β|ρ−p |1/2 xξ0
)
− sn
(
β|ρ−p |1/2 xξ0
)
cn
(
β|ρ−p |1/2 xξ0
)√
|ρ+p |+B2L
√
1 +B2L/|ρ−p |
cn2
(
β|ρ−p |1/2 xξ0
)
− (B2L/|ρ−p |)sn2
(
β|ρ−p |1/2 xξ0
) .
(2.22)
We can again obtain the two other components from g2 by using: g1 = −ζξ0∂xg2/(2ω˜)
and g3 = (α˜p − ω˜2)/(βω˜) + βg22/(2ω˜). Figure 2.4 shows the components g1, g2, g3 of
the quasiclassical matrix Green’s function gˆ, for varying disorder strengths, over a
semi infinite wire with disordered section.
As L→∞, the elliptic functions are replaced by their hyperbolic counterparts,
and we recover Eq. 2.20. This convergence is exponential, so the wire is effectively
infinite when L/(Bξ0) 1. Conversely, if L/(Bξ0) 1, g2(x) decays linearly in the
normal metal. Again, it is the impurity-induced contribution to g3 that is of most
39
Figure 2.4: Components of gˆ,(g1: blue, g2: purple, g3: red) for a wire
with infinite p-wave section and finite disordered section of length L =
5ξ0. Top panel: weak disorder(β = 1/(2τimp∆0) = 0.1). Middle panel:
moderate disorder (β = 1). And bottom panel: strong disorder (β = 10).
The Matsubara frequency is set to ω = ∆0/2.
interest. After analytic continuation we can write the zero-energy limit as
g3(x) =
1
pi
δ()M(x). (2.23)
Here, M(x) describes the x-dependent weight of the zero-energy mode, and can be
extracted from Eq. (2.22):
M(x) = αp
β
+
β
2
BL −
√
αp
2β2
(
1 + β
2B2
2αp
)
sin
(
β|ρ−p |1/2 xξ0
)
cos
(
β|ρ−p |1/2 xξ0
)
cos2
(
β|ρ−p |1/2 xξ0
)
− β2B2
2αp
sin2
(
β|ρ−p |1/2 xξ0
)
2 .
(2.24)
40
It is monotonically decreasing function, and its values at the ends of the wire are
M(0) = 1− BL and M(L) =M(0)− βB2L/2 respectively. Figure 2.5 shows M(x)
in the normal section with length L = 5ξ0, for various disorder strengths. As can
be seen, it becomes peaked closer to x = 0 as the disorder in the normal section
increases. On the other hand, for weak disorder, [M(0) −M(L)]/(L/ξ0) is small,
so the zero-energy peak is delocalized over the entire normal segment.
Figure 2.5: The weight of the zero-energy modeM(x) in a normal section
with length L = 5ξ0 for three disorder strengths (blue: β = 0.1, purple:
β = 1, red: β = 10).
2.6 Conclusion
We presented a quasiclassical description of a quasi-one-dimensional supercon-
ductor. The appropriate Eilenberger equations of the system were solved exactly.
Surprisingly, we discovered that this problem can be mapped to a one-dimensional
classical particle moving in an external potential. In view of the recent interest
41
in superconducting heterostructures, we studied the proximity effects in a normal
segment, attached to a clean p-wave wire. We discovered that despite the presence
of impurities, the proximity-induced superconducting correlations are long-range.
We also found that impurity scattering leads to the appearance of a delocalized
zero-energy peak.
42
Chapter 3: Analogue stochastic gravity in strongly interacting Bose-
Einstein condensates
3.1 Introduction
The idea that a curved spacetime is an emergent structure has a long his-
tory [85, 86] and has been discussed in various physical contexts [87] from classical
fluid mechanics [88,89] and crystals with defects [90] to quantum entanglement [91].
While in the context of fundamental gravity, the emergent scenario remains specu-
lative at this stage, there has been a number of concrete realizations of various as-
pects of general relativity in “analogue gravity” models, where a non-trivial curved
spacetime metric arises naturally in the description of collective modes relative to
a background solution of the field equations [87, 92]. A prominent example of such
analogue theory is a strongly correlated superfluid [93], where the phonon modes,
propagating relative to a (generally inhomogeneous and non-stationary) superflow,
satisfy a wave-equation in an effective curved spacetime
∂µ
(√−ggµν∂νφ) = 0, (3.1)
43
where φ(~r, t) is the phonon field – a small deviation from a “mean-field” configura-
tion, g = detgµν is the determinant and g
µν is the matrix inverse of the metric
gµν =
ρ
c
c2 − v2 ~vT
~v −I3×3
 (3.2)
which is determined by the underlying superflow (~v and c are the superfluid velocity
and the speed of sound, ρ is the density of the fluid including the excitations).
Many exciting general-relativistic effects immediately follow from this observation,
including the formation of sonic horizons and black hole-type physics [92], analogue
Hawking radiation [94], proposed by Unruh [88] and recently reported by Steinhauer
to have been observed in cold-atom Bose-Einstein condensates [95], and a unifying
principle for cosmology and high energy physics discussed extensively by Volovik [93,
96].
This geometric theory of excitations in a superfluid, that this work focuses on
and develops further, is an alternative formulation of the phenomenological Landau-
Khalatnikov two-fluid theory of superfluidity [97, 98], which has been originally de-
veloped as a macroscopic description of superfluid Helium. As the name suggests,
this theory separates the fluid flow into two components – one being the zero entropy,
zero viscosity superflow and the other being the entropy-carrying, dissipative nor-
mal flow. The two-fluid theory relies on the conservation laws for mass, energy, and
momentum in a Galilean-invariant continuum made up of these two components.
In addition to being an accurate macroscopic description of superfluid helium, the
two-fluid theory can be viewed as the first historical example of a long wavelength
hydrodynamic limit of a strongly interacting quantum field theory. The low energy
44
effective field theory paradigm offers a number of powerful techniques to analyze
strongly interacting field theories and the hydrodynamic limit of high energy theo-
ries (e.g., the AdS/CFT and string theory) [99–102].
The main idea of this work relies on a conceptual analogy between the quantum
generalization of the Landau-Khalatnikov two-fluid description and the Caldeira-
Leggett-type theories of “quantum friction”, where a closed system is separated into
two components – a quantum “particle” and a bath to which it is coupled [15,103].
Integrating out the bath leads to classical equations of motion for the particle, which
necessarily feature a friction force and a stochastic Langevin force, connected to each
other via a fluctuation-dissipation theorem. For a strongly correlated BEC, this
analogy associates the superfluid order parameter field with the Caldeira-Leggett
“particle” and the Bogoliubov excitations with the bath. The question we ask here
is what are the corresponding Langevin equations of motion that arise? We develop
and use a combination of the aforementioned geometric theory of excitations and
Keldysh field-theoretical methods in a curved spacetime to answer this question.
The main result is the following stochastic equations of motion
∂tρ+∇ · (ρ~v) = 1
2
∇ ·
[√−g (〈Tˆµν〉+ ξµν) δgµν
δ~v
]
, (3.3a)
∂tθ +
1
2
v2 + µ(ρ) =
√−g
2
(〈
Tˆµν
〉
+ ξµν
) δgµν
δρ
, (3.3b)
where ρ is the density of the fluid, θ is the superfluid flow potential, ~v = ∇θ is the
irrotational flow fluid of the superfluid component, µ(ρ) is the chemical potential of
the fluid, gµν is the matrix inverse of the metric tensor (3.2),
√−g = |det(gµν)|1/2
is the spacetime volume measure, Tˆµν(x) is the stress-energy tensor operator of the
45
phonons (here x is a short-hand for the (3 + 1)-spacetime variable and the Einstein
summation convention is in use) and ξµν(x) is a stochastic tensor field, describing
its fluctuations around the average 〈Tˆµν(x)〉. That is, the statistics of the Gaussian
noise with zero average is determined by the correlator
〈ξµν(x)ξµ′ν′(x′)〉 = 1
2
〈{
tˆµν(x), tˆµ′ν′(x
′)
}〉
,
where tˆµν(x) = Tˆµν(x) − 〈Tˆµν(x)〉 and {·, ·} is the anti-commutator. The averages
here are calculated relative to a deterministic background. What these equations
actually describe are fluctuations in the superfluid, e.g. they yield statistics of
density and velocity fluctuations, which in turn determine a stochastic metric. In
this sense, there is a strong similarity to the stochastic Einstein equations discussed
n the context of stochastic gravity [49,104].
While these stochastic Einstein equations are interesting in and of themselves,
their derivation presents a technical challenge (a non-trivial generalization of the
non-equilibrium Keldysh techniques for a curved spacetime is required) [48] and
gives rise to a number of additional interesting results along the way, as discussed
below. This chapter is structured as follows:
In Sec. 3.2, we discuss the applicability of the metric description of a superfluid
by analyzing the relevant length and energy scales. In analogy with cosmology, the
geometric description breaks down at an effective “Planck energy,” where both the
linear dispersion of phonons and the hydrodynamic description break down.
In Sec. 3.3, we use the background field formalism to write down the Keldysh
quantum field theory of quasiparticles. We emphasize that the Keldysh description
46
is necessary for taking the dynamical fluctuations of the phonon field into account.
In Sec. 3.4, we derive the analogous Einstein equation that governs the back-
ground and the excitations – “matter field.” We establish equivalence of the ana-
logue Einstein equation and the covariant conservation law for the phonons to the
two-fluid conservation laws of Landau and Khalatnikov. We prove the equivalence of
the two descriptions by reducing the covariant conservation law down to the Noether
current of the two-fluid system by using the equations of motion. In Appendix B,
we provide the technical details of this derivation.
In Sec. 3.5, we take the analogy between the superfluid system and general
relativity further to the domain of stochastic fluctuations. We write the response
and dissipation kernels in the covariant language, and give the details of this in
Appendix C. In global thermal equilibrium, we discuss the notion of temperature
on curved spacetime. We prove the fluctuation dissipation relation for a metric with
globally time-like Killing vectors, that is for a flow that can brought to a stationary
form after a Galilean transformation.
Finally in Sec. 3.6, we linearize the stochastic analogue Einstein equation and
obtain a Langevin-type equation for the stochastic corrections to the background.
We show that the symmetries of the flow determine the structure of the Langevin
equation, by considering the Minkowski case.
Throughout this chapter, we will use the Einstein summation convention for
the indices, unless otherwise stated. The spacetime indices are in lower case Greek
letters while the space indices are in lower case Latin letters. We use the sign
convention (+ − −−). In addition, the fluid dynamics equations are written in
47
terms of Cartesian tensors, where there is no distinction between covariant and
contravariant tensors.
3.2 The model and energy scales
In this section, we review the energy scales involved in the analysis of an
interacting system of bosons and its excitations. The analogue“general relativistic”
description is an approximation to the exact theory and its applicability is controlled
by our ability, or lack thereof, to neglect a quantum pressure term discussed below.
The main conclusion of this section is that the stronger the repulsive interactions
between bosons composing the superfluid, the less important the quantum pressure
term and correspondingly the wider the domain of applicability of the general-
relativistic approximation (in the sense of a range of energies and length-scales
where the description applies). We discuss these “Planck” energy and length-scales
below.
Our starting point is just the standard Lagrangian of interacting bosons
L [Φ,Φ∗] = Φ∗i~∂tΦ− ~
2
2m
|∇Φ|2 − ε (|Φ|2) , (3.4)
where Φ(~r, t) ≡ Φ(x) is the boson field, m is the mass of a boson, and the energy
ε(|Φ|2) describes an external potential and density-density repulsion between the
bosons. Though, at this stage we do not specify the external potential and inter-
action potential between bosons, for ε = g
2
|Φ|4 + V |Φ|2, the saddle point of this
Lagrangian, satisfies the Gross-Pitaevskii or non-linear Schro¨dinger equation:
i~∂tΦ = − ~
2
2m
∇2Φ + V (~r, t)Φ + g|Φ|2Φ. (3.5)
48
The first step in deriving the hydrodynamic theory is the Madelung transformation
of the boson field [105], which is a change of variables to polar coordinates in each
point of spacetime:
Φ(x) =
√
ρ(x)
m
eimθ(x)/~. (3.6)
The new variables are the density, ρ, and the phase θ, which in effect is a “flow
potential” for the superfluid, that gives rise to the irrotational flow velocity field
~v = ∇θ.
In terms of these variables, the Lagrangian (3.4) takes the form
−L = ρ∂tθ + 1
2
ρ~v2 + ε(ρ) +
1
8
(
~
m
∇ρ√
ρ
)2
︸ ︷︷ ︸
“quantum pressure”
. (3.7)
Classically, the long-wavelength description of this system in local thermo-
dynamic equilibrium is fluid dynamics. This description can be extended to the
quantum regime, where a macroscopic order exists as in the case of a condensate.
This macroscopic order is a non-trivial vacuum that solves the mean field fluid equa-
tion Eq. (3.8), which is just the Gross-Pitaevskii equation Eq. (3.5) in the Madelung
parametrization:
∂tρ+∇ · (ρ~v) = 0, (3.8a)
∂tθ +
1
2
~v2 +
∂ε
∂ρ
=
1
2
~2
m2
1√
ρ
∇2√ρ. (3.8b)
These are the Euler equations for an ideal, zero entropy fluid, with an additional
energy per unit mass appears in the right-hand side of Eq. (3.8b) due to the quan-
tum potential term ∼ (∇√ρ)2 in Eq. (3.7). Together with the continuity equation
49
Eq. (3.8a), the gradient of (3.8b), when multiplied by the density ρ, produces a
momentum balance equation. In this equation, the quantum potential leads to a
pressure gradient, hence this potential is called “quantum pressure” in Eq. (3.7).
The excitations over this non-trivial vacuum state are the Bogoluibov quasi-
particles. These quasiparticles can be thought of a quantum field propagating on top
of the ground state manifold. When treated semiclassically, the quasiparticles con-
stitute sources to the hydrodynamic equations (3.8), in the spirit of the two-fluid
hydrodynamics. Moreover, the entropy-carrying quasiparticle quantum field acts
as a bath on the background system. In addition to momentum and energy flux
and stresses, the quantum bath creates a noise to the evolution of the background,
leading to a stochastic Langevin component.
We now show, by analyzing the appropriate scales, that when the repulsive
interactions between the bosons are strong, the quantum pressure term is suppressed.
Such a fluid is the basis of the gravitational analogy, as it simulates the spacetime
on which the matter field – that is, the phonon field – propagates [93,96].
If we momentarily ignore the quantum pressure proportional to ∇√ρ, we get
the Lagrangian density of a vortex free perfect fluid with flow velocity ~v = ∇θ. The
excitations of this system are obtained by linearizing the equations of motion and are
the sound waves with speed c2 = ρd2ε/dρ2 at equilibrium density. When quantized,
the excitations have the linear spectrum Ep = ~ck. Therefore, the dimensional
parameters characterizing the vortex free quantum hydrodynamics are the Planck
constant, ~, the equilibrium density ρe, and the equilibrium speed of sound, ce. The
relevant energy scale of this theory is EQ = (~3c5eρe)1/4. Since the energy density
50
is ∼ ρec2e, the characteristic length scale is dQ = (ρece/~)−1/4. Therefore, the mass
scale is ρed
3
Q = MQ = (ρe~3/c3e)1/4 and the time-scale is tQ = dQ/ce = (ρec5e/~)−1/4.
With the addition of the quantum pressure term, the spectrum of Bogoliubov
phonons receives a correction as Ep = ~cek
√
1 + ~2k2/(4m2c2e). Therefore the linear
spectrum of phonons breaks down at a length scale of ξ = ~/(mce), this is the
coherence length of the condensate. At this scale, the phonon energy is of order
ELorentz = mc
2, which can be dubbed the Lorentz violation energy (i.e., where the
phonon spectrum deviates from the linear dispersion). Note that, Lorentz violations
do not necessarily occur at exactly the inter atomic length scale d and thus ELorentz
is generally distinct from the “Planck” energy scale EPlanck = ~ce/d – that is the
energy required to resolve the individual atoms separated by a distance d (at such
length-scales the hydrodynamic description becomes meaningless). Indeed, the ratio
of the coherence length to the inter atomic distance is determined by the strength
of the atomic interactions. Defining g0 = ~2d/m3, and setting gρ2e = ε(ρe) = ρec2e,
we get (ξ/d)2 = g0/g. In summary the relationships between different scales can be
written in terms of the normalized strength of interactions g0/g as:
EPlanck
ELorentz
=
ξ
d
=
√
g0
g
=
(
MQ
m
)4/3
. (3.9)
This ratio determines the relative importance of the quantum pressure term
compared to the interaction energy. The corresponding ratio is (below, L is a length-
scale on which the density changes):
( ~
m
∇√ρ)2
ε(ρ)
∼ ~
2
gm2ρL2
∼ ξ
2
L2
. ξ
2
d2
=
g0
g
.
51
In the weak interaction limit, as in a dilute Bose gas, the coherence length ξ and
the quantum mass scale MQ are large compared to microscopic counterparts d and
m. This signals that the system behaves like a macroscopic quantum object, hence
the condensate fraction is closed to unity. In this regime, Lorentz violation occurs
much before the interatomic scales are reached and the excitations are Bogoluibov
quasiparticles with the non-linear spectrum. In the strong interaction regime, as in
Helium-II or a strongly interacting BEC, the condensate fraction is small, however
the Lagrangian (3.7) is still valid, as it produces correct equations for a zero entropy
dissipationless classical fluid. In this regime, the quantum pressure is negligible
down to the ξ ∼ d scale. This means that the collective excitations are sound
waves all the way up to the effective Planck energy. Therefore, geometric theory of
analogue gravity for the covariant sound waves, that we will summarize in the next
section, applies as long as we stay in the hydrodynamic regime, that is to say at
length scales larger than the inter atomic distance.
3.3 Background field formalism on the Keldysh contour for Bogoli-
ubov quasiparticles
Here, we outline a procedure to extract a field theory of the excitations starting
from Eq. (3.7). We show that an effective curvature and covariance emerge when the
amplitude modes of the excitations are integrated out. Finally, we obtain an effective
action for the superfluid system and the phonon bath. Unless noted otherwise, we
52
Figure 3.1: The Keldysh contour and its forward/backward branches
labeled by the path index s = ±: Each degree of freedom and the
associated sources are defined twice, one on each branch. The values are
matched at tf , for example ρ
+
f = ρ
−
f .
use the units, where
~ = ce = kB = 1. (3.10)
We employ the closed-time path integral or the Keldysh functional integral [43–
46] and the background field formalism [106, 107] to separate the superfluid back-
ground and the quantum field theory of excitations. Note that the procedure, out-
lined below in Eqs. (3.11) – (3.27) is completely general and does not rely on a
particular form of the initial action, but we will apply it specifically to the super-
fluid Lagrangian (3.7).
Generally, given an initial density matrix %ˆ(ti) and the evolution operator Uˆt′,t
that takes the system from t to t′, the density matrix at any point in time t is
%ˆ(t) = Uˆt,ti%ˆ(ti)Uˆti,t. (3.11)
Suppose Oˆ is an observable. In the Schro¨dinger picture, the expectation value of
this operator at time t is
〈O(t)〉 = Tr [O%ˆ(t)] . (3.12)
53
In the Keldysh formalism, the time contour is made up of two branches that
start at an early time ti and meet at tf as shown in Fig 3.1. Each degree of freedom
in the system is defined twice, one on the forward and one on the backward branch of
the time contour labeled by ± respectively. The values of all variables are matched
at the final point tf where the branches meet. The observable can be coupled to
the system via the source currents J±(~r, t) that are added to the Hamiltonian Hˆ →
Hˆ + J±Oˆ. The forward/backward evolution operators associated with the modified
Hamiltonian are denoted as Uˆ (J±). Then the forward/backward expectation values
of the observable can be obtained from the following generating function
Z [J±] = Tr
[
Uˆti,tf (J
−)Uˆtf ,ti(J
+)%ˆ(ti)
]
, (3.13)
by differentiating it with respect to the forward/backward source currents J±
〈
O±(t)
〉
= ±i δ
δJ±(t)
Z [J±]
∣∣∣∣
J±=0
. (3.14)
Note that, if we set J+ = J− from the outset, the forward/backward evolution
operators cancel due to unitarity and
Z [ ~J+ = ~J−] = Tr %ˆ(ti) = 1. (3.15)
This means taking the logarithm of the generating function as in ordinary field
theory, is redundant in Keldysh theory.
The generating function Eq. (3.13) for the boson system in Eq. (3.7) can be
written as a closed time path integral as [15, 47]:
Z [J±ρ , J
±
θ ] =
∫
D [θ±, ρ±]%
(
ρ±i , θ
±
i
)×exp[i∑
C=±
C
∫
dxL (ρC , θC) + JCρ ρ
C + JCθ θ
C
]
.
(3.16)
54
where dx ≡ dtd3r, the sum goes over the upper and lower Keldysh contours, the
factor C is (±1) for the upper/lower contour.
The simplest observables are the mean fields, that is the expectation values of
the fields. Writing
Z [J±ρ , J
±
θ ] = e
iW [J±ρ ,J±θ ], (3.17)
the mean fields are generated by using Eq. (3.14) and Eq. (3.15)
ρ±0 := 〈ρ±〉 = ±
δW
δJ±ρ
∣∣∣∣
J±ρ =0
and θ±0 := 〈θ±〉 = ±
δW
δJ±θ
∣∣∣∣
J±θ =0
. (3.18a)
Now, we can separate the system into a classical background and quantum
excitations around it as follows. Suppose that the mean fields are given. Then one
can express the sources in terms of the mean fields by constructing the effective
action, that is the Legendre transform
Γ[ρ±0 , θ
±
0 ] = W [J
±
ρ , J
±
θ ]−
∑
C=±
C
∫
dx
(
JCρ ρ
C
0 + J
C
θ θ
C
0
)
. (3.19)
Now, the sources can be expressed as
δΓ
δρ±0
= ∓J±ρ and
δΓ
δθ±0
= ∓J±θ . (3.20a)
Exponentiating the effective action and using (3.19) one can eliminate the sources
in (3.16) in favor of the mean fields. If we define the deviations from the mean fields:
ρ˜± = ρ± − ρ±0 , (3.21a)
φ± = θ± − θ±0 , (3.21b)
55
we can write the following integral equation for the effective action
eiΓ[J
±
ρ ,J
±
θ ] =
∫
D [θ±, ρ±]%
(
ρ±i , θ
±
i
)×exp[i∑
C=±
C
∫
dxL (ρC , θC)− δΓ
δρC0
ρ˜C − δΓ
δθC0
φC
]
.
(3.22)
The effective action, Γ, can be solved for iteratively and be expressed as a series
(we restore the Planck constant below to emphasize the semiclassical nature of the
expansion)
Γ = Γ0 + ~Γph + ~2Γ2 + . . . (3.23)
the classical action
S[ρ±, θ±] =
∫
dxL (ρ±, θ±), (3.24)
being the zeroth term:
Γ0 = S[ρ
+
0 , θ
+
0 ]− S[ρ−0 , θ−0 ]. (3.25)
In this work we consider only the first order or one-loop correction Γph to the classical
action. This correction encapsulates the quantum field of Bogoluibov quasiparticles
over the background, that become phonons at long wavelengths (hence we use the
subscript ph, a shorthand for “phonon” corresponding to the first loop correction).
We substitute the lowest-order expression( 3.25) into Γ’s on the right hand
side of Eq. (3.22). On the left-hand side, we substitute Γ0 + Γph, where Γph is an
unknown. Expanding S around ρ0 and θ0, and matching the terms in the equation,
we find the phonon effective action Γph i.e. the leading-order correction in ρ˜ and φ
to Γ. Defining
%˜
(
ρ˜±i , φ
±
i
)
= %
(
ρ±i − ρ±0 , θ±i − θ±0
)
,
56
we write
eiΓph[ρ
±
0 ,θ
±
0 ] =
∫
D [ρ˜±, φ±]%˜
(
ρ˜±i , φ
±
i
) × exp{i∑
C=±
C
∫
dxL C(2)(ρ˜C , φC)
}
, (3.26)
where the L 2, depends on the path index C = ± not only through its arguments
but explicitly as
L ±(2)(ρ˜, θ) = (ρ˜, φ) · δ
2L
δ2(ρ, θ)
∣∣∣∣
ρ±0 ,θ
±
0
· (ρ˜, φ) (3.27)
in multi-index notation. So far, the results are completely general. Now we use the
explicit Lagrangian (3.7) and obtain the phonon effective Lagrangian, L (2). After
suppressing the time-path index C = ±, it is
−L (2) = ρ˜(∂t + ~v0 · ∇)φ+ 1
2
ρ˜
(
c20
ρ0
+ KˆQ
)
ρ˜+
1
2
ρ0(∇φ)2, (3.28)
where we defined the “quantum pressure” operator,
KˆQ =
1
4m2
[
(∇ρ0)2
ρ30
− ∇ρ0
ρ20
· ∇ − ∇
2
ρ0
]
. (3.29)
3.3.1 Covariant phonon action
The one-loop correction can be computed exactly in the strong interaction
limit, as the path integral reduces to a Gaussian integral. As noted in Sec. 3.2, at
energy scales below ELorentz, the operator KˆQ in (3.28), that results from quantum
pressure, can be neglected compared with the term c20/ρ0. This yields the following
Lagrangian
L (2)
KˆQ→0 → −ρ˜Dtφ−
1
2
ρ˜
(
c20
ρ0
)
ρ˜− 1
2
ρ0(∇φ)2, (3.30)
where we defined the material derivative
Dtφ = ∂tφ+ ~v0 · ∇φ.
57
Now, the density fluctuations, ρ˜, can be integrated out. To do the path integral
over ρ˜, we can think of spacetime as divided into cubes with volume ξ4/ce and
discretize the integral. Note that the integrand is a diagonal matrix over spacetime
and the path integral reduces to a product of Gaussian integrals. At this point, we
shorten the notations for the mean-field parameters ρ0, θ0 and ~v0, writing them as
simply ρ, θ and ~v for the sake of brevity. If we define the density matrix as
%φ
(
φ+i , φ
−
i
)
= %˜
( ρ
c2
Dtφ
+
i , φ
+
i ;
ρ
c2
Dtφ
−
i , φ
−
i
)
, (3.31)
integrating out the ρ˜ field produces:
Zph = e
iΓph[ρ
±,θ±] =
∫
D ′[φ±]%φe
i(S+ph[φ+]−S−ph[φ−]). (3.32)
with the measure of the path integral being, again suppressing the ± signs,
D ′[φ] =
∏
dφ
√
ρ
2pic2
, (3.33a)
and the following covariant action for phonons
Sph =
1
2
∫
dt d3x
ρ
c2
[
(Dtφ)
2 − c2 (∇φ)2] . (3.34)
This action can also be obtained through a classical treatment of the La-
grangian Eq. (3.7) [89]. The material derivative Dtφ = ∂tφ+ ~v · ∇φ is the measure
of the time rate of change of φ in a frame comoving with the fluid. This means
the action in Eq. (3.34) describes non-dispersive waves with speed c in the fluid
comoving frame. Galilean invariance of the fluid system requires that the sound
wave velocity ~u = d~x/dt in the lab frame satisfy (~u−~v) = c2. This means the sound
rays with velocity with d~x/dt = ~u are null rays on the manifold with line element
58
ds2 = −ρ
c
[−(c2 − v2)dt2 − 2~v · ~dxdt+ ~dx · ~dx]. (3.35)
This is the line element for any analogue gravity system with background
Galilean symmetry up to a conformal factor, for which the choice of ρ/c allows us
to write the phonon action of Eq (3.34) in the following suggestive form [88,89]
Sph =
1
2
∫
d4x
√−ggµν∂µφ∂νφ. (3.36)
Here, the metric can be read off from the line element Eq. (3.35) by writing ds2 =
gµνdx
µdxν as
gµν =
ρ
c
c2 − v2 ~vT
~v −I3×3
 , (3.37)
here (+−−−) convention is used.
The volume measure factor turns out to be
√−g = ρ2/c . At static equilibrium
the line element Eq. (3.35) is that of Minkowski space.
The measure in Eq. (3.33) is written as
D ′[φ] =
∏
x
(g00)1/2︸ ︷︷ ︸
“anomaly”
(−g)1/4dφ. (3.38)
We note that this measure is manifestly non-covariant due to the coordinate depen-
dent factor g00. For a field in curved spacetime the covariant measure ought to be
(−g)1/4 [108,109]. This means, although the phonon action Eq. (3.34) is covariant,
the path integral is not, leading to a quantum anomaly. We will come back to this
issue in Sec. 3.4.3 where we derive the conservation law of the stress tensor.
59
Figure 3.2: The Keldysh contour for the phonon field. The metric tensor
depends on the background variables ρ±0 and θ
±
0 and therefore is also
defined twice.
Before going into obtaining equations of motion from the effective action of
the Keldysh field theory, it is worthwhile to list a number of properties that Keldysh
theory obeys. We draw Fig. 3.2 to show the forward/backward fields and sources
of the phonon field. We identified Eq. (3.32) with the closed-time-path partition
function of phonons Zph. The first order correction Γph to the classical action is
then given as:
Γph = −i logZph[g+, g−]. (3.39)
As a consequence of unitarity, similar to Eq.(3.15),
Zph[g
± = g] = 1, (3.40)
i.e. when the background is the same on the forward and backward directions,
the product of backward and forward evolution operators is unitary. Also, from
Eq. (3.32),
Zph[g
+, g−; J+, J−] = (Zph[g−, g+; J−φ , J
+
φ ])
∗, (3.41)
where J±φ are additional sources attached in order to compute expectation values of
60
φ. It follows from Eq. (3.40) and Eq. (3.41) that the effective action satisfies
Γph[g, g] = 0, (3.42a)
Γph[g
+, g−] = −Γ∗ph[g−, g+]. (3.42b)
These properties are handy while computing the Keldysh correlation functions.
3.4 Analogue Einstein equations and two-fluid hydrodynamics
In general relativity, the relationship between matter and spacetime curvature
is governed by the Einstein’s equation. Being covariant under coordinate transfor-
mations, the matter field obeys the covariant conservation law. However, this is
not a total conservation law, as there is energy momentum exchange between fields
and the spacetime curvature. There are pseudo-tensor constructions like Einstein
pseudotensor or the Landau-Lifshitz pseudotensor that quantify the stress energy
of the gravitational field [110,111]. These pseudo-tensors, when added to the stress
tensor of matter, become a totally conserved quantity.
In this section, in analogy with general relativity, we will start with the first
loop effective action
Γ = S[ρ+, θ+]− S[ρ−, θ−]− i logZph[g±], (3.43)
where the metric g± is a functional of ρ± and θ± according to Eq. (3.2). The ef-
fective action Eq. (3.43), albeit without using the Keldysh contour, is considered
to compute backreaction corrections to acoustic black holes where the quantum ef-
fects, i.e. Hawking radiation, is important. [112] Unlike Ref. [112], in this chapter,
61
we explicitly integrated out the phonons through methods discussed in Sec. 3.3.
First, this approach revealed the anomalous measure in Eq. (3.38). Secondly, the
controlled series Eq. (3.23), produces higher order loop corrections. The contribu-
tion of these higher loops to the backreaction, especially near the horizon; where
the acoustic approximation becomes less accurate, and the interplay between the
Hawking radiation and covariance anomaly are interesting problems. However, all
of these are subleading corrections at first order. Therefore we will not address these
problems. Moreover, for the sake of computing the stochastic corrections, we extend
the effective action by using the Keldysh formalism.
We will first write the stress-energy tensor for the covariant phonons starting
from the effective action Eq. (3.19). Then we will write down an analogue Einstein
equation that describes the evolution of metric tensor (3.2) due to the stress-energy
of phonons, which play the part of matter. To complete the analogy, we will derive
a total conservation law by using the covariant conservation law and the analogue
Einstein’s equation. Moreover, we will show that the conserved quantity is a canon-
ical Noether current and therefore describes the total conservation of momentum
and energy in the lab frame. This means two fluid hydrodynamics directly follows
from the analogue gravity formalism.
62
3.4.1 Hilbert stress-energy operator of the covariant phonon field
The expectation value of the stress-energy operator can be defined by using
Schwinger’s variational principle [113] as
〈
Tˆµν
〉
= −i 2√−g
δ logZph
δ(g+)µν
∣∣∣∣
g+=g−
=
2√−g
δΓph
δ(g+)µν
∣∣∣∣
g+=g−
. (3.44)
Then, the stress-energy operator is defined (symmetrized for convenience) as
Tˆµν(x) =
1
2
{∂µφˆ(x), ∂νφˆ(x)} − 1
2
gµν∂
αφˆ(x)∂αφˆ(x). (3.45)
This expression is problematic because it contains a product of field operators at
the same spacetime point and generally leads to divergences. These can be cured
through a variety of regularization and renormalization schemes [113]. One of these
methods is called point splitting where the field operators are taken on different
spacetime points, and the limit of coincidence is taken after performing derivatives
and averages. In semiclassical gravity diverging quantities are renormalized into
coupling constants in Einstein’s equation [49,113].
Physically, the zero point quantum fluctuations that add up to an infinite vac-
uum energy. In the strongly interacting analoguesystem, the divergent quantities
are believed to be already accounted for in the background as a part of the internal
energy of the fluid [93]. This ensures the stability of the liquid droplets, by renor-
malizing the equilibrium pressure to zero. Recently, the role of zero point energy
in the formation of stable macroscopic droplets in strongly interacting BEC’s was
investigated both theoretically [114] and experimentally [115]. Therefore, assuming
that the vacuum energy is already renormalized into the background energy, we will
63
formally discard the divergent piece of the stress-energy expectation value. Note
that, in field theory on flat spacetime, the divergence is tacitly discarded through
the normal ordering of operators.
Let 〈Tˆµν(x)〉div represent the divergent piece in the expectation value for the
stress-energy operator. Throughout this chapter we will refer to the renormalized
stress-energy as:
〈
Tˆµν(x)
〉
=
1
2
(
δαµδ
β
ν + δ
α
ν δ
β
µ − gµνgαβ
)×
lim
x′,x′′→x
∂
∂x′α
∂
∂x′′β
〈
Tφˆ(x′)φˆ(x′′)
〉
−
〈
Tˆµν(x)
〉
div
. (3.46)
where T is the time ordering operator. The time ordered correlation function is
equivalent to the forward-forward correlation function of the Keldysh theory [47]
〈
Tφˆ(x′)φˆ(x′′)
〉
=
〈
φˆ+(x′)φˆ+(x′′)
〉
=
δ2Zph[J
+, J−]
δJ+(x′)δJ+(x′′)
∣∣∣∣
J+=J−=0
. (3.47)
3.4.2 Semiclassical analogue Einstein equations and the “phonon mat-
ter”
Having defined the stress tensor of the matter field (phonons), we now write
down the equations of motion for the superfluid and the phonons and argue that it
is analogous to the semi-classical Einstein’s equation.
After dropping the quantum pressure, and using Eq. (3.44), the Euler-Lagrange
equations that follow from the effective action Eq. (3.43) in the limit (ρ±, θ±)→ (ρ,
64
θ) are the fluid equations of motion Eq. (3.8) with semiclassical source terms
∂tρ+∇ · (ρ~v) = 1
2
∇ ·
(√−g 〈Tˆµν〉 δgµν
δ~v
)
, (3.48a)
∂tθ +
1
2
v2 + µ(ρ) =
1
2
(√−g 〈Tˆµν〉 δgµν
δρ
)
. (3.48b)
Here µ(ρ) = ∂ε/∂ρ is the local chemical potential for the superfluid.
Given the initial density operator of phonons and initial values and boundary
conditions for the background fields ρ and θ, one can compute the metric everywhere
by solving Eq. (3.48) and plugging the solutions into the definition Eq. (3.2). The
source terms on the right hand side must be self-consistent with this solution, as the
metric appears both explicitly in (3.48) and also inside the definition of the stress
tensor in Eq. (3.46). This is because the metric tensor determines how the sound
field propagates, classically expressed as Eq. (3.1). In this respect, the Eqs. (3.48)
resembles Einstein’s equations where the Eulerian left hand sides provide dynamics
to the metric tensor and are analogous to the Einstein tensor and the stress-tensor
corresponds to the matter whose motion is dictated by the curvature.
3.4.3 Canonical versus covariant conserved currents
Here, we show that the covariant conservation law and the analogue Einstein
equations Eq. (3.48) leads to a canonical conservation law for energy-momentum.
Classically, the stress tensor in Eq. (3.44), obeys the covariant conservation
law
∇µT µν = ∂µT µν + T θν Γµθµ − T µθ Γθµν = 0, (3.49)
owing to the fact that it is derived from the effective action [113], where ∇µ denotes
65
the covariant derivative. Writing the definitions of Christoffel symbols Eq. (B.1) in
terms of the metric and using Eq. (B.2) and Eq. (B.3), we get:
∇µT µν =
1√−g∂µ
(
T µν
√−g)+ 1
2
Tαβ ∂νg
αβ = 0. (3.50)
The partial derivative is streamlined with comma notation whenever convenient,
i.e. for any quantity A, ∂µA = A,µ. The Lagrangian density for phonons can be
extracted from Eq. (3.36) as
Lph =
1
2
√−ggµν∂µφ∂νφ. (3.51)
Then the first term in Eq. (3.50) is related to the canonical stress tensor of phonons,
and by inspecting the classical version of the Hilbert stress-energy tensor Eq. (3.45),
it can be written as
√−gT µν =
∂Lph
∂φ,µ
φ,ν −Lphδµν . (3.52)
By using the chain rule and the definition Eq. (3.44), the second piece in Eq. (3.50)
reduces to
1
2
√−gTαβ ∂νgαβ =
∂Lph
∂θ,µ
θ,µν +
∂Lph
∂ρ
ρ,ν . (3.53)
Let Lcl denote the Lagrangian density of the background after the quantum
pressure term is dropped
−Lcl = ρ∂tθ + 1
2
ρ~v2 + ε(ρ). (3.54)
By using the equations of motion Eq. (3.48), in the Euler-Lagrange form, as shown
in Appendix B write can write Eq. (3.50) as the conservation of the following current
Tµν =
∂Lph
∂φ,µ
φ,ν +
∂(Lph +Lcl)
∂θ,µ
θ,ν − (Lph +Lcl) δµν . (3.55)
66
If we define the total Lagrangian of the background-phonon composite system
Lsys = Lcl +Lph,
and noticing that ∂Lcl/∂ρ,µ and ∂Lcl/∂φ,µ both vanish, T in Eq. (3.55) is is precisely
the conserved Noether current
Tµν =
∂Lsys
∂(ρ, θ, φ),µ
· (ρ, θ, φ),ν −Lsysδµν .
due to the spacetime translation invariance of the overall system represented by the
effective action Eq. (3.43).
We define T , the stress energy tensor of the analoguegravitational field as
√−gT µν =
∂(Lph +Lcl)
∂θ,µ
θ,ν −Lclδµν . (3.56)
This expression generates the familiar energ and momentum density in the back-
ground. For example the energy density of the background in the laboratory frame
follows from Eq. (3.56) as
√−gT 00 =
1
2
ρ~v2 + ε(ρ). (3.57)
Similarly, the laboratory frame momentum density of the background follows from
Eq. (3.56) as
√−gT 0i = −ρvi. (3.58)
The Noether current Eq. (3.55) can be written as the current due to the
background and the excitations as Eq. (3.49) as the
Tµν =
√−gT µν +
√−gT µν . (3.59)
67
This means the mixed canonical stress tensor of the excitations correspond to energy-
momentum corrections due to excitations in the laboratory frame.
However, because of the covariance anomaly of the quantum phonon field that
manifests itself in the path integral measure Eq. (3.38), the covaraint derivative of
the expectation value of the quantum stress operator must be equal to an anomalous
current, i.e. ∇µ〈Tˆ µν 〉 = Janomν . Furthermore this current should be Galilean covari-
ant due to the overall Galilean invariance of the system. This is the second type of
anomaly in the analogue gravity system, the first being the trace anomaly due to
Hawking radiation, which occurs when there is a sonic horizon in the system [94].
In this work we assume that no sonic horizon exists and that the quantum pressure
term is weak everywhere. We defer the rigorous analysis and computation of the
anomalous current to another publication. Since anomalies are necessarily quantum
effects, in the regimes we work, they should be washed out by thermal contributions.
Then for the expectation
Tµν =
√−g
〈
Tˆ µν
〉
+
√−g
〈
T µν [φˆ]
〉
, (3.60)
the conservation law
∂µT
µ
ν = 0 (3.61)
holds. In the next section we rewrite Eq. (3.61) and Eq. (3.48) in the two-fluid
variables.
68
3.4.4 Mass-energy-momentum balance and the covariant conserva-
tion of stress-energy operator
In this section, we complete the connection of the analogue Einstein equation
Eq. (3.48) and the covariant conservation law Eq. (3.50) and two-fluid hydrodynam-
ics by identifying the normal and superfluid components in terms of the background
fields and the stress tensor.
Inspecting the conservation law Eq. (3.60) that we derived in Sec. 3.4.3, we
define the momentum density ~P , energy density E and energy flux ~Q due to phonons,
in the lab frame
Pi : = −
√−g
〈
Tˆ 0i
〉
, (3.62a)
E : =
√−g
〈
Tˆ 00
〉
, (3.62b)
Qi : =
√−g
〈
Tˆ i0
〉
. (3.62c)
Here, the quantities on the left hand side are momentum density, momentum flux
tensor and energy density. Being Cartesian tensors, their indices are uppered/lowered
by using the Kronecker delta function. By Galilean transforming the quantities on
the right hand side according to the usual tensor transformation rules, we observe
that Pi is Galilean invariant while E and Qi are not.
We define the following symmetric, Galilean invariant, Cartesian momentum
flux tensor
piij :=
c
ρ
√−g
〈
Tˆij
〉
=
c
ρ
√−g
〈
Tˆ µj
〉
giµ. (3.63)
By using the expression for the metric Eq. (3.2) and the definitions in Eq. (3.62),
69
the mixed momentum flux tensor due to phonons is
−√−g
〈
Tˆ ij
〉
= piij + Pjv
s
i , (3.64)
The equations of two fluid hydrodynamics are conservation laws for the den-
sity, momentum and energy for the superfluid and normal components of the system
denoted by the superscripts s and n respectively. We make the following identifica-
tion of velocity and density in Eq. (3.48) as the velocity of the superfluid component
and the total density (i.e. superfluid plus normal)respectively.
~v := ~vs, (3.65a)
ρ := ρs + ρn. (3.65b)
Consequently, ~vs = ∇θ.
Now, we identify the analogue Einstein equations Eq. (3.48) as mass conser-
vation and superflow equations respectively, by writing the phonon contribution in
terms of energy and momenta defined in Eq. (3.62).
The inverse metric in Eq. (3.48) follows from Eq. (3.2) as
gµν =
1
ρc
 1 ~vTs
~vs −c2I3×3 + ~vs~vTs
 . (3.66)
If A is any tensor with two covariant indices contracted and κ = ∂ ln c/∂ ln ρ
characterizes the logarithmic derivative of the energy of a linearly dispersing sound
wave with respect to the density of the medium, then the derivative of the metric
70
Eq. (3.66) obeys the following rules
δgµν
δρ
A ...µν... =
1
ρ
(κ− 1)Aµ ...µ... − 2κcA00 ...... , (3.67a)
δgµν
δvis
A ...µν... = A
0 ...
i... + A
0 ...
i ... . (3.67b)
The source current can be written as quasi-particle momentum by making the
identification
− 1
2
√−g
〈
Tˆµν
〉 δgµν
δ~vs
= −√−g
〈
Tˆ 0i
〉
= Pi. (3.68)
Then the first analogue Einstein equation Eq. (3.48a) is recast as the continuity
equation for mass
∂tρ+∇ · (ρ~vs + ~P ) = 0. (3.69)
The source term in Eq. (3.48b) reads
1
2
√−g
〈
Tˆµν
〉 δgµν
δρ
=
√−g
2ρ
[κ− 1]
〈
Tˆ µµ
〉
anom
− κ
ρ
ρc
√−g
〈
Tˆ 00
〉
:= −κ
ρ
[E − ~vs · ~P ] = −κ
ρ
E0 = −∂E0
∂ρ
. (3.70)
Here, the trace of the stress-operator is zero in the absence of an acoustic horizon
where the Hawking radiation creates a trace anomaly. The frame shifted E0, by
virtue of its tensorial expression being manifestly Galilean invariant, is the comoving
frame energy density of quasiparticles. In these variables the Bernoulli Eq. (3.48b)
takes its familiar form
∂tθ +
1
2
v2s + µ(ρ) +
∂E0
∂ρ
= 0. (3.71)
A corrolary of this equation is that the classical Lagrangian L of the background is
related to the isotropic pressure in the system
Lcl = −ρ∂tθ − ρ1
2
v2s − ε(ρ) = [ρµ(ρ)− ε(ρ)] + ρ
∂E0
∂ρ
− E0 + E0 = p+ E0. (3.72)
71
From Eq. (3.61), the momentum conservation equation reads
∂µT
µ
i = ∂tT
0
i + ∂jT
j
i = 0. (3.73)
Writing T as in Eq. (3.60), computing T according to Eq. (3.56) and using Eqs. (3.62),
Eq. (3.64) and Eq. (3.72) we get
∂t(ρvsi + Pi) + ∂j([ρvsi + Pi]vsj + Pjvsi + piij) + ∂i(p + E0) = 0. (3.74)
where repeated indices are summed.
Similarly, From Eq. (3.61), the energy conservation equation reads
∂µT
µ
0 = ∂tT
0
0 + ∂jT
j
0 = 0. (3.75)
Writing T as in Eq. (3.60), computing T according to Eq. (3.56) and using Eqs. (3.62)
and Eq. (3.71), we get
∂t
(
1
2
ρv2 + ε(ρ) + E
)
+ ∂i
(
Qi + [ρvsi + Pi]
[
1
2
v2s + µ(ρ) + ∂E0/∂ρ
])
= 0 (3.76)
In summary, the analogue Einstein equation Eq. (3.48), the covariant conser-
vation law of stress tensor Eq. (3.49) and the consequent conservation law Eq. (3.61)
leads to the continuity of mass Eq. (3.69), Bernouilli equation Eq. (3.71) and the
conservation of momentum Eq. (3.74). If we define the current density ~J , energy
density E , momentum flux Π and the superfluid potential G as
Ji := ρvsi + Pi, (3.77a)
E :=
1
2
ρv2 + ε(ρ) + E, (3.77b)
Πij := ρvsivsj + Pivsj + Pjvsi + piij + [p+ E0]δij , (3.77c)
G :=
1
2
v2s + µ(ρ) +
∂E0
∂ρ
. (3.77d)
72
we rewrite Eq. (3.69), Eq. (3.74), Eq. (3.76) and Eq. (3.71) as the Landau- Khalat-
nikov equations for the superfluid
∂tρ+∇ · ~J = 0, (3.78a)
∂tJi +∇jΠij = 0, (3.78b)
∂tE +∇ · ( ~Q+ ~JG) = 0, (3.78c)
∂t~vs +∇G = 0. (3.78d)
The Eqs. (3.78a) and (3.78d) share the manifest Galilean invariance of Euler
equations, because the Pi and E0 are comoving frame momentum and energy of
phonons. The Galilean invariance of Eqs. (3.78b) and (3.78c) follow immediately,
because they are derived using the covariant conservation law Eq. (3.49), that is
valid in all frames, and the analogue Einstein equations, which in the two-fluid
language become Eqs. (3.78a) and (3.78d).
Finally, note that the dissipative effects due to the normal fluid are taken into
account in the heat flux ~Q and the momentum flux tensor piij. The dissipative
components of these tensors can be computed according to linear response theory
which we explain in the next section. In the limit ~vs → 0, the Eqs. (3.78) become
that of a normal fluid with ρ → ρn is the normal density and Pi → ρnvni defines
the velocity of normal component. Assuming Stoke’s constitutive law for stress, the
momentum balance equation (3.78b) becomes the Navier-Stokes equation.
73
3.5 Stochastic analogue Einstein equations
Starting from the Lagrangian Eq. (3.7), we first derived the Euler equations
Eq. (3.8) for the background and then compute the phonon contribution and called
the overall system of equations the analogue Einstein (order one in metric pertur-
bation) equations Eq. (3.48).
In this section we will go to second order in the metric perturbations and
obtain dissipation and noise kernels due to phonons through a generalized linear
response method. We will write an effective stochastic action for phonons that cap-
ture this noise and derive the stochastic analogue Einstein equation. This equation
when linearized around a deterministic solution for the background fields ρ and θ,
describes the motion of the stochastic corrections to the solution. Finally we show
that in thermodynamic equilibrium the stochastic forcing term is balanced by the
dissipation, that is the fluctuation-dissipation relation holds.
3.5.1 Linear response and the covariant stress-energy correlator
To see the semi-classical expansion due to metric perturbations, we first define
the vector in time-path space composed of forward/backward metric tensors
~g =
(
g+µν(x), g−µν(x)
)
. (3.79)
Then the expansion of phonon effective action up to second order in metric pertur-
bations is
Γph =
δΓph
δ~g
· δ~g + 1
2
δ~g · δ
2Γph
δ~gδ~g
· δ~g +O(δ~g3). (3.80)
74
Here the boldface symbol δ is for variation with respect to the element g+µν is
defined as
δΓ
δg+µν
=
1√−g(x) δΓ[g+(x), g−(x)]δg+µν(x)
∣∣∣∣
g+=g−=g
, (3.81)
and the product of two time-path vectors (~A · ~B) sums over all path and tensor
indices and integrates over position by using the metric g+ = g− = g, namely,
~A · ~B =
∫
d4x
√
−g(x)
(
A+µν(x)B+µν(x) + A
−µν(x)B−µν(x)
)
. (3.82)
Using Eq. (3.40), (3.41) and (3.42), it follows that the first order variation of
the phonon effective action is
δΓph
δ~g
=
1
2
(
T ,−T
)
, (3.83)
where
T =
〈
Tˆµν(x)
〉
. (3.84)
We define the deviations of the stress energy operator from its expectation
value
tˆµν(x) = Tˆµν(x)−
〈
Tˆµν(x)
〉
. (3.85)
Following Martin and Verdaguer [116] we define the local stress deviation K, the
causal response kernel H and the noise kernel N that are bi-tensors made of corre-
lators of tˆ
Hˆµναβ(x, y) = −iθ(x0 − y0)
〈[
tˆµν(x), tˆαβ(y)
]〉
, (3.86a)
Nˆµναβ(x, y) =
1
2
〈{
tˆµν(x), tˆαβ(y)
}〉
, (3.86b)
Kˆµναβ(x, y) =
−4√
g(x)g(y)
〈
δ2Sph[φˆ]
δgµν(x)δgαβ(y)
〉
. (3.86c)
75
These tensors are symmetric in the following way. If A is any of the above kernels
than
Aµναβ = Aνµαβ = Aµνβα (3.87)
In Eq. (3.86a), H is manifestly causal and HS,A are symmetric/anti-symmetric parts
of H respectively,
HS,Aµναβ(x, y) = ±HS,Aαβµν(y, x). (3.88)
and they are
HˆSµναβ(x, y) = Im
〈
T∗
{
tˆµν(x)tˆαβ(y)
}〉
, (3.89a)
HˆAµναβ(x, y) =
−i
2
〈[
tˆµν(x), tˆαβ(y)
]〉
. (3.89b)
As we show in the Appendix C, the second order variation is a matrix in
time-path space is
δ2Γph
δ~gδ~g
=
1
4
iN−HS −K −iN−HA
−iN + HA iN + HS + K
 . (3.90)
All of these kernels are real. HA, the anti-symmetric component of the causal
response function H, changes sign in Eq. (3.90) when the forward/backward branches
are switched. Therefore it breaks time-reversal symmetry and therefore gives rise
to dissipative effects.
3.5.2 Global thermal equilibrium and fluctuation dissipation relation
Here we show, at global thermal equilibrium, that the noise kernel N and the
dissipation kernel H are related in a stationary flow. For example, a flow through a
76
tube with varying cross section, but constant in time for each point in the tube and
the temperature measured in the lab is uniform throughout the system.
The stationary flow with global thermal equilibrium is especially important
because in the analogue gravity language it maps onto a hot curved spacetime at
global thermal equilibrium. Such a space contains globally time-like Killing vectors
κµ. [113,117] The local temperature in the spacetime varies according to the norm
of the time-like Killing vector and is dictated by Tolman’s law:
TTol
√
gµνκµκν = constant. (3.91)
Once there are globally time-like Killing vectors, modes that solve the covariant wave
equation Eq. (3.1) are classified as positive/negative Killing frequencies according
to the value of their directional (Lie) derivative along the Killing vector, that is
Lκun = −iωnun, (3.92a)
Lκu
∗
n = iωiu
∗
n, ω > 0. (3.92b)
Now the field operator is written in the second quantization language and reads
φˆ =
∑
n
aˆ†nu
∗
n + aˆnun, (3.93)
where the vacuum state is uniquely defined. The Hamiltonian operator with eigen-
values equal to the Killing frequencies can be written in terms of the positive energy
mode operators
Hˆ =
∑
n
ωnaˆ
†
naˆn. (3.94)
As long as the conserved energy ω and the associated temperature is used, the
methods of thermal field theory in flat spacetime is easily generalized to the curved
77
spacetime case [116]. In the fluid system, this is not a surprise, as in a stationary
case, the Hamiltonian associated with the Lagrangian in Eq. (3.34) is time indepen-
dent and second quantized in the form of Eq. (3.94).
In analogue gravity systems, this energy ω coincides with the lab frame energy
of the mode [118]. Note that, this is not the comoving frame energy or the energy
measured by observers in curved spacetime, which is not conserved. This means
the temperature in the comoving frame and the lab frame and the different and
are related to each other through the Tolman law [93]. Stationary flow means the
Killing vector is κ = (1, 0, 0, 0) In global thermodynamic equilibrium, the positive
energe modes ui are distributed thermally according to the lab frame temperature
so that we have
TTol
√
g00 = TTol
√
ρc(1− v2/c2) = Tlab = 1
β
. (3.95)
Note that this analogue Tolman law is a purely classical effect that stems from the
emergent Lorentz invariance of excitations in the strongly correlated superfluid. It
is to be distinguished from the Unruh effect, where an accelerating observer detects
a thermal radiation of phonons even if the lab frame’s temperature is zero.
In equilibrium the eigenstates are distributed according to the lab frame tem-
perature, with the density operator of defined in Eq. (3.31) that reads
%ˆφ = exp(−βHˆ)/Tr
{
%ˆφ
}
. (3.96)
Then, the operator averages are best handled in imaginary time (it = τ) formalism.
For example for two operators Oˆ1 and Oˆ2 we have, suppressing the space indices,
78
〈
Oˆ1(τ)Oˆ2(τ
′)
〉
= Tr
(
Oˆ1(τ)Oˆ2(τ
′)ρˆ
)
= Tr
(
U(iτ)Oˆ1U(−iτ)U(iτ ′)Oˆ2U(−iτ ′)U(−iβ)
)
= Tr
(
Oˆ2(τ
′ + β)Oˆ1(τ)ρˆ
)
=
〈
Oˆ2(τ
′ + β)Oˆ1(τ)
〉
. (3.97)
Stationarity allows the use of Fourier transformation in time domain, hence this
equation can be written as
〈
Oˆ1(ω)Oˆ2(−ω)
〉
= eβω
〈
Oˆ2(−ω)Oˆ1(ω)
〉
. (3.98)
Now we show that a fluctuation dissipation relation exists. Inspection of
Eq. (3.104), (3.90) and (3.106) reveals that the time- reversal odd piece HA of
the causal response causes dissipation and the fluctuation kernel N causes noise.
Defining
Fµναβ(ω, ~x, ~y) =
∫
dω
2pi
ei(t−t
′) 〈tˆµν(t, ~x)tˆαβ(t′, ~y)〉 , (3.99)
and rewriting the definitions in Eq. (3.86) in frequency domain and using Eq. (3.98)
we get:
HˆAµναβ(ω, ~x, ~y) = −
i
2
F (ω, ~x, ~y)
(
1− e−βω) , (3.100a)
Nˆµναβ(ω, ~x, ~y) =
1
2
F (ω, ~x, ~y)
(
1 + e−βω
)
. (3.100b)
The fluctuation dissipation relation immediately follows as:
HˆAµναβ(ω, ~x, ~y) = −i tanh(βω/2)Nˆµναβ(ω, ~x, ~y). (3.101)
79
3.5.3 Hydrodynamic fluctuations around a deterministic solution
Suppose ρ¯ and θ¯ are solutions to the analogue Einstein equations Eq. (3.48).
Define the classical and quantum corrections to the solutions
ρ± = ρ¯+ ρ± = ρ¯+ ρc ± ρq (3.102a)
θ± = θ¯ + θ± = θ¯ + θc ± θq. (3.102b)
Let the corresponding perturbations to the metric tensor be hc + hq and hc − hq so
that
~g = (g¯µν + (hq)µν + (hq)µν , g¯µν + (hc)µν − (hq)µν). (3.103a)
Breaking the deviations into classical and quantum parts amounts to (Keldysh)
rotating the matrix in Eq. (3.90). The classical deviation can be thought of as a
real displacement while the quantum deviation as a virtual displacement. Variation
with respect to virtual displacement (hq) gives the classical equations of motion with
one loop corrections as in Eq. (3.48). To find the fluctuations of the stress-energy,
we can expand the effective actions as a Taylor series in the classical and quantum
deviations to second order as
Γph[~g] = T · hq − 1
2
hq · (H + K) · hc + i
2
hq ·N · hq +O(h3), (3.104)
where, all the kernels are functionals of the unperturbed metric g and ~h is decom-
posed as (hc,hq) and the dot product on the right hand side contracts spacetime
indices with g and integrates over spacetime. We notice that the equation (3.104)
80
is in the same form as the Keldysh action for a quantum particle in a bath [15].
The well established technique to derive Langevin type equation from this action
is to define an auxiliary stochastic tensor ξ to decouple the O((hq)2) term (the
Hubbard-Stratanovich decomposition).
exp (iΓph[~g]) =
1√
Det(2piN)
∫
D [ξ]e−
1
2
ξ·N−1·ξeiRe{Γph}+iξ·h
q
=
〈
eiRe{Γph}+iξ·h
q〉
ξ
= 〈exp (iΓph[~g; ξ])〉ξ . (3.105)
The mean and correlation function of the noise tensor field ξ is defined by the
above Gaussian path integral, using the definition Eq. (3.106) of the noise kernel N,
which yields
〈ξµν(x)〉ξ = 0, (3.106a)
〈ξµν(x)ξαβ(y)〉ξ =
1
2
〈{tˆµν(x), tˆαβ(y)}〉 . (3.106b)
The new phonon effective action
Γph[~g; ξ[g¯]] = (T + ξ) · hq − 1
2
hq · (H + K) · hc +O(h3) (3.107)
depends on the noise tensor ξ.
Then the background plus phonon effective action in Eq. (3.43) as a functional
of noise tensor
Γ[ρ±, θ±, ξ] = S[ρ+, θ+]− S[ρ−, θ−] + Γph[~g; ξ[g¯]]. (3.108)
81
3.5.4 Stochastic analogue Einstein equation
The Euler-Lagrange equations for the stochastic effective action Γ[ρ±, θ±, ξ] in
the limit (ρ±, θ±) → (ρ, θ) and hence hq → 0 are the following stochastic analogue
Einstein equations for the two-fluid hydrodynamics
∂tρ+∇ · (ρ~v) = 1
2
∇ ·
(√−g [〈Tˆµν〉+ ξµν] δgµν
δ~v
)
; (3.109a)
∂tθ +
1
2
v2 + µ(ρ) =
1
2
(√−g [〈Tˆµν〉+ ξµν] δgµν
δρ
)
. (3.109b)
In this equation, the noise source ξ[g¯] is computed by using the metric g¯(ρ¯, v¯) that
solves Eq. (3.48), while ρ and θ suffer deviations ρc, θc from the solution of Eq. (3.48)
due to the existence of noise. The energy momentum tensor is also computed by
taking these deviations in to account. Next we summarize the procedure to find the
stochastic deviations ρc and θc of the background.
3.5.5 Procedure to compute background metric fluctuations from the
analogue Einstein equation
Below, we outline the procedure to determine fluctuations of the metric, given
initial data that consists of the initial conditions, boundary conditions, an initial
state of the phonon density operator, and external sources, if any.
First, is to compute the deterministic solution (ρ¯, v¯) and the associated metric
g¯µν [ρ¯, θ¯] as defined in Eq. (3.2), by solving the analogue Einstein equation (3.48)
self consistently. In general this is equivalent to solving the two-fluid equations
82
Eq. (3.78) and is a difficult task. However, one can approach this problem pertur-
batively. If one has a solution to the Euler equations that describes the background
without the phonons (Eq. (3.8) with no quantum pressure), then one can compute
the expectation value of the stress-energy operator Eq. (3.46) by using the tech-
niques in Ref. [113] and references therein. One then continues this process up to
a desired order, so that the background is consistent with the metric on which the
sources are computed.
Second, is to compute the correlators of the stress tensor in Eq. (3.86) to form
the response and noise kernels. For the methods required to perform this, we refer
the reader to Ref.[ 119, 120] and references therein.
Third, is to linearize the stochastic analogue Einstein equation (3.109) around
the deterministic solution ρ¯, θ¯. The resulting equation is linear in the stochastic
corrections ρc and θc, to the background variables. The associated first-order cor-
rections to the deterministic metric g¯ follow from the chain rule as
(hc)µν = ρc
δg¯µν
δρ
+ ~vc · δg¯
µν
δ~v
. (3.110)
Then, the linearized stochastic analogue Einstein equation, that we dub the analogue
Einstein-Langevin equation, follows from Eq. (3.109), where we linearize the stress
tensor by using Eq. (3.104), as
∂tρ
c+∇·(ρ¯~vc)+∇·(ρcv¯) = 1
2
∇·
(√−g¯ δg¯µν
δv¯
×
[
−1
2
∫
d4y(H +K)µναβ(x, y)(h
c)αβ(y) + ξµν
])
(3.111)
83
and
∂tθ
c+v¯ ·~vc+ c¯
2
ρ¯
ρc =
1
2
(√−g¯ δg¯µν
δρ
×
[
−1
2
∫
d4y(H +K)µναβ(x, y)(h
c)αβ(y) + ξµν
])
.
(3.112)
Fourth and final, is to write down correlators of hc. To do this, one first
expresses the correlators of the background corrections ρc and θc in terms of the
noise correlator N by using the Green’s functions of the linear equations Eq. (3.111)
and Eq. (3.112). Then the correlators of hc follow from Eq. (3.110).
3.6 Equilibrium ‘Minkowski’ fluctuations
In this section we illustrate the procedure of Sec. 3.5.5 for the static (~vs = 0)
thermodynamic equilibrium focusing on a qualitative analysis (detailed technical cal-
culations and results for the correlators are cumbersome and will be presented else-
where). The metric tensor for the static background reduces to that of Minkowski,
if one sets
ρ¯ = ρe = ce = c¯ = 1 (3.113)
in addition to the choice of units in Eq. (3.10).
This renders Step 1 in Sec. 3.5.5 trivial. The analogue Einstein equations
Eq. (3.48) and the covariant conservation law Eq. (3.49) are trivially satisfied. Nev-
ertheless, it is instructive to outline how the expectation value of the stress-tensor
is computed.
The isotropy and homogeneity of Minkowski space places strong restrictions
on the stress-energy tensor and its correlators. For the stress tensor, homogeneity
84
means all components are constant in spacetime. The trace of the expectation value
of stress energy vanishes for a massless scalar field. The energy density is positive.
There is no isotropic Cartesian vector; therefore, the momentum density vanishes.
The only isotropic rank-2 Cartesian tensor is the Kronecker delta; therefore, the
spatial components are isotropic. The only rank-2 Minkowski tensor with the above
properties is
〈
Tˆ µν
〉
=
pi2T 4
45

3 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1

, (3.114)
where the overall factor is the black body energy density and can be obtained from
the standard momentum space integral that arise in Eq. (3.46), after dropping the
infinite vacuum energy.
Similarly to the stress tensor, the symmetries greatly reduce the complexity of
calculating its correlates as well. First of all, the homogeneity of spacetime requires
that any function f(x, y) = f(x−y). The stress tensor correllators can be expressed
as momentum space integrals. In thermal equilibrium, all non-local behavior is due
to the factor coth(βω/2), which behaves like
coth(βω/2)
kBT~ω−−−−−→ 2kBT
~ω
(3.115)
in the high-temperature limit and all kernels become local (memoryless) operators
in spacetime.
In this limit, we linearize Eq. (3.109) around ρ¯ and θ¯ to get Eq. (3.111) and
85
(3.112) for the static equilibrium. This linearized equation is in the Langevin form.
In Fourier domain it reads−iω −|~k|2
1 −iω

ρc
θc
 =
R11 R12
R21 R22

ρc
θc
+
ζ1
ζ2
 . (3.116)
where Rˆ’s are local operators, hence polynomials in ~k and ω and ζ’s are stochastic
sources.
The stochastic sources ζ’s are deduced from ξµν by applying Eq. (3.67a), as
ζ1 = −ikjξ0j (3.117a)
ζ2 =
1
2
(κ− 1)ξµµ − κξ00 (3.117b)
where κ = ∂ ln c/∂ ln ρ is the change of energy of phonons with density.
The diagonal terms R11 and R22 in Eq. (3.116) are equal and the symmetric
part of the response kernel does not contribute to them:
R11 = R22 = −i(1− κ)
2
kj(HA)0 µj µ − iκkj(HA)0 00j . (3.118)
Whereas, the antisymmetric part of the response kernel does not contribute to the
off diagonal components
R12 = k
ikj(HS +K)0 0i j (3.119)
and
R21 = −(1− κ)
2
4
(HS +K)ν µν µ− κ2(HS +K)0000 + κ(κ− 1)(HS +K)µ 00µ . (3.120)
Now one can solve Eq. (3.116) and express the correlators of stochastic correc-
tions ρc and θc in terms of the noise correlators and the metric fluctuation follows
by using Eq. (3.110).
86
We also note that, to lowest order in k, introducing some constants α, β, ν we
have
R11 = R22 → −ν|~k|2 (3.121a)
R12 → αijkikj (3.121b)
R21 → −β (3.121c)
and the Eq. (3.116) takes the form of the linearized Navier-Stokes and continuity
equations with stochastic forcing terms
∂tρ
c + (δij + αij)∇i~vcj − ν∇2ρc = ζ1, (3.122a)
∂t~v
c − ν∇2~vc = −(1 + β)∇ρc +∇ζ2. (3.122b)
The coefficient of viscosity ν can be extracted as
ν =
1
3
∂
∂kj
[
(1− κ)
2
(HA)0 µj µ − κ(HA)0 00j
] ∣∣∣∣
~k=0
. (3.123)
3.7 Conclusions and outlook
In this work, we develop a geometric generalization of the Landau-Khalatnikov
two-fluid hydrodynamics for a strongly correlated bosonic superfluid and derive a
system of coupled equations describing the motion of the superfluid background and
the entropy-carrying normal fluid of phonon excitations. By exploiting the emergent
covariance of the phonon field, we draw analogies between general relativity and the
87
two-fluid system including stochastic effects due to the fluctuations of the phonon
“matter” field.
We emphasize that the methods and analogies used here are applicable to a
variety of systems. Indeed, weakly interacting quasiparticles excited over a ground
state is a common picture in many-body physics. Hydrodynamics captures the long-
wavelength behavior of such systems. Galilean invariance requires that quasiparticle
dynamics is defined on the frame comoving with the background fluid that represents
the ground state manifold. This means, when referred to the lab frame, that the
quasiparticle field experiences shifts due to the background flow – an effect captured
in the definition of the metric tensor (3.2). From thereon, the quasiparticles can
be described by quantum field theory on curved spacetime, thereby justifying the
geometric formulation of the problem.
We would like to conclude with a brief summary of our work and promising
directions for further research
In Sec. 3.2, following an earlier work [93], we discussed the applicability of
the metric description for fluid by analyzing the length- and energy-scales. In anal-
ogy with cosmology [96,121], the geometric description breaks down at an effective
“Planck energy,” where the hydrodynamic description becomes inapplicable. How-
ever, the quantum description of the boson system is known at all energies contrary
to the case in cosmology. This paves the way to understanding the cosmological
phenomena through condensed matter systems [122–124].
In Sec. 3.3, we used the background field formalism to write down the Keldysh
theory of quasiparticles. We use the Keldysh contour because the dynamical fluc-
88
tuations of the phonon field follow naturally from the Keldysh effective action that
is “designed” to account for non-equilibrium phenomena. One interesting finding of
this analysis is an additional “gravitational anomaly” , not present in the original
work of Unruh [88] and Stone [89] : that is, despite the action for the phonons is
covariant, as already noted in the literature, the measure of the path integral is not
covariant. Hence, in contrast to classical phonon action, their full quantum theory
is not covariant. We believe this anomaly should be taken into account, whenever
quantum effects such as Hawking radiation, is important. It is curious to see if this
anomaly is related to gravitational anomalies in high-energy phenomena. These
interesting issues will be discussed in a future publication.
In Sec. 3.4, inspired by Ref. 93, we derive the analogue Einstein equation
that governs the background and the excitations (matter). We establish the equiv-
alence of the analogue Einstein equation and the covariant conservation law for the
phonons, to the two-fluid conservation laws. Instead of assuming a specific form
for the stress-energy tensor as in Ref. [96], we prove the equivalence of the two de-
scriptions by reducing the covariant conservation law down to the Noether current
of the two-fluid system and by using the equations of motion. We believe that the
proof presented here is more general, than the specific superfluid model we study,
and should be applicable to any emergent general-relativistic theory, which may be
derived in the hydrodynamic, long-wave-length limit of a “parent” condensed matter
system.
In Sec. 3.5, we take the analogy between the superfluid system and general
relativity further to the domain of stochastic fluctuations. This allows the appli-
89
cation of a variety of methods that have been invented for stochastic gravity [49]
to condensed matter systems.We write the response and dissipation kernels in the
covariant language. In global thermal equilibrium, we discuss the notion of temper-
ature on curved spacetime and show the appearance of analogue Tolman’s law. We
prove the fluctuation-dissipation relation for a metric with globally time-like Killing
vectors – that is, for a flow that can be brought to a stationary form after a Galilean
transformation. Finally, we outline a procedure to calculate correlators of various
observables, including fluctuations of the metric.
One interesting direction would be to consider the effects of higher order cor-
relators of the stress-energy tensor and build a hierarchy of equations similar to the
BBGKY hierarchy.
Finally, we linearize the stochastic analogue Einstein equation and obtain a
Langevin-type equation for the stochastic corrections to the background. We show
that the symmetries of the flow determine structure of the Langevin equation, by
considering the Minkowski case. Classification of the flow induced spacetimes, the
stress tensor and its fluctuations on the basis of the symmetry of the flow is a
well defined mathematical problem. In general relativity these correspond to the
Petrov classification of spacetimes and Segre classification of symmetric tensors [125].
Together with the machinery of general relativity such as conformal transformations
and general coordinate transformations, the classification of the solutions of the two-
fluid equations can be carried out in a similar spirit.
The main fundamental physical consequence of our analysis is the finding that
sound waves propagating on a superfluid background at a finite temperature experi-
90
ence a random, stochastic metric on top a dynamical but deterministic background
metric determined by the classical superfluid flow. Even in the absence of any flows
and in equilibrium, where the average metric is that of flat Minkowski space, the
phonon “rays” would experience random deviations from the flat background. This
may lead to analogue gravitational lensing of acoustic waves due to the phenomenon
called ‘intermittency’. In Ref. [126], Zeldovich considered light rays propagating in
a random medium with a fluctuating metric, and showed that even if the average
metric is flat, an observer receiving two distant rays would see the rays bend and the
corresponding object shrink, due to the stochasticity. The effect seems unobserv-
able in actual general relativity due to very weak metric fluctuations, if any [127].
However, similar fluctuations of “synthetic metric” can be greatly enhanced in a
superfluid and it is conceivable that the corresponding analogue Zeldovich effect –
bending of phonon “rays” propagating through a thermal superfluid – could become
observable.
91
Chapter 4: Dynamical many-body localization in an integrable model
4.1 Introduction
Recently, there has been a lot of interest and progress in understanding Anderson-
type localization properties of disordered, interacting many-body systems. No-
tably, the remarkable phenomenon of many-body localization (MBL) was discov-
ered [32–39]. In an isolated system, MBL manifests itself in the localization of all
eigenstates and leads to the breakdown of ergodicity and violation of the eigenvalue
thermalization hypothesis [29,31], forcing to revisit the very foundations of quantum
statistical mechanics [128,129].
In this work, we ask whether a driven interacting system can be dynamically
many-body localized. We answer this question in the affirmative and present an
exactly solvable model of a kicked chain of interacting linear rotors, which shows
both dynamical MBL and delocalized regimes.
A quantum kicked-rotor is a canonical model of quantum chaos [25,130] which
exhibits dynamical localization in momentum space. The time dependent Schro¨dinger
equation for a general kicked rotor is given by (here and below, we set ~ = 1 and
92
the driving period T = 1),
i∂tψ(θ, t) = [2piα(−i∂θ)l +K(θ)δ(t− n)]ψ(θ, t). (4.1)
In the ground-breaking paper [25], Fishman, Prange, and Grempel proved that
the eigenvalue problem for the Floquet operator of a kicked rotor is equivalent to
that of a particle hopping in a (quasi)periodic potential ((ir)rational α) in one-
dimension [28] given by,
∑
r
Wm+rur + tan[ω − 2piαml]um = Eum. (4.2)
This mapping traced the origin of the dynamical localization in driven systems
to Anderson localization in time-independent settings. While the quadratic rotor
(l = 2) is nonintegrable, the linear rotor model (l = 1) in Eq. (4.1) was exactly
solved in Refs. [25, 27, 28, 41, 131]. The corresponding integrable lattice version
in Eq. (4.2) with l = 1 is dubbed Maryland model (MM) [132, 133] (See section
A.1 for technical details of this mapping). For the linear rotor, both classical and
quantum dynamics is integrable, and the dynamical localization (absence of chaos)
is due to the existence of a complete set of integrals of motion [41]. However, the
incommensurate MM is an Anderson insulator with no classical interpretation [28]
even though the linear rotor manifests classical integrability. Thus, the dynamical
localization for the quantum version of both linear (l = 1) and quadratic rotor
(l = 2) seems to stem from the Anderson mechanism [28].
In this work, we generalize the linear kicked rotor model by considering an
interacting chain of driven rotors in order to understand the anatomy of many-body
93
localization in the dynamical space. One of the remarkable features of this many-
particle model is that it manifests both localization and delocalization in dynamical
space depending on whether the components of ~α are irrational or rational.
Let us emphasize from the outset that the model we consider is integrable for
all parameters owing to the first order differential operator. This leads to integrals
of motion (IOMs), which are local in the spatial (angular) variables. The underlying
integrability is a special, non-universal feature of our model (4.3), which allows us to
solve it exactly. However, the existence of the local-in-θ IOMs has no direct relation
to dynamical MBL, which occurs in momentum space. The non-interacting version
of these emergent IOM’s for the linear kicked rotor was pointed out by Berry in
Ref. ( [41]).
As shown below, dynamical MBL is accompanied by the appearance of ad-
ditional integrals of motion bounded in momentum space – a central result of this
work. As argued below, these “emergent” IOMs and the dynamical MBL are univer-
sal phenomena, which would survive in non-integrable generalizations of the model
(which however is not analytically solvable).
To capture the dynamical localization for this interacting model, we monitor
three indicators: energy growth at long times, (momentum degrees of freedom at
long times, and the existence of integrals of motion in momentum space. Below, we
first present the details of our model in Section 4.2. The main analytical results are
outlined in Section 4.3, and their numerical analysis and key conclusions are given in
Section 4.4. An outline of technical derivation of the results is given in Section 4.5.
The experimental proposal to realize our model (4.3), is explained in Section 4.6.
94
In section 4.7 we list our conclusions. Finally, Appendix A.1 and Appendix A.2 are
devoted to the summary of the connection of the rotor problem to a lattice model
and the correspondence between our model and a d-dimensional lattice, respectively.
Throughout the text, in all the summations
∑
i ... or
∑
ij ... we only consider
i 6= j. So that, expressions like ∑j 1/(αi − αj) are not divergent. The denominator
vanishes only when there is a resonance, such as αi → αj.
4.2 The Model
We consider a many-body interacting generalization,
Hˆ(t) = Hˆ0 + Vˆ
∞∑
n=−∞
δ(t− n), with Vˆ =
d∑
i=1
K(θˆi),
Hˆ0 = 2pi
d∑
i=1
αipˆi +
1
2
d−1∑
i 6=j
Jij(θˆi − θˆj) (4.3)
of the linear rotor model.
Hˆ0 is the static Hamiltonian describing d particles on a ring, each rotating
2piαi radians per one period of the kick. θˆi is the position operator for the ith
particle on the ring and pˆi is its momentum. These d particles interact through a
translationally invariant two-body potential Jij(θˆi− θˆj). This form of the interaction
ensures conservation of momentum. The rotors are driven by Vˆ (t), which contains
periodic delta function impulses, where the strength of the impulse is given by a
general periodic one-body potential K(θˆi). The local potentials are periodic with 2pi
and therefore the most general form can be written as K(θj) =
∑
m kme
imθj . Here,
km is the mth Fourier component of the potential that acts on the jth particle. The
95
periodic form of the interaction potential J can be written as Jij =
∑
m b
ij
me
im(θi−θj).
Here, bijm is the mth Fourier component of the interaction potential between the
ith and jth particle. Reversing i and j and replacing m with −m in this Fourier
expansion has no effect on the components; therefore, bijm = b
ji
−m. This property will
be handy while deriving formulas throughout the text.
Note that in our model the localization is a consequence of incommensurate
driving period and angular velocities ~α. This situation is different from recent works
using Floquet analysis to probe dynamical properties of MBL states of disordered
Hamiltonians [134–137] (in these papers, a Floquet perturbation is imposed on a
state, many-body localized in coordinate space, while our goal is to induce dynamical
many-body localization in momentum space by the Floquet perturbation).
4.3 Main Results
4.3.1 Energy dynamics
Following the conjecture of D’Alessio and Polkovnikov [138], we test the dy-
namical localization by computing the energy growth as a function of time at long
times. The average energy after N kicks (equivalent to time) can be written as,
E(N) = 〈ψN |Hˆ0|ψN〉. To compute it, we write
|ψN〉 = UˆNF |ψ0〉, UˆF = e−iVˆ e−iHˆ0 , (4.4)
where UˆF is the Floquet evolution operator, which captures the state of the sys-
tem after each kick. Owing to the linear dependence of the Hamiltonian on the
96
momentum, we can explicitly compute this expectation.
E(N) = E(0) +
d∑
i=1
∑
m
2piαi〈Γˆmi〉0 sin(mNpiαi)
sin(mpiαi)
. (4.5)
In the above expression, E(0) = 〈ψ0|H0|ψ0〉 corresponds to the average many-body
energy over the initial state. Γˆmi = −imkmeim(θˆi+piαi[N+1]) depends on the chosen
form of K(θˆ) averaged over the initial state and due to its periodic nature is a
bounded function of the number of kicks N . The growth of energy for large N is
then completely dependent on the nature of αi appearing in the ratio
sin(mNpiαi)
sin(mpiαi)
.
Note that this expression is completely independent of interactions, which we prove
in Section 4.5.1. This is a consequence of momentum conserving interactions and
this property is no longer valid when the translational invariance of interactions is
broken.
4.3.2 Momentum dynamics
Another indicator for dynamical localization is the spread in the momentum
degrees of freedom. The ith momentum after N kicks, pi(N) = 〈ψN |pˆi|ψN〉 is given
by the following expression,
pi(N) = 〈pˆi〉0 +
∑
m
〈Γˆmi〉0 sin(mNpiαi)
sin(mpiαi)
+
∑
mj
〈Γˆintmij〉0
sin(mNpi∆αij)
mpi∆αij
.
We have defined ∆αij = αi − αj. In the above expression, the first term is the ith
momentum in the initial eigenstate 〈pi〉0 = 〈ψ0|pˆi|ψ0〉. The second term corresponds
to the kicking potential as defined in Eq. (4.5). The last term depends explicitly
on the form of interaction via Γˆintmij = −imbijmeim(θˆi−θˆj+piN∆αij) and is a bounded
97
function of N . The growth of momenta at long times corresponding to the last term
is completely determined by the ratio
sin(mNpi∆αij)
mpi∆αij
.
4.3.3 Integrals of motion
Recent works have shown that the existence of integrals of motion (IOM) can
be used as a diagnostic to quantify both non-interacting Anderson localization [139]
and many-body localization [40, 140]. In the context of dynamical localization for
the model at hand, we work in the momentum basis and search for existence of
IOMs in this basis. We begin by constructing IOMs by identifying operators Cˆi
that satisfies [Cˆi, UˆF ] = 0 and [Cˆi, Cˆj] = 0. The existence of these IOMs encode
information about dynamical localization. Operators that commute with UˆF satisfy
the property 〈Cˆi〉N+1 = 〈Cˆi〉N and are given by
Cˆi = pˆi +
1
2
∑
m
mkme
im(θˆi+piαi)
sin(mpiαi)
+
1
2
∑
mj
bijme
im(θˆi−θˆj)
pi(αi − αj) . (4.6)
This expression is a generalization of the constant of motion given by Berry [41]. The
derivation of this expression is given in Sec. 4.5.2. Since Cˆi is an IOM, pˆi is bounded
in time as long as the series in the last two terms converges. The delocalization of
pˆi with time will occur due to the diverging denominators in the θˆ dependent terms.
We reiterate that the model always contains d integrals of motion, Bˆi =
θˆi/αi(mod 2pi) ([Bˆi, UˆF ] = 0) which results in integrability for both localized and
delocalized cases. For example, if αi = ri/si is rational, the existence of Bˆi results
in integrability even though Cˆ1...d do not exist. Since Bˆi is momentum independent,
its existence cannot bind pi(N). For the fully localized case, we have additional
98
0 20 40 60 80 1000
5
10
15
20
25
N
Σ
HE
L
(AI)
~α ∈ irrationals
αi 6= αj
0 20 40 60 80 100
0
5
10
15
20
N
Σ
HE
L
(BI)
~α ∈ irrationals
α1 = α2
0 20 40 60 80 100
50
100
150
200
N
Σ
HE
L
(CI)
α1 = 1/2,
α2,..,10 ∈ irrat.
0 20 40 60 80 1000
5
10
15
20
N
Σ
Hp
1L,
.
.
,
Σ
Hp
10
L
(AII)
~α ∈ irrationals
αi 6= αj
0 20 40 60 80 100
0
10
20
30
40
50
N
Σ
Hp
1L,
.
.
,
Σ
Hp
10
L
(BII)
~α ∈ irrationals
α1 = α2
0 20 40 60 80 100
0
10
20
30
40
50
60
70
N
Σ
Hp
1L,
.
.
,
Σ
Hp
10
L
(CII)
α1 = 1/2,
α2,..,10 ∈ irrat.
Figure 4.1: (Top panel) Plots showing evolution of the root mean square
deviation of energy, σ(E) =
√〈H20 〉 − 〈H0〉2, as a function of time, la-
beled by the number of kicks, N . (Bottom Panel) Plots showing evo-
lution of the root mean square deviation of individual momenta in an
ensemble of 10 particles σ(pi) =
√〈p2i 〉 − 〈pi〉2 as a function of time,
labeled by the number of kicks, N . The initial states of the rotors are
assumed to be definite momentum states. The total number of particles
in this interacting ensemble is d = 10 and circular boundary conditions
apply. We consider two body interaction term to be Jij = cos(θi − θj).
The periodic kicking potential is given by K(θ) =
∑∞
m=1 km cos(mθ) for
all particles. The m-th Fourier coefficient is fixed by km = z˜/m
2, where
z˜ is a random complex number with real and imaginary parts uniformly
distributed in the interval [0,1]. We make sure that K(θ) is real. We
choose ~α corresponding to three scenarios: Figs. (AI, AII): If ϕ denotes
the golden ratio, αj = (j/d)ϕ− (1/2)ϕ are all irrationals for j = 1 . . . 10.
Figs. (BI, BII): We consider αj distinct irrationals as in Case A, but set
α1 = α2, which results in the resonant growth of σ(p1) and σ(p2), while
σ(E) is bounded. Figs. (CI, CII): We consider αj as in Case A and set
α1 = 1/2, which results in the resonant growth of σ(E) and σ(pi).
99
d IOM’s Cˆ1...d given in Eq. (4.6) that explicitly depend of pˆi and thus constrain
it. Thus, Cˆi’s can be thought of as “emergent” IOM’s constraining the momentum
growth, resulting in dynamical MBL and “emergent” superintegrability.
4.4 Dynamical many-body localization and the structure of ~α
Equipped with three analytical expression for the energy growth, momentum
growth and IOMs [Eqs. (4.5), (4.6), and (4.6)], we now diagnose the dynamical
localization for three distinct cases with varying structure of ~α. We note that, there
are cases where mαi can get arbitrarily close to integers (Liouville numbers) and
total energy and momentum are no longer bounded. In the single rotor case, such
a situation yields interesting consequences like marginal resonance and a mobility
edge in the momentum lattice, [41, 141]. In this study we exclude this possibility
and refer to generic irrationals only.
4.4.1 Case I: α1 . . . αd are distinct generic irrationals
For irrational values of αi, the total energy in Eq. (4.5) is always a bounded
function of N since mαi /∈ integer. In Fig. 4.1(AI), we fix initial states to be
momentum eigenstates ψ0(θ) = e
ip0θ. For momentum eigenstates, 〈Γˆmi〉0 ∼ 0. Thus,
we plot the root mean square (RMS) of system’s energy σ(E) =
√
〈Hˆ20 〉 − 〈Hˆ0〉2 as
a function of N . Fig. 4.1 (AI) shows boundedness in the spread of the energy as a
function N . The momentum growth shown in Eq. (4.6) has contributions from both
interactions and the kicking potential. In Fig. 4.1 (AII) we plot RMS deviation of
100
momenta σ(p1) . . . σ(pd) (we used d = 10 for the specific simulation) and show that
the spread in the momenta is bounded as a function of N . For this case the integrals
of motion in Eq. (4.6) exist and convergent. Thus, all the diagnostics for this case
point towards a true many-body dynamical localization.
4.4.2 Case II: α1 = α2 and α2 . . . αd are distinct generic irrationals
For this case, the energy remains a bounded function of N since mαi are not
integers. Thus, the RMS deviation σ(E) is bounded as shown in Fig. (4.1 BI).
However, the second term in the momentum expression (Eq. (4.6)) develops a res-
onance (for the momenta p1 and p2) since α1 → α2. Due to this resonance term,
sin(mNpi∆α12)/(mpi∆α12) ∼ N as α1 → α2. This resonant growth is reflected in the
RMS deviation of σ(p1) and σ(p2) growing linearly with N , while the σ(p3) . . . σ(p10)
remain bounded as shown in Fig. 4.1 (BII). This is a striking scenario where the
resonant momenta are not localized even if the total energy is bounded for large
N . However, the delocalization of p1 and p2 in time is reflected in the break down
of IOMs Cˆ1 and Cˆ2 due to diverging denominators in the resonant limit α1 → α2.
For this case, the bounded total energy fails to diagnose delocalization as shown in
Fig. 4.1 (BI, BII). We note that this scenario has no analogue in the non-interacting
limit. Notice that interactions we considered possess translational invariance, i.e.
in the form J(θi − θj) given in Eq. (4.3) and therefore interactions conserve mo-
mentum. The dichotomy between the energy growth and momentum growth is a
result of conservation of momentum and the linear dependence of energy on mo-
101
mentum. If we allow interactions that break translational invariance, momentum is
no longer conserved and resonance due to interactions triggers unbounded growth
(delocalization) of both momenta and the total energy.
4.4.3 Case III: α1 = 1/2 and α2 . . . αd are distinct generic irrationals
For rational α1 = 1/2, the system develops a different kind of resonance com-
pared to Case II. We consider a kicking potential with (k2 6= 0). The resonance
condition mα1 ∈ integer can be satisfied for m = 2 and the ratio sin(mNpiα1)sin(mpiα1) grows
as N . It results in energy growth and delocalization of momenta p1 as shown in
the RMS deviations in Fig. 4.1 (CI, CII). This is a converse situation to Case II
where the energy delocalizes with the delocalization of p1 even though the momenta
p2, p3, . . . p10 are bounded. This situation is again captured by the IOMs, where Cˆ1
does not exist, while Cˆ2 . . . Cˆ10 are well defined and convergent as seen in Eq. (4.6).
We can generalize the above representative cases. For each rational αi, its
integral of motion Cˆi breaks down and the corresponding momentum and energy
diverge with time. For each pair αi = αj both Cˆi and Cˆj break down and the
corresponding momenta diverge in opposite directions, therefore Cˆi + Cˆj is still an
IOM and the total momentum and energy of the pair are bounded.
Other than the behaviour of energy in Case II in Section 4.4.2, the rest of our
analysis apply equally to the interactions that break translational symmetry.
102
4.5 Derivation of main results
Having established the physical understanding of localization for this model,
we now sketch the brief derivation leading to the final results in Eq. (4.5), Eq. (4.6)
and Eq. (4.6). In the following we consider the expectation of a generic operator as a
function of time, X(N) = 〈ψN |Xˆ|ψN〉. We write the explicit evolution of the many-
body wave function between two successive kicks, |ψN〉 = e−iVˆ e−iHˆ0|ψN−1〉. Notice
that Hˆ0 = 2pi~α·~ˆp+ 12
∑d−1
i 6=j Jij(θˆi−θˆj) contains the many-body interaction term which
may seem daunting, however, the linear momentum term allows a factorization in the
Floquet operator. The Baker-Campbell-Hausdorff formula Zˆ = ln(eXˆeYˆ ) becomes
tractable when [Xˆ, Yˆ ] = sYˆ . In this case, the result simply reads Zˆ = Xˆ + sYˆ /(1−
e−s). Now fix m and let Xˆ = i2pi(α1pˆ1 + α2pˆ2) and Yˆ = ib˜12m exp(im[θˆ1 − θˆ2])
where ∆α12 = α1 − α2. The result of the commutator reads [Xˆ, Yˆ ] = i2pi∆α12mYˆ ,
precisely the tractable case discussed above. Replacing s with i2pi∆α12m in the
formula Zˆ = Xˆ + sYˆ /(1 − e−s) and inverting both sides of the equality, we obtain
the following factorization
e−i2pi~α·~p−ib˜
12
m exp(im[θ1−θ2]) = e−ib
12
m exp(im[θ1−θ2])e−i2pi~α·~p. (4.7)
For this to hold, the Fourier coefficients must satisfy
b˜ijm =
sin(pim∆αij)
pim∆αij
bijme
−impi∆αij . (4.8)
This argument can be generalized to more particles and Fourier coefficients. Due to
linearity summations over particle indices i, j and Fourier indices m are introduced.
103
All in all, we can factorize the evolution operator for our model in Eq. (4.3) as,
|ψN〉 = e−iVˆ− i2
∑
ij J˜ij(θˆi−θˆj)e−i2pi~α·~ˆp|ψN−1〉, (4.9)
where we have defined the modified interaction term as,
J˜ij =
∑
m
b˜ijme
im(θi−θj) (4.10a)
=
∑
m
sin(pim∆αij)
pim(∆αij)
bijme
im(θi−θj−pi∆αij). (4.10b)
The advantage of this factorization is that the operator e−i2pi~α·~ˆp is a trans-
lation operator in the position basis. We can rewrite Eq. (4.9) in the position
basis by acting with 〈θ| from the left. We define 〈θ|ψN〉 = ψN(θ) and express
〈θ|e−i2pi~α·~p|ψN−1〉 = ψN−1(~θ − 2pi~α). For a single kick we then have,
ψN(~θ) = e
−iV (~θ)−i 1
2
∑
ij J˜ij(θi−θj)ψN−1(~θ − 2pi~α). (4.11)
The above equation can be recursively iterated to yield,
ψN(~θ) = e
−i
N−1∑
n=0
[V (~θ−2pin~α)+
∑
ij J˜ij(
~θ−2pin~α)]
ψ0(~θ − 2piN~α). (4.12)
4.5.1 Derivation of the evolution of energy and momentum averages
Now consider a generic operator Xˆ ≡ X(pˆ1...pˆd; θˆ1...θˆd). The expectation value
of this operator after N kicks is
X(N) =
∫
d~θ ψ∗N(~θ)X
(
∂
dθ1
...
∂
dθd
; θ1...θd
)
ψN(~θ). (4.13)
If we substitute Xˆ = pˆk, we get
pl(N) = 〈pˆl〉0 −
N∑
n=1
〈
∂lV (~θ + 2pin~α) + ∂l
1
2
∑
ij
J˜ij(~θ + 2pin~α)
〉
0
.
104
Here, ∂l ≡ ∂/∂θl and 〈...〉0 ≡
∫
d~θ...|ψ0|2. The contribution from the kicking poten-
tial is:
〈
−∂l
∑
n
V (~θ + 2pin~α)
〉
0
=
∑
m
〈Γˆml〉0 sin(mNpiαl)
sin(mpiαl)
. (4.14)
The contribution from the interaction potential is
〈
−∂l
∑
n
1
2
∑
ij
J˜ij(θˆi − θˆj + 2pin[αi − αj])
〉
0
=
∑
jm
〈Γˆintmlj〉0
sin(mNpi∆αli)
pim∆αli
.
Putting together above expressions, we obtain the expression for the momentum
dynamics presented in Sec. (4.3.2).
pi(N) = 〈pˆi〉0 +
∑
m
〈Γˆmi〉0 sin(mNpiαi)
sin(mpiαi)
+
∑
mji
〈Γˆintmij〉0
sin(mNpi∆αij)
mpi∆αij
. (4.15)
Now we derive the expression for the energy growth. Substituting Xˆ = Hˆ0 in
Eq. (4.13), we have
E(N) = E(0) +
∑
mi
2piαi〈Γˆmi〉0 sin(mNpiαl)
sin(mpiαl)
+
1
2
∑
ij
〈Jij(θˆi − θˆj + 2piN∆αij)− Jij(θˆi − θˆj)〉0
+
∑
ijm
2piαi〈Γˆintmij〉0
sin(mNpi∆αij)
mpi∆αij
.
(4.16)
In the above equation, a cancellation occurs between the interaction terms in the
last two lines of Eq. (4.16) owing to the following relation,
1
2
∑
ij
〈Jij(θˆi − θˆj + 2piN∆αij)− Jij(θi − θj)〉0
= −
∑
ijm
2piαi〈Γˆintmij〉0
sin(mNpi∆αij)
mpi∆αij
. (4.17)
105
This completes the derivation of the energy growth shown in Sec. (4.16)
E(N) = E(0) +
d∑
i=1
∑
m
2piαi〈Γˆmi〉0 sin(mNpiαi)
sin(mpiαi)
. (4.18)
The dropping out of interaction terms from the total energy can be interpreted
in the following way. As seen from Eq. (4.15), the contribution of interaction to the
momentum average picks up a negative sign when i and j are interchanged. This
means interaction transfers momentum from one particle to the other at each kick,
in other words momentum is conserved for each couple of rotors. Since the energy
is linear in momenta, when the momenta of rotors are summed, the contribution of
interactions vanishes. We emphasize that when the interactions break translational
invariance, they no longer conserve momentum and in that case energy growth
depends on interactions too.
4.5.2 Construction of integrals of motion
In this section, we outline the derivation involved in the construction of in-
tegrals of motion. By inspecting Eq. (4.14) and using the identity sin(mpiα) =
[exp(impiα)− exp(−impiα)]/(2i), we can write
pl(N + 1)− pl(N) = 1
2
∑
m
mkm
sin(mpiαl)
(〈
eim(θˆl+piαl)
〉
N
−
〈
eim(θˆl+piαl)
〉
N+1
)
+
1
2
∑
mj
bljm
pi∆αlj
(〈
eim(θˆl−θˆj)
〉
N
−
〈
eim(θˆl−θˆj)
〉
N+1
)
, (4.19)
noting that, the expression is valid whenever the denominators sin(mpiαl) and
∆αlj = αl − αj are non-zero, in other words, whenever the resonances are avoided.
106
The above expression can be organized in a way that it manifests the IOMs,〈
pˆi +
1
2
∑
m
mkme
im(θˆi+piαi)
sin(mpiαi)
+
1
2
∑
mj
bijme
im(θˆi−θˆj)
pi(αi − αj)
〉
N+1
=
〈
pˆi +
1
2
∑
m
mkme
im(θˆi+piαi)
sin(mpiαi)
+
1
2
∑
mj
bijme
im(θˆi−θˆj)
pi(αi − αj)
〉
N
(4.20)
If we use the definition in Eq.(4.6) we get 〈Cˆl〉N+1 = 〈Cˆl〉N , thereby proving
that Cˆl is an integral of motion.
4.5.3 Floquet Hamiltonian and quasienergy eigenstates
A special case of our model is when there are no resonances, namely, all IOMs
associated with the momentum localization Cˆ1..d are intact. In this case, of particular
importance is the following combination of the integrals of motion
HˆF =
∑
i
2piαiCˆi. (4.21)
Where HˆF is known as the Floquet Hamiltonian and is defined as,
e−iHF = e−iVˆ e−iHˆ0 = UˆF . (4.22)
The quasienergy wave functions ψω are simultaneous eigenstates of HˆF and Cˆi’s.
The quasienergy-ω state centered around momenta 〈~ˆp〉 = ~M is
ψω(~θ) = (2pi)
−N/2 exp
{
i ~M · ~θ −
∑
jm
kme
im(θj+piαj)
2 sin(mpiαj)
−
∑
ijm
bijm
4pim(αi − αj)e
im(θi−θj)
}
.
(4.23)
This satisfies Cˆiψω = Miψω. By writing HˆFψω = ωψω and using Eq. (4.21),
we see that the eigenvalue equation is satisfied when, ω = 2pi~α · ~M(mod 2pi).
107
We can also compute the momentum-momentum correlator, 〈pˆipˆj〉 − 〈pˆi〉〈pˆj〉
for i 6= j over quasienergy eigenstates. The correlator over this state follows as
〈pˆipˆj〉 − 〈pˆi〉〈pˆj〉 = −1
4
∑
m
|bijm|2
pi2(αi − αj)2 . (4.24)
In the case of a resonance αi → αj, this correlator clearly diverges, while in the
localized case it remains finite.
4.6 Experimental proposal
A natural venue for realizing the interacting kicked rotor model is supercon-
ducting grains. The Hamiltonian, Eq. (4.3) could be implemented using a chain
of voltage-biased superconducting grains coupled to each other using Josephson
junctions. Consider a chain of grains gated by a ground plane which resistively
connected to ground (see, Fig. 4.2). We then supply a gate voltage Vi to each grain,
and connect them to each other by Josephson junctions Jij.
Because the voltage on the i-th grain is locked to be Vi, its phase winds with
an angular velocity φ˙i = 2eVi/~. Therefore, the resistance and gate capacitor can
be ignored when writing the effective stationary part of the Hamiltonian:
H0 =
∑
i
qiVi −
∑
ij
Jij cos(φi − φj). (4.25)
In addition, the “kick” term is obtained by connecting each grain to a common
macroscopic superconductor (which is itself grounded), for a short time and through
a strong Josephson coupling.
V (t) = −
∑
i, n
K(t− nT ) cosφi, (4.26)
108
Vi-1 Vi Vi+1
C
R
C
R
C
R
Macroscopic Superconductor
Ki-1 Ki Ki+1
Ji-1 Ji Ji+1
Stroboscopic 
Josephson Junction 
Coupling 
Figure 4.2: Schematic of chain of voltage-biased (Vi) superconducting
grains coupled to each other (Ji) using Josephson junctions. Each grain
is connected (Ki) to a grounded common macroscopic superconductor
stroboscopically through a strong Josephson coupling.
with K(t) = K when |t| < τ . To make the kick term as close to a delta function as
possible, we must have 2eViτ  ~2pi, and Kτ ∼ ~, and τ  T . A diagram of the
circuit for a nearest-neighbor interaction is shown in Fig. 4.2. Identifying, φi = θi,
2piαi = 2eVi/~ and pi = ~qi/2e we see that we indeed obtain the Hamiltonian of
Eq. (4.3).
4.7 Conclusions
In this work, we introduced the concept of dynamical many-body localization
and presented an exactly-solvable model of driven linear rotors, which exhibits this
phenomenon. Although the model possesses a full set of integrals of motion, it is
shown that dynamical MBL is accompanied by the emergence of additional integrals
109
of motion, local in momentum space. We believe that this observation has important
general implications for understanding dynamics of interacting many-body systems.
We have shown that these integrals of motion break down due to two types
of resonances indicating delocalization in momentum space. One type of resonance
originates from commensuration of the external driving period with the parameters
of the system and the other from the static interactions. An interesting feature of
this model is that the total energy in the system, that is linear in momenta, fails
to be a good indicator of dynamical localization, since when momentum conserving
interactions delocalize momentum, momenta of interacting pair grow in opposite
directions.
Moreover, we have shown, by utilizing the lattice mapping introduced by Fish-
man and Grempel and Prange, that our model maps into a disordered lattice with
as many dimensions as there are rotors. Based on this observation, we argue that
what is observed is an Anderson type localization, particularly of the type seen in
correlated disorder systems.
We also have proposed an experimental setup composed of Josephson junctions
and superconducting grains to realize the model Hamiltonian.
Finally, we emphasize that the results presented here can apply to more generic
non-integrable systems. For example, a recent work [142] considers interacting
kicked Dirac particles with individual Hamiltonians, H0 = 2piασxp + Mσz, and
provides a simple argument that this non- integrable system also exhibits MBL.
First, this model also exhibits localization when α’s are generic distinct irrationals.
Second, although the number of interacting particles that can be considered numer-
110
ically is limited, note that at large momenta the Dirac model crosses over to the
linear model considered here. This suggests that MBL should be robust to a class
of non-integrable generalizations, for any number of interacting rotors.
111
Chapter 5: Conclusions
Classical mechanics is built on assumptions that hold for macroscopic bodies,
therefore, is applicable to a diverse set of phenomena that includes those that appear
in anthropic scales, which makes the theory appealing to our intuition. For this
very reason it historically precedes quantum mechanics that lies at the heart of our
current understanding of nature.
Classical models have enormous utility for being well understood and intuitive.
In this thesis we put forward three models for many-body quantum systems, solve
them under certain assumptions and employ a classical analogy to help understand
the underlying physics and/or make inference about a broader class of systems.
From a different perspective, these analogies contribute to resolve the problem
of how classical dynamics emerge from quantum mechanics. Although the emer-
gence of classical mechanics from quantum mechanics, commonly referred to as the
classical limit, is controlled by a single parameter, the Planck constant, the quali-
tative picture the two theories paint is fundamentally different. The measurement
problem, the failure of conventional quantization schemes for the gravitational field
and the emergence of chaos from unitary quantum dynamics are all different cases
of the ‘classical limit’ problem that plagues almost all fields of modern physics in
112
addition to rising foundational questions. All three of these issues are related to
how we extract a set of observables (what we refer to as the ‘system’) from a very
large number of interacting quantum degrees of freedom, a majority of which make
up the ‘environment’ for the system. Therefore, the afore-mentioned issues can be
considered in the context of the ‘quantum many-body problem’ as we argued in the
introduction.
Below is a summary of the models considered in these thesis, the corresponding
classical analogy, its advantages and/or its relation to a broader problem say of
classicalization of particles/fields, emergence of spacetime or chaos.
In Part 2, we considered a quasi-one-dimensional superconductor. In the su-
perconducting phase of matter, the complete many-body quantum field theory can
be reduced to the solution of a PDE for the superconducting order parameter and
the two-point function for the quasiparticles. This PDE parametrically depends on
the disorder in the system. In the long wavelength limit, the appropriate Eilen-
berger equations of the system were solved. We discovered that this problem can
be mapped to a one-dimensional classical particle moving in an external potential.
In view of the recent interest in superconducting heterostructures, we studied the
proximity effects in a normal segment, attached to a clean p-wave wire. We discov-
ered that despite the presence of impurities, the proximity-induced superconducting
correlations are long-range. We also found that impurity scattering leads to the ap-
pearance of a delocalized zero-energy peak that hints to the long sought zero-energy
Majorana mode. The methods we employ can be applied to analyze a variety of sit-
uations with quasi one dimensional disordered superconductor and metal junctions,
113
which will form the building block of topological quantum computers.
In Part 3, we considered a strongly interacting Bose system. Similarly, the
superfluid phase is described as a classical object: a perfect fluid. The environment
is the gas of phonons which effect the background flow by applying stresses. This
picture is called the two-fluid model. The phonons, possessing Lorentz symmetry,
can be shown to travel on an emergent spacetime metric defined by the perfect fluid.
With this observation we drew analogies between general relativity and the two-fluid
system including stochastic effects due to the fluctuations of the phonon “matter”
field. This gives an idea of how classical spacetime might emerge from quantum
degrees of freedom in the low energy limit. Moreover, based on this analogy, we
conjectured a lensing for a phonon wave packet due to stochastic fluctuations of the
background fluid, similar to lensing of light rays due to a fluctuating metric field.
In Part 4 we considered the analogy between localization of electron states in
a lattice and the quantum bounds to classical chaos, known as the Maryland model.
To understand if the quantum suppression of chaos persists despite interactions,
we introduced the concept of dynamical many-body localization and presented an
exactly-solvable model of driven and interacting linear rotors, which exhibits this
phenomenon. For each rotor, the rest of the rotors is the environment. The ex-
istence of dynamical MBL depends on whether the environment restores chaos or
not. Although the model possesses a full set of integrals of motion, it is shown that
dynamical MBL is accompanied by the emergence of additional integrals of motion,
local in momentum space. In both cases the quantum evolution equations closely
follow their classical counter parts. The additional integrals of motion break down
114
when interactions become resonant causing phase space diffusion or delocalization.
We also have proposed an experimental setup composed of Josephson junctions
and superconducting grains to realize the model Hamiltonian. Finally, we emphasize
that the results presented here can apply to more generic non-integrable systems.
For example, a later work [142] considers interacting kicked Dirac particles with
individual Hamiltonians, H0 = 2piασxp + Mσz, and provides a simple argument
that this non- integrable system also exhibits MBL. This suggests that MBL should
be robust to a class of non-integrable generalizations, for any number of interacting
rotors.
115
Appendix A: Mapping between Quantum Kicked Rotor and the tight-
binding model
In the first part of this section, we derive the known lattice mapping of the
one dimensional kicked rotor [28]. Once we establish this derivation, we show that
the many-body linear kicked rotor 4.3 also admits a lattice mapping of a particle
on a d-dimensional lattice. We emphasize that existence of such a mapping in the
many-body case is limited to the case of linear p model (l = 1).
A.1 Lattice model of single quantum kicked rotor
In the introduction we considered the time dependent Schro¨dinger equation
for a kicked rotor. The kinetic part was considered to be pˆl. For the quadratic
kicked rotor case l = 2 and the linear kicked rotor that we consider in this work is
l = 1. Also ~ = 1 and kicking period T = 1 are used in this equation.
i∂tψ(θ, t) = [2piα(−i∂θ)l +K(θ)
∑
n
δ(t− n)]ψ(θ, t). (A.1)
The above equation can be solved for ψ(θ, t) = e−iωtu(θ, t), where the function u
has the unit periodicity of the driving. Let u± defines the state just before and after
116
the kick and are connected by the evolution operator in the following way,
u+ = e−iK(θ)u−, u− = ei(ω−2piαpˆ
l)u+ (A.2)
Define the following: u¯ = u
++u−
2
, exp(−iKˆ(θ)) = (1 − iWˆ (θ))−1(1 + iWˆ (θ)),
exp(−i[2piαpˆl − ω]) = (1 − iTˆ (θ))−1(1 + iTˆ (θ)). Based on the above definitions,
u± = (1∓ iTˆ (θ))u¯ and
[Tˆ (θ) + Wˆ (θ)]u¯ = 0 (A.3)
is obtained.
Fourier transforming the above expression, we get the following tight binding
model,
∑
n 6=m
Wm−nun + Tmum = Eum. (A.4)
Here, the energies and hoppings are:
Wm−n = −Eδm,n−
1
2pi
∫ 2pi
0
e−i(m−n)θ
{
tan
(
K(θ)
2
)}
dθ,
(A.5a)
Tm = tan
(
1
2
[ω − 2piαml]
)
. (A.5b)
This completes the derivation for the lattice mapping for the single rotor case. In
the following section, we generalize this derivation to demonstrate the existence of
a lattice mapping for the interacting rotor model of Eq. (4.3).
A.2 d-dimensional lattice model
In this section we show that there exists a d− dimensional lattice model corre-
sponding to the d particle interacting version of the kicked rotor model in Eq. (4.3).
117
Such a mapping has been previously identified for the case of a d rotors driven by
an interaction potential [131]. Notice that in our case, the interactions are encoded
in the stating Hamiltonian Hˆ0. However, we show that for the linear momentum
dependence of the kinetic term, static interactions can expressed as the driven inter-
actions and the rest of the lattice mapping simple follows from Ref. ( [131]). In order
to establish this mapping, we use the factorization of the Floquet operator discussed
in the Section. (4.5). The time dependent Hamiltonian in Eq. (4.3) produces the
same Floquet operator as,
Hˆ(t) = 2pi
d∑
i=1
αipˆi + Vˆ
F
∞∑
n=−∞
δ(t− n), (A.6)
where we have defined,
Vˆ F =
d∑
i=1
K(θˆi) +
1
2
∑
i 6=j
J˜ij(θˆi − θˆj) (A.7)
=
d∑
j=1
∑
m
kme
imθˆj +
1
2
∑
i 6=j
∑
m
b˜ijme
im(θˆi−θˆj). (A.8)
The factorization enables to treat on site and interaction potentials on equal footing.
The b˜ij is defined in Eq. (4.8). Moreover, we write:
Vˆ F =
∑
~m
V F~m e
i~m·~θ. (A.9)
Here, ~m is a d-dimensional vector characterized by the integer m. The com-
ponents of the potential V F~m are as follows. For fixed i and j, for vectors ~m parallel
to a coordinate axis, i.e. ml = mδlj, V
F
~m takes the value km. For vectors ~m such
that ml = m(δli − δlj), V F~m takes the value b˜ijm/2. For any other vector, V F~m van-
ishes. Treating α’s and p’s as d-dimensional vectors as well, the Hamiltonian can be
succinctly as:
118
Hˆ(t) = 2pi~α · ~ˆp+
∑
~m
V~me
i~m·~ˆθ
∞∑
n=−∞
δ(t− n). (A.10)
Following Ref. ( [131]), the above driven Hamiltonian can be mapped on to a d-
dimensional lattice model, which is closely related to the lattice mapping outlined
in the previous section for the single rotor case.
H~m,~nu~n = T~mu~m +
∑
~n
W~m,~nu~n = Eu~m. (A.11)
Here, ~m and ~n are vectors that contain integers that correspond to the quantized
eigenvalues of the angular momentum operator. The hopping and onsite terms are
defined as,
W~m−~n = −Eδ~m,~n+
1
(2pi)d
∫ 2pi
0
e−i(~m−~n)·
~θ
{
− tan
(
V F (~θ)
2
)}
d~θ
, (A.12a)
T~m = tan
(
1
2
[ω − 2pi~α · ~m]
)
. (A.12b)
Here, we defined E = −(1/[2pi]d) ∫ tan(V F (~θ)/2) so as to make W0 = 0.
119
Appendix B: Derivation of the canonical conservation law
Here we derive the canonical conservation laws in Part 3, by using the covari-
ant conservation law and the equations of motion for the two-fluid system. The
Chrystoffel symbols are
Γµνα =
1
2
gµβ (gβν,α + gβα,ν − gνα,β) , (B.1)
where the comma notation means
gβν,α ≡ ∂
∂xα
gβν .
For a symmetric tensor T µθ , with the application of chain rule, we have
T µθ Γ
θ
µν = −
1
2
Tγθ
(
gγµ,νδ
θ
µ + g
γµ
,µδ
θ
ν − gθµ,µδγν
)
= −1
2
Tγθ
(
gγµ,νδ
θ
µ
)
= −1
2
Tγµ g
γµ
,ν .
(B.2)
Similarly, we have
Γµνµ =
1
2
gµβ
(
gβν,µ + gβµ,ν − gνµ,β
)
=
1
2
gµβgβµ,ν =
1
2g
g,ν = ∂ν log
√−g. (B.3)
Using Eq. (B.2) and Eq. (B.3) and relabeling dummy indices as required, we
write the covariant conservation law Eq. (3.49) in the main text as Eq. (3.50).
120
Now, by using Eq. (3.52) and Eq. (3.53), the covariant conservation law
Eq. (3.50) can be recast in the Noether current form that appears in Eq. (3.55)
as follows.
√−g∇µT µν =
(
∂Lph
∂φ,µ
φ,ν −Lphδµν
)
,µ
+
∂Lph
∂θ,µ
θ,µν +
∂Lph
∂ρ
ρ,ν = 0. (B.4)
Using the Euler-Lagrange equation (classical limit of Eq. (3.48a))
∂(Lph +Lcl)
∂ρ
=
(
∂(Lph +Lcl)
∂ρ,µ
)
,µ
= 0,
and using the chain rule
∂νLcl =
∂Lcl
∂ρ
∂νρ+
∂Lcl
∂θ,µ
θ,µν ,
the covaraint derivative in Eq. (B.4) becomes
√−g∇µT µν =
(
∂Lph
∂φ,µ
φ,ν −Lphδµν
)
,µ
+
∂(Lph +Lcl)
∂θ,µ
θ,µν − ∂µLclδµν = 0. (B.5)
Now using, the second Euler-Lagrange equation (classical limit of Eq. (3.48b)),
(
∂(Lph +Lcl)
∂θ,µ
)
,µ
=
∂(Lph +Lcl)
∂θ
= 0.
the Eq. (B.5) reduces to
√−g∇µT µν =
(
∂Lph
∂φ,µ
φ,ν − (Lcl +Lph)δµν
)
,µ
+
(
∂(Lph +Lcl)
∂θ,µ
θ,ν
)
,µ
= 0. (B.6)
The total derivative on the right hand side can be written as the divergence of the
Noether current Eq. (3.55).
121
Appendix C: Noise and response kernels
Here, we write the first and second order variations of the effective phonon
action in Part 3 as tensors and bi-tensor kernels. For example consider the second
order variations of the phonon effective action with respect to the forward metric:
δ2Γph
δ(g+)µν(x)δ(g+)αβ(y)
∣∣∣∣
g+=g−
=
−i
4
√
−g(x)
〈
Tˆµν(x)
〉√
−g(y)
〈
Tˆµν(y)
〉
− i δ
2Zph
δ(g+)µν(x)δ(g+)αβ(y)
∣∣∣∣
g+=g−
. (C.1)
In the last part we used the unitarity of the partition function in Eq. (3.40).
The second order variation of the phonon partition function contains the ex-
pectation value of a product of stress-energy operators.
δ2Zph
δ(g+)µν(x)δ(g+)αβ(y)
∣∣∣∣
g+=g−
=
− 1
4
√
−g(x)
√
−g(y)
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
+ i
〈
δ2Sph[φˆ]
δgµν(x)δgαβ(y)
〉
. (C.2)
Here, the Weyl or T∗ ordered average of the product of stress operators is understood
in the following way. The stress-energy operator contains terms quadratic in the
field and its derivatives. It can be regularized by using the point splitting method
[104,113]. In this regularization scheme, to take an average as in Eq. (C.2), one first
122
writes the quadratic operator average as
lim
x′,x′′→x
〈
T∗
{
∂µφˆ(x
′)∂νφˆ(x′′)
}〉
= lim
x′,x′′→x
∂
∂x′µ
∂
∂x′′ν
〈
T
{
φˆ(x′)φˆ(x′′)
}〉
. (C.3)
Then, the time ordered operator average is computed and renormalized by sub-
tracting the divergent terms. Lastly, the operators are acted on the result and the
coincidence limit is taken.
Weyl ordering inherits all properties of time or path ordering, for example it
is possible to rewrite the Weyl ordered average as
2
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
=
〈{
Tˆµν(x), Tˆαβ(y)
}〉
+
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
−
〈
T˜∗
{
Tˆµν(x)Tˆαβ(y)
}〉
. (C.4)
Here, {, } is the anti-commutator and T˜ is the reverse Weyl ordering computed by
reversing the time ordering in Eq. (C.3). Using Eq. (3.41), we can decompose the
Weyl ordered average into its real and imaginary parts
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
=
1
2
〈{
Tˆµν(x), Tˆαβ(y)
}〉
+ iIm
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
.
(C.5)
Following Martin and Verdaguer [116] we define the (bi-tensor) real kernels.
HˆSµναβ(x, y) = Im
〈
T∗
{
Tˆµν(x)Tˆαβ(y)
}〉
, (C.6a)
HˆAµναβ(x, y) =
−i
2
〈[
Tˆµν(x), Tˆαβ(y)
]〉
, (C.6b)
Hˆµναβ(x, y) = Hˆ
A
µναβ(x, y) + Hˆ
S
µναβ(x, y), (C.6c)
= −iθ(x0 − y0)
〈[
Tˆµν(x), Tˆαβ(y)
]〉
(C.6d)
123
Nˆµναβ(x, y) =
1
2
〈{
tˆµν(x), tˆαβ(y)
}〉
, (C.7a)
where tˆµν(x) = Tˆµν(x)−
〈
Tˆµν(x)
〉
, (C.7b)
Kˆµναβ(x, y) =
−4√−g(x)√−g(y)
〈
δ2Sph[φˆ]
δgµν(x)δgαβ(y)
〉
. (C.7c)
The kernels HS and HA are the symmetric and anti-symmetric bi-tensor parts of
H, i.e.
HS,Aµναβ(x, y) = ±HS,Aαβµν(y, x). (C.8)
With these, the second order variational derivative of the effective action on
the forward branch reads
4√−g(x)√−g(y) δ2Γphδ(g+)µν(x)δ(g+)αβ(y)
∣∣∣∣
g+=g−
= iNˆµναβ(x, y)−HˆSµναβ(x, y)−Kˆµναβ(x, y).
(C.9)
Similarly, the second order variation with respect to the other combinations
of the forward and backward values of the metric are
4√−g(x)√−g(y) δ2Γphδ(g+)µν(x)δ(g−)αβ(y)
∣∣∣∣
g+=g−
= −iNˆµναβ(x, y)− HˆAµναβ(x, y),
(C.10)
4√−g(x)√−g(y) δ2Γphδ(g−)µν(x)δ(g+)αβ(y)
∣∣∣∣
g+=g−
= −iNˆµναβ(x, y) + HˆAµναβ(x, y),
(C.11)
4√−g(x)√−g(y) δ2Γphδ(g−)µν(x)δ(g−)αβ(y)
∣∣∣∣
g+=g−
= iNˆµναβ(x, y)+Hˆ
S
µναβ(x, y)+Kˆµναβ(x, y).
(C.12)
The Eq. (3.90) is a compact way to write the above second order variations
Eqs. (C.9)– (C.12).
124
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