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arXiv:1002.1147 · Journal of the Physical Society of Japan

Spectral Properties of Interacting One-Dimensional Spinless Fermions

Masanori Kohno, Mitsuhiro Arikawa, Jun Sato, Kazumitsu Sakai

Published Mar 25, 2010• 10 citations
arXivHTMLPDF

Abstract

The spectral properties of the spinless fermion model with nearest-neighbor repulsive interactions on a one-dimensional lattice are investigated using the Bethe ansatz. Although its bulk quantities are exactly the same as those of the spin-1/2 XXZ chain, the difference in the statistics of particles causes substantial effects on spectral features, such as gapless points of dispersion relations and line shapes of spectral functions. In this Letter, we clarify the origin of the differences in spectral features between fermionic and bosonic systems in terms of Bethe ansatz solutions. We also confirm that the two-string solutions have considerable spectral weights in the high-energy regime.

DOI
10.1143/jpsj.79.043707

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The spectral properties of the spinless fermion model with nearest-neighbor repulsive interactions on a one-dimensional lattice are investigated using the Bethe ansatz. Although its bulk quantities are exactly the same as those of the spin-1/2 XXZ chain, the difference in the statistics of particles causes substantial effects on spectral features, such as gapless points of dispersion relations and line shapes of spectral functions. In this Letter, we clarify the origin of the differences in spectral features between fermionic and bosonic systems in terms of Bethe ansatz solutions. We also confirm that the two-string solutions have considerable spectral weights in the high-energy regime.
Using the Bethe-ansatz analysis as reformulated by Edwards, we calculate the spectral properties of a particle interacting with a bath of fermions in one dimension for the case of equal particle-fermion masses. These are directly related to singularities apparent in optical experiments in one-dimensional systems. The orthogonality catastrophe for the case of an infinite particle mass survives in the limit of equal masses. We find that the exponent \ensuremath{\beta} of the quasiparticle weight Z\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$ is different for the two cases and proportional to their respective phase shifts at the Fermi surface; we present a simple physical argument for this difference. We also show that these exponents describe the low-energy behavior of the spectral function for repulsive as well as attractive interactions.
We study the spectral function of interacting one-dimensional fermions for an integrable lattice model away from half-filling. The divergent power-law singularity of the spectral function near the single-particle or single-hole energy is described by an effective x-ray edge type model. At low densities and for momentum near the zone boundary, we find a second divergent singularity at higher energies which is associated with a two-particle bound state. We use the Bethe ansatz solution of the model to calculate the exact singularity exponents for any momentum and for arbitrary values of chemical potential and interaction strength in the critical regime. We relate the singularities of the spectral function to the long-time decay of the fermion Green's function and compare our predictions with numerical results from the time-dependent density-matrix renormalization group (tDMRG). Our results show that the tDMRG method is able to provide accurate time decay exponents in the cases of power-law decay of the Green's function. Some implications for the line shape of the dynamical structure factor away from half-filling are also discussed. In addition, we show that 85% of the total spectral weight in the zero field Heisenberg model near wave-vector $q=\ensuremath{\pi}$ is given by the two-spinon contribution.
We examine zero-temperature one-particle spectral functions for the one-dimensional two-band spinless fermions with different velocities and general forward-scattering interactions. By using the bosonization technique and diagonalizing the model to two Tomonaga-Luttinger liquid Hamiltonians, we obtain general expressions for the spectral functions, which are given in terms of the Appell hypergeometric functions. For the case of identical two-band fermions, corresponding to the SU(2) symmetric spin-1/2 fermions with repulsive interactions, the spectral functions can be expressed in terms of the Gauss hypergeometric function and are shown to recover the double-peak structure suggesting the well-known ``spin-charge'' separation. By tuning the difference in velocities for the two-band fermions, we clarify the crossover in spectral functions from the spin-charge separation to the decoupled fermions. We discuss the relevance of our results to the spin-1/2 Hubbard model under a magnetic field, which can be mapped onto two-band spinless fermions.
A one-dimensional gas mixture consisting of bosons and fermions without spin interacting via a repulsive [Formula: see text]-function potential is considered. The model is integrable and soluble via two nested Bethe ansatz, if all particles are assumed to have equal masses and the interaction strength between the bosons and among the bosons and fermions is the same. The low energy excitation spectrum is a two-component Luttinger liquid and can be parametrized by a conformal field theory with conformal charges c = 1. In the low-energy limit, where the band curvature terms in the dispersion can be neglected, the linear dispersion of a Luttinger liquid is asymptotically exact. The spectral function, however, displays deviations from the Luttinger behavior for higher energy excitations. In the neighborhood of the single-particle (hole) energy, the spectral function is represented by an effective X-ray edge type model. Expressions of the critical exponents for the single-hole Green’s function are obtained using the Bethe ansatz solution in the limit of the bosonic gas. The results could be of relevance in the context of ultracold atoms confined to an elongated optical trap.
According to the exact analytical Bethe-ansatz solution of the one-dimensional spin-1/2 XXZ model, we perform a numerical study of one-dimensional spinless fermions in an optical lattice in the presence of harmonic potential by using a local density approximation. We study the density profile, and obtain the -u phase diagram (here =NfV2/t is the characteristic dimensionless density and u=V/t is the interaction strength scaled in units of the hopping parameter t). With the increases of particle density and nearest-neighbour interaction, the system undergoes five different quantum phases. Through calculating the thermodynamic stiffness S, we find that it can be used as an order parameter. Its singular points can measure the quantum phase transitions in such a confined system.
The single-particle spectral properties near the Mott transition in the one-dimensional Hubbard model are investigated by using the dynamical density-matrix renormalization group method and the Bethe ansatz. The pseudogap, hole-pocket behavior, spectral-weight transfer, and upper Hubbard band are explained in terms of spinons, holons, antiholons, and doublons. The Mott transition is characterized by the emergence of a gapless mode whose dispersion relation extends up to the order of hopping t (spin exchange J) in the weak (strong) interaction regime caused by infinitesimal doping.