One-particle spectral function singularities in a one-dimensional gas of spin-1/2 fermions with repulsive delta-function interaction

Abstract

The momentum, fermionic density, spin density, and interaction dependencies of the exponents that control the (k, ω)-plane singular features of the one-fermion spectral functions of a onedimensional gas of spin-1/2 fermions with repulsive delta-function interaction both at zero and finite magnetic field are studied in detail. Our results refer to energy scales beyond the reach of the low-energy Tomonaga-Luttinger liquid and rely on the pseudofermion dynamical theory for integrable models. The one-fermion spectral weight distributions associated with the spectral functions studied in this paper may be observed in systems of spin-1/2 ultra-cold fermionic atoms in optical lattices.PACS numbers: I. INTRODUCTIONThe one-dimensional (1D) continuous fermionic gas with repulsive delta-function interaction, which in this paper we call 1D repulsive fermion model, was one of the first quantum problems solved by the Bethe ansatz (BA) [1]. This was achieved by Yang [2] and by Gaudin [3]. Yang's solution of the 1D repulsive fermion model was actually the precursor of the BA solution of the lattice 1D Hubbard model by . That the latter is the simplest condensed-matter toy model for the description of the role of correlations in the exotic properties of 1D and quasi-1D lattice condensed matter systems [8,9] justifies why for several decades it had attracted more attention than its continuous cousin, the fermionic gas with repulsive delta-function interaction. This refers both to its metallic and Mott-Hubbard insulator phases [10], the latter not existing in the case of the continuous 1D repulsive fermion model.However, in the last years the interest in that Yang-Baxter integrable model has been renewed by its new found impact on experiments in both condensed matter physics and ultra-cold atomic gases [11]. The latter have provided new opportunities for studying 1D systems of spin-1/2 fermions with repulsive interaction [12,13]. The present model can indeed be implemented with ultra-cold atoms [11][12][13][14][15][16][17]. Ultra-cold Fermi gases trapped inside a tight atomic waveguide offer for instance the opportunity to measure the spin-drag relaxation rate that controls the broadening of a spin packet. It has been found that while the propagation of long-wavelength charge excitations is essentially ballistic, spin propagation is intrinsically damped and diffusive [18,19]. A related interesting problem is the force applied to a spin-flipped fermion in a gas, which may lead to Bloch oscillations of the fermion's position and velocity. The existence of such oscillations has been found crucially to depend on the viscous friction force exerted by the rest of the gas on the spin excitation [20].The ground-state energy of the relative motion of a system of two fermions with spin up and spin down interacting via a delta-function potential in a 1D harmonic trap has been calculated by combining the BA with the variational principle [21]. Recently, related ground-state properties of a 1D repulsive Fermi gas subjected to a commensurate per...