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Journal of Physics Conference Series

Topological superconductivity in the one-dimensional interacting Creutz model

Doru Sticlet, L. F. Oleiro Seabra, Frank Pollmann, J. Cayssol

Published Mar 18, 2015• 1 citation
Publisher

Abstract

We consider one-dimensional topological insulators characterized by zero energy end states. In presence of proximity induced pairing, those end states can become Majorana states. We study here the fate of those various end states when Hubbard electron-electron repulsive interactions are added, using a combination of mean-field theory and density matrix renormalization group techniques.

DOI
10.1088/1742-6596/592/1/012133

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We consider one-dimensional topological insulators characterized by zero energy end states. In presence of proximity induced pairing, those end states can become Majorana states. We study here the fate of those various end states when Hubbard electron-electron repulsive interactions are added, using a combination of mean-field theory and density matrix renormalization group techniques.
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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.
Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), ppp-wave, spinless superconductor, in the absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper pairs, or a second gapless phase. In contrast to the LL of Cooper pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-eee excitations. The three phases are separated from each other by phase transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase transitions.
Majorana modes can arise as zero energy bound states in a variety of solid state systems. A lattice of these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. This thesis is devoted to the study of interactions between the Majorana zero modes when such a lattice has a triangular geometry. We map the phase diagram of this model using a combination of mean field theory and numerical simulation of ladders through the density-matrix renormalization-group technique. Our analysis suggests that interactions drive a topologically non-trivial gapped phase into a gapless Luttinger liquid.
Majorana modes can arise as zero energy bound states in a variety of solid state systems. A lattice of these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. This thesis is devoted to the study of interactions between the Majorana zero modes when such a lattice has a triangular geometry. We map the phase diagram of this model using a combination of mean field theory and numerical simulation of ladders through the density-matrix renormalization-group technique. Our analysis suggests that interactions drive a topologically non-trivial gapped phase into a gapless Luttinger liquid.
By decreasing the transversal confinement potential in interacting quantum wires and populating the second energetically lowest sub-band, electron systems transition into a quasi-one-dimensional state with dominant superconducting correlations and one gapless mode. By combining effective field theory approach and numerical density matrix renormalization group simulations we show that this quasi-one-dimensional state is a topological state that hosts Majorana edge modes. We also study the single-particle correlations across the interface between this quasi-one-dimensional and single-channel states.