Relaxation and localization in interacting quantum maps

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems, for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but-quantally shows resonant transport, no saturation, and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.     

Going beyond the self-consistent mean-field with the symmetry-restored generator coordinate method

Starting from successful self-consistent mean-field models, this paper discusses why and how to go beyond the mean field approximation. To include long-range correlations from fluctuations in collective degrees of freedom, one has to consider symmetry restoration and configuration mixing, which give access to ground-state correlations and spectroscopy.     

Reconciling phase diffusion and Hartree–Fock approximation in condensate systems

Despite the weakly interacting regime, the physics of Bose–Einstein condensates is widely affected by particle–particle interactions. They determine quantum phase diffusion, which is known to be the main cause of loss of coherence. Studying a simple model of two interacting Bose systems, we show how to predict the appearance of phase diffusion beyond the Bogoliubov approximation, providing a self-consistent treatment in the framework of a generalized Hartree–Fock–Bogoliubov perturbation theory.     

Self-consistent chaotic transport in fluids and plasmas

Self-consistent chaotic transport is the transport of a field F by a velocity field v according to an advection-diffusion equation in which there is a dynamical constrain between the two fields, i.e., O(F,v)=0 where O is an integral or differential operator, and the Lagrangian trajectories of fluid particles exhibit sensitive dependence on initial conditions. In this paper we study self-consistent chaotic transport in two-dimensional incompressible shear flows. In this problem F is the vorticity zeta, the corresponding advection-diffusion equation is the vorticity equation, and the self-consistent constrain is the vorticity-velocity coupling z nabla xv=zeta. To study this problem we consider three self-consistent models of intermediate complexity between the simple but limited kinematic chaotic advection models and the approach based on the direct numerical simulation of the Navier-Stokes equation. The first two models, the vorticity defect model and the single wave model, are constructed by successive simplifications of the vorticity-velocity coupling. The third model is an area preserving self-consistent map obtained from a space-time discretization of the single wave model. From the dynamical systems perspective these models are useful because they provide relatively simple self-consistent Hamiltonians (streamfunctions) for the Lagrangian advection problem. Numerical simulations show that the models capture the basic phenomenology of shear flow instability, vortex formation and relaxation typically observed in direct numerical simulations of the Navier-Stokes equation. Self-consistent chaotic transport in electron plasmas in the context of kinetic theory is also discussed. In this case F is the electron distribution function in phase space, the corresponding advection equation is the Vlasov equation and the self-consistent constrain is the Poisson equation. This problem is closely related to the vorticity problem. In particular, the vorticity defect model is analogous to the Vlasov-Poisson model and the single wave model and the self-consistent map apply equally to both plasmas and fluids. Also, the single wave model is analogous to models used in the study of globally coupled oscillator systems. (c) 2000 American Institute of Physics.     

Evidence for a metallic–like state in the T=0 K phase diagram of a high temperature superconductor

We examine the effects of a phenomenological pseudogap on the T=0 K phase diagram of a high temperature superconductor within a self-consistent model which exhibits a d-wave pairing symmetry. At the mean-field level the presence of a pseudogap in the normal phase of the high temperature superconductor is proved to be essential for the existence of a metallic–like state in the density versus interaction phase diagram. In the small density limit, at high attractive interaction, bosonic–like degrees of freedom are likely to emerge. Our result should be relevant for underdoped high temperature superconductors, where there is a strong evidence for the presence of a pseudogap in the excitation spectrum of the normal state quasiparticles.     

Quantum phase transition in capacitively coupled double quantum dots

We investigate two equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. With increasing interdot coupling, a rich range of behavior is uncovered: first a crossover from spin- to charge-Kondo physics, via an intermediate SU(4) state with entangled spin and charge degrees of freedom, followed by a quantum phase transition of Kosterlitz-Thouless type to a non-Fermi-liquid "charge-ordered" phase with finite residual entropy and anomalous transport properties. Physical arguments and numerical renormalization group methods are employed to obtain a detailed understanding of the problem.     

Robust transport barriers resulting from strong Kolmogorov-Arnold-Moser stability

Hamiltonian systems that locally violate the twist condition arise in many applications. Numerical simulations reveal that, when systems of this type are perturbed, the degenerate or nontwist tori are remarkably stable. This phenomenon, which we refer to as strong Kolmogorov-Arnold-Moser (KAM) stability, is shown to be linked to very small resonance widths near degenerate tori. Quantitative estimates of degenerate resonance widths are derived and bifurcations of degenerate resonances are described. Strong KAM stability leads to robust transport barriers, which are important in all of the many applications in which Hamilitonians with the nontwist property arise.     

High-Moment-Low-Moment Description of Magnetovolume Effects in Y(Mn x Al 1− x ) 2 and Y x Sc 1− x Mn 2

Based on the assumption of a high-moment-low-moment instability of the Mn atom, we construct a simple spin model with coupled magnetic and spatial degrees of freedom to describe the Laves phase systems Y(Mn x Al 1 m x ) 2 and Y x Sc 1 m x Mn 2 . Monte Carlo simulations of this model qualitatively reproduce anomalies observed in these materials like a discontinuous giant volume change and anomalous thermal expansion behavior.     

Magnetic-field probing of an SU(4) Kondo resonance in a single-atom transistor

Semiconductor devices have been scaled to the point that transport can be dominated by only a single dopant atom. As a result, in a Si fin-type field effect transistor Kondo physics can govern transport when one electron is bound to the single dopant. Orbital (valley) degrees of freedom, apart from the standard spin, strongly modify the Kondo effect in such systems. Owing to the small size and the s-like orbital symmetry of the ground state of the dopant, these orbital degrees of freedom do not couple to external magnetic fields which allows us to tune the symmetry of the Kondo effect. Here we study this tunable Kondo effect and demonstrate experimentally a symmetry crossover from an SU(4) ground state to a pure orbital SU(2) ground state as a function of magnetic field. Our claim is supported by theoretical calculations that unambiguously show that the SU(2) symmetric case corresponds to a pure valley Kondo effect of fully polarized electrons.     

Renormalization and transition to chaos in area preserving nontwist maps

The problem of transition to chaos, i.e. the destruction of invariant circles or KAM (Kolmogorov-Arnold-Moser) curves, in area preserving nontwist maps is studied within the renormalization group framework. Nontwist maps are maps for which the twist condition is violated along a curve known as the shearless curve.In renormalization language this problem is that of finding and studying the fixed points of the renormalization group operator that acts on the space of maps. A simple period-two fixed point of , whose basin of attraction contains the nontwist maps for which the shearless curve exists, is found. Also, a critical period-12 fixed point of , with two unstable eigenvalues, is found. The basin of attraction of this critical fixed point contains the nontwist maps for which the shearless curve is at the threshold of destruction. This basin defines a new universality class for the transition to chaos in area preserving maps.     

A kinetic theory and benchmark predictions for polymer-dispersed, semi-flexible macromolecular rods or platelets

The goals of this paper are: to present a mean-field kinetic theory for the hydrodynamics of macromolecular high aspect ratio rods or platelets dispersed in a polymeric solvent; and, to apply the formalism to predict the impact due to a polymeric versus viscous solvent on the classical Onsager isotropic-nematic equilibrium phase diagram and on the monodomain response to imposed steady shear. The kinetic theory coupling between the nanoscale rods or platelets and the polymeric solvent is incorporated through a mean-field potential that reflects the enormous particle-polymer surface area and the particle-polymer interactions across this interfacial area. To determine predictions of this theory on the equilibrium and sheared monodomain phase diagrams, we present a reduction procedure which approximates the coupled Smoluchowski equations for the polymer chain probability distribution function (PDF) and the nano-particle orientational PDF in favor of a coupled system of equations for the rank 2 second-moment tensors for each PDF. The reduced model consists of an 11-dimensional dynamical system, which we solve using continuation software (AUTO) to predict the modified Onsager equilibrium phase diagram and the modified Doi-Hess shear phase diagram due to the physics of polymer-particle surface interactions.     

Semiclassical theory of spin-orbit interactions using spin coherent states

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.     

Meanders and reconnection-collision sequences in the standard nontwist map

New global periodic orbit collision and separatrix reconnection scenarios exhibited by the standard nontwist map are described in detail, including exact methods for determining reconnection thresholds, methods that are implemented numerically. Results are compared to a parameter space breakup diagram for shearless invariant curves. The existence of meanders, invariant tori that are not graphs, is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space. Implications for transport are discussed.     

An approach for characterizing coupling in dynamical systems

The study of coupling in dynamical systems dates back to Christian Hyugens who, in 1665, discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other. In that case the observed synchronous motion was out of phase. In this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems. The measure is based on quantifying the active degrees of freedom (e.g. correlation dimension) of the coupled system and the constituent subsystems. The time-delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system. We use the coupled Lorenz, Rossler and Hénon model systems with a coupling strength variable for evaluation of the proposed approach. Results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied. Furthermore, using this approach the direction of coupling can be determined.     

Stochastically Stable Globally Coupled Maps with Bistable Thermodynamic Limit

We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium.     

The coupling of valence shell and particle-hole degrees of freedom in a partial random phase approximation

It is well known that the random phase approximation breaks down in the absence of a substantial energy gap between occupied and unoccupied single-particle states. Particle-hole excitations are then inevitably accompanied by substantial rearrangements of the particles in the neighbourhood of the Fermi surface. To accommodate this situation, a partial RPA is introduced which corresponds to replacing only the particle-hole degrees of freedom by bosons but leaving the valence space degrees of freedom intact. The PRPA is therefore a mapping of the many-fermion dynamics into the dynamics of a coupled boson-valence space. In application of the PRPA, algebraic methods, of either a fermionic or Lie algebra type, can be introduced, if desired, to facilitate the treatment of the valence space degrees of freedom. Results of applications are presented in which the valence space particles are treated in the rotational and SU(3) models, and are coupled strongly to giant dipole and quadrupole resonances.     

Nonlinear Topological Effects in Optical Coupled Hexagonal Lattice

Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson–hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.