Cyclotron toroidal braids: A cure for portrayal composite fermions on a torus

We present an inspection of the statistics of particles including composite fermions on a torus starting from a braid group analysis. For this purpose we considered a system of electrons confined to the surface of a torus under the influence of a strong magnetic field and interacting through a general rotational invariant potential. An explanation of the appearance of the cyclotron braids as an effect of restriction imposed by magnetic field on braid trajectories which in analyzed case reduces the full braid group to one of its subgroups (i.e. cyclotron subgroups), is given. The modified Feynman path-integral method is also reproduced with some minor enhancements. We improve known results concerning on braid groups on a torus in two directions: we obtain new estimates in terms of cyclotron braid subgroups and cyclotron band generator, respectively; we demonstrate that only multi-loop generators are accessible in the fractional quantum regime well and we also formally explain the unique statistic of composite fermions by construct trial wave function for the system on a torus, based on this idea. The topological oddness of torus geometry can be driven by shifting of electrons between the two different group of generators allowed for an explanation in satisfactory accordance the both compact commensurability condition and some numerical calculations in toroidal geometry. Besides, our approach may shed some new light on few interesting aspects in better understanding the fractional quantum Hall effect.     

Inhomogeneous atomic Bose-Fermi mixtures in cubic lattices

We determine the ground state properties of inhomogeneous mixtures of bosons and fermions in cubic lattices and parabolic confining potentials. For finite hopping we determine the domain boundaries between Mott-insulator plateaux and hopping-dominated regions for lattices of arbitrary dimension within mean-field and perturbation theory. The results are compared with a new numerical method that is based on a Gutzwiller variational approach for the bosons and an exact treatment for the fermions. The findings can be applied as a guideline for future experiments with trapped atomic Bose-Fermi mixtures in optical lattices.     

Analytical models for valence fermions in isotropic traps

For isotropic confining Ioffe-Pritchard or TOP potentials, a valence fermion trapped with a closed core of other fermions can be described by an analytical effective one-particle model with a physical eigenspectrum. Related constructions exist for Paul and Penning traps. The analytical models arise from quantum-mechanical supersymmetry.     

Fermionic Zero Modes in Self-dual Vortex Background on a Torus

We study fermionic zero modes in the self-dual vortex background on an extra two-dimensional Riemann surface in （5＋1） dimensions. Using the generalized Abelian-Higgs model, we obtain the inner topological structure of the self-dual vortex and establish the exact self-duality equation with topological term. Then we analyze the Dirac operator on an extra torus and the effective Lagrangian of four-dimensional fermions with the self-dual vortex background. Solving the Dirac equation, the fermionic zero modes on a torus with the self-dual vortex background in two simple cases are obtained.     

Unitary quantum three-body problem in a harmonic trap

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.     

Trapped fermions with density imbalance in the Bose-Einstein condensate limit

We analyze the effects of imbalancing the populations of two-component trapped fermions, in the Bose-Einstein condensate limit of the attractive interaction between different fermions. Starting from the gap equation with two fermionic chemical potentials, we derive a set of coupled equations that describe composite bosons and excess fermions. We include in these equations the processes leading to the correct dimer-dimer and dimer-fermion scattering lengths. The coupled equations are then solved in the Thomas-Fermi approximation to obtain the density profiles for composite bosons and excess fermions, which are relevant to the recent experiments with trapped fermionic atoms.     

Majorana fermions in equilibrium and in driven cold-atom quantum wires

We introduce a new approach to create and detect Majorana fermions using optically trapped 1D fermionic atoms. In our proposed setup, two internal states of the atoms couple via an optical Raman transition-simultaneously inducing an effective spin-orbit interaction and magnetic field-while a background molecular BEC cloud generates s-wave pairing for the atoms. The resulting cold-atom quantum wire supports Majorana fermions at phase boundaries between topologically trivial and nontrivial regions, as well as "Floquet Majorana fermions" when the system is periodically driven. We analyze experimental parameters, detection schemes, and various imperfections.     

Dirac returns: Non-Abelian statistics of vortices with Dirac fermions

Topological superconductors classified as type D admit zero-energy Majorana fermions inside vortex cores, and consequently the exchange statistics of vortices becomes non-Abelian, giving a promising example of non-Abelian anyons. On the other hand, types C and DIII admit zero-energy Dirac fermions inside vortex cores. It has been long believed that an essential condition for the realization of non-Abelian statistics is non-locality of Dirac fermions made of two Majorana fermions trapped inside two well-separated vortices as in the case of type D. Contrary to this conventional wisdom, however, we show that vortices with local Dirac fermions also obey non-Abelian statistics.     

Atom interferometry with trapped fermi gases

We realize an interferometer with an atomic Fermi gas trapped in an optical lattice under the influence of gravity. The single-particle interference between the eigenstates of the lattice results in macroscopic Bloch oscillations of the sample. The absence of interactions between fermions allows a time-resolved study of many periods of the oscillations, leading to a sensitive determination of the acceleration of gravity. The experiment proves the superiority of noninteracting fermions with respect to bosons for precision interferometry and offers a way for the measurement of forces with microscopic spatial resolution.     

Graphene-like physics in optical lattices

Graphene has attracted enormous attention over the past years in condensed matter physics. The most interesting feature of graphene is that its low-energy excitations are relativistic Dirac fermions. Such feature is the origin of many topological properties in graphene-like physics. On the other hand, ultracold quantum gas trapped in an optical lattice has become a unique setting for quantum simulation of condensed matter physics. Here, we mainly review our recent work on quantum simulation of graphene-like physics with ultracold atoms trapped in a honeycomb or square optical lattice, including the simulation of Dirac fermions and quantum Hall effect with and without Landau levels. We also present the related experimental advances.     

Neutral excitation and bulk gap of fractional quantum Hall liquids in disk geometry

For the numerical simulation of the fractional quantum Hall(FQH) effects on a finite disk, the rotational symmetry is the only symmetry that is used in diagonalizing the Hamiltonian. In this work, we propose a method of using the weak translational symmetry for the center of mass of the many-body system. With this approach, the bulk properties, such as the energy gap and the magneto-roton excitation are consistent with those in the closed manifolds like the sphere and torus. As an application, we consider the FQH phase and its phase transition in the fast rotated dipolar fermions. We thus demonstrate the disk geometry having versatility in analyzing the bulk properties beside the usual edge physics.     

Phase Diagram and Phase Separation of a Trapped Interacting Bose-Fermi Gas Mixture

In six different regimes for a spatial phase diagram of a trapped interacting Bose-Fermi gas mixture at low temperatures, we present the conditions for the spatial demixing and separation of bosons and fermions. Starting from a semiclassically thermodynamic model for the local density functional of thermal bosons and fermions,the explicit analytical expressions for the fugacities of bosons and fermions are derived in different regimes by means of a first-order perturbation method in a local-density approximation. The critical values of the fermionboson interaction strength as a function of the fractional composition of fermions have a general feature: increase,extreme and decrease with increasing the fermionic composition slightly above Bose-Einstein critical temperature.     

Persistent currents for Coulomb interacting electrons on 2d disordered lattices

A Wigner crystal structure of the electronic ground state is induced by strong Coulomb interactions at low temperature in clean or disordered two-dimensional (2d) samples. For fermions on a mesoscopic disordered 2d lattice, being closed to a torus, we study the persistent current in the regime of strong interaction at zero temperature. We perform a perturbation expansion starting from the Wigner crystal limit which yields power laws for the dependence of the persistent current on the interaction strength. The sign of the persistent current in the strong interaction limit is independent of the disorder realization and strength. It depends only on the electro-statically determined configuration of the particles in the Wigner crystal. Received 14 March 2000     

Confinement and superfluidity in one-dimensional degenerate fermionic cold atoms

The physical properties of arbitrary half-integer spins F = N - (1/2) fermionic cold atoms trapped in a one-dimensional optical lattice are investigated by means of a low-energy approach. Two different superfluid phases are found for F > or = (3/2) depending on whether a discrete symmetry is spontaneously broken or not: an unconfined BCS pairing phase and a confined molecular-superfluid instability made of 2N fermions. We propose an experimental distinction between these phases for a gas trapped in an annular geometry. The confined-unconfined transition is shown to belong to the Z(N) generalized Ising universality class. We discuss the possible Mott phases at (1/2) filling.     

Static and dynamic properties of trapped fermionic Tonks-Girardeau gases

We investigate some static and dynamic properties of harmonically trapped fermionic Tonks-Girardeau gases in tight de Broglie waveguides with attractive p-wave interactions induced by a Feshbach resonance. An exact solution for the one-body density matrix is analyzed in terms of its natural orbitals, with the surprising result that for odd, but not for even, numbers of fermions the maximally occupied orbital coincides with the ground harmonic oscillator orbital. A dynamical consequence of this is that when the interactions are turned off the on-axis density remains a maximum for an odd number of fermions, whereas a minimum develops for an even number.     

Expansion of an interacting fermi gas

We study the expansion of a dilute ultracold sample of fermions initially trapped in an anisotropic harmonic trap. The expansion of the cloud provides valuable information about the state of the system and the role of interactions. In particular, the time evolution of the deformation of the expanding cloud behaves quite differently depending on whether the system is in the normal or in the superfluid phase. For the superfluid phase, we predict an inversion of the deformation of the sample, similar to what happens with Bose-Einstein condensates. Vice versa, in the normal phase, the inversion of the aspect ratio is never achieved, if the mean field interaction is attractive and collisions are negligible.     

Multicomponent Strongly Interacting Few-Fermion Systems in One Dimension

The paper examines a trapped one-dimensional system of multicomponent spinless fermions that interact with a zero-range two-body potential. We show that when the repulsion between particles is very large the system can be approached analytically. To illustrate this analytical approach we consider a simple system of three distinguishable particles, which can be addressed experimentally. For this system we show that for infinite repulsion the energy spectrum is sixfold degenerate. We also show that this degeneracy is partially lifted for finitely large repulsion for which we find and describe corresponding wave functions.     

Study of the interaction between two electrons in the single band Hubbard model

A single-band Hubbard model has been investigated for two case (1) the 1-D Hubbard ring, (2) the 2-D Hubbard square lattice on a torus. In both cases it is found that the interaction between two electrons is always repulsive in the limit of infiniteU. In 1-D, the pair correlation function is naturally similar to that of spin-less fermions, whilst in 2-D it is quite different.     

Trapping of neutral mercury atoms and prospects for optical lattice clocks

We report vapor-cell magneto-optical trapping of Hg isotopes on the (1)S(0)-(3)P(1) intercombination transition. Six abundant isotopes, including four bosons and two fermions, were trapped. Hg is the heaviest nonradioactive atom trapped so far, which enables sensitive atomic searches for "new physics" beyond the standard model. We propose an accurate optical lattice clock based on Hg and evaluate its systematic accuracy to be better than 10;{-18}. Highly accurate and stable Hg-based clocks will provide a new avenue for the research of optical lattice clocks and the time variation of the fine-structure constant.