Inaugural dissertationof the ,

presented by

from Greece

Supervisor of the doctoral thesis:

of the

Inaugural dissertationof the ,

presented by

from Greece

Supervisor of the doctoral thesis:

of the

Accepted by the .

Bern,  August 27, 2024The Dean  Prof. Dr. Marco Herwegh

To the love of my life, Helena, for always being there for me, and to my parents for all their sacrifices.

Whereof one cannot speak, thereof one must be silent.

Ludwig Wittgenstein

The aim of this thesis is to systematically and consistently study strongly coupled bosonic and fermionic conformal field theories using the large quantum number expansion. The idea behind it is to study sectors of conformal field theories that are characterised by large quantum numbers under global symmetries. In this limit, the conformal field theories, even if they initially were strongly coupled and interacting, can now be written in terms of an effective field theory that is weakly coupled. Some common effective field theories that appear in the literature are the bosonic conformal superfluid and the Fermi sphere, condensed matter systems characterised by a high particle density, making the study of such systems a cross-disciplinary matter.

In the first part of the thesis, we start by reviewing concepts of quantum field theories, conformal field theories and of the large charge expansion that are essential for the subsequent chapters.

The second part of the thesis is devoted to the analysis of the \(O(2)\) vector model in the large charge expansion, where we start by reviewing the classical treatment of the model, and we continue by examining its quantum behaviour. Following that, we compute three and four-point correlation functions of large charge primaries with insertions of the conserved current \(\mathcal{J}^\mu\) and/or the energy momentum tensor \(T^{\mu \nu}\) and in the final part of the chapter, we compute three and four-point correlation functions of spinning charged primary operators \(\Vpp[\Qp][\ell m]\) with the insertion of light charged spinning primary operators \(\mathscr{O}^q\) with \(q \ll \Qp\) in the middle.

In the third part of the thesis, we use the resurgence methodology to study the asymptotics of the \(O(N)\) vector model. We start by introducing the \(O(N)\) model at large charge and large-\(N\) and we study its asymptotics on the torus \(\setT^2\) and the sphere \(\setS^2\) using the resurgence framework. Then, we derive a worldline interpretation of the heat-kernel trace that replicates the previously computed results, and we deduce that its geometric nature is robust enough to allow us to extrapolate the results to finite \(N\). Finally, we compare the resurgent results with the small-charge regime which is accessible at the doubling-scaling limit, while we theorise that our conjunctions hold for finite values of \(N\), and we find the value of optimal truncation.

In the last part of the thesis, we study various fermionic models with a four-fermion interaction at large charge and large-\(N\). We start by presenting the models that we will examine in the subsequent sections and afterwards, we discuss the symmetry breaking pattern of the models, and we investigate the appearance of a condensate. We continue by studying the spectrum of the fluctuations for the models, and we compute the scaling dimension of the gn model at large charge and of the njl model both at large and small-charge regime.

My personal belief has always been that for every problem, there exists a rational
answer. Every human being is born with some sort of curiosity about their existence and
place in the universe. Growing up, I had my set of philosophical questions as
well, and for me, the scientific method has always been an ideal way to tackle such
questions. My passion for physics and especially theoretical physics is motivated
strongly by the fact that it not only explains every day phenomena but also allows us
to imagine things unseen through thought experiments and mathematical reasoning.

This line of questioning eventually led me to do my PhD at the  at the  under the supervision of Susanne Reffert. Therefore, the first obvious, but sincere, acknowledgements are to her. I am truly grateful for believing in me, choosing me as a PhD student, and giving me the chance to (hopefully) attain the highest academic degree. During these three and a half years, Susanne has been an immense help in the pandemic and post-pandemic period where the lives of all of us drastically changed, and we managed to overcome the difficulties that arose and have a meaningful and fruitful collaboration. Her broad knowledge of physics, the clarity of her thoughts, and her incredible ability to accurately express complex concepts and ideas are inspirational. I am grateful for helping me be a better physicist and being part of a great team in such a prestigious institute. For what matters, sincerely, thanks!

At this point, I want to take the opportunity to personally thank every member of the team, starting with Domenico Orlando. His innovative ideas and his passion for physics have been stimulating. I also want to thank him for being his teaching assistant for the last three years, during which we had an excellent collaboration. He was available at all times to help and answer all possible questions, and he was always a joy to be around.

Next, I want to thank my collaborators, Nicola Dondi, Rafael Moser, Vito Pellizzani, Fabio Apruzzi and Donald Youmans, who besides co-workers have also become friends. Nicola, thank you for sharing the office with me, and working with me on all of my projects. I learned so much from you, and I wish you the best in your future academic career, which you truly deserve. Donald, thank you for our conversations, for unofficially sharing the office with me and Nicola and all the great times we had at the office and outside. Vito and Rafael, thank you for sharing the demanding experience of doing a PhD, and for our conversations about physics and more. I wish you both the best to your future career plans. Especially Rafael, I took great pleasure in our collaboration on most of our projects, I admired your talent and brightness from the time that you were my student, and it was an honour working with you. Last but not least, I want to thank Thiago Araujo for our short time together, I am genuinely sorry that the pandemic did not let me get to know you better, and we did not get the opportunity to write a paper together that we both wanted so much.

Finally, I want to thank three people I worked with during these years, Matthias Blau, Simeon Hellerman and Gilberto Colangelo, I learned a lot from you, and it was an honour working together.

Ultimately, it has been six years since I left Greece, but in all those years my friends from back home were always there for me and for this, I am truly grateful. Doing a PhD would have been impossible if I did not have the help and support of my family and especially my parents, who sacrificed so much for me to be here. Words cannot describe my gratitude and appreciation for all that you have offered me and how you raised me. The last person I want to thank is the one closest to my heart. Helena, thank you for all your sacrifices all these years and all the sleepless nights together during the writing of this thesis. Thank you for being there for me and helping me overcome all the adversities that we have faced together. Thank you for supporting me in spirit and in body during the pandemic and after it. If it wasn't for you, I do not know if I would have been able to reach thus far. You are my anchor and the love of my life, and for that, I feel truly blessed.

QFT     Quantum Field Theory

CFT     Conformal Field Theory

EFT     Effective Field Theory

LCE     Large Charge Expansion

SSB     Spontaneous Symmetry Breaking

WF     Wilson-Fisher

RG     Renormalisation Group

RGE     Renormalisation Group Equation

IR     Infrared

UV     Ultraviolet

SCTs     Special Conformal Transformations

OPE     Operator Product Expansion

EOM     Equation Of Motion

1PI     One Particle Irreducible

MS     Minimal-Subtraction

PDE     Partial Differential Equation

CCWZ     Coleman-Callan-Wess-Zumino

BCS     Bardeen-Cooper Schrieffer

VEV     VacuumExpectation Value

LSM     Linear Sigma Model

NLSM     Non Linear Sigma Model

DOF     Degrees Of Freedom

PDE     Partial Differential Equation

ODE     Ordinary Differential Equation

WL     Worldline

NJL     Nambu–Jona–Lasinio

GN     Gross-Neveu

GNY     Gross-Neveu-Yukawa

PG     Pauli-Gürsey

BCS     Bardeen-Cooper-Schrieffer

NLO     Next-to-Leading-Order

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

American Physical Society

journaltitlePhys. Rev. B
titleRenormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture

American Physical Society

journaltitlePhys. Rev. B
titleRenormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior

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titleThe constraints of conformal symmetry on RG flows

journaltitlePhysical Review B
titleCritical behavior of the three-dimensional XY universality class

journaltitlePhysical Review B
titleCritical exponents and equation of state of the three-dimensional Heisenberg universality class

journaltitleJETP Lett.
titleConformal symmetry of critical fluctuations

journaltitleNuclear Physics B
titleCovariant expansion of the conformal four-point function

journaltitleNuclear Physics B
titleInfinite conformal symmetry in two-dimensional quantum field theory

journaltitleNuclear Physics B
titleConformal four point functions and the operator product expansion

journaltitleNuclear Physics B
titleConformal partial waves and the operator product expansion

journaltitleJournal of High Energy Physics
titleBounding scalar operator dimensions in 4D CFT

titleTASI Lectures on the Conformal Bootstrap
High Energy Physics - Theory (hep-th),Statistical Mechanics (cond-mat.stat-mech),Strongly Correlated Electrons (cond-mat.str-el),FOS: Physical sciences,FOS: Physical sciences

Elsevier BV

journaltitlePhysics Reports
titleQuantum field theory in the large-\(N\) limit: a review

Oxford university press

titleQuantum field theory and critical phenomena

journaltitlePhysics Reports
titleThe renormalization group and the \(\varepsilon\) expansion

IOP Publishing

journaltitleJournal of High Energy Physics
titleComments on operators with large spin

journaltitleJournal of High Energy Physics
titleConvexity and liberation at large spin

journaltitleJournal of High Energy Physics
titleThe analytic bootstrap and AdS superhorizon locality

journaltitleJournal of High Energy Physics
titleOn the CFT operator spectrum at large global charge

journaltitleJournal of High Energy Physics
titleSemiclassics, Goldstone bosons and CFT data

journaltitlePhys. Rept.
titleSelected topics in the large quantum number expansion

journaltitleJournal of High Energy Physics
titleThe epsilon expansion meets semiclassics

journaltitlePhysics Letters B
titleA note on the large charge expansion in 4d CFT

titleLarge charge, semiclassics and superfluids : from broken symmetries to conformal field theories

journaltitleJournal of High Energy Physics
titleSpontaneously broken boosts in CFTs

journaltitlePhysical review letters
titleConformal dimensions via large-charge expansion

journaltitlePhysical review letters
titleConformal dimensions in the large charge sectors at the \(O(4)\) Wilson-Fisher fixed point

journaltitlePhysical Review D
titleSubleading conformal dimensions at the \(O(4)\) Wilson-Fisher fixed point

journaltitleNuclear Physics B
titleSpinning correlators in large-charge CFTs

journaltitleJournal of High Energy Physics
titleCompensating strong coupling with large charge

journaltitleJournal of High Energy Physics
titleLarge charge at large N

journaltitleJournal of High Energy Physics
titleConvexity, large charge and the large-N phase diagram of the $\varphi$4 theory

journaltitleJournal of High Energy Physics
titleOn the large charge sector in the critical \(O(N)\) model at large \(N\)

journaltitlePhysical Review D
titleCharging the O (N) model

journaltitlearXiv preprint arXiv:2006.10078
titleCharging the Walking \(U (N) \times U (N)\) Higgs Theory as a Complex CFT

journaltitlePhysical Review D
titleAnomalous dimensions at large charge for \(U (N)\times U (N)\) theory in three and four dimensions

journaltitlePhysical Review D
titleScaling dimensions at large charge for cubic $\phi^3$ theory in six dimensions

journaltitleJournal of High Energy Physics
titleConformal bootstrap at large charge

journaltitleJournal of High Energy Physics
titleCorrelation functions in scalar field theory at large charge

journaltitlearXiv preprint arXiv:2203.00059
titleLarge-charge conformal dimensions at the $ O (N) $ Wilson-Fisher fixed point

journaltitleJournal of High Energy Physics
titleLarge charges on the Wilson loop in \(\mathcal{N} = 4\) SYM: matrix model and classical string

journaltitlearXiv preprint arXiv:2202.07627
titleLarge charges on the Wilson line in \(N= 4\) SYM: II. Quantum fluctuations, OPE, and spectral curve

journaltitleJournal of High Energy Physics
titleOperator dimensions from moduli

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titleOn the Large R-charge Expansion in \(\mathcal{N}=2\) Superconformal Field Theories

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titleA limit for large R-charge correlators in \(\mathcal{N}=2\) theories

journaltitleJournal of High Energy Physics
titleUniversal correlation functions in rank 1 SCFTs

Springer Science
Business Media LLC

journaltitleJournal of High Energy Physics
titleS-duality and correlation functions at large R-charge

journaltitlearXiv preprint arXiv:1809.06280
titleOn the large R-charge N= 2 chiral correlators and the Toda equation (2018)

journaltitlearXiv preprint arXiv:2103.05642
titleLarge R-charge EFT correlators in \(N= 2\) SQCD

journaltitlearXiv preprint arXiv:2103.09312
titleOn the exponentially small corrections to \(\mathcal{N}=2\) superconformal correlators at large R-charge

journaltitleReviews of modern physics
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Amer. Math. Soc.

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journaltitleNuclear Physics B
titleCalculation of gauge couplings and compact circumferences from self-consistent dimensional reduction

World Scientific

journaltitleAnalysis and Applications
titleThe Barnes $\zeta$-function, sphere determinants and Glaisher–Kinkelin–Bendersky constants

American Physical Society

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titleDivergence of Perturbation Theory in Quantum Electrodynamics

American Physical Society

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titleLarge-Order Behavior of Perturbation Theory

Université de Paris-Sud, Département de Mathématique,

titleLes fonctions résurgentes:(en trois parties)

journaltitlearXiv preprint physics/0605038
titleConsequences of Dirac theory of the positron

World Scientific

booktitleFrom Fields to Strings: Circumnavigating Theoretical Physics: Ian Kogan Memorial Collection (In 3 Volumes)
titleHeisenberg–Euler effective Lagrangians: basics and extensions

journaltitlePhysical Review D
titleWorldline instantons and pair production in inhomogenous fields

journaltitlePhysical Review D
titleWorldline instantons and the fluctuation prefactor

journaltitleSciPost Physics
titleThermalization and hydrodynamics of two-dimensional quantum field theories

journaltitleJournal of High Energy Physics
titleYukawa interactions at large charge

journaltitleJournal of High Energy Physics
titleThe large-charge expansion for Schrodinger systems

journaltitleJournal of High Energy Physics
titleNonrelativistic conformal field theories in the large charge sector

journaltitleJournal of High Energy Physics
titleThe spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals

journaltitlePhysical Review D
titleNear-Schrodinger dynamics at large charge

journaltitlearXiv preprint arXiv:2010.07967
titleDroplet-edge operators in nonrelativistic conformal field theories

journaltitlePhysical Review D
titleOperator spectrum of nonrelativistic CFTs at large charge

journaltitleJournal of High Energy Physics
titleNonrelativistic CFTs at large charge: Casimir energy and logarithmic enhancements

journaltitlePhysical Review D
titleDynamical symmetry breaking in asymptotically free field theories

journaltitlePhysical review
titleDynamical model of elementary particles based on an analogy with superconductivity I

journaltitlePhysical review
titleDynamical model of elementary particles based on an analogy with superconductivity II

journaltitlearXiv preprint arXiv:2211.15318
titleFermionic CFTs at large charge and large N

journaltitleIl Nuovo Cimento (1955-1965)
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journaltitleIl Nuovo Cimento (1955-1965)
titleRelation of charge independence and baryon conservation to Pauli’s transformation

journaltitlePhysics Letters B
titleTwo phase transitions in chiral Gross-Neveu model in \(2+ \varepsilon\) dimensions at low \(N\)

journaltitlePhysical Review D
titleCompetition and duality correspondence between chiral and superconducting channels in \((2+ 1)\)-dimensional four-fermion models with fermion number and chiral chemical potentials

Cambridge University Press

titleThe Quantum Theory of Fields: Volume 1, Foundations

Cambridge University Press

titleThe Quantum Theory of Fields: Volume 2, Modern Applications

Cambridge University Press

titleThe Quantum Theory of Fields: Volume 3, Supersymmetry

Reading, USA

titleAn Introduction to quantum field theory

Cambridge Univ. Press

titleQuantum Field Theory

titleLectures on Conformal Field Theory

Cambridge University Press

titleGauge/Gravity Duality: Foundations and Applications

titleConformal field theory for particle physicists
High Energy Physics - Theory (hep-th),High Energy Physics - Phenomenology (hep-ph),FOS: Physical sciences,FOS: Physical sciences

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleQuantum Fields in Curved Space

titleWhy Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?
High Energy Physics - Theory (hep-th),General Relativity and Quantum Cosmology (gr-qc),FOS: Physical sciences,FOS: Physical sciences

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleThe Large Scale Structure of Space-Time

Springer New York

seriesGraduate Texts in Mathematics
titleMathematical Methods of Classical Mechanics

American Mathematical Society

numberv. 1
seriesQuantum Fields and Strings: A Course for Mathematicians
titleQuantum Fields and Strings: A Course for Mathematicians, Part 1

American Mathematical Society

numberv. 2
seriesQuantum Fields and Strings: A Course for Mathematicians
titleQuantum Fields and Strings: A Course for Mathematicians, Part 2

Informa UK Limited

journaltitleTransport Theory and Statistical Physics
titleInvariant variation problems

CRC Press

titleFeynman lectures on gravitation

ETH Zurich

titleQuantum Field Theory I

titlePath Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics

journaltitlePhys. Z. Sowjetunion
titleThe Lagrangian in quantum mechanics

World Scientific

titleFeynman's Thesis: A New Approach to Quantum Theory

Dover Publications

seriesDover Books on Physics
titleQuantum Mechanics and Path Integrals

journaltitleZeitschrift für Physik
titleKonfigurationsraum und zweite Quantelung

seriesInternational series in pure and applied physics
titleRelativistic Quantum Fields

Springer New York

seriesTexts in Applied Mathematics
titleIntroduction to Perturbation Methods

seriesAdvanced Mathematical Methods for Scientists and Engineers
titleAdvanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory

American Physical Society

journaltitlePhys. Rev.
titleBroken Symmetries

seriesITS: Publications
titleRelativityé Groups And Topology:

journaltitleNuovo Cim.
titleRelativistic field theories with symmetry breaking solutions

Cornell University

titleLectures on Quantum Gravity and Black Holes
Physics; Relativity; Black holes (Astronomy); General relativity (Physics)

University of Copenhagen

titleBlack Holes and Holographic Quantum Mechanics

American Physical Society

journaltitlePhys. Rev.
titleProperties of Bethe-Salpeter Wave Functions

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titleAxioms for Euclidean Green's Functions

journaltitleCommun. Math. Phys.
titleAxioms for Euclidean Green's Functions II

journaltitleNucl. Phys. B
titleRegularization and Renormalization of Gauge Fields

World Scientific

titleZeta Regularization Techniques with Applications

IOP Publishing

journaltitleJournal of Physics A: Mathematical and Theoretical
titleFunctional determinants in quantum field theory

Dover Publications

seriesApplied mathematics series
titleHandbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
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journaltitleNuclear Physics B
titleDimensional regularization and the renormalization group

American Physical Society

journaltitlePhys. Rev. D
titleNew methods for the renormalization group

World Scientific Publishing Company

titleCritical Properties of \(\phi^4\) - Theories

American Physical Society

journaltitleRev. Mod. Phys.
titleThe renormalization group: Critical phenomena and the Kondo problem

American Physical Society

journaltitlePhys. Rev. Lett.
titleCritical Exponents in 3.99 Dimensions

American Physical Society

journaltitlePhysics Physique Fizika
titleScaling laws for ising models near ${T}_{c}$

WORLD SCIENTIFIC

booktitlePhysics In D $\geq$ 4 Tasi 2004
titleEffective Field Theory and Heavy Quark Physics

Grefe & Tiedemann

titleBeitrag zur theorie des ferro-und paramagnetismus

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleSuperstring Theory: Volume 1, Introduction

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleSuperstring Theory: Volume 2, Loop Amplitudes, Anomalies and Phenomenology

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleString Theory: Volume 1, An Introduction to the Bosonic String

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleString Theory: Volume 2, Superstring Theory and Beyond

Princeton University Press

seriesIn a Nutshell
titleString Theory in a Nutshell: Second Edition

journaltitleAdv. Theor. Math. Phys.
titleThe Large N limit of superconformal field theories and supergravity

titleAnti De Sitter Space And Holography
High Energy Physics - Theory (hep-th),FOS: Physical sciences,FOS: Physical sciences

Elsevier BV

journaltitlePhysics Letters B
titleGauge theory correlators from non-critical string theory

titleDimensional Reduction in Quantum Gravity
General Relativity and Quantum Cosmology (gr-qc),FOS: Physical sciences,FOS: Physical sciences

journaltitleJournal of Mathematical Physics
titleThe world as a hologram

journaltitlePhys. Rept.
titleLarge N field theories, string theory and gravity

World Scientific

titleGroup Theory in Physics

Springer Science
Business Media LLC

journaltitleThe European Physical Journal Plus
titleOn the equivalence of two definitions of conformal primary fields in d > 2 dimensions

Springer International Publishing

titleEPFL Lectures on Conformal Field Theory in D $\geq$ 3 Dimensions

American Physical Society

journaltitlePhys. Rev. D
titleSolving the 3D Ising model with the conformal bootstrap

journaltitleJournal of High Energy Physics
titleA semidefinite program solver for the conformal bootstrap

journaltitlePhysical Review D
titleOPE meets semiclassics

journaltitlearXiv preprint arXiv:2207.08919
titleIdentifying Large Charge Operators

journaltitlePhysical Review D
titleRotating superfluids and spinning charged operators in conformal field theory

journaltitleJournal of High Energy Physics
titleSuperfluids, vortices and spinning charged operators in 4d CFT

journaltitleJournal of High Energy Physics
titleLarge charge sector of 3d parity-violating CFTs

journaltitleJournal of High Energy Physics
titleGiant Vortices and the Regge Limit

journaltitleIl Nuovo Cimento (1955-1965)
titleIntroduction to complex orbital momenta

journaltitleJournal of High Energy Physics
titleA note on inhomogeneous ground states at large global charge

journaltitleJournal of High Energy Physics
titleObservables in inhomogeneous ground states at large global charge

journaltitlePhysical Review
titleBound electron pairs in a degenerate Fermi gas

journaltitlePhysical Review
titleMicroscopic theory of superconductivity

journaltitlearXiv preprint hep-th/9210046
titleEffective field theory and the Fermi surface

journaltitleReviews of Modern Physics
titleRenormalization-group approach to interacting fermions

American Association for the Advancement of Science

titleAnomalous criticality in the electrical resistivity of La2–x Sr x CuO4

journaltitlePhysics Today
titleQuantum Criticality

journaltitleJournal of High Energy Physics
titleResurgence of the large-charge expansion

journaltitleJournal of High Energy Physics
titleExtremal correlators and random matrix theory

journaltitlePhysical Review D
titleCharging non-Abelian Higgs theories

journaltitlePhysical Review D
titleMore on the cubic versus quartic interaction equivalence in the O (N) model

journaltitlePhysical Review D
titleTwo-dimensional model with chiral condensates and Cooper pairs having a QCD-like phase structure

journaltitlePhysical Review D
titleSuperconducting phase transitions induced by chemical potential in (2+ 1)-dimensional four-fermion quantum field theory

journaltitleMoscow University Physics Bulletin
titleSuperconductivity in chiral-asymmetric matter within the (2+ 1)-dimensional four-fermion model

journaltitleJournal of High Energy Physics
titleThe large charge expansion at large N

journaltitlePhysical Review D
title2D CFTs: Large charge is not enough to control the dynamics

journaltitleNuclear Physics B
titleOn how to count Goldstone bosons

journaltitlePhysical Review D
titleNonrelativistic effective lagrangians

journaltitlePhysical Review X
titleEffective Lagrangian for nonrelativistic systems

journaltitlePhysical review
titleUniqueness property of the twofold vacuum expectation value

journaltitlePhysical Review
titleExistence of long-range order in one and two dimensions

journaltitlePhysical Review Letters
titleAbsence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models

journaltitleCommunications in Mathematical Physics
titleThere are no Goldstone bosons in two dimensions

journaltitlePhysical Review D
titleNumber of Nambu-Goldstone bosons and its relation to charge densities

journaltitlePhysical Review Letters
titleCounting rule for Nambu-Goldstone modes in nonrelativistic systems

journaltitlePhysical Review D
titleCritical \(O(N)\) models in 6 - $\varepsilon$ dimensions

journaltitleZeitschrift für Physik
titleEinige folgerungen aus der schrödingerschen theorie für die termstrukturen

journaltitleReviews of modern Physics
titleThe application of group theory to the quantum dynamics of monatomic systems

journaltitlePhysical Review D
titlePartition function on spheres: how to use zeta function regularization

journaltitlePhysical Review Letters
titleMassive Nambu-Goldstone Bosons

Annual Reviews

journaltitleAnnual Review of Condensed Matter Physics
titleCounting rules of Nambu–Goldstone modes

journaltitleRelativity, groups and topology
titleDynamical theory of groups and fields

Elsevier BV

journaltitleAnnals of Physics
titleAn introduction to resurgence, trans-series and alien calculus

Elsevier BV

journaltitlePhysics Reports
titleA primer on resurgent transseries and their asymptotics

titlePhase transitions and critical phenomena

Academic Press

journaltitlePhase transition and critical phenomena
titleField theoretical approach to critical phenomena

journaltitlePhysical Review Letters
titleCalculation of partition functions

booktitleSoviet Physics Doklady
titleOn a method of calculating quantum distribution functions

journaltitlePhysics Letters B
titleFollowing the flow for large N and large charge

Cambridge University Press

seriesCambridge Monographs on Mathematical Physics
titleFinite-Temperature Field Theory: Principles and Applications

titleTen physical applications of spectral zeta functions

titleSpectral functions in mathematics and physics

American Association of Physics Teachers

titleHandbook of mathematical functions with formulas, graphs, and mathematical tables

American Association of physics teachers

titleMathematical methods for physicists

National Acad Sciences

journaltitleProceedings of the National Academy of Sciences
titleOn Epstein's zeta function (I)

Cambridge University Press

titleInstantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory

titleTransseries for beginners

journaltitlearXiv preprint math/0407288
titleSelberg's trace formula: an introduction

journaltitleBull. Amer. Math. Soc
titleZeta functions and their asymptotic expansions for compact locally symmetric spaces of negative curvature

Cambridge University Press

booktitleMathematical Proceedings of the Cambridge Philosophical Society
titleAn asymptotic expansion for

journaltitleNachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
titleÜber die asymptotische Verteilung der Eigenwerte

Cambridge University Press

titleThe Laplacian on a Riemannian manifold: an introduction to analysis on manifolds

journaltitlePhysics reports
titleHeat kernel expansion: user's manual

Wiley Online Library

journaltitleJournal of Mathematics and Physics
titleNon-linear transformations of divergent and slowly convergent sequences

booktitleAnnales scientifiques de l'École Normale Supérieure
titleÉtude mathématique du mouvement brownien de rotation

journaltitleCommunications in Mathematical Physics
titleNonperturbative ambiguities and the reality of resurgent transseries

journaltitleJournal of High Energy Physics
titleDecoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

journaltitlePhysical Review
titleMathematical formulation of the quantum theory of electromagnetic interaction

journaltitlePhysical Review
titleAn operator calculus having applications in quantum electrodynamics

journaltitlePhysics Reports
titlePerturbative quantum field theory in the string-inspired formalism

journaltitleNuclear Physics B
titleLoop calculations in quantum-mechanical non-linear sigma models

Cambridge University Press

titlePath integrals and anomalies in curved space

Oxford University Press, USA

titleThe global approach to quantum field theory

journaltitlePhysics Reports
titleHarmonic analysis and propagators on homogeneous spaces

New York

editionPearson new international edition
titleSpacetime and geometry, an introduction to general relativity
general relativity; geometry; spacetime

School on Spinning Particles in Quantum Field Theory: Worldline Formalism, Higher Spins,
Conformal Geometry

titleLectures on the Worldline Formalism

journaltitleNuclear Physics B
titleField theory without Feynman diagrams: one-loop effective actions

World Scientific

titlePath integrals, hyperbolic spaces and Selberg trace formulae

journaltitleCentral European Journal of Physics
titleThe Maslov correction in the semiclassical Feynman integral

journaltitleJournal of High Energy Physics
titleQuantum mechanical path integrals in curved spaces and the type-A trace anomaly

journaltitleAMS/IP Stud. Adv. Math
titleAnalytic continuation of Chern-Simons theory

journaltitleProceedings of the American Mathematical Society
titleOn Minakshisundaram-Pleijel zeta functions of spheres

journaltitleJournal of High Energy Physics
titleA safe CFT at large charge

journaltitlePhysical Review D
titleUntangling scaling dimensions of fixed charge operators in Higgs theories

journaltitlePhysics Letters B
titleFeynman diagrams and the large charge expansion in 3- $\varepsilon$ dimensions

journaltitleJournal of High Energy Physics
titleAccessing large global charge via the \(\epsilon\)-expansion

journaltitleJournal of High Energy Physics
titleThe large charge limit of scalar field theories, and the Wilson-Fisher fixed point at = 0

journaltitleJournal of High Energy Physics
titleOn the UV completion of the \(O (N)\) model in \(6 - \varepsilon\) dimensions: a stable large-charge sector

journaltitleJournal of High Energy Physics
titleGapped Goldstones at the cut-off scale: a non-relativistic EFT

journaltitlePhysical Review D
titleSpontaneous symmetry breaking in the linear sigma model at finite chemical potential: One-loop corrections

journaltitlePhysical Review Letters
titleImplications of relativity on nonrelativistic Goldstone theorems: gapped excitations at finite charge density

IOP Publishing

journaltitleJournal of High Energy Physics
titleNonperturbative effects and nonperturbative definitions in matrix models and topological strings

journaltitleJournal of High Energy Physics
titleTransmutation of a trans-series: the Gross-Witten-Wadia phase transition

journaltitleNuclear Physics B
titleThe theory of non-renormalizable interactions: The large N expansion

journaltitleNuclear Physics B
titleFour-fermion interaction near four dimensions

journaltitlearXiv preprint arXiv:1607.05316
titleYukawa CFTs and emergent supersymmetry

journaltitlePhysical Review D
titleQuantum critical behavior in three dimensional lattice Gross-Neveu models

journaltitlePhysical Review D
titleCompetition of color ferromagnetic and superconductive states in a quark-gluon system

journaltitleAnnals of Physics
titleFour-Fermi theories in fewer than four dimensions

University of Chicago press

titleGeneral relativity

Cambridge University Press

titleGeneral Relativity; an Einstein Centenary Survey

Chicago, IL

University of Chicago Press

seriesChicago Lectures in Physics
titleQuantum Field Theory in Curved Space-Time and Black Hole Thermodynamics

journaltitleHelv. Phys. Acta
titleOn the definition of the Renormalization Constants in Quantum Electrodynamics

journaltitleIl Nuovo Cimento (1943-1954)
titleÜber Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder

Springer New York

titleConformal Field Theory

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[pages=-]Extra/CV academic.pdf


# Scalable Data Balancing
for Unlabeled Satellite Imagery

Deep Patel
<EMAIL_ADDRESS>
Erin Gao11footnotemark: 1
<EMAIL_ADDRESS>
Anirudh Koul11footnotemark: 1
<EMAIL_ADDRESS>
Siddha Ganju
Meher Anand Kasam
Equal Contribution; Work done as researchers at Space ML Ganju et al. (2020)

##  1 Introduction

Data imbalance is a ubiquitous problem in machine learning. In large scale
collected and annotated datasets, data imbalance is either mitigated manually
(by undersampling frequent classes and oversampling rare classes), or planned
for with imputation and augmentation techniques. In both cases balancing data
requires labels. In other words, only annotated data can be balanced.
Collecting fully annotated datasets is challenging, especially for large scale
satellite systems such as the unlabeled NASA’s 35 PB Earth Imagery dataset
Seeley et al. (2020).
Although the NASA Earth Imagery dataset is unlabeled, there are implicit
properties of the data source that we can rely on to hypothesize about its
imbalance - such as distribution of land and water in the case of Earth’s
imagery. Since over 70% of Earth’s surface is covered by water bodies, large
scale satellite data is expected to have a much higher percentage of water
imagery. On the other hand, phenomena like hurricanes would have rare
occurrences. A naive ML model can classify everything as “ocean” and still
achieve a 70% accuracy.
We present a new iterative method to balance unlabeled data - our method
utilizes image embeddings as a proxy for image labels that can be used to
balance data, and ultimately when trained increases overall accuracy.

##  2 Diversely Selecting Training Samples

Starting with unbalanced and unlabeled data, we use self-supervised learning
methods (including autoencoder and SimCLR) to train a model, then use the
trained model (an embedder) to generate embeddings for the entire dataset.
Clusters are identified based on these embeddings and an equal number of
points per cluster are chosen to generate a new balanced training dataset.
Again, a self-supervised model is trained with this dataset which is further
used to generate a more balanced training dataset. Each iteration improves the
representative power of the embeddings, thereby improving the definition of
the next cluster, balancing data further, and interestingly, leads to much
higher accuracy on the same dataset size compared to a random dataset. Our
experiments on a sample unbalanced dataset which when balanced by the
algorithm gives 90% accuracy, a considerable 50% improvement over a baseline
with 40% accuracy and 2.5x the number of training images.
The method presented is generalizable and can be extended beyond image
modalities to text and video data where data imbalance is similarly
ubiquitous. Since our annotation technique is largely automatic, our method is
scalable and generalizable and especially useful for unexplored fields with
abundant unlabeled data.

Algorithm 1 Diversely Select N Embeddings from embedder, data, n

$features\leftarrow embedder(data)$

$result\leftarrow[random.choice(data)]$

$distances\leftarrow[]*len(data)$

while $n\neq 0$ do

$i\leftarrow 0$

while $i<len(features)$ do

$dist\leftarrow dist(features[i],result[-1])$

if $distances[i]>dist$ then

$distances[i]=dist$

end if

end while

$idx\leftarrow distances.index(max(distances))$

$result.append(data[idx])$

$delfeatures[idx]$

$n\leftarrow n-1$

end while

$return\ result$

##  3 Experiments

To understand the benefits of selecting embeddings with this iterative method,
we analyze the benefits with experiments on two different datasets. With a
Baised Circles Dataset, we analyze how selecting 500 training points with this
method improves model accuracy over a baseline of selecting points randomly.
With a biased sampling of the UC Merced satellite imagery dataset, we view how
diversely selecting training samples improves model accuracy, especially as
this is process is done iteratively over multiple model iterations.

###  3.1 Circles Dataset

500 points were selected, via two different methods, from the Biased Circles
dataset (Figure 1) above. The dataset is composed of 4500 points, which
consists of 10 rings. With each circle numbered from 1 to 10 from outer circle
to inner circle, each odd numbered circle containing eight times as many
elements as its adjacent inner circle. The colors of the rings correlate with
which class the points belong to where each circle is its own class of points

Figure 1: Biased Circles Dataset

Figure 2 is an image of the data points selected randomly whereas Figure 3
consists of data points chosen using diverse embedding selection. The latter
set of data points are far more structured than the randomly selected points
as they do not retain nearly as much of the imbalance in the original dataset.

Figure 2: Randomly chosen embeddings

Figure 3: Diversely chosen embeddings

After classifying on a randomly selected test set, it can be seen in Figure 4
that diversely chosen points provide a little over a 40% increase in model
accuracy from the baseline model that was trained on randomly chosen training
points which is a very significant improvement.

Selection Method | No. Training Points | Avg Accuracy
---|---|---
Random | $500$ | $0.484\pm 0.003$
Diverse | $500$ | $0.906\pm 0.011$
Figure 4: Training Model on 500 Points

Figure 5: Classification Accuracies for Larger Training Sets

Figure 6: Class-size Standard Deviations from Selected Training Sets

Furthermore, in Figure 5, we see that these results are consistent when the
size of the selected training points increase. The difference is mitigated
when the difference between the random and diversely selected training sets is
small at larger trainings sets. However, it takes about 250% more randomly
selected training data to reach a trained model accuracy as good as one
trained on a diversely selected training set. This is expected as more points
selected makes the diversely selected points more similar to the randomly
selected points. Regardless, on the circles dataset, it can be seen that when
training on smaller datasets, diversly selected training sets can greatly
improve model performance. In Figure 6, it can be seen that selecting features
using this strategy allows for a training dataset that is more balanced
compared to the the randomly selected training set. This is noticable because
the standard deviation between the classes is far less for diversly-selected
training sets. This means the number of features in each training set are far
closer than the randomly selected dataset. This result shows that diversely
selecting features allows for much better representation from each class in
the overall training set.

###  3.2 UC Merced Dataset

Selecting training points diversely is primarily is intended to work best with
unbalanced data; however, the UC Merced satellite imagery dataset is a
balanced dataset. As a result, we subset this dataset into one that produces
an unbalance across the UC Merced dataset classes. The dataset that we use in
this experiment can be seen in Figure 7.

Figure 7: Unbalanced Subset of UC Merced Dataset

For this experiment, we first train an autoencoder to generate a list of
diverse or random features on the unbalanced dataset. Then, we iteratively
train a model on the most these features, use the model to make an encoder,
obtain diverse or random features, and repeat. At each model iteration, we
expect the models to improve, produce a better encoding representation of the
dataset, find a better balanced training set, and train an even better model.
The results of the model performance and the class representation of the
training set (represented by the standard deviation) over each model iteration
can be found in Figure 8 and Figure 9, respectively.

Figure 8: Training Accuracies from Iteratively Training Models

Figure 9: Class-size Standard Deviations from Selected Training Sets

The results show that there is a small improvement in the model test
accuracies throughout most of the iterations; however, this difference appears
to be mitigated at later iterations. The results of this experiment show that
diversely selecting features still allows for some improvement over models
trained on randomly selected features. Similarly, the representation of
classes between training sets is similar, but the diversly selected features
have slightly better balance. It is important to notice that these
improvements here are not as drastic as the experiments with the circles
dataset. As a result, it is clear that the improvements from diversely
selecting a training set can be dependent on the structure and features in the
dataset.

##  4 Conclusions

As a result, through the experiments with the circles and UC Merced dataset,
we see that the iterative method to create balanced data is viable and can
provide great improvements to class representation in a dataset. As a summary,
these are the high-level takeaways that can be obtained from these
experiments:

  * •

The iterative method to diversely balance data is effective at balancing
unbalaced data

  * •

Using datasets that are produced from diversly selected features tends to
result in better models due to the better balanced data

  * •

The benefits of diversely selecting features is likely to be more dependent on
the dataset and more effective on datasets with higher unbalances.

The results provide justification for a variety of many other experiments on
the process of balancing unlabeled and unbalanced data.

The effectiveness of this iterative selection method has appeared to be
contingent on the structure of the dataset as seen by the difference in the
results of the experiments of circles and unbalanced UC Merced dataset. This
may require furthur experimentation as it would be important to determine what
datasets would be effective for using this method. For example, running
similar experiments on varying datasets depending on feature variation,
dataset size, and overall structure.

Furthermore, varying the shape of the dataset across classes would provide
insight into what kinds of datasets would benefit from this iterative method
for feature selection. Other experiments include varying the structure and
methods for training and creating the initial self-supervised model to obtain
different representations of the feature data.

Finally, this method was originally intended to work on larger datasets than
those shown in the paper. As a result, researching a concurrent implementation
of this algorithm to see how well this method can perform and balance
extremely large datasets would shed light on the effectiveness in the
practical use of this algorithm.

## References

  * Ganju et al. (2020) Siddha Ganju, Anirudh Koul, Alexander Lavin, Josh Veitch-Michaelis, Meher Kasam, and James Parr.  Learnings from Frontier Development Lab and SpaceML - AI Accelerators for NASA and ESA, 2020.
  * Seeley et al. (2020) M Seeley, F Civilini, N Srishankar, Satyarth Praveen, Anirudh Koul, Anamarea Berea, and Hosam El-Askary.  Knowledge Discovery Framework, 2020.