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arXiv:cond-mat/0311206 · Journal of Physics B Atomic Molecular and Optical Physics

A density-functional approach to fermionization in the 1D Bose gas

Joachim Brand

Published Mar 24, 2004• 27 citations
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Abstract

A time-dependent Kohn–Sham scheme for 1D bosons with contact interaction is derived based on a model of spinor fermions. This model is specifically designed for the study of the strong interaction regime close to the Tonks gas. It allows us to treat the transition from the strongly interacting Tonks–Girardeau to the weakly interacting quasicondensate regime and provides an intuitive picture of the extent of fermionization in the system. An adiabatic local-density approximation is devised for the study of time-dependent processes. This scheme is shown to yield not only accurate ground-state properties but also overall features of the elementary excitation spectrum, which is described exactly in the Tonks-gas limit.

DOI
10.1088/0953-4075/37/7/073

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A time-dependent Kohn–Sham scheme for 1D bosons with contact interaction is derived based on a model of spinor fermions. This model is specifically designed for the study of the strong interaction regime close to the Tonks gas. It allows us to treat the transition from the strongly interacting Tonks–Girardeau to the weakly interacting quasicondensate regime and provides an intuitive picture of the extent of fermionization in the system. An adiabatic local-density approximation is devised for the study of time-dependent processes. This scheme is shown to yield not only accurate ground-state properties but also overall features of the elementary excitation spectrum, which is described exactly in the Tonks-gas limit.
While the one-dimensional (1D) Bose gas appears to exhibit superfluid response under certain conditions, it fails the Landau criterion according to the elementary excitation spectrum calculated by Lieb. The apparent riddle is solved by calculating the dynamic structure factor of the Lieb-Liniger 1D Bose gas. In a dual representation, strongly interacting bosons are mapped into weakly interacting fermions, which are treated in the Hartree-Fock and generalized random-phase approximations. The dynamic structure factor is calculated analytically in this approximation, which is valid to first order in $1∕\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is the dimensionless interaction strength of the model. The results clearly indicate a suppression of superfluidity-breaking umklapp excitations near the Tonks-gas limit, which should be observable by Bragg scattering in current experiments.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
We investigate local quantum field theories for one-dimensional (1D) Bose and Fermi gases with contact interactions, which are closely connected with each other by Girardeau's Bose-Fermi mapping. While the Lagrangian for bosons includes only a two-body interaction, a marginally relevant three-body interaction term is found to be necessary for fermions. Because of this three-body coupling, the three-body contact characterizing a local triad correlation appears in the energy relation for fermions, which is one of the sum rules for a momentum distribution. In addition, we apply in both systems the operator product expansion to derive large-energy and momentum asymptotics of a dynamic structure factor and a single-particle spectral density. These behaviors are universal in the sense that they hold for any 1D scattering length at any temperature. The asymptotics for the Tonks-Girardeau gas, which is a Bose gas with a hardcore repulsion, as well as the Bose-Fermi correspondence in the presence of three-body attractions are also discussed.
Building on the recent experimental achievements obtained with scanning electron microscopy on ultracold atoms, we study one-dimensional Bose gases in the crossover between the weakly (quasicondensate) and the strongly interacting (Tonks-Girardeau) regime. We measure the temporal two-particle correlation function and compare it with calculations performed using the time-evolving block decimation algorithm. More pronounced antibunching is observed when entering the more strongly interacting regime. Even though this mimics the onset of a fermionic behavior, we highlight that the exact and simple duality between one-dimensional bosons and fermions does not hold when such a dynamical response is probed. The onset of fermionization is also reflected in the density distribution, which we measure in situ to extract the relevant parameters and to identify the different regimes. Our results show agreement between experiment and theory and give insight into the dynamics of strongly correlated many-body systems.
Two of the most popular quantum mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a 1D double-well potential, a model of atoms in a quasi-1D optical lattice or electrons of a Hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly-correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density functional theory (DFT) functional and 2-site Hubbard models. A 1D adiabatic LDA kernel is constructed for use in time-dependent density functional theory (TDDFT), and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant (LC) density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably with the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with non-adiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.