On topological properties of massless fermions in a magnetic field

Make more fluid: In condensed matter systems, electrons can acquire unusual properties from their interaction with the atomic lattice. In some examples, they can behave as massless particles, mimicking the relativistic behavior of photons. This thesis is dedicated to the study of such massless electronic excitations, focusing on systems exhibiting Majorana, Weyl, and Dirac fermions. In this thesis, we show how new states can arise in the presence of a magnetic field, find new signatures of such states, and present new methods that can be used to study them.Quantum Matter and Optic

Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special "staggered" grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form ${\mathcal H}\psi=E {\mathcal P}\psi$, with Hermitian tight-binding operators ${\mathcal H}$, ${\mathcal P}$, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.Comment: 16 pages, 3 figure

Method to preserve the chiral-symmetry protection of the zeroth Landau level on a two-dimensional lattice

The spectrum of massless Dirac fermions on the surface of a topological insulator in a perpendicular magnetic field $B$ contains a $B$-independent "zeroth Landau level", protected by chiral symmetry. If the Dirac equation is discretized on a lattice by the method of "Wilson fermions", the chiral symmetry is broken and the zeroth Landau level is broadened when $B$ has spatial fluctuations. We show how this lattice artefact can be avoided starting from an alternative nonlocal discretization scheme introduced by Stacey. A key step is to spatially separate the states of opposite chirality in the zeroth Landau level, by adjoining $+B$ and $-B$ regions.Comment: Contribution to a special issue of Annals of Physics in memory of Kostya Efeto

Chiral charge transfer along magnetic field lines in a Weyl superconductor

We identify an effect of chirality in the electrical conduction along magnetic vortices in a Weyl superconductor. The conductance depends on whether the magnetic field is parallel or antiparallel to the vector in the Brillouin zone that separates Weyl points of opposite chirality.Theoretical Physic

Dynamical simulation of the injection of vortices into a Majorana edge mode

The chiral edge modes of a topological superconductor can transport fermionic quasiparticles, with Abelian exchange statistics, but they can also transport non-Abelian anyons: Majorana zero-modes bound to a {\pi}-phase domain wall that propagates along the boundary. Such an edge vortex is injected by the application of an h/2e flux bias over a Josephson junction. Existing descriptions of the injection process rely on the instantaneous scattering approximation of the adiabatic regime, where the internal dynamics of the Josephson junction is ignored. Here we go beyond that approximation in a time-dependent many-body simulation of the injection process, followed by a braiding of the mobile edge vortex with an immobile Abrikosov vortex in the bulk of the superconductor. Our simulation sheds light on the properties of the Josephson junction needed for a successful implementation of a flying Majorana qubit.Comment: 13 pages 12 figure

Magnetic breakdown spectrum of a Kramers-Weyl semimetal

We calculate the Landau levels of a Kramers-Weyl semimetal thin slab in a perpendicular magnetic field $B$. The coupling of Fermi arcs on opposite surfaces broadens the Landau levels with a band width that oscillates periodically in $1/B$. We interpret the spectrum in terms of a one-dimensional superlattice induced by magnetic breakdown at Weyl points. The band width oscillations may be observed as $1/B$-periodic magnetoconductance oscillations, at weaker fields and higher temperatures than the Shubnikov-de Haas oscillations due to Landau level quantization. No such spectrum appears in a generic Weyl semimetal, the Kramers degeneracy at time-reversally invariant momenta is essential.Comment: 13 pages, 18 figure

Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling

I. Introduction II. Two-dimensional lattice fermions III. Methods to avoid fermion doubling (sine dispersion, sine plus cosine dispersion, staggered lattice dispersion, linear sawtooth dispersion, tangent dispersion) IV. Topologically protected Dirac cone V. Application: Klein tunneling (tangent fermions on a space-time lattice, wave packet propagation) VI. Application: Strong antilocalization (transfer matrix of tangent fermions, topological insulator versus graphene) VII. Application: Anomalous quantum Hall effect (gauge invariant tangent fermions, topologically protected zeroth Landau level) VIII. Application: Majorana metal (Dirac versus Majorana fermions, phase diagram) IX. OutlookComment: review article, 26 pages, 13 figures; V2: added three appendices, and provided code for the various implementation

Localization landscape for Dirac fermions

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schr\"{o}dinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function $u(r)$ is defined on a lattice as a solution of the differential equation $\overline{{H}}u(r)=1$, where $\overline{{H}}$ is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.Comment: 6 pages, 6 figure

Supercell symmetry modified spectral statistics of Kramers-Weyl fermions

We calculate the spectral statistics of the Kramers-Weyl Hamiltonian H = v n-ary sumation ( alpha ) sigma ( alpha ) sin p ( alpha ) + t sigma (0) n-ary sumation ( alpha )cos p ( alpha ) in a chaotic quantum dot. The Hamiltonian has symplectic time-reversal symmetry (H is invariant when spin sigma ( alpha ) and momentum p ( alpha ) both change sign), and yet for small t the level spacing distributionP(s) proportional to s ( beta ) follows the beta = 1 orthogonal ensemble instead of the beta = 4 symplectic ensemble. We identify a supercell symmetry of H that explains this finding. The supercell symmetry is broken by the spin-independent hopping energy proportional to t cos p, which induces a transition from beta = 1 to beta = 4 statistics that shows up in the conductance as a transition from weak localization to weak antilocalization.Theoretical Physic

Chirality inversion of Majorana edge modes in a Fu-Kane heterostructure

Fu and Kane have discovered that a topological insulator with induced s-wave superconductivity (gap Delta(0), Fermi velocity v (F), Fermi energy mu) supports chiral Majorana modes propagating on the surface along the edge with a magnetic insulator. We show that the direction of motion of the Majorana fermions can be inverted by the counterflow of supercurrent, when the Cooper pair momentum along the boundary exceeds Delta(2)(0)/mu v(F) . The chirality inversion is signaled by a doubling of the thermal conductance of a channel parallel to the supercurrent. Moreover, the inverted edge can transport a nonzero electrical current, carried by a Dirac mode that appears when the Majorana mode switches chirality. The chirality inversion is a unique signature of Majorana fermions in a spinful topological superconductor: it does not exist for spinless chiral p-wave pairing.Theoretical Physic