362 research outputs found
Scattering theory of topological phases in discrete-time quantum walks
One-dimensional discrete-time quantum walks show a rich spectrum of
topological phases that have so far been exclusively analysed in momentum
space. In this work we introduce an alternative approach to topology which is
based on the scattering matrix of a quantum walk, adapting concepts from
time-independent systems. For gapped quantum walks, topological invariants at
quasienergies 0 and {\pi} probe directly the existence of protected boundary
states, while quantum walks with a non-trivial quasienergy winding have a
discrete number of perfectly transmistting unidirectional modes. Our
classification provides a unified framework that includes all known types of
topology in one dimensional discrete-time quantum walks and is very well suited
for the analysis of finite size and disorder effects. We provide a simple
scheme to directly measure the topological invariants in an optical quantum
walk experiment.Comment: 12 pages. v2: minor correction
Universal nonequilibrium signatures of Majorana zero modes in quench dynamics
The quantum evolution after a metallic lead is suddenly connected to an
electron system contains information about the excitation spectrum of the
combined system. We exploit this type of "quantum quench" to probe the presence
of Majorana fermions at the ends of a topological superconducting wire. We
obtain an algebraically decaying overlap (Loschmidt echo) for large times after the quench, with
a universal critical exponent =1/4 that is found to be remarkably
robust against details of the setup, such as interactions in the normal lead,
the existence of additional lead channels or the presence of bound levels
between the lead and the superconductor. As in recent quantum dot experiments,
this exponent could be measured by optical absorption, offering a new signature
of Majorana zero modes that is distinct from interferometry and tunneling
spectroscopy.Comment: 9 pages + appendices, 4 figures. v3: published versio
Andreev reflection from a topological superconductor with chiral symmetry
It was pointed out by Tewari and Sau that chiral symmetry (H -> -H if e
h) of the Hamiltonian of electron-hole (e-h) excitations in an N-mode
superconducting wire is associated with a topological quantum number
Q\in\mathbb{Z} (symmetry class BDI). Here we show that Q=Tr(r_{he}) equals the
trace of the matrix of Andreev reflection amplitudes, providing a link with the
electrical conductance G. We derive G=(2e^2/h)|Q| for |Q|=N,N-1, and more
generally provide a Q-dependent upper and lower bound on G. We calculate the
probability distribution P(G) for chaotic scattering, in the circular ensemble
of random-matrix theory, to obtain the Q-dependence of weak localization and
mesoscopic conductance fluctuations. We investigate the effects of chiral
symmetry breaking by spin-orbit coupling of the transverse momentum (causing a
class BDI-to-D crossover), in a model of a disordered semiconductor nanowire
with induced superconductivity. For wire widths less than the spin-orbit
coupling length, the conductance as a function of chemical potential can show a
sequence of 2e^2/h steps - insensitive to disorder.Comment: 10 pages, 5 figures. Corrected typo (missing square root) in
equations A13 and A1
Emergence of massless Dirac fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by
the moir\'e super- lattice of graphene on an hexagonal crystal substrate, is
known to exhibit gapped Dirac cones. We show that the gap can be closed by
slightly misaligning the substrate, producing a hierarchy of conical
singularities (Dirac points) in the band structure at rational values Phi =
(p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a
switch of the topological quantum number in the connected component of the
quantum Hall phase diagram. Model calculations reveal the scale invariant
conductivity sigma = 2qe^2 / pi h and Klein tunneling associated with massless
Dirac fermions at these connectivity switches.Comment: 4 pages, 6 figures + appendix (3 pages, 1 figure
Quantized conductance at the Majorana phase transition in a disordered superconducting wire
Superconducting wires without time-reversal and spin-rotation symmetries can
be driven into a topological phase that supports Majorana bound states. Direct
detection of these zero-energy states is complicated by the proliferation of
low-lying excitations in a disordered multi-mode wire. We show that the phase
transition itself is signaled by a quantized thermal conductance and electrical
shot noise power, irrespective of the degree of disorder. In a ring geometry,
the phase transition is signaled by a period doubling of the magnetoconductance
oscillations. These signatures directly follow from the identification of the
sign of the determinant of the reflection matrix as a topological quantum
number.Comment: 7 pages, 4 figures; v3: added appendix with numerics for long-range
disorde
Wigner-Poisson statistics of topological transitions in a Josephson junction
The phase-dependent bound states (Andreev levels) of a Josephson junction can
cross at the Fermi level, if the superconducting ground state switches between
even and odd fermion parity. The level crossing is topologically protected, in
the absence of time-reversal and spin-rotation symmetry, irrespective of
whether the superconductor itself is topologically trivial or not. We develop a
statistical theory of these topological transitions in an N-mode quantum-dot
Josephson junction, by associating the Andreev level crossings with the real
eigenvalues of a random non-Hermitian matrix. The number of topological
transitions in a 2pi phase interval scales as sqrt(N) and their spacing
distribution is a hybrid of the Wigner and Poisson distributions of
random-matrix theory.Comment: 12 pages, 15 figures; v2 to appear in PRL, with appendix in the
supplementary materia
Geodesic scattering by surface deformations of a topological insulator
We consider the classical ballistic dynamics of massless electrons on the
conducting surface of a three-dimensional topological insulator, influenced by
random variations of the surface height. By solving the geodesic equation and
the Boltzmann equation in the limit of shallow deformations, we obtain the
scattering cross section and the conductivity {\sigma}, for arbitrary
anisotropic dispersion relation. At large surface electron densities n this
geodesic scattering mechanism (with {\sigma} propto sqrt{n}) is more effective
at limiting the surface conductivity than electrostatic potential scattering.Comment: 9 pages, 5 figure
Quantum Hall effect in a one-dimensional dynamical system
We construct a periodically time-dependent Hamiltonian with a phase
transition in the quantum Hall universality class. One spatial dimension can be
eliminated by introducing a second incommensurate driving frequency, so that we
can study the quantum Hall effect in a one-dimensional (1D) system. This
reduction to 1D is very efficient computationally and would make it possible to
perform experiments on the 2D quantum Hall effect using cold atoms in a 1D
optical lattice.Comment: 8 pages, 6 figure
Quantum point contact as a probe of a topological superconductor
We calculate the conductance of a ballistic point contact to a
superconducting wire, produced by the s-wave proximity effect in a
semiconductor with spin-orbit coupling in a parallel magnetic field. The
conductance G as a function of contact width or Fermi energy shows plateaus at
half-integer multiples of 4e^2/h if the superconductor is in a topologically
nontrivial phase. In contrast, the plateaus are at the usual integer multiples
in the topologically trivial phase. Disorder destroys all plateaus except the
first, which remains precisely quantized, consistent with previous results for
a tunnel contact. The advantage of a ballistic contact over a tunnel contact as
a probe of the topological phase is the strongly reduced sensitivity to finite
voltage or temperature.Comment: 6 pages, 6 figures; corrected App.
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