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.

Aug 23, 20200 citations

Triangular lattice Majorana-Hubbard model: Mean-field theory and DMRG on a width-4 torus

Tarun Tummuru, Alberto Nocera, Ian K. Affleck

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 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.

arXiv:2008.09963Physical review. B./Physical review. BMar 15, 20212 citations

Phase diagram of the interacting Majorana chain model

Armin Rahmani, Xiaoyu Zhu, Marcel Franz +1 more

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.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.

arXiv:1505.03966Physical Review BDec 14, 2015117 citations

Quantum phases of a one-dimensional Majorana-Bose-Hubbard model

Ananda Roy, Johannes Hauschild, Frank Pollmann

Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), $p$ -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- $e$ 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 zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), $p$-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-$e$ 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.

arXiv:1911.08120Physical review. B./Physical review. BFeb 18, 202010 citations

Majorana-Hubbard model on the honeycomb lattice

Chengshu Li, Marcel Franz

Phase diagram of a Hubbard model for Majorana fermions on the honeycomb lattice is explored using a combination of field theory, renormalization group, and mean-field arguments, as well as exact numerical diagonalization. Unlike the previously studied versions of the model we find that even weak interactions break symmetries and lead to interesting topological phases. We establish two topologically nontrivial phases at weak coupling, one gapped with chiral edge modes and the other gapless with antichiral edge modes. At strong coupling a mapping onto a novel frustrated spin- $\frac{1}{2}$ model suggests a highly entangled spin liquid ground state.Phase diagram of a Hubbard model for Majorana fermions on the honeycomb lattice is explored using a combination of field theory, renormalization group, and mean-field arguments, as well as exact numerical diagonalization. Unlike the previously studied versions of the model we find that even weak interactions break symmetries and lead to interesting topological phases. We establish two topologically nontrivial phases at weak coupling, one gapped with chiral edge modes and the other gapless with antichiral edge modes. At strong coupling a mapping onto a novel frustrated spin-$\frac{1}{2}$ model suggests a highly entangled spin liquid ground state.

arXiv:1806.06092Physical review. B./Physical review. BSep 14, 201821 citations

Majorana-Hubbard Ladders

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

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.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.

Bulletin of the American Physical SocietyMar 5, 20190 citations

Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D),

p

-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-

e

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.

\frac{1}{2}

model suggests a highly entangled spin liquid ground state.