70 research outputs found
Majorana Fermions and Non-Abelian Statistics in Three Dimensions
We show that three dimensional superconductors, described within a Bogoliubov
de Gennes framework can have zero energy bound states associated with pointlike
topological defects. The Majorana fermions associated with these modes have
non-Abelian exchange statistics, despite the fact that the braid group is
trivial in three dimensions. This can occur because the defects are associated
with an orientation that can undergo topologically nontrivial rotations. A new
feature of three dimensional systems is that there are "braidless" operations
in which it is possible to manipulate the groundstate associated with a set of
defects without moving or measuring them. To illustrate these effects we
analyze specific architectures involving topological insulators and
superconductors.Comment: 4 pages, 2 figures, published versio
Discrete Dynamical Systems Embedded in Cantor Sets
While the notion of chaos is well established for dynamical systems on
manifolds, it is not so for dynamical systems over discrete spaces with
variables, as binary neural networks and cellular automata. The main difficulty
is the choice of a suitable topology to study the limit . By
embedding the discrete phase space into a Cantor set we provided a natural
setting to define topological entropy and Lyapunov exponents through the
concept of error-profile. We made explicit calculations both numerical and
analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running
top to bottom in figures, to appear in J. Math. Phy
Topological Defects and Gapless Modes in Insulators and Superconductors
We develop a unified framework to classify topological defects in insulators
and superconductors described by spatially modulated Bloch and Bogoliubov de
Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with
adiabatic parameters r surrounding the defect and belong to any of the ten
symmetry classes defined by time reversal symmetry and particle-hole symmetry.
The topological classes for such defects are identified, and explicit formulas
for the topological invariants are presented. We introduce a generalization of
the bulk-boundary correspondence that relates the topological classes to defect
Hamiltonians to the presence of protected gapless modes at the defect. Many
examples of line and point defects in three dimensional systems will be
discussed. These can host one dimensional chiral Dirac fermions, helical Dirac
fermions, chiral Majorana fermions and helical Majorana fermions, as well as
zero dimensional chiral and Majorana zero modes. This approach can also be used
to classify temporal pumping cycles, such as the Thouless charge pump, as well
as a fermion parity pump, which is related to the Ising non-Abelian statistics
of defects that support Majorana zero modes.Comment: 27 pages, 15 figures, Published versio
Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables
This paper is devoted to the specific class of pseudoconformal mappings of
quaternion and octonion variables. Normal families of functions are defined and
investigated. Four criteria of a family being normal are proven. Then groups of
pseudoconformal diffeomorphisms of quaternion and octonion manifolds are
investigated. It is proven, that they are finite dimensional Lie groups for
compact manifolds. Their examples are given. Many charactersitic features are
found in comparison with commutative geometry over or .Comment: 55 pages, 53 reference
Schroedinger equation for joint bidirectional motion in time
The conventional, time-dependent Schroedinger equation describes only
unidirectional time evolution of the state of a physical system, i.e., forward
or, less commonly, backward. This paper proposes a generalized quantum dynamics
for the description of joint, and interactive, forward and backward time
evolution within a physical system. [...] Three applications are studied: (1) a
formal theory of collisions in terms of perturbation theory; (2) a
relativistically invariant quantum field theory for a system that kinematically
comprises the direct sum of two quantized real scalar fields, such that one
field evolves forward and the other backward in time, and such that there is
dynamical coupling between the subfields; (3) an argument that in the latter
field theory, the dynamics predicts that in a range of values of the coupling
constants, the expectation value of the vacuum energy of the universe is forced
to be zero to high accuracy. [...]Comment: 30 pages, no figures. Related material is in quant-ph/0404012.
Differs from published version by a few added remarks on the possibility of a
large-scale-average negative energy density in spac
Quantization and spacetime topology
We consider classical and quantum dynamics of a free particle in de Sitter's
space-times with different topologies to see what happens to space-time
singularities of removable type in quantum theory. We find analytic solution of
the classical dynamics. The quantum dynamics is solved by finding an
essentially self-adjoint representation of the algebra of observables
integrable to the unitary representations of the symmetry group of each
considered gravitational system. The dynamics of a massless particle is
obtained in the zero-mass limit of the massive case. Our results indicate that
taking account of global properties of space-time enables quantization of
particle dynamics in all considered cases.Comment: Class. Quantum Grav. 20 (2003) 2491-2507; no figures, RevTeX
Chiral non-linear sigma-models as models for topological superconductivity
We study the mechanism of topological superconductivity in a hierarchical
chain of chiral non-linear sigma-models (models of current algebra) in one,
two, and three spatial dimensions. The models have roots in the 1D
Peierls-Frohlich model and illustrate how the 1D Frohlich's ideal conductivity
extends to a genuine superconductivity in dimensions higher than one. The
mechanism is based on the fact that a point-like topological soliton carries an
electric charge. We discuss a flux quantization mechanism and show that it is
essentially a generalization of the persistent current phenomenon, known in
quantum wires. We also discuss why the superconducting state is stable in the
presence of a weak disorder.Comment: 5 pages, revtex, no figure
PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations
For any central potential V in D dimensions, the angular Schroedinger
equation remains the same and defines the so called hyperspherical harmonics.
For non-central models, the situation is more complicated. We contemplate two
examples in the plane: (1) the partial differential Calogero's three-body model
(without centre of mass and with an impenetrable core in the two-body
interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic
oscillator (with one or two impenetrable barriers). These examples are solvable
due to the presence of the barriers. We contemplate a small complex shift of
the angle. This creates a problem: the barriers become "translucent" and the
angular potentials cease to be solvable, having the sextuple-well form for
Calogero model and the quadruple or double well form otherwise. We mimic the
effect of these potentials on the spectrum by the multiple, purely imaginary
square wells and tabulate and discuss the result in the first nontrivial
double-well case.Comment: 21 pages, 5 figures (see version 1), amendment (a single comment
added on p. 7
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