Conductance of a spin-1 quantum dot: the two-stage Kondo effect

We discuss the physics of a of a spin-1 quantum dot, coupled to two metallic leads and develop a simple model for the temperature dependence of its conductance. Such quantum dots are described by a two-channel Kondo model with asymmetric coupling constants and the spin screening of the dot by the leads is expected to proceed via a two-stage process. When the Kondo temperatures of each channel are widely separated, on cooling, the dot passes through a broad cross-over regime dominated by underscreened Kondo physics. A singular, or non-fermi liquid correction to the conductance develops in this regime. At the lowest temperatures, destructive interference between resonant scattering in both channels leads to the eventual suppression of the conductance of the dot. We develop a model to describe the growth, and ultimate suppression of the conductance in the two channel Kondo model as it is screened successively by its two channels. Our model is based upon large-N approximation in which the localized spin degrees of freedom are described using the Schwinger boson formalism.Comment: 16 pages, 10 figure

SOME REMARKS ON THE SMARANDACHE FUNCTION

Remarks on a Function in the Number Theory

On Electrostatic Positron Acceleration In The Accretion Flow Onto Neutron Stars

As first shown by Shvartsman (1970), a neutron star accreting close to the Eddington limit must acquire a positive charge in order for electrons and protons to move at the same speed. The resulting electrostatic field may contribute to accelerating positrons produced near the star surface in conjunction with the radiative force. We reconsider the balance between energy gains and losses, including inverse Compton (IC), bremsstrahlung and non--radiative scatterings. It is found that, even accounting for IC losses only, the maximum positron energy never exceeds $\approx 400$ keV. The electrostatic field alone may produce energies $\approx 50$ keV at most. We also show that Coulomb collisions and annihilation with accreting electrons severely limit the number of positrons that escape to infinity.Comment: 9 pages plus 3 postscript figures, to be published in Ap

Quasi-exactly solvable quartic: real algebraic spectral locus

We describe the real quasi-exactly solvable spectral locus of the PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure

Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice

We propose theoretically an experimentally realizable method to demonstrate the Lyapunov instability and to extract the value of the largest Lyapunov exponent for a chaotic many-particle interacting system. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We suggest to use imperfect time-reversal of system's dynamics known as Loschmidt echo, which can be realized experimentally by reversing the sign of the Hamiltonian of the system. The routine involves tracking and then subtracting the noise of virtually any observable quantity before and after the time-reversal. We support the theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent can indeed be extracted from the Loschmidt echo routine. We also discuss possible values of experimental parameters required for implementing this proposal

Building Blocks of Topological Quantum Chemistry: Elementary Band Representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In Nature 547, 298-305 (2017), we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, which we introduced in Nature 547, 298-305 (2017), we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.Comment: 15+9 pages, 4 figures; v2: minor corrections; v3: updated references (published version

Constraints on the small-scale power spectrum of density fluctuations from high-redshift gamma-ray bursts

Cosmological models that include suppression of the power spectrum of density fluctuations on small scales exhibit an exponential reduction of high-redshift, non-linear structures, including a reduction in the rate of gamma ray bursts (GRBs). Here we quantify the constraints that the detection of distant GRBs would place on structure formation models with reduced small-scale power. We compute the number of GRBs that could be detectable by the Swift satellite at high redshifts (z > 6), assuming that the GRBs trace the cosmic star formation history, which itself traces the formation of non-linear structures. We calibrate simple models of the intrinsic luminosity function of the bursts to the number and flux distribution of GRBs observed by the Burst And Transient Source Experiment (BATSE). We find that a discovery of high-z GRBs would imply strong constraints on models with reduced small-scale power. For example, a single GRB at z > 10, or 10 GRBs at z > 5, discovered by Swift during its scheduled two-year mission, would rule out an exponential suppression of the power spectrum on scales below R_c=0.09 Mpc (exemplified by warm dark matter models with a particle mass of m_x=2 keV). Models with a less sharp suppression of small-scale power, such as those with a red tilt or a running scalar index, n_s, are more difficult to constrain, because they are more degenerate with an increase in the power spectrum normalization, sigma_8, and with models in which star-formation is allowed in low-mass minihalos. We find that a tilt of \delta n_s ~ 0.1 is difficult to detect; however, an observed rate of 1 GRB/yr at z > 12 would yield an upper limit on the running of the spectral index, alpha = d(n_s)/d(ln k) > -0.05.Comment: 10 pages, 6 figures; Minor changes to match version published in Ap

Graph Theory Data for Topological Quantum Chemistry

Topological phases of noninteracting particles are distinguished by global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties which are local (at high symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [B. Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and so identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed BANDREP program on the Bilbao Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented in arXiv:1703.02050 v2: Accepted version. References updated, figures improve

Topological quantum chemistry

The past decade's apparent success in predicting and experimentally discovering distinct classes of topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000 stoichiometric compounds extant in material databases, only several hundred of them are topologically nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? To address this, we propose a new and complete electronic band theory that highlights the link between topology and local chemical bonding, and combines this with the conventional band theory of electrons. Topological Quantum Chemistry is a description of the universal global properties of all possible band structures and materials, comprised of a graph theoretical description of momentum space and a dual group theoretical description in real space. We classify the possible band structures for all 230 crystal symmetry groups that arise from local atomic orbitals, and show which are topologically nontrivial. We show how our topological band theory sheds new light on known TIs, and demonstrate the power of our method to predict a plethora of new TIs.Comment: v1: 8 pages + 40 pages supplemenetary material. Previously submitted v2: ~ Published version. 11 pages + 79 pages supplementary material. Descriptions of the data used in this paper can be found in arXiv:1706.08529 and arXiv:1706.09272. All data can be accessed via the Bilbao Crystallographic Server (http://cryst.ehu.es). Two additional papers elaborating on the general theory currently in pre