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# Topological insulators and semimetals in the presence of interactions

#### Park, Moon Jip

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https://hdl.handle.net/2142/101764

## Description

- Title
- Topological insulators and semimetals in the presence of interactions
- Author(s)
- Park, Moon Jip
- Issue Date
- 2018-06-22
- Director of Research (if dissertation) or Advisor (if thesis)
- Gilbert, Matthew J.
- Doctoral Committee Chair(s)
- Stone, Michael
- Committee Member(s)
- Mason, Nadya
- Ravaioli, Umberto

- Department of Study
- Physics
- Discipline
- Physics
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- Topological insulator
- Weyl semimetal

- Abstract
- Topological insulators and semimetals possess the exotic gapless excitations that are governed by relativistic quantum mechanics. Due to the growing interests in these phases, the number of the materials predicted and shown to have the topological characters are continuously increasing. Nevertheless, the physical behaviors of the topological materials in the presence of various types of interactions are not well-understood. In this thesis, we study the quantum mechanical responses of the topological insulators and the topological semimetals in the presence of the interactions. In chapter. 1, we begin our discussion by introducing the general concepts of the topological materials as a prerequisite to understanding the research in the subsequent chapters. We introduce the low energy Hamiltonian and the unique physical responses of the topological insulators and the topological semimetals. In addition, we construct the tight-binding models that are used in the next chapters. In chapter. 2, we study the superconducting proximity effect in the 3D time-reversal invariant topological insulator(TI). The 3D TI proximity coupled with s-wave superconductor is predicted to host Majorana fermions. To experimentally detect the Majorana fermions, the 3D TI should be thinner than the decay length of the proximity effect to have the fully superconducting TI surface. We study the superconducting proximity effect on the thin film TI, which has a finite hybridization gap between the top and the bottom surface. By calculating the induced superconducting order parameter in the TI as a function of the hybridization gap, Zeeman energy, and chemical potential, we determine the relevant experimental parameters that harbor the topological superconductivity. Our results offer the relevant experimental parameters in searching for the topological superconductivity and the Majorana fermions in the 3D TI-superconductor proximity systems. In chapter. 3, we investigate the unconventional superconducting pairing symmetry that may occur in the 3D TI-superconductor proximity coupled system. In the presence of the in-plane Zeeman effect to the TI surface, we find that Fulde-Ferrell(FF) state can be induced in the conventional superconductor. This occurs when the inverse proximity effect(IPE) of the TI to the superconductor is large enough that the normal band of the superconductor possesses a proximity induced spin-orbit coupling and magnetization. We compare the ground state energies of the FF pairing and the BCS pairing to determine the relevant parameters of the system that energetically favor the FF pairing. When we increase the thickness of the superconductor film, we find that the BCS pairing is more favored than the FF pairing. This is because of the increased number of the metallic bands near the Fermi surface that originally favor the BCS pairing. Our result indicates that the FF pairing can only be found in the thin-film limit of the superconductor. In chapter 4, we turn our attention from the TI to the superconducting state of Weyl semimetal(WSM). Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and point nodal BCS state are known to be the possible superconducting states in inversion symmetric doped WSM. To experimentally determine the preferred pairing state, we propose a Josephson junction transport method that shows a unique quantum interference pattern in the presence of FFLO. The Josephson junction consists of a doped WSM and a conventional s-wave superconductor. By applying an external transverse current in the s-wave superconductor, the s-wave order parameter effectively gains a momentum. When the momentum of the s-wave superconductor matches with the momentum of the FFLO states, we find the enhancement of the Josephson current, that serves as an indicator of FFLO states in doped WSMs. In chapter 5, we study the effect of Anderson-type disorder on inversion symmetric Weyl semimetal. In general, the WSMs can be classified into two types by considering the underlying symmetries: type-I WSMs, that have broken inversion or time-reversal symmetry, and type-II WSMs, that additionally breaks Lorentz invariance. Using the Born approximation, we find that the Anderson disorder renormalizes the topological mass of the WSM. The renormalization of the topological mass induces a quantum phase transition from type-I WSMs to type-II WSMs. The phase transition occurs since the renormalization of the topological mass effectively reduces the Fermi velocity of the Weyl node. We also confirm the disorder-induced phase transitions using the numerical exact diagonalization method on the tight-binding model of the WSM. In chapter 6, we study the gravitational anomaly of the Weyl and the Dirac fermions. One possible means of studying the topological phases of matter is to examine the quantum anomalies in the boundary theory, which indicates the presence of the non-trivial topology in the higher dimensional bulk. This approach is based on the fact that if the edge theory has a non-trivial response to certain transformations, then the edge theory cannot be consistent on its manifold and will manifest itself as the edge of a higher dimensional system. As such, we calculate the responses of the Weyl fermions on torus under modular transformations, known as large gravitational anomaly. In (d+1)-D torus, the modular transformations form PSL(Z,d+1) group. Using both analytical and numerical regularization methods that support the analytical calculation, we find that both Dirac fermions in (2+1)-D and Weyl fermions in (3+1)-D are anomaly free under PSL(Z,3) and PSL(Z,4) groups respectively. Yet, we find that the Weyl fermion still has the gravitational anomaly when external magnetic field is coupled. We conclude that this is a modification of a mixed chiral anomaly of the Weyl fermion.
- Graduation Semester
- 2018-08
- Type of Resource
- text
- Permalink
- http://hdl.handle.net/2142/101764
- Copyright and License Information
- Copyright 2018 Moon Jip Park

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