We consider the propagation of electrons in a lattice with an anisotropic dispersion in the x -y plane (lattice constant a), such that it supports open orbits along the x axis in an out-of-plane... Show moreWe consider the propagation of electrons in a lattice with an anisotropic dispersion in the x -y plane (lattice constant a), such that it supports open orbits along the x axis in an out-of-plane magnetic field B. We show that a point source excites a "breathing mode," a state that periodically spreads out and refocuses after having propagated over a distance . pound = (eaB/h)-1 in the x direction. Unlike known magnetic focusing effects, governed by the classical cyclotron radius, this is an intrinsically quantum mechanical effect with a focal length oc h over bar. Show less
The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple... Show moreThe symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space-time lattice that removes both obstructions is introduced. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single-cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time-evolution operator is discontinuous at Brillouin zone boundaries. Show less
Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as... Show moreMassless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as 'Klein tunneling', may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach. Show less
We identify a mapping between two-dimensional (2D) electron transport in a minimally twisted graphene bilayer and a one-dimensional (1D) quantum walk, where one spatial dimension plays the role of... Show moreWe identify a mapping between two-dimensional (2D) electron transport in a minimally twisted graphene bilayer and a one-dimensional (1D) quantum walk, where one spatial dimension plays the role of time. In this mapping, a magnetic field B perpendicular to the bilayer maps onto an electric field. Bloch oscillations due to the periodic motion in a 1D Bloch band can then be observed in purely DC transport as magnetoconductance oscillations with periodicity set by the Bloch frequency. Show less
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... Show moreWe 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. Show less
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... Show moreThe 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 H psi = EP psi, with Hermitian tight-binding operators H, 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. Show less
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... Show moreFu 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. Show less
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... Show moreWe 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. Show less
A spatially oscillating pair potential Delta(r) = Delta(0)e(2iK center dot r) with momentum K > Delta(0)/hv drives a deconfinement transition of the Majorana bound states in the vortex cores of... Show moreA spatially oscillating pair potential Delta(r) = Delta(0)e(2iK center dot r) with momentum K > Delta(0)/hv drives a deconfinement transition of the Majorana bound states in the vortex cores of a Fu-Kane heterostructure (a 3D topological insulator with Fermi velocity v, on a superconducting substrate with gap Delta(0), in a perpendicular magnetic field). In the deconfined phase at zero chemical potential the Majorana fermions form a dispersionless Landau level, protected by chiral symmetry against broadening due to vortex scattering. The coherent superposition of electrons and holes in the Majorana Landau level is detectable as a local density of states oscillation with wave vector root K-2 - (Delta(0)/hv)(2). The striped pattern also provides a means to measure the chirality of the Majorana fermions. Show less
We calculate the current-voltage (I-V) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap).... Show moreWe calculate the current-voltage (I-V) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap). The phase phi across the junction becomes time dependent in response to a DC current bias. Rabi oscillations in the Andreev levels produce a staircase I-V characteristic. The number of voltage steps counts the number of Rabi oscillations per 2 pi increment of phi, providing a way to probe the coherence of the qubit in the absence of any external AC driving. The phenomenology is the same as the Majorana-induced DC Shapiro steps in topological Josephson junctions of Choi et al. [Phys. Rev. B 102, 140501(R) (2020)]-but now for a nontopological Andreev qubit. Show less
The chiral edge modes of a topological superconductor support two types of excitations: fermionic quasiparticles known as Majorana fermions and pi-phase domain walls known as edge vortices. Edge... Show moreThe chiral edge modes of a topological superconductor support two types of excitations: fermionic quasiparticles known as Majorana fermions and pi-phase domain walls known as edge vortices. Edge vortices are injected pairwise into counter-propagating edge modes by a flux bias or voltage bias applied to a Josephson junction. An unpaired edge mode carries zero electrical current on average, but there are time-dependent current fluctuations. We calculate the shot noise power produced by a sequence of edge vortices and find that it increases logarithmically with their spacing - even if the spacing is much larger than the core size so the vortices do not overlap. This nonlocality produces an anomalous V ln V increase of the shot noise in a voltage-biased geometry, which serves as a distinguishing feature in comparison with the linear-in-V Majorana fermion shot noise. Show less
