Exact zero-temperature correlation functions for two-leg Hubbard ladders and carbon nanotubes

Abstract

Motivated by recent work of Lin, Balents, and Fisher [Phys. Rev. B 58, 1794 (1998)], we compute correlation functions at zero temperature for weakly coupled two-leg Hubbard ladders and $(N,N)$ armchair carbon nanotubes. In this paper it was argued that such systems renormalize towards the SO(8) Gross-Neveu model, an integrable theory. We exploit this integrability to perform the computation at the SO(8) invariant point. Any terms breaking the SO(8) symmetry can be treated systematically in perturbation theory, leading to a model with the same qualitative features as the integrable theory. Using said correlators, we determine the optical conductivity, the single-particle spectral function, and the $I\ensuremath{-}V$ curve for tunneling into the system from an external metallic lead. The frequency, \ensuremath{\omega}, dependent optical conductivity is determined exactly for $\ensuremath{\omega}<3m$ $(m$ being the fermion particle mass in the SO(8) Gross-Neveu model). It is characterized by a sharp ``exciton'' peak at $\ensuremath{\omega}=\sqrt{3}m,$ followed by the onset of the particle-hole continuum beginning at $\ensuremath{\omega}=2m.$ Interactions modify this onset to $\ensuremath{\sigma}(\ensuremath{\omega}+2m)\ensuremath{\sim}{\ensuremath{\omega}}^{1/2}$ and not the ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ one would expect from the van Hove singularity in the density of states. Similarly, we obtain the exact single-particle spectral function for energies less than $3m.$ The latter possesses a $\ensuremath{\delta}$ function peak arising from single-particle excitations, together with a two-particle continuum for $\ensuremath{\omega}>~2m.$ The final quantity we compute is the tunneling $I\ensuremath{-}V$ curve to lowest nonvanishing order in the tunneling matrix elements. For this quantity, we present exact results for voltages, $V<(1+\sqrt{3})m.$ The resulting differential conductance is marked by a finite jump at $\ensuremath{\omega}=2m,$ the energy of the onset of tunneling into the continuum of two-particle states. Through integrability, we are able to characterize this jump exactly. All calculations are done through form-factor expansions of correlation functions. These give exact closed form expressions for spectral functions because the SO(8) Gross-Neveu model is massive: each term in the expansion has an energy threshold below which it does not contribute. Thus, we obtain exact results below certain thresholds by computing a finite number of terms in this series. Previous to this paper, the only computed form factor of SO(8) Gross-Neveu was the two-particle form factor of an SO(8) current with two fundamental fermions. In this paper we compute the set of all one- and two-particle form factors for all relevant fields, the currents as well as the kinks and fermions.