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Register a new userDensity profile of interacting Fermions in a one-dimensional optical trap
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The density distribution of the one-dimensional Hubbard model in a harmonic trapping potential is investigated in order to study the effect of the confining trap. Strong superimposed oscillations are always present on top of a uniform density cloud, which show universal scaling behavior as a function of increasing interactions. An analytical formula is proposed on the basis of bosonization, which describes the density oscillations for all interaction strengths. The wavelength of the dominant oscillation changes with interaction, which indicates the crossover to a spin-incoherent regime. Using the Bethe ansatz the shape of the uniform fermion cloud is analyzed in detail, which can be described by a universal scaling form.
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We outline a procedure for using matrix mechanics to compute energy eigenvalues and eigenstates for two and three interacting particles in a confining trap, in one dimension. Such calculations can bridge a gap in the undergraduate physics curriculum between single-particle and many-particle quantum systems, and can also provide a pathway from standard quantum mechanics course material to understanding current research on cold-atom systems. In particular we illustrate the notion of fermionization and how it occurs not only for the ground state in the presence of strong repulsive interactions, but also for excited states, in both the strongly attractive and strongly repulsive regimes.
Mott insulators with both spin and orbital degeneracy are pertinent to a large number of transition metal oxides. The intertwined spin and orbital fluctuations can lead to rather exotic phases such as quantum spin-orbital liquids. Here we consider two-component (spin 1/2) fermionic atoms with strong repulsive interactions on the $p$-band of the optical square lattice. We derive the spin-orbital exchange for quarter filling of the $p$-band when the density fluctuations are suppressed, and show it frustrates the development of long range spin order. Exact diagonalization indicates a spin-disordered ground state with ferro-orbital order. The system dynamically decouples into individual Heisenberg spin chains, each realizing a Luttinger liquid accessible at higher temperatures compared to atoms confined to the $s$-band.
We compute exactly the average spatial density for $N$ spinless noninteracting fermions in a $2d$ harmonic trap rotating with a constant frequency $Omega$ in the presence of an additional repulsive central potential $gamma/r^2$. We find that, in the large $N$ limit, the bulk density has a rich and nontrivial profile -- with a hole at the center of the trap and surrounded by a multi-layered wedding cake structure. The number of layers depends on $N$ and on the two parameters $Omega$ and $gamma$ leading to a rich phase diagram. Zooming in on the edge of the $k^{rm th}$ layer, we find that the edge density profile exhibits $k$ kinks located at the zeroes of the $k^{rm th}$ Hermite polynomial. Interestingly, in the large $k$ limit, we show that the edge density profile approaches a limiting form, which resembles the shape of a propagating front, found in the unitary evolution of certain quantum spin chains. We also study how a newly formed droplet grows in size on top of the last layer as one changes the parameters.
We simulate a balanced attractively interacting two-component Fermi gas in a one-dimensional lattice perturbed with a moving potential well or barrier. Using the time-evolving block decimation method, we study different velocities of the perturbation and distinguish two velocity regimes based on clear differences in the time evolution of particle densities and the pair correlation function. We show that, in the slow regime, the densities deform as particles are either attracted by the potential well or repelled by the barrier, and a wave front of hole or particle excitations propagates at the maximum group velocity. Simultaneously, the initial pair correlations are broken and coherence over different sites is lost. In contrast, in the fast regime, the densities are not considerably deformed and the pair correlations are preserved.
We evaluate the dynamic structure factor $S(q,omega)$ of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of $S(q,omega)$. The sharp peak $Spropto qdelta(omega-uq)$, characteristic for the Tomonaga-Luttinger model, broadens up; $S(q,omega)$ for a fixed $q$ becomes finite at arbitrarily large $omega$. The main spectral weight, however, is confined to a narrow frequency interval of the width $deltaomegasim q^2/m$. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number $q$.
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