On the Coefficients of Quasiconformality for Convex Functions

Let f be holomorpic and univalent in the unit disc E and f(E) be convex. We consider the conformal radius R = R(D,z) = {pipe;} f′(ζ){pipe;}(1-ζ̄) of D = f(E) at the point z = f(ζ). In [3] and [4] the coefficient kf(r), r ∈ (0,1), of quasiconformality has been defined by the equation, In this paper the authors computed the quantity kf(r) for some convex functions. These examples led them to the conjecture that kf (r) ≤ r2 for any convex function holomorphic in E. The function f(ζ) = log((1 + ζ)/(1 -ζ)), which was among their examples, shows that this bound is sharp for any r∈ (0,1). In the present article, we will prove that the above conjecture is true and that the the above example is essentially the only one for which equality is attained. © 2010 Pleiades Publishing, Ltd

Fermion propagators in space-time

The one- and the two-particle propagators for an infinite non-interacting Fermi system are studied as functions of space-time coordinates. Their behaviour at the origin and in the asymptotic region is discussed, as is their scaling in the Fermi momentum. Both propagators are shown to have a divergence at equal times. The impact of the interaction among the fermions on their momentum distribution, on their pair correlation function and, hence, on the Coulomb sum rule is explored using a phenomenological model. Finally the problem of how the confinement is reflected in the momentum distribution of the system's constituents is briefly addressed.Comment: 26 pages, 9 figures, accepted for publication on Phys. Rev.

Microscopically-constrained Fock energy density functionals from chiral effective field theory. I. Two-nucleon interactions

The density matrix expansion (DME) of Negele and Vautherin is a convenient tool to map finite-range physics associated with vacuum two- and three-nucleon interactions into the form of a Skyme-like energy density functional (EDF) with density-dependent couplings. In this work, we apply the improved formulation of the DME proposed recently in arXiv:0910.4979 by Gebremariam {\it et al.} to the non-local Fock energy obtained from chiral effective field theory (EFT) two-nucleon (NN) interactions at next-to-next-to-leading-order (N$^2$LO). The structure of the chiral interactions is such that each coupling in the DME Fock functional can be decomposed into a cutoff-dependent coupling {\it constant} arising from zero-range contact interactions and a cutoff-independent coupling {\it function} of the density arising from the universal long-range pion exchanges. This motivates a new microscopically-guided Skyrme phenomenology where the density-dependent couplings associated with the underlying pion-exchange interactions are added to standard empirical Skyrme functionals, and the density-independent Skyrme parameters subsequently refit to data. A Mathematica notebook containing the novel density-dependent couplings is provided.Comment: 28 pages, 12 figures. Mathematica notebook provided with submission

Asymptotic behavior of impurity-induced bound states in low-dimensional topological superconductors

We study theoretically the asymptotic behavior of the Shiba bound states associated with magnetic impurities embedded in both 2D and 1D anomalous superconductors. We calculate analytically the spatial dependence of the local density of states together with the spin polarization associated with the Shiba bound states. We show that the latter quantity exhibits drastic differences between s-wave and different types of p-wave superconductors. Such properties, which could be measured using spin-polarized STM, offer therefore a way to discriminate between singlet and triplet pairing in low-dimensional superconductors, as well as a way to estimate the amplitude of the triplet pairing in these systems.Comment: 18 pages, 5 figure

Impurity Induced Quantum Phase Transitions and Magnetic Order in Conventional Superconductors: Competition between Bound and Quasiparticle states

We theoretically study bound states generated by magnetic impurities within conventional $s$-wave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is overwhelmingly dominated by the quasiparticle contribution, the ground state of the system is determined by the bound state energies. Self-consistently calculating the superconducting order parameter, we find a discontinuity when the system undergoes a quantum phase transition as indicated by the bound state energies.Comment: 8 pages, 7 figure

Topological p_x+ip_y Superfluid Phase of Fermionic Polar Molecules

We discuss the topological p_x+ip_y superfluid phase in a 2D gas of single-component fermionic polar molecules dressed by a circularly polarized microwave field. This phase emerges because the molecules may interact with each other via a potential V_0(r) that has an attractive dipole-dipole 1/r^3 tail, which provides p-wave superfluid pairing at fairly high temperatures. We calculate the amplitude of elastic p-wave scattering in the potential V_0(r) taking into account both the anomalous scattering due to the dipole-dipole tail and the short-range contribution. This amplitude is then used for the analytical and numerical solution of the renormalized BCS gap equation which includes the second order Gor'kov-Melik-Barkhudarov corrections and the correction related to the effective mass of the quasiparticles. We find that the critical temperature T_c can be varied within a few orders of magnitude by modifying the short-range part of the potential V_0(r). The decay of the system via collisional relaxation of molecules to dressed states with lower energies is rather slow due to the necessity of a large momentum transfer. The presence of a constant transverse electric field reduces the inelastic rate, and the lifetime of the system can be of the order of seconds even at 2D densities ~ 10^9 cm^{-2}. This leads to T_c of up to a few tens of nanokelvins and makes it realistic to obtain the topological p_x+ip_y phase in experiments with ultracold polar molecules.Comment: 15 pages, 9 figures, published versio