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.
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.
We have applied the real space quantum renormalization group approach to study the topological quantum phase transition in the one-dimensional chain of a spinless p-wave superconductor. We investigate the behavior of local compressibility and ground-state fidelity of the Kitaev chain. We show that the topological phase transition is signaled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point have been used as a proper indicators for the topological quantum phase transition, which signals the appearance of Majorana fermions. We also present the scaling analysis of ground-state fidelity near the critical point that manifests the universal information about the topological phase transition.
Finite-length one-dimensional topological superconductor wires host localized Majorana zero modes at their ends. In realistic models, these appear only after a topological quantum critical point is crossed by external tuning of parameters. Thus, there is a universal finite-size scaling, governed by the critical point, that dictates the evolution of the energy of the Majorana modes near the transition. We first describe this scaling, then apply it in detail to an explicit synthetic topological superconductor model. Our work not only connects Ising quantum criticality with realistic nanowires in the presence of spin-orbit coupling, Zeeman splitting, and superconductivity, but also provides a viable experimental route for discerning the existence of the topological quantum critical point.
We study the topological properties of a one-dimensional quasiperiodic-potential-modulated mosaic trimer lattice. To begin with, we first investigate the topological properties of the model in the clean limit free of quasiperiodic disorder based on analytical derivation and numerical calculations of the Zak phase
Z and the polarization
P. Two nontrivial topological phases corresponding to the
1/3 filling and
2/3 filling, respectively, are revealed. Then we incorporate the mosaic modulation and investigate the influence of quasiperiodic disorder on the two existing topological phases. Interestingly, it turns out that quasiperiodic disorder gives rise to multiple distinct effects for different fillings. At
2/3 filling, the topological phase is significantly enhanced by the quasiperiodic disorder and topological Anderson insulator emerges. Based on the calculations of polarization and energy gap, we explicitly present corresponding topological phase diagram in the
λ−J plane. While for the
1/3 filling case, % the topological phase is dramatically suppressed by the same quasiperiodic disorder. the quasiperiodic disorder dramatically compresses the topological phase, and strikingly, further induces the emergence of reentrant topological phase transitions instead. Furthermore, we verify the topological phase diagrams by computing the many-body ground state fidelity susceptibility for both the
1/3 filling and
2/3 filling cases. Our work exemplifies the diverse roles of quasiperiodic disorder in the modulation of topological properties, and will further inspire more research on the competitive and cooperative interplay between topological properties and quasiperiodic disorder.