New perspectives on fractional quantum Hall physics

Despite having been discovered nearly four decades ago, fractional quantum Hall (FQH) states continue to provide platforms for the discovery of novel physical phenomena. In this thesis, I present a study of these remarkable phases of matter from three perspectives: (1) the role of an underlying lattice and its symmetries in engendering novel FQH states, (2) the description of non-Abelian FQH states using recently developed field theory dualities, and (3) the use of entanglement entropy in characterizing interfaces of FQH states. In the first part, we examine phases of matter known as fractional Chern insulators (FCIs), lattice analogues of FQH states. We begin in Chapter 2 by formulating a composite fermion theory for FCI states in a kagome lattice model, making use of a recently developed lattice Chern-Simons theory to effect the flux attachment. We identify sequences of Abelian states, including states for which the Hall conductance does not match the filling fraction, which we characterize as realizing distinct translational symmetry fractionalization classes. Next, we apply this formalism in Chapter 3 to identify paired states of composite fermions in a square-lattice Hofstadter model. Magnetic translation symmetry is found to enforce finite-momentum pairing of the composite fermions, yielding pair-density wave (PDW) states with daughter charge-density wave order, analogous to the PDWs conjectured to describe the high-T c cuprate superconductors. This constitutes a novel example of intertwined orders, in which topological order and broken symmetry order arise from a common microscopic origin. In the second part, we apply a recently proposed web of Chern-Simons-matter theory dualities to develop effective field theories for a large class of non-Abelian FQH states. First, in Chapter 4, we demonstrate how these dualities can be used to construct bosonic Landau-Ginzburg theories of the Read-Rezayi and generalized non-Abelian spin singlet states by introducing interlayer interactions in a multilayer Abelian FQH system. Next, we extend this construction in Chapter 5 to develop a field theory and motivate a trial wave function for the elusive Fibonacci FQH state, which is the minimal model for realizing a universal topological quantum computer. We subsequently examine in Chapter 6 dual fermionic non-Abelian Chern-Simons-matter theories, allowing us to develop composite fermion descriptions of the Blok-Wen FQH states and a series of states which may be understood as arising from pairing in a dual Abelian composite fermion theory. Our analysis reveals that dual fermionic theories can predict distinct ground states in a magnetic field, demonstrating the utility of dualities in mapping out regions of the phase diagram of electrons at fractional filling. In the final part of this thesis, which comprises Chapter 7, we characterize interfaces of non-Abelian Moore-Read FQH states using entanglement entropy. We first employ a cut-and-glue approach to obtain the expected topological entanglement entropy (TEE) for a uniform Moore-Read state on the torus in each topological sector. This involves approximating the entanglement as arising purely from the coupled 1D chiral edge degrees of freedom at the entanglement cut. We next consider interfaces of distinct generalized Moore-Read states, identify when the interfaces can be gapped using an anyon condensation picture, construct explicit gapping interactions, and then compute the TEE for an entanglement cut along the interface. It is found that the value of the TEE is related to the total quantum dimension of a “parent” topological phase of the two generalized Moore-Read states between which the interface is formed.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Sohal, Ramanjit

Contributors dc:contributor

• Fradkin, Eduardo
• Stone, Michael
• Madhavan, Vidya
• Gadway, Bryce

• quantum Hall effect
• fractional Chern insulators
• topological order
• field theory dualities
• entanglement entropy

Statement dc:rights

• Copyright 2021 Ramanjit Sohal

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113123

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/113123