# DENSITY-FUNCTIONAL THEORY AND DIFFUSION MONTE CARLO METHOD FOR FRACTIONAL QUANTUM HALL EFFECT

Restricted (Penn State Only)

- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 18, 2018
- Committee Members:
- Jainendra Jain, Dissertation Advisor
- Jainendra Jain, Committee Chair
- Reka Z Albert, Committee Member
- Eric W Hudson, Committee Member
- Ismaila Dabo, Outside Member

- Keywords:
- fractional quantum Hall effect
- diffusion Monte Carlo
- density functional theory

- Abstract:
- The fractional quantum Hall effect (FQHE) is fundamentally interesting in theoretical physics as being one of the very few almost exact solutions to the many-body Schrodinger equation where the Coulomb interaction dominates the behavior. Fractional quantum Hall Effect further inspired the discovery of topological phases of matter – a great step since Landau’s phase transition theory. Finally, FQHE provided an alternative and peculiar way to possibly achieve quantum computation. The first milestone in FQH theory was landed by Dr. Robert Laughlin who proposed a wave function for the 1/3 and 1/5 FQH plateaus. Later on, Dr. Jainendra Jain came up with the idea of “composite fermion” that makes it possible to explain almost all filling factors (1/3, 2/5, 3/7, 1/5, etc.). Therefrom a new theoretical field opened where great efforts have been made in explaining experimental results of 2D systems in a strong magnetic field. Many triumphs have been achieved. My work focuses on two branches of the composite-fermion (CF) theory. In the first paper we for the first time we brought the methodology of density-functional theory (DFT) into the field of composite fermion theory. It resulted in a publication in Physical Review Letters which was also selected as Editor’s Choice. A DFT framework using CFs was carefully built up in this work and the phenomenon called “edge reconstruction” was demonstrated by the first real-space density profile. In the second paper we solved a long-lasting puzzle about the re-entrant electronic insulating phase using a numerical method called fixed-phase diffusion Monte Carlo. The foundation of this method is that the imaginary-time Schrodinger equation describes a diffusion equation and only the lowest-energy state will survive in the long time limit. Simulating the equation with Monte Carlo provides a nonperturbative solution regarding the effect of finite magnetic field through the momentum term (called “diffusion”). Fixed-phase approximation is finally implemented for the time-reversal asymmetry system. The phase diagram demonstrates the role of Landau-level mixing for the appearance of insulating electronic phase at nu=0.37 for p-doped but not for p-doped nonetheless. The arrangement of this thesis is as following. In chapter 1 I present an overview of the fractional quantum Hall effect and the fundamentals of the composite fermion theory. In chapter 2 I present details of the fixed-phase diffusion Monte Carlo method and how it is applied to composite fermion systems to realize a non-perturbative treatment to Landau-level mixing. In Chapter 3 I present fundamentals of the density functional theory, from the Hohenberg-Kohn theoretical framework, to Kohn-Sham theory and to Mermin’s density functional theory at finite temperature. I also introduce two pioneering papers that applied the density function theory to FQH system. In Chapter 4 I reproduced my first paper -- Density-Functional Theory of the Fractional Quantum Hall Effect where I developed the density functional theory based on composite fermion theory and further applied it to investigate edge reconstruction and to plot the density profile near filling factor ½ in the presence of a triangular crystalline antidot background. In Chapter 5 I present my second publication -- Landau-level-mixing induced crystallization in the fractional quantum Hall regime. I use fixed-phase diffusion Monte Carlo to research the role played by Landau-level mixing in developing the reentrant quantum crystal phase between filling factor 1/3 and 2/5 and between 1/5 and 2/9. In Chapter 6 I present a brief conclusion and outlook.