Fermi Acceleration in Anti-Integrable Limits of the Standard Map

We consider a dynamical system on the semi-infinite cylinder which models the high energy dynamics of a family of mechanical models. We provide conditions under which we ensure that the set of orbits undergoing Fermi acceleration has measure zero.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Persistently expansive geodesic flows

We prove thatC ¹-persistently expansive geodesic flows of compact, boundaryless Riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set. In the case of compact surfaces we deduce that the geodesic flow isC ¹-persistently expansive if and only if it is an Anosov flow.     

Thermodynamics of Small Fermi Systems: Quantum Statistical Fluctuations

We investigate the probability distribution of the quantum fluctuations of thermodynamic functions of finite, ballistic, phase-coherent Fermi gases. Depending on the chaotic or integrable nature of the underlying classical dynamics, on the thermodynamic function considered, and on temperature, we find that the probability distributions are dominated either (i) by the local fluctuations of the single-particle spectrum on the scale of the mean level spacing, or (ii) by the long-range modulations of that spectrum produced by the short periodic orbits. In case (i) the probability distributions are computed using the appropriate local universality class, uncorrelated levels for integrable systems, and random matrix theory for chaotic ones. In case (ii) all the moments of the distributions can be explicitly computed in terms of periodic orbit theory and are system-dependent, nonuniversal, functions. The dependence on temperature and on number of particles of the fluctuations is explicitly computed in all cases, and the different relevant energy scales are displayed.     

On the Periodic Orbits of the Static,Spherically Symmetric Einstein-Yang-Mills Equations

In this paper we analyze the existence of the periodic orbits of the static, spherically symmetric Einstein–Yang–Mills Equations by using the qualitative theory of the ordinary differential equation. We prove that there are no periodic orbits restricted to some invariant set of codimension 1. Furthermore if there is a periodic orbit out of this invariant set, then there must be other periodic orbits, which are symmetric to the first one. We also have results on the non–existence of periodic orbits when the cosmological constant is negative.     

Magnetic breakdown induced peierls transition

We predict a new type of phase transition in a quasi-one-dimensional system of interacting electrons at high magnetic fields, the stabilization of a density wave which transforms a two-dimensional open Fermi surface into a periodic chain of large pockets with small distances between them. We show that quantum tunneling of electrons between the neighboring closed orbits enveloping these pockets transforms the electron spectrum into a set of extremely narrow energy bands and gaps that decreases the total electron energy, thus leading to a magnetic breakdown induced density wave ground state analogous to the well-known instability of the Peierls type.     

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.     

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.     

Noise can reduce disorder in chaotic dynamics

We evoke the idea of representation of the chaotic attractor by the set of unstable periodic orbits and disclose a novel noise-induced ordering phenomenon. For long unstable periodic orbits forming the strange attractor the weights (or natural measure) is generally highly inhomogeneous over the set, either diminishing or enhancing the contribution of these orbits into system dynamics. We show analytically and numerically a weak noise to reduce this inhomogeneity and, additionally to obvious perturbing impact, make a regularizing influence on the chaotic dynamics. This universal effect is rooted into the nature of deterministic chaos.     

Exact solution of the one-dimensional deterministic fixed-energy sandpile

By reason of the strongly nonergodic dynamical behavior, universality properties of deterministic fixed-energy sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the nonergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction's basins of the different periodic orbits lead to a complete characterization of the activity vs energy density phase diagram in the limit of large system's size.     

Fractionalization and Fermi-surface volume in heavy-fermion compounds: the case of YbRh2Si2

We establish an effective theory for heavy-fermion compounds close to a zero temperature antiferromagnetic (AFM) transition. Coming from the heavy Fermi liquid phase across to the AFM phase, the heavy electron fractionalizes into a light electron, a bosonic spinon, and a new excitation: a spinless fermionic field. Assuming this field acquires dynamics and dispersion when one integrates out the high energy degrees of freedom, we give a scenario for the volume of its Fermi surface through the phase diagram. We apply our theory to the special case of YbRh2(Si1-xGex)2 where we recover, within experimental resolution, several low temperature exponents for transport and thermodynamics.     

Generalization of Lagrangian dynamics

We speculate on a generalized dynamics described by an integral over action functionals that is a generalization of the standard functional integral. In a simple Gaussian case we obtain a certain differential equation for the measure of Feynman integral. We prove that the equation is satisfied for the spin zero field in one space-time dimension.     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Casimir interaction among objects immersed in a fermionic environment

Using ensembles of two, three, and four spheres immersed in a fermionic background we evaluate the (integrated) density of states and the Casimir energy. We thus infer that for sufficiently smooth objects, whose various geometric characteristic lengths are larger then the Fermi wave length one can use the simplest semiclassical approximation (the contribution due shortest periodic orbits only) to evaluate the Casimir energy. We also show that the Casimir energy for several objects can be represented fairly accurately as a sum of pairwise Casimir interactions between pairs of objects.     

Escape dynamics in collinear atomic-like three mass point systems

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise interaction. Specifically, we focus on the asymptotic behaviour for systems with non-negative total energy.On the zero energy level set there are two distinct asymptotic states, called 1+1+1escape configurations, where all the three separations infinitely increase as t→∞. We show that 1+1+1 escapes are improbable by proving that the set of initial conditions leading to such asymptotic configurations has zero Lebesgue measure. When the outer mass points are of the same kind we deduce the existence of a heteroclinic orbit connecting the 1+1+1 escape configurations. We further prove that this orbit is stable under parameter perturbation.In the positive energies’ case, we show that the set of initial conditions leading to 1+1+1 escape configurations has positive Lebesgue measure.     

Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems

In this paper we study the dynamics near resonant elliptic equilibria in three-degree-of-freedom Hamiltonian systems. The resonances we consider have multiplicity two, and the corresponding local normal form for the equilibrium is integrable at cubic order. We prove the existence of families of 3-tori and whiskered 2-tori with nearby chaotic dynamics in the quartic normal form. The whiskers of the 2-tori intersect in a non-trivial way giving rise to multi-pulse homoclinic and heteroclinic connections. These connections survive in the full system as orbits homoclinic to invariant 3-spheres.     

Electronic structure of La (0001) thin films on W (110) studied by photoemission spectroscopy and first principle calculations

Surface states that have a dz2 symmetry around the center of the surface Brillouin zone(BZ)have been regarded common in closely-packed surfaces of rare-earth metals.In this work,we report the electronic structure of dhcp La(0001)thin films by ultrahigh energy resolution angle-resolved photoemission spectroscopy(ARPES)and first principle calculations.Our first principle analysis is based on the many-body approach,therefore,density function theory(DFT)combined with dynamic mean-field theory(DMFT).The experimentally observed Fermi surface topology and band structure close to the Fermi energy qualitatively agree with first principle calculations when using a renormalization factor of between 2 and 3 for the DFT bands.Photon energy dependent ARPES measurements revealed clear kZ dependence for the hole-like band around the BZ center,previously regarded as a surface state.The obtained ARPES results and theoretical calculations suggest that the major bands of dhcp La(0001)near the Fermi level originate from the bulk La 5d orbits as opposed to originating from the surface states.     

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of 3-dimensional billiards. Interestingly, we find that for some track billiards, the mechanism generating hyperbolicity is not the defocusing one, which requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.     

Drift of slow variables in slow-fast Hamiltonian systems

We study the drift of slow variables in a slow-fast Hamiltonian system with several fast and slow degrees of freedom. Keeping the slow variables frozen, for any periodic trajectory of the fast subsystem we define an action. For a family of periodic orbits, the action is a scalar function of the slow variables and can be considered as a Hamiltonian function which generates some slow dynamics. These dynamics depend on the family of periodic orbits.Assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.