Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States

Lorenz curve, Gini coefficient, family income

Energy spectrum of graphene multilayers in a parallel magnetic field

We study the orbital effect of a strong magnetic field parallel to the layers on the energy spectrum of the Bernal-stacked graphene bilayer and multilayers, including graphite. We consider the minimal model with the electron tunneling between the nearest sites in the plane and out of the plane. Using the semiclassical analytical approximation and exact numerical diagonalization, we find that the energy spectrum consists of two domains. In the low- and high-energy domains, the semiclassical electron orbits are closed and open, so the spectra are discrete and continuous, correspondingly. The discrete energy levels are the analogs of the Landau levels for the parallel magnetic field. They can be detected experimentally using electron tunneling and optical spectroscopy. In both domains, the electron wave functions are localized on a finite number of graphene layers, so the results can be applied to graphene multilayers of a finite thickness.Comment: 11 pages, 13 figures. Added to v.2: Appendix A, Fig. 13, Refs. [18-23]. V.3: minor stylistic corrections from the published versio

Hopf Term for a Two-Dimensional Electron Gas

In this Comment on the paper by W. Apel and Yu. A. Bychkov, cond-mat/9610040 and Phys. Rev. Lett. 78, 2188 (1997), we draw attention to our prior microscopic derivations of the Hopf term for various systems and to shortcomings of the Apel-Bychkov derivation. We explain how the value of the Hopt term prefactor $\Theta$ is expressed in terms of a topological invariant in the momentum space and the quantized Hall conductivity of the system. (See also related paper cond-mat/9703195)Comment: RevTeX, 1 page, no figure

Curie law, entropy excess, and superconductivity in heavy fermion metals and other strongly interacting Fermi liquids

Low-temperature thermodynamic properties of strongly interacting Fermi liquids with fermion condensate are investigated. We demonstrate that the spin susceptibility of these systems exhibits the Curie-Weiss law, and the entropy contains a temperature-independent term. The excessive entropy is released at the superconducting transition, enhancing the specific heat jump Delta C and rendering it proportional to the effective Curie constant. The theoretical results are favorably compared with the experimental data on the heavy fermion metal CeCoIn5, as well as He-3 films.Comment: 4 pages, 2 figures. V.2: a reference added; minor changes as in the published versio

Detecting D-Wave Pairing and Collective Modes in Fermionic Condensates with Bragg Scattering

We show how the appearance of d-wave pairing in fermionic condensates manifests itself in inelastic light scattering. Specifically, we calculate the Bragg scattering intensity from the dynamic structure factor and the spin susceptibility, which can be inferred from spin flip Raman transitions. This information provides a precise tool with which we can identify nontrivial correlations in the state of the system beyond the information contained in the density profile imaging alone. Due to the lack of Coulomb effects in neutral superfluids, this is also an opportunity to observe the Anderson-Bogoliubov collective mode