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Topological Phases and Majorana Fermions

Abstract : In this thesis, we study theoretically different aspects of topological systems. These models present resilient properties due to a non-trivial topology of their band structures, and in particular exotic edge excitations such as Majorana fermions.Entanglement entropy and entanglement spectrum have been fundamental to the study of these systems and of gapless systems in general, but are difficult to measure experimentally. Bipartite charge fluctuations were proposed as a weak measurement of this entanglement, in particular for one-dimensional gapless phases. We extend previous results on standard Luttinger Liquids to generic families of one- and two-dimensional non-interacting topological systems. Through exact computations, we show that their critical points are characterized by universal coefficients that reveal the topological aspect of the transitions. In two dimensions, the Dirac cones give quantized contributions to the fluctuations and various correlation functions. These contributions depend on their winding numbers, allowing for a precise determination of the topological structure of the gapless points. A volume law is also present and linked to the Quantum Fisher information, with characteristic non-analyticities at the phase transitions.In a second time, we include interactions and show that some of these signatures are preserved in topological superconductors even in their presence. Through analytical (bosonization, renormalization group) and numerical (exact diagonalization and DMRG) methods, we study the phase diagram of two Coulomb-coupled topological superconducting wires. We are interested in their behavior when the interactions are strong enough to break the topological protection: the interplay between unconventional superconductivity and interactions leads to exotic phases. We show the appearance of phases spontaneously breaking the time-reversal symmetry, with non-trivial orbital currents, and of an unusual gapless phase that is the extension of two critical interacting Majorana modes.Finally, we are interested in electronic transport mediated by Majorana fermions. We study a floating superconducting island carrying several such impurities. This device is thought to be a potential building block for a quantum computer. The Majorana fermions affect the statistics of the charge carriers, which leads to very resilient fractionalized transport. We extend previous studies to the charge degenerate case, where the total number of fermions in the island is not fixed, and map it to the well-known Multi-Channel Kondo model at large interaction. We reinterpret this standard model in terms of a particle moving in a highly dimensional, dissipative lattice.
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Submitted on : Wednesday, November 29, 2017 - 10:52:07 AM
Last modification on : Wednesday, February 5, 2020 - 8:04:39 AM


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  • HAL Id : tel-01651575, version 1


Loïc Herviou. Topological Phases and Majorana Fermions. Superconductivity [cond-mat.supr-con]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLX036⟩. ⟨tel-01651575⟩



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