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Periodic Report Summary 1 - INTTOPSUP (Transport Properties of Emergent Edge States in Interacting Topological Superconductors)

The goal of the proposed research is to study topological phases of interacting superconductors, the nature of their emergents end states and the way they manifest themselves in experimentally measurable quantities, with the focus on one and two dimensional realisations. Topological superconducting wires are known to harbor Majorana zero modes at their ends. Majorana zero modes obey a unique type of quantum statistics known as non-abelian statistics. This property makes Majorana zero modes promising candidates in realising fault tolerant quantum memory devices. While their existence is known to be fairly stable to imperfect conditions such as impurities and weak electron-electron interactions, aspirations to observe and manipulate Majorana zero modes requires a detailed study of these effects. Yet aside from the implications for ongoing and future experiments and applications, the implications of interactions on the physical properties of the topological phase is predominantly a fundamental one.

The aim of objective 1 was to understand the nature of the bound states that emerge at the ends of a multiple channel topological superconductor with interactions. Work towards this objective was initiated following the writing of the proposal. The results of this work was published prior to the funding approval, in a paper that studied an interacting topological superconducting wire connected to a non interaction lead [D. Meidan, A. Romito, and P. W. Brouwer, “Scattering Matrix Formulation of the Topological Index of Interacting Fermions in One-Dimensional Superconductors”,Phys. Rev. Lett. 113, 057003 (2014)]. In this work we have identified an interacting topological phase that supports emergent many-body end states, which we identify to be a topologically protected Kondo-like resonance.

The study of open systems, and formulation of the topological index in terms of a scattering matrix, is an important benchmark towards charting the experimental signatures of interacting topological phases. The second research objective was to establish the experimental signatures of the emergent end states of interacting topological states. This objective was achieved in a succeeding publication [D. Meidan, A. Romito, P. W. Brouwer, ”Transport signatures of interacting fermions in quasi-one-dimensional topological superconductors”,Phys. Rev. B 93, 125433 (2016)]. Here we showed that the Kondo-like resonance that appears at the edge of the interacting topological superconductor exhibits distinctive experimental features such as an anomalous temperature dependence of the zero-bias conductance and an anomalous Fano factor. As part of a future project, I intend to consolidate the insight gained in this work towards the third research objective by studying the pumping properties of the emergent Kondo like resonance.

The aim of objectives 3 and 4 was to study interacting topological phases in two-dimension, with a focus on fractional topological phases that are not adiabatically connected to non interacting systems. Work towards this challenging task has followed several complementary routes. Below I will describe two such projects that are currently in final stages of completion.

The aim of project 1 was to study topological phases of a system constructed by coupling a large number of superconducting chains with interactions. In particular we study the transport along the edge of a multi-chain system, in the macroscopically large chain limit [A. Romito (Lancaster University), Z. Liu (Free University Berlin), E. Bergholtz (Stockholm University) and D. Meidan, in preparation ]. This work shows that transport along the edge of the stacked structure shows a distinct signature for different number of chains N, even in the large N limit. These differences can be detected in a two terminal setup. As a future work we intend to study the thermalization of the boundary system by looking at the effect of random interactions on the many-body level statistics, and how these manifest in measurable quantities.

The aim of the second project was to classify gapped phases of one dimensional structures embedded in fractionalized two dimensional systems [E. Berg, D. Meidan, A. Stern, in preparation]. In this work we identified a composite Haldane phase whose Kramers degenerate end states carry fractionalized spin 1/3, as well as a topological phase with emergent Majorana fermions end states, in a system whose constituent particles are bosons, or in a system without any form of pairing.

In addition, as dictated by advances in the field, recent developments in the study of many body localisation has led us to study the stability of the topological phase of superconducting wires in the presence of disorder and interactions [G. Kells, N. Moran and D. Meidan, in preparation].
In particular we study how short range interactions and potential disorder affect the topologically protected Majorana modes in a one dimensional Kitaev chain. We identify two regimes of parameters where disorder has strikingly different effects: 1) The disorder-obstructed-topological-order - where disorder extends the coherence length of the many body Majorana states, effectively reducing the degree to which our topological information is tolerant against error, and 2) The localised-assisted-topological-order, where interactions introduce decay transitions that alter the zero mode occupation. In this region, disorder induces a systematic reduction in both the residual energy and multi-particle content of the near-zero mode thus recovering the exponential decay that is a key requirement of a topological mode. Building on the insight gained in this work, we plan to study the manifestation of the two different behaviours in experimentally measurable quantities, and in particular in tunnelling experiments.

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