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Bulletin of the American Physical Society

Majorana-Hubbard Ladders

Armin Rahmani, Dmitry I. Pikulin, Ian K. Affleck

Published Mar 5, 2019• 0 citations
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Abstract

Models of interacting Majorana modes may be realized in vortex lattices in superconducting films in contact with topological insulators and may be tuned to the strong interaction regime by adjusting the chemical potential. Extending the results on one- and two-dimensional Majorana-Hubbard models, here we study two- and four-leg ladders using both field theory and the {density-matrix renormalization group (DMRG)} methods, finding a phase diagram largely consistent with that proposed for the two-dimensional model on the square lattice.

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Bulletin of the American Physical Society•Jan 1, 2019•0 citations
Models of interacting Majorana modes may be realized in vortex lattices in superconducting films in contact with topological insulators and may be tuned to the strong interaction regime by adjusting the chemical potential. Extending the results on one- and two-dimensional Majorana-Hubbard models, here we study two- and four-leg ladders using both field theory and the {density-matrix renormalization group (DMRG)} methods, finding a phase diagram largely consistent with that proposed for the two-dimensional model on the square lattice.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.
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.
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.
We study the fully gapped chiral Mott insulator (CMI) of frustrated Bose-Hubbard models on ladders and two-dimensional lattices by perturbative strong-coupling analysis and density matrix renormalization group (DMRG). First we show the existence of a low-lying exciton state on several geometries carrying the correct quantum numbers responsible for the condensation of excitons and formation of the CMI in the intermediate interaction regime. Then we perform systematic DMRG simulations on several two-leg ladder systems with $\ensuremath{\pi}$ flux and carefully characterize the two quantum phase transitions. We discuss the possibility to extend the generally very small CMI window by including repulsive nearest-neighbor interactions or changing density and coupling ratios.
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.