Applications of Kohn-Sham Density Functional Theory to Topological Particles
Open Access
- Author:
- Hu, Yayun
- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 05, 2021
- Committee Members:
- Jainendra Jain, Chair & Dissertation Advisor
Dezhe Jin, Major Field Member
Gerald Knizia, Outside Unit & Field Member
Cuizu Chang, Major Field Member
Nitin Samarth, Program Head/Chair - Keywords:
- fractional quantum Hall effect
density functional theory
composite fermion
anyon - Abstract:
- The interplay between interaction and topology plays an important role in condensed matter physics, which leads to novel phenomena, including the fractional quantum Hall effect (FQHE) and exotic particles such as anyons that obey fractional statistics and are thought to be relevant for quantum computation. However, extensive study of the strongly interacting topological systems requests an upgrade in the modern toolbox, among which one example is the density functional theory (DFT). While the DFT is an indispensable tool for treating problems of interaction, it finds little success when applied to the FQHE or anyons over the past four decades. My research during the Ph.D. period connects these three classic topics in interacting particles, namely the DFT, the FQHE, and anyons. It opens a new direction to attack the FQHE and anyons, and also provides a paradigm of applying DFT to explore the topology encompassed in the strong interaction. In this thesis, I first summarize some important facts of the fractional quantum Hall effect and the abelian anyons. Then I will explain my Ph.D. research works that apply DFT to FQHE and abelian anyons, followed by a brief conclusion and an outlook of future research directions. The arrangement of this thesis is as follows. In chapter 1, the important concepts in the quantum Hall effect and in the Kohn-Sham density functional theory are reviewed. This chapter is supposed to provide the basic knowledge of both fields. In chapter 2, I reproduce the first paper [“Kohn-Sham Theory of the Fractional Quantum Hall Effect”, Yayun Hu and J. K. Jain, Phys. Rev. Lett. 123, 176802(2019)] of my Ph.D. We make a breakthrough by mapping the original electron problem into a reference problem of “non-interacting” composite fermions (CFs), which are electrons bound with quantum vortices, and thus experience a density-dependent effective magnetic field. One subtlety of this construction is that CFs are not really non-interacting but interact through the long-range gauge interaction due to their attached flux. Therefore, the long-range exchange-correlation (XC) effect, which is crucial for FQHE, is automatically implemented. Our numerical results demonstrate that our formulation captures not only ground state density and energy but also topological properties such as fractional charge and fractional braid statistics for the excitations. Chapter 3 is a reproduction of my second paper [``Kohn-Sham Density Functional Theory of Abelian Anyons”, Yayun Hu, G. Murthy, S. Rao, J. K. Jain, Phys. Rev. B 103, 035124 (2021)], which develops a density functional treatment of the non-interacting abelian anyons. A sufficient understanding of the abelian anyons is still lacking despite the efforts for nearly half a century. Our work is capable, in principle, of dealing with a large number of anyons in arbitrary external potentials. To demonstrate the power of our method, we look at two essential aspects of anyons: (i) superconductivity and (ii) quantum Hall effect (QHE). (a) It remains an open question whether anyons can trigger superconductivity. Mean-field theory predicts that anyons in uniform systems show “Meissner expulsion” of the external magnetic field when they form superconductors. We find that when the density is nonuniform, this prediction is not valid in finite systems; however, the “Meissner expulsion” restores only in the thermodynamic limit when the edge effects are negligible. (b) Even non-interacting anyons form strongly correlated states to satisfy the statistic boundary conditions, which give rise to emergent excitations that are distinct from the constituent anyons. In the QHE of anyons, we show the emergent topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, using our DFT method. In chapter 4, I reproduce another work [``Crystalline Solutions of Kohn-Sham Equations in the Fractional Quantum Hall Regime”, Yayun Hu, Yang Ge, Jian-Xiao Zhang, and J. K. Jain, arXiv:2102.12603]. We upgrade the numeric code to deal with systems in irregular potentials that do not have rotational symmetry. We find that a crystalline solution that spontaneously breaks the rotational symmetry of the system is stabilized for strong XC between CFs. Our results favor a Wigner-crystal rather than a Hall crystal because of the development of an energy gap around the Fermi energy as well as an average filling factor of one electron per unit cell at the lowest temperatures that we studied. The effects of temperature and disorder on the Kohn-Sham solutions are also discussed. This work pushes our DFT method a step further to explore various problems of interest by enabling realistic modelings of the FQHE edges, disorder, spin physics, screening, and mesoscopic devices. In chapter 5, I discuss some possible future research directions along the line of my Ph.D. study. In particular, I discuss the possibility of modeling the recent experiments that detect evidence of anyons in the Fabry-Perot interferometer in fractional quantum Hall regime.