Abstract

Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.

Committee Chair

Alexander Seidel

Committee Members

Aliakbar Daemi; Erik Henriksen; Li Yang; Zohar Nussinov

Degree

Doctor of Philosophy (PhD)

Author's Department

Physics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

4-28-2026

Language

English (en)

Author's ORCID

https://orcid.org/0000-0002-0275-1102

Download

Share

COinS