(PHD, 2020)

Abstract:

My various projects in graduate school have centered around a common theme: harnessing relatively well-understood phases of matter and combining them to create exotic physics. They also involve Majoranas, or more accurately, defects that bind Majorana zero modes and are the centerpiece for topological quantum computation. We exploit and enrich this Majorana zero mode by employing topological superconductors, time crystals, and quantum dots and combining them together. Our first project involved joining Majorana nanowires and quantum dots to simulate the SYK model, a zero-dimensional strongly interacting phase with connections to black holes and holography. We follow by explaining how to combine spontaneous symmetry-breaking with topological superconductivity to recover parafermion physics in one dimension. We explain an exact mapping that relates fermions to parafermions, illustrating a deep connection between different one-dimensional phases of matter. We finally show that enhancing the topological superconductor with a time crystal, a phase of matter that spontaneously breaks time-translation symmetry, creates an anomalous zero mode that displays 4*T*periodicity in the Floquet drive. By combining these different phases in judicious ways we achieve exotic physics unattainable by the constituent parts. Our work thus illustrates profitable directions for harnessing Majorana zero modes to study the physics of exotic matter.

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(PHD, 2019)

Abstract:

Topological materials have been a fastest growing research topic in the recent decade. Out of the numerous new phases proposed and/or discovered, “topological insulators” (TIs) are one of the most promising materials that could lead to further advances in high-performance electronics and to applications in quantum computing. Similar to the ordinary semiconductors, TIs have a bulk gap; yet they host robust edge/surface states which are protected from non-magnetic disorder and interactions while the gap remains open. This feature is a manifestation of the non-trivial topology of TIs, the crucial feature that distinguishes them from ordinary semiconductors. Although the search for more topological materials continues, discovered TI currently are limited by practical difficulties that prevent industrialization.

In this thesis, we study graphene, which is the first proposed TI candidate in the history, and its derivatives. With the intrinsic spin-orbital coupling (SOC) on graphene, one can open a topologically nontrivial band gap at the Dirac cones, although the SOC of the carbon atoms is exceedingly small for topological insulation to be observed in experiments. Many proposals exist to enhance the SOC on graphene by doping with adatoms, changing the functionality of the surface, placing graphene on top of other strong SOC materials, etc. However, few proposed TI signatures have been found experimentally. Furthermore, measuring these intrinsic SOCs through magnetoconductance is challenging due to their relatively weak signatures in transport. This work addresses the challenges in transport measurements from both analytical and numerical approaches on various graphene-based materials. Graphene’s Dirac band structure and open geometry underlie its exciting prospects for engineering new physics via impurity-induced spin-orbit coupling. As a tantalizing example, previous theory works predicted a robust quantum-spin-Hall phase in graphene covered with dilute heavy adatoms such as In, Tl, and Os, although experiments to date have not detected the required enhancement of spin-orbit coupling. Motivated by these experiments, we explore the consequences of adatom-generated spin-orbit couplings on magneto-transport in graphene. We attack the problem using diagrammatic techniques and the Landauer-Buttiker transport simulation informed by microscopics, and study various coverages, chemical potentials, and disorder types. We find that the induced spin-orbit couplings can contribute to magneto-conductance differently from conventional intrinsic and Rasbha spin-orbit couplings. Our results provide a possible rationale for the absence of spin-orbit signatures in recent experiments, and also highlight a roadmap for their discovery in future work.

In addition to the adatom-dedoped graphene, we also study graphene placing on top of strong SOC substrate, WS_{2}, by jointing theory, numerics, and experiment. We demonstrate, in experiment, a clear weak anti-localization (WAL) effect arising from induced Rashba spin–orbit coupling (SOC) in WS_{2}-covered single-layer and bilayer graphene devices. Contrary to the uncovered region of a shared single layer graphene flake, WAL in WS_{2}-covered graphene occurs over a wide range of carrier densities on both the electron and hole sides.

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(PHD, 2018)

Abstract:

We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if ** C** is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies

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(PHD, 2015)

Abstract:

Topological superconductors are particularly interesting in light of the active ongoing experimental efforts for realizing exotic physics such as Majorana zero modes. These systems have excitations with non-Abelian exchange statistics, which provides a path towards topological quantum information processing. Intrinsic topological superconductors are quite rare in nature. However, one can engineer topological superconductivity by inducing effective p-wave pairing in materials which can be grown in the laboratory. One possibility is to induce the proximity effect in topological insulators; another is to use hybrid structures of superconductors and semiconductors.

The proposal of interfacing s-wave superconductors with quantum spin Hall systems provides a promising route to engineered topological superconductivity. Given the exciting recent progress on the fabrication side, identifying experiments that definitively expose the topological superconducting phase (and clearly distinguish it from a trivial state) raises an increasingly important problem. With this goal in mind, we proposed a detection scheme to get an unambiguous signature of topological superconductivity, even in the presence of ordinarily detrimental effects such as thermal fluctuations and quasiparticle poisoning. We considered a Josephson junction built on top of a quantum spin Hall material. This system allows the proximity effect to turn edge states in effective topological superconductors. Such a setup is promising because experimentalists have demonstrated that supercurrents indeed flow through quantum spin Hall edges. To demonstrate the topological nature of the superconducting quantum spin Hall edges, theorists have proposed examining the periodicity of Josephson currents respect to the phase across a Josephson junction. The periodicity of tunneling currents of ground states in a topological superconductor Josephson junction is double that of a conventional Josephson junction. In practice, this modification of periodicity is extremely difficult to observe because noise sources, such as quasiparticle poisoning, wash out the signature of topological superconductors. For this reason, We propose a new, relatively simple DC measurement that can compellingly reveal topological superconductivity in such quantum spin Hall/superconductor heterostructures. More specifically, We develop a general framework for capturing the junction’s current-voltage characteristics as a function of applied magnetic flux. Our analysis reveals sharp signatures of topological superconductivity in the field-dependent critical current. These signatures include the presence of multiple critical currents and a non-vanishing critical current for all magnetic field strengths as a reliable identification scheme for topological superconductivity.

This system becomes more interesting as interactions between electrons are involved. By modeling edge states as a Luttinger liquid, we find conductance provides universal signatures to distinguish between normal and topological superconductors. More specifically, we use renormalization group methods to extract universal transport characteristics of superconductor/quantum spin Hall heterostructures where the native edge states serve as a lead. Interestingly, arbitrarily weak interactions induce qualitative changes in the behavior relative to the free-fermion limit, leading to a sharp dichotomy in conductance for the trivial (narrow superconductor) and topological (wide superconductor) cases. Furthermore, we find that strong interactions can in principle induce parafermion excitations at a superconductor/quantum spin Hall junction.

As we identify the existence of topological superconductor, we can take a step further. One can use topological superconductor for realizing Majorana modes by breaking time reversal symmetry. An advantage of 2D topological insulator is that networks required for braiding Majoranas along the edge channels can be obtained by adjoining 2D topological insulator to form corner junctions. Physically cutting quantum wells for this purpose, however, presents technical challenges. For this reason, I propose a more accessible means of forming networks that rely on dynamically manipulating the location of edge states inside of a single 2D topological insulator sheet. In particular, I show that edge states can effectively be dragged into the system’s interior by gating a region near the edge into a metallic regime and then removing the resulting gapless carriers via proximity-induced superconductivity. This method allows one to construct rather general quasi-1D networks along which Majorana modes can be exchanged by electrostatic means.

Apart from 2D topological insulators, Majorana fermions can also be generated in other more accessible materials such as semiconductors. Following up on a suggestion by experimentalist Charlie Marcus, I proposed a novel geometry to create Majorana fermions by placing a 2D electron gas in proximity to an interdigitated superconductor-ferromagnet structure. This architecture evades several manufacturing challenges by allowing single-side fabrication and widening the class of 2D electron gas that may be used, such as the surface states of bulk semiconductors. Furthermore, it naturally allows one to trap and manipulate Majorana fermions through the application of currents. Thus, this structure may lead to the development of a circuit that enables fully electrical manipulation of topologically-protected quantum memory. To reveal these exotic Majorana zero modes, I also proposed an interference scheme to detect Majorana fermions that is broadly applicable to any 2D topological superconductor platform.

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