We present a comprehensive theoretical study of interacting and disordered topological phases of coupled Kitaev wires, which may support further realistic applications of Majorana fermions. We develop a variety of analytical, mathematical, and numerical methods for one- and two-coupled wires, associated with a topological marker accessible from real-space correlation functions on the wire(s). We verify the stability of the topological superconducting phase and quantify disorder effects close to the quantum phase transitions, e.g., through two-point correlation functions or using a renormalization group analysis of disorder. We show that the double critical Ising (DCI) phase---a fractional Majorana liquid characterized by a pair of half central charges and topological numbers---is stabilized by strong interactions against disorder, which respects the inversion symmetry between the wires (i.e., parity conservation on each wire). In the presence of an interwire hopping term, the DCI phase turns into a protected topological phase with a bulk gap. We study the localization physics developing along the critical line for weaker interactions.
We study the condition for a topological superconductor (TS) phase with edge Majorana fermions to appear when a quasiperiodic lattice modulation is applied to a one-dimensional quantum wire with strong spin-orbit interaction situated under a magnetic field and in proximity to a superconductor. By density-matrix renormalization group analysis we find that multiple topological phases with Majorana edge modes are realized in finite ranges of the filling factor, showing a sequence of reentrant transitions. An on-site interaction stabilizes the topological phases, as observed for systems without lattice modulation in [E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, Phys. Rev. B 84, 014503 (2011)]. The stability of the TS in the presence of a harmonic trap potential is also discussed.
Recent tunneling experiments on InSb hybrid superconductor-semiconductor devices have provided hope for a stabilization of Majorana edge modes in a spin-orbit quantum wire subject to a magnetic field and superconducting proximity effect. Connecting the experimental scenario with a microscopic description poses challenges of a different kind, such as accounting for the effect of interactions on the tunneling properties of the wire. We develop a density matrix renormalization group (DMRG) analysis of the tunneling spectra of interacting Majorana chains, which we explicate for the Kitaev chain model. Our DMRG approach allows us to calculate the spectral function down to zero frequency, where we analyze how the Majorana zero-bias peak is affected by interactions. For topological phase transitions between the topological and trivial superconducting phase in the Majorana wire, the bulk gap closure generically affects the proximity peaks and the Majorana peak.
In this paper, we investigate Majorana correlations in quantum impurities coupled to a topological superconducting wire. By employing a density matrix renormalization group approach, we have calculated the Majorana correlation functions in the ground state of the system for different positions of the impurities along the wire. We observe that the correlations decrease exponentially with the distance between the impurities and the ends of the wire. Moreover, different electron-electron interactions on the wire and on the impurities were also analyzed, and the results showed a stronger effect as the electron occupation in the wire is increased. In addition, while changing the occupation of the impurities by tuning their energy levels, we observe that, for specific values of $\ensuremath{\varepsilon}$, the absolute value of correlation is highly peaked when the chemical potential lies within the window of the topological regime. These peaks are associated with Coulomb blockade phenomena in the impurities, revealing very clearly its effect on the Majorana bound states in the topological phase of the system.
We study the combined effect of interactions and disorder on topological order in one dimension. Toward that end, we consider a generalized Kitaev chain including fermion-fermion interactions and disorder in the chemical potential. We determine the phase diagram by performing density-matrix renormalization-group calculations on the corresponding spin-1/2 chain. We find that moderate disorder or repulsive interactions individually stabilize the topological order, which remains valid for their combined effect. However, both repulsive and attractive interactions lead to a suppression of the topological phase at strong disorder.
Using the density-matrix renormalization group, we determine the different topological phases and low-energy excitations of a time-reversal invariant topological superconducting (TRITOPS) wire with extended
s-wave superconductivity, Rashba spin-orbit coupling (SOC) and on-site repulsion
U, under an externally applied Zeeman field
J. For the case in which
J is perpendicular to the SOC, the model describes a chain of Shiba impurities on top of a superconductor with extended superconductor pairing. We identify the different topological phases of the model at temperature
T=0, and in particular we study the stability of the TRITOPS phase against the Zeeman field
J and the chemical potential $\ensuremath{\mu}$, for different values of
U. In the case where the magnetic field
J is perpendicular to the SOC axis, the pair of Kramers degenerate Majorana zero modes at the edges of the system that exist for
J=0, remain degenerate until a critical value of the magnetic field is reached. For
J parallel to the SOC and up to moderate values of
U, the fractional spin projection $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}=1/4$ at the ends, found for noninteracting wires at
U=0, is recovered. In addition, the analytic expression that relates $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}$ with
J for finite noninteracting chains is shown to be universal up to moderate values of
U.