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

Quasi Exactly Solvable Difference Equations

Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.Comment: LaTeX with amsfonts, no figure, 17 pages, a few typos corrected, a reference renewed, 3/2 pages comments on hermiticity adde

Holographic Derivation of Entanglement Entropy from AdS/CFT

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.Comment: 5 pages, 3 figures, Revtex, references adde

Growth of Magnetic Fields Induced by Turbulent Motions

We present numerical simulations of driven magnetohydrodynamic (MHD) turbulence with weak/moderate imposed magnetic fields. The main goal is to clarify dynamics of magnetic field growth. We also investigate the effects of the imposed magnetic fields on the MHD turbulence, including, as a limit, the case of zero external field. Our findings are as follows. First, when we start off simulations with weak mean magnetic field only (or with small scale random field with zero imposed field), we observe that there is a stage at which magnetic energy density grows linearly with time. Runs with different numerical resolutions and/or different simulation parameters show consistent results for the growth rate at the linear stage. Second, we find that, when the strength of the external field increases, the equilibrium kinetic energy density drops by roughly the product of the rms velocity and the strength of the external field. The equilibrium magnetic energy density rises by roughly the same amount. Third, when the external magnetic field is not very strong (say, less than ~0.2 times the rms velocity when measured in the units of Alfven speed), the turbulence at large scales remains statistically isotropic, i.e. there is no apparent global anisotropy of order B_0/v. We discuss implications of our results on astrophysical fluids.Comment: 16 pages, 18 figures; ApJ, accepte

Holographic classification of Topological Insulators and its 8-fold periodicity

Using generic properties of Clifford algebras in any spatial dimension, we explicitly classify Dirac hamiltonians with zero modes protected by the discrete symmetries of time-reversal, particle-hole symmetry, and chirality. Assuming the boundary states of topological insulators are Dirac fermions, we thereby holographically reproduce the Periodic Table of topological insulators found by Kitaev and Ryu. et. al, without using topological invariants nor K-theory. In addition we find candidate Z_2 topological insulators in classes AI, AII in dimensions 0,4 mod 8 and in classes C, D in dimensions 2,6 mod 8.Comment: 19 pages, 4 Table

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

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

Field-driven topological glass transition in a model flux line lattice

We show that the flux line lattice in a model layered HTSC becomes unstable above a critical magnetic field with respect to a plastic deformation via penetration of pairs of point-like disclination defects. The instability is characterized by the competition between the elastic and the pinning energies and is essentially assisted by softening of the lattice induced by a dimensional crossover of the fluctuations as field increases. We confirm through a computer simulation that this indeed may lead to a phase transition from crystalline order at low fields to a topologically disordered phase at higher fields. We propose that this mechanism provides a model of the low temperature field--driven disordering transition observed in neutron diffraction experiments on ${\rm Bi_2Sr_2CaCu_2O_8\, }$ single crystals.Comment: 11 pages, 4 figures available upon request via snail mail from [email protected]

The Effect of the Random Magnetic Field Component on the Parker Instability

The Parker instability is considered to play important roles in the evolution of the interstellar medium. Most studies on the development of the instability so far have been based on an initial equilibrium system with a uniform magnetic field. However, the Galactic magnetic field possesses a random component in addition to the mean uniform component, with comparable strength of the two components. Parker and Jokipii have recently suggested that the random component can suppress the growth of small wavelength perturbations. Here, we extend their analysis by including gas pressure which was ignored in their work, and study the stabilizing effect of the random component in the interstellar gas with finite pressure. Following Parker and Jokipii, the magnetic field is modeled as a mean azimuthal component, $B(z)$, plus a random radial component, $\epsilon(z) B(z)$, where $\epsilon(z)$ is a random function of height from the equatorial plane. We show that for the observationally suggested values of $^{1/2}$, the tension due to the random component becomes important, so that the growth of the instability is either significantly reduced or completely suppressed. When the instability still works, the radial wavenumber of the most unstable mode is found to be zero. That is, the instability is reduced to be effectively two-dimensional. We discuss briefly the implications of our finding.Comment: 10 pages including 2 figures, to appear in The Astrophysical Journal Letter

Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of "discrete" quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are the Askey-Wilson polynomials. This is an extension of our previous study [1,2], which established the "discrete analogue" of the well-known fact; The equilibrium positions of the Calogero systems are described by the Hermite and Laguerre polynomials, whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the Hermite and Laguerre polynomials.Comment: 14 pages, 1 figure. The outline of derivation of the result in section 2 is adde