We present an extensive numerical study of ground-state properties of confined repulsively interacting fermions on one-dimensional optical lattices. Detailed predictions for the atom-density profiles are obtained from parallel Kohn-Sham density-functional calculations and quantum Monte Carlo simulations. The density-functional calculations employ a Bethe-Ansatz-based local-density approximation for the correlation energy, which accounts for Luttinger-liquid and Mott-insulator physics. Semi-analytical and fully numerical formulations of this approximation are compared with each other and with a cruder Thomas-Fermi-like local-density approximation for the total energy. Precise quantum Monte Carlo simulations are used to assess the reliability of the various localdensity approximations, and in conjunction with these allow to obtain a detailed microscopic picture of the consequences of the interplay between particle-particle interactions and confinement in one-dimensional systems of strongly correlated fermions.

We theoretically study a one-dimensional quasi-periodic Fermi system with topological p-wave superfluidity, which can be deduced from a topologically non-trivial tight-binding model on the square lattice in a uniform magnetic field and subject to a non-Abelian gauge field. The system may be regarded a non-Abelian generalization of the well-known Aubry-André-Harper model. We investigate its phase diagram as functions of the strength of the quasi-disorder and the amplitude of the p-wave order parameter, through a number of numerical investigations, including a multifractal analysis. There are four distinct phases separated by three critical lines, i.e., two phases with all extended wave-functions (I and IV), a topologically trivial phase (II) with all localized wavefunctions and a critical phase (III) with all multifractal wave-functions. The phase I is related to the phase IV by duality. It also seems to be related to the phase II by duality. Our proposed phase diagram may be observable in current cold-atom experiments, in view of simulating non-Abelian gauge fields and topological insulators/superfluids with ultracold atoms.

The Luther-Emery liquid is a state of matter that is predicted to occur in one-dimensional systems of interacting fermions and is characterized by a gapless charge spectrum and a gapped spin spectrum. In this Letter we discuss a realization of the Luther-Emery phase in a trapped cold-atom gas. We study by means of the density-matrix renormalization-group technique a twocomponent atomic Fermi gas with attractive interactions subject to parabolic trapping inside an optical lattice. We demonstrate how this system exhibits compound phases characterized by the coexistence of spin pairing and atomic-density waves. A smooth crossover occurs with increasing magnitude of the atom-atom attraction to a state in which tightly bound spin-singlet dimers occupy the center of the trap. The existence of atomic-density waves could be detected in the elastic contribution to the light-scattering diffraction pattern.

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Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional ͑Q1D͒ harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms.

Interacting two-component Fermi gases loaded in a one-dimensional ͑1D͒ lattice and subject to harmonic trapping exhibit intriguing compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a random potential on these ground-state phases. Within a densityfunctional scheme we show that disorder has two main effects: ͑i͒ it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and ͑ii͒ it induces an anomaly in the compressibility at low density from quenching of percolation.

We investigate the quench dynamics of a one-dimensional incommensurate lattice described by the Aubry-André model by a sudden change of the strength of incommensurate potential ∆ and unveil that the dynamical signature of localization-delocalization transition can be characterized by the occurrence of zero points in the Loschmit echo. For the quench process with quenching taking place between two limits of ∆ = 0 and ∆ = ∞, we give analytical expressions of the Loschmidt echo, which indicate the existence of a series of zero points in the Loschmidt echo. For a general quench process, we calculate the Loschmidt echo numerically and analyze its statistical behavior. Our results show that if both the initial and post-quench Hamiltonian are in extended phase or localized phase, Loschmidt echo will always be greater than a positive number; however if they locate in different phases, Loschmidt echo can reach nearby zero at some time intervals.

We develop the continuum mechanics of quantum many-body systems in the linear response regime. The basic variable of the theory is the displacement field, for which we derive a closed equation of motion under the assumption that the time-dependent wave function in a locally comoving reference frame can be described as a geometric deformation of the ground-state wave function. We show that this equation of motion is exact for systems consisting of a single particle, and for all systems at sufficiently high frequency, and that it leads to an excitation spectrum that has the correct integrated strength. The theory is illustrated by simple model applications to oneand two-electron systems.The dynamics of quantum many-particle systems, as displayed in electromagnetic transitions, chemical reactions, ionization and collision processes, poses a major challenge to computational physicists and chemists. Whereas the calculation of ground-state properties can be tackled by powerful computational methods such as the quantum Monte Carlo, [1] the development of similar methods for time-dependent properties has been slow. One of the most successful methods to date is the timedependent density functional theory (TDDFT), or its more recent version -time-dependent current density functional theory (TDCDFT).[2] In the common KohnSham implementation of this method [3,4] the formidable problem of solving the time-dependent Schrödinger equation for the many-body wave function is replaced by the much simpler problem of determining N single-particle orbitals. However, even this simplified problem is quite complex, and furthermore there are features such as multi-particle excitations [5] and dispersion forces [6] that are very difficult to treat within the conventional approximation schemes.An alternative approach, which actually dates back to the early days of quantum mechanics [7,8,9], attempts to calculate directly the collective variables of interestdensity and current. This approach we call "quantum continuum mechanics" (QCM), because in analogy with classical theories of continuous media it attempts to describe the quantum many-body system without explicit reference to the individual particles.[10]The possibility of a QCM formulation of the quantum many-body problem is guaranteed by the very same theorems that lay down the foundation of TDDFT and TDCDFT. [11,12] Let us consider a system of particles described by the time-dependent HamiltonianwhereĤ 0 =T +Ŵ +V 0 is the sum of kinetic energy (T ), interaction potential energy (Ŵ ), and the potential energy associated with an external static potential (V 0 ). n(r) is the particle density operator and V 1 (r, t) is an external time-dependent potential. The exact Heisenberg equation of motion for the current density operator, averaged over the quantum state, leads to the Euler equationHere m is the mass of the particles and repeated indices are summed over. The key quantity on the right hand side of Eq. (2) is the stress tensor P µν (r, t) -a symmetric tensor whose divergence yields th...