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

Casimir effect of electromagnetic field in D-dimensional spherically symmetric cavities

Eigenmodes of electromagnetic field with perfectly conducting or infinitely permeable conditions on the boundary of a D-dimensional spherically symmetric cavity is derived explicitly. It is shown that there are (D-2) polarizations for TE modes and one polarization for TM modes, giving rise to a total of (D-1) polarizations. In case of a D-dimensional ball, the eigenfrequencies of electromagnetic field with perfectly conducting boundary condition coincides with the eigenfrequencies of gauge one-forms with relative boundary condition; whereas the eigenfrequencies of electromagnetic field with infinitely permeable boundary condition coincides with the eigenfrequencies of gauge one-forms with absolute boundary condition. Casimir energy for a D-dimensional spherical shell configuration is computed using both cut-off regularization and zeta regularization. For a double spherical shell configuration, it is shown that the Casimir energy can be written as a sum of the single spherical shell contributions and an interacting term, and the latter is free of divergence. The interacting term always gives rise to an attractive force between the two spherical shells. Its leading term is the Casimir force acting between two parallel plates of the same area, as expected by proximity force approximation.Comment: 28 page

Finite Temperature Casimir Effect in Randall-Sundrum Models

The finite temperature Casimir effect for a scalar field in the bulk region of the two Randall-Sundrum models, RSI and RSII, is studied. We calculate the Casimir energy and the Casimir force for two parallel plates with separation $a$ on the visible brane in the RSI model. High-temperature and low-temperature cases are covered. Attractiveness versus repulsiveness of the temperature correction to the force is discussed in the typical special cases of Dirichlet-Dirichlet, Neumann-Neumann, and Dirichlet-Neumann boundary conditions at low temperature. The Abel-Plana summation formula is made use of, as this turns out to be most convenient. Some comments are made on the related contemporary literature.Comment: 33 pages latex, 2 figures. Some changes in the discussion. To appear in New J. Phy

Spin texture on the Fermi surface of tensile strained HgTe

We present ab initio and k.p calculations of the spin texture on the Fermi surface of tensile strained HgTe, which is obtained by stretching the zincblende lattice along the (111) axis. Tensile strained HgTe is a semimetal with pointlike accidental degeneracies between a mirror symmetry protected twofold degenerate band and two nondegenerate bands near the Fermi level. The Fermi surface consists of two ellipsoids which contact at the point where the Fermi level crosses the twofold degenerate band along the (111) axis. However, the spin texture of occupied states indicates that neither ellipsoid carries a compensating Chern number. Consequently, the spin texture is locked in the plane perpendicular to the (111) axis, exhibits a nonzero winding number in that plane, and changes winding number from one end of the Fermi ellipsoids to the other. The change in the winding of the spin texture suggests the existence of singular points. An ordered alloy of HgTe with ZnTe has the same effect as stretching the zincblende lattice in the (111) direction. We present ab initio calculations of ordered Hg_xZn_1-xTe that confirm the existence of a spin texture locked in a 2D plane on the Fermi surface with different winding numbers on either end.Comment: 8 pages, 8 figure

Dirac semimetal in three dimensions

In a Dirac semimetal, the conduction and valence bands contact only at discrete (Dirac) points in the Brillouin zone (BZ) and disperse linearly in all directions around these critical points. Including spin, the low energy effective theory around each critical point is a four band Dirac Hamiltonian. In two dimensions (2D), this situation is realized in graphene without spin-orbit coupling. 3D Dirac points are predicted to exist at the phase transition between a topological and a normal insulator in the presence of inversion symmetry. Here we show that 3D Dirac points can also be protected by crystallographic symmetries in particular space-groups and enumerate the criteria necessary to identify these groups. This reveals the possibility of 3D analogs to graphene. We provide a systematic approach for identifying such materials and present ab initio calculations of metastable \beta-cristobalite BiO_2 which exhibits Dirac points at the three symmetry related X points of the BZ.Comment: 6 pages, 4 figure

Casimir effect of electromagnetic field in Randall-Sundrum spacetime

We study the finite temperature Casimir effect on a pair of parallel perfectly conducting plates in Randall-Sundrum model without using scalar field analogy. Two different ways of interpreting perfectly conducting conditions are discussed. The conventional way that uses perfectly conducting condition induced from 5D leads to three discrete mode corrections. This is very different from the result obtained from imposing 4D perfectly conducting conditions on the 4D massless and massive vector fields obtained by decomposing the 5D electromagnetic field. The latter only contains two discrete mode corrections, but it has a continuum mode correction that depends on the thicknesses of the plates. It is shown that under both boundary conditions, the corrections to the Casimir force make the Casimir force more attractive. The correction under 4D perfectly conducting condition is always smaller than the correction under the 5D induced perfectly conducting condition. These statements are true at any temperature.Comment: 20 pages, 4 figure

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

Optimal control strategies for tuberculosis treatment: a case study in Angola

We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.Comment: This is a preprint of a paper whose final and definite form will appear in the international journal Numerical Algebra, Control and Optimization (NACO). Paper accepted for publication 15-March-201

Axially symmetric rotating traversable wormholes

This paper generalizes the static and spherically symmetric traversable wormhole geometry to a rotating axially symmetric one with a time-dependent angular velocity by means of an exact solution. It was found that the violation of the weak energy condition, although unavoidable, is considerably less severe than in the static spherically symmetric case. The radial tidal constraint is more easily met due to the rotation. Similar improvements are seen in one of the lateral tidal constraints. The magnitude of the angular velocity may have little effect on the weak energy condition violation for an axially symmetric wormhole. For a spherically symmetric one, however, the violation becomes less severe with increasing angular velocity. The time rate of change of the angular velocity, on the other hand, was found to have no effect at all. Finally, the angular velocity must depend only on the radial coordinate, confirming an earlier result.Comment: 17 pages, AMSTe

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010