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

The goal of the project is to study topological phases of interacting superconductors, the nature of the emergents end states and the way they manifest themselves in experimentally measurable quantities, with the focus on one and two dimensional realisations. In particular, the first and second objective deal with emergent many body end states of a topological superconducting wire, in the presence of time reversal symmetry. Here objective 1 is aimed at unveiling the physical nature of the emergent many body end states, while Objective 2 is aimed at studying their experimental signatures.The third and forth objective are focused on emergent end states that manifest in two dimensional structures. As an intermediate milestone, Objective 3 was aimed at identifying topological superconducting pumps in systems with a gap due to interactions. Once the topological classification of the pumps was achieved, Objective 4 aimed to construct the two-dimensional system that is the analog of the interacting pump.

During the first period of the research project we have addressed research Objectives 1 and 2. The aim of Objective 1 was to understand the physical properties of the end states of the seven non trivial phases in interacting superconducting wires with time reversal symmetry. This goal was achieved 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 the open system, and formulation of the topological index in terms of a scattering matrix, was an important benchmark towards the the second research objective, whose goal was to study the experimental signatures of these interacting topological states. The results of the second research objective were reported 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.

These results motivated us to study the ground state and transport along a one dimensional chain Majorana bound states, which interact through a local quartic interaction [Zhao Liu, Emil J. Bergholtz, Alessandro Romito, and Dganit Meidan Phys. Rev. B 96, 205442 (2017)]. This model describes for example the edge physics of a quasi one dimensional stack of Kitaev chains with modified time-reversal symmetry. This project therefore links the two parts of the proposals that deal with topological superconducting phases in one dimensions and two dimensional structures. We found that for finite but large chain length, the scattering matrix partially reflects the topological periodicity, and the chain exhibits strikingly different transport properties for different chain lengths. Our analysis in the above article indicates that in the thermodynamic limit the Majorana chain has a two-fold degenerate ground state. Regarding the Majorana chain as an effective model that emerges on the edge of stacked superconducting chains, the presence of a robust zero mode in the thermodynamic limit indicates that the two-dimensional bulk system is a weak interacting topological phase. This is quite remarkable given that its non-interacting analog is topologically trivial.

We were able to make considerable progress in research objectives 3 and 4 by presenting a comprehensive classification of gapped phases of a chain of parafermions [D. Meidan, E. Berg, Ady Stern, Phys. Rev. B 95, 205104 (2017)]. Such chains can be realized in one dimensional structures embedded in fractionalized two dimensional states of matter, e.g. on the edges of a fractional quantum spin Hall system, where counter propagating modes may be gapped either by back-scattering or by coupling to a superconductor. This work 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, recent developments in fabrication techniques and new experimental platforms aimed at controlling and manipulating Majorana zero modes led us to study experimental signatures of Majorana manipulations. We have identified a new relation between the dynamical manipulation of Majorana modes and quantized heat currents. This remarkable property can be used for better experimental control of the exchange process and can provide a new potential tool for engineering heat pumping device [Dganit Meidan, Tal Gur, and Alessandro Romito Phys. Rev. B 99, 205101(2019)].

Finally, we have addressed the stability of topological features in Majorana nanowire with interactions and disorder by introducing a powerful new variational method based on Matrix-Product-State/Operator technology [G. Kells, N. Moran, and D. Meidan Phys. Rev. B 97, 085425 (2018)]. This method can resolve the full position space structure of the multi-particle Majorana modes as well as extract statistical information about the entire many-body spectrum. Our analysis identified two regimes in which disorder exhibits opposing effects on the stability of the zero modes. In particular, we identify a regime of parameters where many body effects dominates, and disorder reduces the multi-particle content of the zero mode, which in tern enhances the stability of the zero modes.

On a different front, we study transport in topological semi-metals. The non-trivial topology of Weyl semimetal can be revealed by applying an external magnetic field, which leads to the formation of Landau levels (LL). The Weyl nodes result in the emergence of chiral zero Landau levels, protected against scattering for a sufficiently smooth disorder. In a recent work we studied the effect of inelastic processes on the magneto-transport [P. G. Matveeva, D. N. Aristov, D. Meidan, and D. B. Gutman Phys. Rev. B 96, 165406 (2017)]. We show that by measuring shot noise as a function of a magnetic field, for different applied voltage, one can estimate the electron-electron inelastic length. Unlike in three dimensional materials where these Weyl nodes are stable to any (sufficiently weak) perturbation, robust Weyl semimetals in two dimension do not exist. The closest analog was recently proposed by Young and Kane, who considered two dimensional lattice with non-symmorphic symmetries. They show that by breaking of time reversal or inversion symmetry one can split the Dirac cones into Weyl node. We found that this splitting is accompanied by the formation of gapless edge states [P. G. Matveeva, D. N. Aristov, D. Meidan, and D. B. Gutman Phys. Rev. B 99, 075409 (2019)]. The type and strength of applied deformation control the location and Weyl nodes and the composition of the respective edge states.

During the first period of the research project we have addressed research Objectives 1 and 2. The aim of Objective 1 was to understand the physical properties of the end states of the seven non trivial phases in interacting superconducting wires with time reversal symmetry. This goal was achieved 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 the open system, and formulation of the topological index in terms of a scattering matrix, was an important benchmark towards the the second research objective, whose goal was to study the experimental signatures of these interacting topological states. The results of the second research objective were reported 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.

These results motivated us to study the ground state and transport along a one dimensional chain Majorana bound states, which interact through a local quartic interaction [Zhao Liu, Emil J. Bergholtz, Alessandro Romito, and Dganit Meidan Phys. Rev. B 96, 205442 (2017)]. This model describes for example the edge physics of a quasi one dimensional stack of Kitaev chains with modified time-reversal symmetry. This project therefore links the two parts of the proposals that deal with topological superconducting phases in one dimensions and two dimensional structures. We found that for finite but large chain length, the scattering matrix partially reflects the topological periodicity, and the chain exhibits strikingly different transport properties for different chain lengths. Our analysis in the above article indicates that in the thermodynamic limit the Majorana chain has a two-fold degenerate ground state. Regarding the Majorana chain as an effective model that emerges on the edge of stacked superconducting chains, the presence of a robust zero mode in the thermodynamic limit indicates that the two-dimensional bulk system is a weak interacting topological phase. This is quite remarkable given that its non-interacting analog is topologically trivial.

We were able to make considerable progress in research objectives 3 and 4 by presenting a comprehensive classification of gapped phases of a chain of parafermions [D. Meidan, E. Berg, Ady Stern, Phys. Rev. B 95, 205104 (2017)]. Such chains can be realized in one dimensional structures embedded in fractionalized two dimensional states of matter, e.g. on the edges of a fractional quantum spin Hall system, where counter propagating modes may be gapped either by back-scattering or by coupling to a superconductor. This work 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, recent developments in fabrication techniques and new experimental platforms aimed at controlling and manipulating Majorana zero modes led us to study experimental signatures of Majorana manipulations. We have identified a new relation between the dynamical manipulation of Majorana modes and quantized heat currents. This remarkable property can be used for better experimental control of the exchange process and can provide a new potential tool for engineering heat pumping device [Dganit Meidan, Tal Gur, and Alessandro Romito Phys. Rev. B 99, 205101(2019)].

Finally, we have addressed the stability of topological features in Majorana nanowire with interactions and disorder by introducing a powerful new variational method based on Matrix-Product-State/Operator technology [G. Kells, N. Moran, and D. Meidan Phys. Rev. B 97, 085425 (2018)]. This method can resolve the full position space structure of the multi-particle Majorana modes as well as extract statistical information about the entire many-body spectrum. Our analysis identified two regimes in which disorder exhibits opposing effects on the stability of the zero modes. In particular, we identify a regime of parameters where many body effects dominates, and disorder reduces the multi-particle content of the zero mode, which in tern enhances the stability of the zero modes.

On a different front, we study transport in topological semi-metals. The non-trivial topology of Weyl semimetal can be revealed by applying an external magnetic field, which leads to the formation of Landau levels (LL). The Weyl nodes result in the emergence of chiral zero Landau levels, protected against scattering for a sufficiently smooth disorder. In a recent work we studied the effect of inelastic processes on the magneto-transport [P. G. Matveeva, D. N. Aristov, D. Meidan, and D. B. Gutman Phys. Rev. B 96, 165406 (2017)]. We show that by measuring shot noise as a function of a magnetic field, for different applied voltage, one can estimate the electron-electron inelastic length. Unlike in three dimensional materials where these Weyl nodes are stable to any (sufficiently weak) perturbation, robust Weyl semimetals in two dimension do not exist. The closest analog was recently proposed by Young and Kane, who considered two dimensional lattice with non-symmorphic symmetries. They show that by breaking of time reversal or inversion symmetry one can split the Dirac cones into Weyl node. We found that this splitting is accompanied by the formation of gapless edge states [P. G. Matveeva, D. N. Aristov, D. Meidan, and D. B. Gutman Phys. Rev. B 99, 075409 (2019)]. The type and strength of applied deformation control the location and Weyl nodes and the composition of the respective edge states.