Perturbations of geodesic flows on surface of constant negative curvature and their mixing properties

We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the Hamilton-Jacobi equation (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, ) group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time (up to logarithms).Part of this work was performed while the first and third authors were in residence at the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455, USASupported by the Mathematics Dept. of Princeton University of by Stiftung Volkswagenwerk through IHES, and IMA     

Magnetic moment of a 2D electron gas with spin-orbit interaction

An explicit analytic expression is derived for the magnetic moment of a 2D electron gas taking into account the spin-orbit interaction in the Rashba model with T = 0. The cases of constant chemical potential and number of electrons are investigated. The magnetic field and temperature dependences of the magnetic moment are analyzed. The results are compared with the results of experimental studies of magnetization.     

Magnetic moment of a two-dimensional degenerate electron gas in mesoscopic samples

The orbital magnetism of two-dimensional electrons in mesoscopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquette, bracelet with hard wall confinement and also a confinement with a parabolic potential. We calculate the average magnetic moment which means an average with respect to size fluctuations and de Haas-van Alphen oscillations which arise in the case of a sharp Fermi cutoff. We see three distinct ranges in the magnetic field: (i) small field region where perturbation theory applies; (ii) moderate fields where edge currents play a prominent role; and (iii) the high field range with a Landau type susceptibility. In a quasiclassical picture, the electronic orbits are not qualitatively changed by a magnetic field in (i); skipping orbits are important in (ii); and in (iii), the cyclotron radius is smaller than the sample size. As a rule, we find an enhancement of the magnetic response which increases with k_FL, that is, with sample size divided by the Fermi wave length. Also, we have found out that the quasiclassical approximation fails in the calculation of the magnetic properties; on the other hand, we have seen no essential differences between the canonical ensemble (fixed particle number) and the grand canonical ensemble (chemical potential given). In the case of plaquettes, in particular for samples in the form of squares, we have found agreement with experimental results by Lévy, Reich, Pfeiffer and West.     

Magnetic field correlation tensor in spaces of constant curvature

The concept of statistical homogeneity and isotropy for vector fields in spatial sections of constant curvature was analyzed. Solenoidality conditions for a corresponding correlation tensor were obtained for positive and negative curvature. It was shown that these conditions differ from the corresponding condition for fields in Euclidean space.     

Darboux related quantum integrable systems on a constant curvature surface

We consider integrable deformations of the Laplace–Beltrami operator on a constant curvature surface, obtained through the action of first-order Darboux transformations. Darboux transformations are related to the symmetries of the underlying geometric space and lead to separable potentials which are related to the KdV equation. Eigenfunctions of the corresponding operators are related to highest weight representations of the symmetry algebra of the underlying space.     

New measurement of the electron magnetic moment and the fine structure constant

A measurement using a one-electron quantum cyclotron gives the electron magnetic moment in Bohr magnetons, g/2=1.001 159 652 180 73 (28) [0.28 ppt], with an uncertainty 2.7 and 15 times smaller than for previous measurements in 2006 and 1987. The electron is used as a magnetometer to allow line shape statistics to accumulate, and its spontaneous emission rate determines the correction for its interaction with a cylindrical trap cavity. The new measurement and QED theory determine the fine structure constant, with alpha{-1}=137.035 999 084 (51) [0.37 ppb], and an uncertainty 20 times smaller than for any independent determination of alpha.     

Casimir effect for gauge fields in spaces with negative constant curvature

We consider gauge theories based on abelian p-forms on real compact hyperbolic spaces. Using the zeta-function regularization method and the trace tensor kernel formula, we determine explicitly an expression for the vacuum energy (Casimir energy) corresponding to skew-symmetric tensor fields. It is shown that the topological component of the Casimir energy for co-exact forms on even-dimensional spaces, associated with the trivial character, is always negative. We infer on the possible cosmological consequences of this result.Received: 14 September 2004, Revised: 14 November 2004, Published online: 23 December 2004PACS: 04.70.Dy, 11.25.Mj     

The surface energy of an electron gas in model crystals

The surface energy of an electron gas in a crystal is considered. The results obtained for a quadratic spectrum are generalized to an arbitrary energy spectrum in certain crystal models. The surface energy of an electron gas with a quadratic spectrum is found for a sample with a rough boundary when the height of irregularities is small compared with the electron wavelength.     

Magnetic moment of relativistic quark gas in intense magnetic field

Magnetic properties of a system of noninteracting relativistic quark gas in presence of an external intense magnetic field has been calculated assuming that quarks obey parastatistics of order three. It is found, although the magnetization is greater for quark gas than that for normal hadronic system, it is unlikely that the spontaneous magnetization will take place in a quark star.     

Magnetic response of an electron gas in a quantum ring of non-zero width

An investigation of the magnetic moment of an electron gas in a quantum ring of non-zero width is made. Analytic expressions are obtained for the magnetic moment. For the magnetic moment of the system, the dependence on temperature and parameters of the ring are found and investigated in detail. De Haas–van Alphen and Aharonov–Bohm oscillations are investigated.     

Superintegrable systems on spaces of constant curvature

Construction and classification of two-dimensional (2D) superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of constant curvature and separable in the so-called geodesic polar coordinates are presented. The method proposed is applicable to any value of curvature including the case of Euclidean plane, sphere and hyperbolic plane. The main result is a generalization of Bertrand’s theorem on 2D spaces of constant curvature and covers most of the known separable and superintegrable models on such spaces (in particular, the so-called Tremblay–Turbiner–Winternitz (TTW) and Post–Winternitz (PW) models which have recently attracted some interest).     

Influence of the intrinsic magnetic moment of an electron on the filling of energy levels of a nonrelativistic degenerate electron gas in a quantizing magnetic field

The effect of a quantizing magnetic field on the filling of energy subbands of a degenerate nonrelativistic electron gas, whose magnetic moments are parallel and antiparallel to the direction of the field, is studied. Calculations are made under the assumption that the Fermi kinetic energy of the electron system increases as a result of Pauli's paramagnetism as compared to the kinetic energy calculated without taking this effect into account. Numerical values of the electron concentration are calculated as a function of the magnetic moment direction for magnetic fields under which the electrons are in states with given numbers of the Landau quantum level.A. S. Pushkin State Education Institute, Brest. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 9–13, June, 1992.     

Note on the correlation energy of an electron gas

The residual ring diagram contribution which is due to the use of approximate eigenvalues and a momentum cutoff is evaluated and the terms of orderr _s in the correlation energy are given explicitly. The result is exact to orderr _s within neglect of the third order exchange contribution and improves the results of Du Bois, and Carr and Maradudin. The correlation energy plotted againstr _s connects rather smoothly to the low density results obtained recently by Stevens and Pokrant based on an entirely different variational method.This work was supported by the National Science Foundation     

Crystallization of an electron gas

The ground state energy of an electron gas that is evaluated by the method of Padé approximants shows a singularity which may be associated with the formation of the Wigner lattice. The energy curve in the vicinity of this singularity is examined, and the pressure and the correlation energy are evaluated. Permanent address: Department of Physics, State University of New York at Buffalo, N.Y., U.S.A.     

Magnetic ratchet effects in a two-dimensional electron gas

The effect of the magnetic field on the generation of an electric current in a two-dimensional electronic ratchet is theoretically studied. Mechanisms of the formation of magnetically induced photocurrent are proposed for a structure with a two-dimensional electron gas (quantum well, graphene, or topological insulator) with a lateral asymmetric superlattice consisting of metallic strips on the external surface of the structure. The ratchet with the spatially oscillating magnetic field generated by the ferromagnetic lattice, as well as the nonmagnetic ratchet placed in the uniform magnetic field both classically weak and strong quantizing, is considered. It is established that the ratio of the amplitude of the magnetic oscillations of photocurrent to the ratchet photocurrent in zero field can exceed two orders of magnitude.     

Magnetic phase diagram in a quasi-two-dimensional electron gas

Using spin density functional theory within the framework of the local spin density approximation with Perdew-Zunger type exchange-correlation energy, ferromagnetism in a quasi-two-dimensional electron gas (Q-2DEG) is studied. The electronic and magnetic structures of a thin film are calculated as a function of film thickness and electron density. Ferromagnetism in the Q-2DEG is found to appear at a higher electron density than in the three-dimensional electron gas. Unless a film is very thin, with decreasing electron density, a magnetic phase transition occurs from a spin-unpolarized fluid to a Wigner film with surface magnetism, in which the spin polarization localizes only in the neighborhood of surfaces. Further decreasing density induces another transition to a fully spin-polarized ferromagnetic Wigner film.     

Magnetic moment of an acceptor center in cubic semiconductors

The magnetic field-induced splitting of the ground-state spin levels of an acceptor center, described by superposition of the Coulomb and the central-cell potentials, has been calculated for diamond-type. An analytic expression for the g factor obtained in the zero-radius potential approximation depends only on the light to heavy hole mass ratio. It is shown that the differences between the values of the g factor for the limiting cases of purely Coulomb and zero-radius potentials do not exceed 5%, thus permitting one to use this analytic expression for estimates. Fiz. Tverd. Tela (St. Petersburg) 39, 58–60 (January 1997)     

Imaging performance of an isotropic negative dielectric constant slab

The influence of material and thickness on the subwavelength imaging performance of a negative dielectric constant slab is studied. Resonance in the plane-wave transfer function produces a high spatial frequency ripple that could be useful in fabricating periodic structures. A cost function based on the plane-wave transfer function provides a useful metric to evaluate the planar slab lens performance, and using this, the optimal slab dielectric constant can be determined.     

The doubling of the anomalous magnetic moment of electron in a very strong constant homogeneous electric field

Calculated by the author previously [8], the anomalous magnetic moment (AMM) of the electron in an intense constant electric field changes nonmonotonically as the field increases, passing through a minimum and tending to the doubled Schwinger value for very strong fields. In the present paper, it is supposed that the AMM is related by the Lande factor to the angular momentum of a virtual electron accompanied by a virtual photon. This factor changes its effective value because of the influence of the external field on the motion of the virtual electron and its self-action. With increase of the electric field strength, the virtual electron can successively occupy the excited states l = 1, j = 1/2 and l = 1, j = 3/2 in addition to the original state with the orbital angular momentum l = 0 and the total angular momentum j = 1/2. The first of these excited states decreases the AMM and the second increases and doubles it if only this state is occupied for a very strong field. The latter condition is equivalent to the alignment of the spin and the orbital angular momentum of the electron along the field, while the total angular momentum of the entire system of the virtual electron and the virtual photon remains equal to 1/2.