We study the condition for a topological superconductor (TS) phase with end 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 end modes are realized in finite ranges of the filling factor, showing a sequence of reentrant transitions as the chemical potential is tuned. The locations of these phases reflect the structure of bands in the noninteracting case, which exhibits a distinct self-similar structure. The stability of the TS in the presence of an on-site interaction or a harmonic trap potential is also discussed.
Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree--Fock techniques, we demonstrate that while interactions degrade the bulk gap---consistent with recent results of Gangadharaiah et al.---they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.
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.
The one-dimensional (1D)
J1−J2 quantum spin model can be viewed as a strong-coupling analogue of the Schrieffer-Su-Heeger model with two inequivalent alternating Ising couplings along the wire, associated to the physics of resonating valence bonds. Similar to the quantum Ising model, which differently presents a long-range N\' eel ordered phase, this model also maps onto a p-wave superconducting wire which shows a topological phase transition with the emergence of low-energy Majorana fermions. We show how signatures of the topological phase transition for the p-wave superconducting wire, i.e. a half Skyrmion, are revealed through local (short-range) spin observables and their derivatives related to the capacitance of the pairing fermion model. Then, we present an edge correspondence through the edge spin susceptibility in the
J1−J2 model revealing that the topological phase transition is a metal of Majorana fermions. We justify that the spin magnetization at an edge at very small transverse magnetic field is a good marker of the topological invariant and of Majorana zero modes. We identify a correspondence between the quantum information of resonating valence bonds and the charge fluctuations in a p-wave superconductor through our method "the bipartite fluctuations". Physical properties of this 1D model are in fact robust when including additional interactions, which is optimistic for practical applications e.g. in quantum circuits.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective one-dimensional (1D) lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization-group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of two-dimensional topological superconductors when strong disorder and interactions are present.
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.