Using a reformulated Kubo formula we calculate the zero-energy minimal conductivity of bilayer graphene taking into account the small but finite trigonal warping. We find that the conductivity is independent of the strength of the trigonal warping and it is 3 times as large as that without trigonal warping and 6 times larger than that in single layer graphene. Although the trigonal warping of the dispersion relation around the valleys in the Brillouin zone is effective only for low-energy excitations, our result shows that its role cannot be neglected in the zero-energy minimal conductivity.

The Lagrangian of W = 1 supergravity is dimensionally reduced to one (timelike) dimension assuming spatial homogeneity of any Bianchi type within class A of the classification of Ellis and McCallum. The algebra of the supersymmetry generators, the Lorentz generators, the diffeomorphism generators, and the Hamiltonian generator is determined and found to close. In contrast to earlier work, infinitely many physical states with nonvanishing even fermion number are found to exist in these models, indicating that minisuperspace models in supergravity may be just as useful as in pure gravity. PACS numbers: 04.60.Kz, 04.65.+e Minisuperspace models have long served as an enormously useful testing ground for new ideas in quantum gravity, ranging from explorations of its mathematical structure to investigation in the many open problems of quantum cosmology (see, e.g. , Refs. [1 -3]). More recently, also the quantum theory of supersymmetric minisuperspace models has attracted the interest of many authors (see, e.g., Refs. [4 -16]) for similar reasons. Our own interest was first aroused by the chaotic classical nature of the Bianchi type IX models and the discovery [7] that supersymmetric versions have simple explicit analytical solutions in the empty and filled fermion sectors which can be interpreted as wormhole states [9] and, in other cases, as Hartle-Hawking no-boundary states

The collective excitations of Bose condensates in anisotropic axially symmetric harmonic traps are investigated in the hydrodynamic and Thomas-Fermi limit. We identify an additional conserved quantity, besides the axial angular momentum and the total energy, and separate the wave equation in elliptic coordinates. The solution is thereby reduced to the algebraic problem of diagonalizing finite dimensional matrices. The classical quasi-particle dynamics in the local density approximation for energies of the order of the chemical potential is shown to be chaotic. 03.75.Fi,05.30.Jp,32.80.Pj,67.90.+z The new Bose condensates of alkali atoms in magnetic traps [1][2][3] offer a unique way to investigate the low-lying collective excitations in Bose condensates [4][5][6][7][8][9][10][11][12][13][14][15][16]. Experimentally collective modes with a given symmetry have been excited by time-dependent modulations of the trapping potential, and their evolution has been followed in real time by measurements of the resulting shape oscillations of the condensates. The measurements performed so far have involved turning off the trap after a given time [4][5][6], but in the future they could even be performed non-destructively by elastic off-resonant light scattering [7]. Theoretically the collective modes have been analyzed by using the Bogoliubov-equations or by linearizing the time-dependent Gross-Pitaevskii equation around the time-independent condensate and solving these equations numerically [8,9] or analytically in various approximations [10][11][12][13][14][15][16]. Very good agreement between the numerical and the experimental results has been found.In a seminal paper Stringari [16] has shown how the coupled wave equations for the collective excitations are simplified in the hydrodynamic limit to become a single second-order wave equation for density waves, and he obtained analytical solutions for all its modes in spherically symmetric harmonic traps, and, remarkably, also for some of its modes in axially symmetric harmonic traps. The latter are particularly important, because all experiments have been performed with traps of this symmetry [4][5][6].In the present paper it is our goal to study in more detail by analytical means the hydrodynamic wave equation in the axially symmetric case. We wish to find an explanation why at least some analytical solutions have been possible in this case and intend to use this insight to construct more solutions in a systematic way.In principle the collective mode problem looks very different for isotropic and for axially symmetric traps: In the isotropic case the rotational symmetry ensures that angular momentum conservation gives two good quantum numbers, and therefore the wave equation is separable in spherical coordinates. For axial symmetry, however, only the axial component of angular momentum remains a good quantum number, besides the energy, and one may expect that the system, having three degrees of freedom, is not integrable. In fact, this expectation is born out for collect...

The mode frequencies of a weakly interacting Bose gas in a magnetic trap are studied as a function of the anisotropy of the trap. As in earlier works the generalized Hartree-Fock-Bogoliubov equations within the Popov approximation (HFB-Popov) are used for our calculations. The new feature of our work is the combined use of a mode expansion in a finite basis and a semiclassical approximation of the highly excited states. The results are applied to check the accuracy of the recently suggested equivalent zero-temperature condensate (EZC) approximation which involves a much simpler model. 03.75.Fi,05.30.Jp,67.40.Db

We present predictions for the temperature dependent shifts and damping rates. They are obtained by applying the dielectric formalism to a simple model of a trapped Bose gas. Within the framework of the model we use lowest order perturbation theory to determine the first order correction to the results of Hartree-Fock-Bogoliubov-Popov theory for the complex collective excitation frequencies, and present numerical results for the temperature dependence of the damping rates and the frequency shifts. Good agreement with the experimental values measured at JILA are found for the m = 2 mode, while we find disagreements in the shifts for m = 0. The latter point to the necessity of a non-perturbative treatment for an explanation of the temperature-dependence of the m=0 shifts. 03.75.Fi,05.30.Jp,67.40.Db