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Gauge Theory and Topological Order: Superconducting Chains

Mohammadi, Fatemeh | 2023

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 55852 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Kargarian, Mehdi
  7. Abstract:
  8. Notwithstanding massive efforts put forward in the last few decades, the experimental realization of quantum states with topological orders in a controlled way and accessible conditions has remained elusive. The realization of such states in fractional quantum Hall states requires extreme conditions like the cryogenic temperatures and strong magnetic fields. Also, the existence of a pure topological order in spin liquid materials is limited by disorder, subleading interactions, or lack of enough experimental information. Alternatively, we may think of extrinsic models to simulate the exotic phases by combining a set of naturally accessible quantum systems. Our approach involves the superconducting Kondo lattice model in which a gapped electronic system of noninteracting fermions is coupled to local moments via the exchange interaction. The electronic system is composed of one-dimensional topological superconductors and when Kondo coupled to an otherwise lattice of free magnetic ions, induces Abelian topological orders in the magnetic system in the weak coupling limit. The topological superconductors are currently accessible in semiconductor nanowires proximitized to conventional superconductors. To connect the latter model to more experimentally accessible settings, we show that topological orders can be engineered in architecture patterns of semiconductor nanowires hosting Majorana bound states. The Majorana states are allowed to tunnel to quantum dots sitting on the vertices of the lattices. In the limit of strong onsite Coulomb interactions, where the charge fluctuations are suppressed, the magnetic moments of dots are described by topologically ordered spin models like Z_2 topological order and Z_2× Z_2 color code model which has enhanced computational capability. Then we show that the latter topological order can also be realized in a network of purely Majorana zero modes in the absence of coupling to quantum dots. In all of these models in the limit of weak Kondo coupling J_K, a simple degenerate perturbation theory generates a spin model whose low-energy states are described by topologically ordered phases. In the following, we investigate the persistence of these exotic phases even in the stronger values of Kondo coupling by studying the phase diagram and possible quantum phase transitions of the superconducting Kondo lattice model. Using slave-particle representation of spins and exact numerical calculations, we obtain the phase diagram of the model in terms of Kondo coupling J_K and diagnose a quantum phase transition occurring at a rather large value of Kondo coupling J_K^c, so the topological order phase is rather robust beyond the perturbation limit. We recognize the trivial phase at J_K>J_K^c as an invertible phase where the spinons become a topological superconductor characterized by an invariant with the opposite sign of the superconducting electrons. Since the invariant of the electrons and spinons vanishes together, the total system is trivial. The phase transition can be understood as a Mott insulating transition from the invertible phase to a topological order which can be used for the classification of topological phases in the future. Furthermore, we show that in the regime J_K
  9. Keywords:
  10. Topological Order ; Gauge Theory ; Topological Superconductor ; Proximity Effect ; Majorana Fermion ; Quantum Phase Transition ; Electron Fractionalization ; Superconducting Kondo Lattice Model ; Surface Codes

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