Novel Quantum Phenomena in Fractal Band Structures Open Access

Abstract

This dissertation investigates novel quantum phenomena in fractal electronic energy bands also known as Hofstadter bands. In two-dimensional lattices subject to an external perpendicular magnetic field, Hofstadter bands arise when the semi-classical cyclotron radius of the electron becomes comparable to the lattice constant. Hofstadter lattices exhibit a strong interplay between topological electronic states and lattice effects, suggesting rich quantum critical phenomena and novel phases of electronic matter beyond conventional quantum Hall physics.   

However, this regime had remained inaccessible due to limitations in accessing large magnetic flux per unit cell in conventional 2D materials at laboratory-scale fields, causing Hofstadter bands to collapse into degenerate Landau levels which have been much explored in the quantum Hall effect. 

In recent years, the emergence of new 2D quantum platforms with nanometer scale unit cells, such as moire materials and nano patterned superlattices, has led to experimental realization of Hofstadter bands. Consequently, Hofstadter physics has experienced a renaissance, emerging as an active frontier to investigate new fundamental questions in quantum matter.

Exploiting these opportunities, this thesis develops a theoretical investigation of novel quantum phenomena in Hofstadter quantum materials, concerned with three central themes: (1) a study of the nature of topological quantum phase transitions, (2) a new framework to classify the poorly understood regime of superconductivity in fractal electronic systems and (3) an investigation of the nature of competing electronic orders, uncovering mechanisms to realize unconventional symmetry-broken and topological electronic orders driven entirely by repulsive interactions.

The theoretical framework established in this thesis opens a new path toward the realization of novel quantum states, unleashing new paradigms in Hofstadter quantum materials.

Table of Contents

Introduction 1

Hofstadter Model: Background 14

Quantum Phase Transitions in Hofstadter Systems 21

Theory of Hofstadter Superconductors 46

Competing Orders in Hofstadter Quantum Materials 112

Discussion and Outlook 144

Bibliography 149

About this Dissertation

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School	Laney Graduate School
Department	Physics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Physics, Condensed Matter
Keyword	quantum phase transition Hubbard-Hofstadter model Ginzburg-Landau theory Chern insulator Hofstadter system reentrant superconductivity magnetic translation group topological superconductor quantum Hall effect renormalization group
Committee Chair / Thesis Advisor	Santos, Luiz, Emory University
Committee Members	Boettcher, Stefan, Emory University Burton, Justin, Emory University Sa de Melo, Carlos, Georgia Institute of Technology Srivastava, Ajit, Emory University

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