Three-dimensional topological phase on the diamond lattice

An interacting bosonic model of Kitaev type is proposed on the three-dimensional diamond lattice. Similarly to the two-dimensional Kitaev model on the honeycomb lattice which exhibits both Abelian and non-Abelian phases, the model has two (``weak'' and ``strong'' pairing) phases. In the weak pairing phase, the auxiliary Majorana hopping problem is in a topological superconducting phase characterized by a non-zero winding number introduced in A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, arXiv:0803.2786. The topological character of the weak pairing phase is protected by a discrete symmetry.Comment: 7 pages, 5 figure

Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics

The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.Comment: 43 pages, no figures, LaTeX2e, with amsmath, amssym

Non-polynomial extensions of solvable potentials a la Abraham-Moses

Abraham-Moses transformations, besides Darboux transformations, are well-known procedures to generate extensions of solvable potentials in one-dimensional quantum mechanics. Here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the Abraham-Moses transformations for typical solvable potentials, e.g. the radial oscillator, the Darboux-P\"oschl-Teller and some others. These seed solutions are simple generalisations of the virtual state wavefunctions, which are obtained from the eigenfunctions by discrete symmetries of the potentials. The virtual state wavefunctions have been an essential ingredient for constructing multi-indexed Laguerre and Jacobi polynomials through multiple Darboux-Crum transformations. In contrast to the Darboux transformations, the virtual state wavefunctions generate non-polynomial extensions of solvable potentials through the Abraham-Moses transformations.Comment: 29 page

Orthogonal Polynomials from Hermitian Matrices

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.Comment: 53 pages, no figures. Several sentences and a reference are added. To be published in J. Math. Phy

Discovery of K-Shell Emission Lines of Neutral Atoms in the Galactic Center Region

The K-shell emission line of neutral irons from the Galactic center (GC) region is one of the key for the structure and activity of the GC. The origin is still open question, but possibly due either to X-ray radiation or to electron bombarding to neutral atoms. To address this issue, we analyzed the Suzaku X-ray spectrum from the GC region of intense neutral iron line emission, and report on the discovery of Kalpha lines of neutral argon, calcium, chrome, and manganese atoms. The equivalent widths of these Kalpha lines indicate that the metal abundances in the GC region should be ~1.6 and ~4 of solar value, depending on the X-ray and the electron origins, respectively. On the other hand, the metal abundances in the hot plasma in the GC region are found to be ~1-2 solar. These results favor that the origin of the neutral Kalpha lines are due to X-ray irradiation.Comment: 7 pages, 5 figures, accepted for publication in PASJ (Vol.62, No.2, pp.423--429

Disorder-induced metal-insulator transitions in three-dimensional topological insulators and superconductors

We discuss the effects of disorder in time-reversal invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field theory analysis (the large-$N_f$ expansion where $N_f$ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a non-trivial critical point separating a metallic phase and a Dirac semi-metal. A similar field theory approach is developed for disorder-driven transitions in symmetry class AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the non-linear sigma model supplemented with a topological term which is a three-dimensional analogue of the Pruisken term in the integer quantum Hall effect.Comment: 19 pages, 5 figure