Novel Quantum Phenomena in Fractal Band Structures Open Access

Wang, Jian (Summer 2022)

Permanent URL:


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

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
  • English
Research Field
Committee Chair / Thesis Advisor
Committee Members
Last modified

Primary PDF

Supplemental Files