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arXiv:1711.03632 · Physical review. B./Physical review. B

Majorana stripe order on the surface of a three-dimensional topological insulator

Yoshitomo Kamiya, Akira Furusaki, Jeffrey C. Y. Teo, Gia-Wei Chern

Published Oct 26, 2018• 9 citations
arXivHTMLPDF

Abstract

The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping, we demonstrate that the interaction induces a Majorana stripe state, a gapped state spontaneously breaking lattice translational and rotational symmetries, as opposed to the previously conjectured topological quantum criticality. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization.

DOI
10.1103/physrevb.98.161409

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The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping, we demonstrate that the interaction induces a Majorana stripe state, a gapped state spontaneously breaking lattice translational and rotational symmetries, as opposed to the previously conjectured topological quantum criticality. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization.
We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.
Dirac semimetals, with their protected Dirac points, present an ideal platform for realizing intrinsic topological superconductivity. In this work, we investigate superconductivity in a two-dimensional, square-lattice nonsymmorphic Dirac semimetal. In the normal state near half-filling, the Fermi surface consists of two distinct pockets, each enclosing a Dirac point at a time-reversal invariant momentum (X=(π,0)\textbf{X}=(π,0)X=(π,0) and Y=(0,π)\textbf{Y}=(0,π)Y=(0,π)). Considering an on-site repulsive and nearest-neighbor attractive interaction, we use self-consistent mean-field theory to determine the ground-state pairing symmetry. We find that an even-parity, spin-singlet dx2−y2d_{x^{2}-y^{2}}dx2−y2​-wave pairing is favored as it gives rise to a fully gapped superconducting state. Since the pairing amplitude has opposite signs on the two Dirac Fermi pockets, the superconducting state is identified as a second-order topological superconductor. The hallmark of this topological phase is the emergence of Majorana zero modes at the system's boundaries. Notably, the positions of these Majorana modes are highly controllable and can be manipulated simply by tailoring the boundary sublattice terminations. Our results highlight the promise of nonsymmorphic Dirac semimetals for realizing and manipulating Majorana modes.
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.
Topological superconductors represent one of the key hosts of Majorana-based topological quantum computing.Typical scenarios for one-dimensional (1D) topological superconductivity assume a broken gauge symmetry associated to a superconducting state.However, no interacting 1D many-body system is known to spontaneously break gauge symmetries.Here, we show that zero modes emerge in a many-body system without gauge symmetry breaking and in the absence of superconducting order.In particular, we demonstrate that Majorana zero modes of the symmetry-broken superconducting state are continuously connected to these zero-mode excitations, demonstrating that zero-bias anomalies may emerge in the absence of gauge symmetry breaking.We demonstrate that these many-body zero modes share the robustness features of the Majorana zero modes of symmetry-broken topological superconductors.We further show that the interface between the interacting model and a 1D topological superconductor does not support Majorana modes.We introduce a bosonization formalism to analyze these excitations and show that a ground state analogous to a topological superconducting state can be analytically found in a certain limit.Our results demonstrate that robust Majorana-like zero modes may appear in a many-body system without gauge symmetry breaking, thus introducing a family of protected excitations with no single-particle analogs.
Majorana modes can arise as zero-energy bound states in a variety of solid state systems. A two-dimensional phase supporting these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. At low energies, such a setup can be modeled by Majorana modes that interact with each other on the Abrikosov lattice. In experiments, the lattice is usually triangular. Motivated by the practical relevance, we explore the phase diagram of this Hubbard-like Majorana model using a combination of mean-field theory and numerical simulation of thin torus geometries through the density matrix renormalization group algorithm. Our analysis indicates that attractive interactions between Majoranas can drive a phase transition in an otherwise gapped topological state.