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arXiv:cond-mat/0005108 · Physical Review Letters

Finite-Frequency Optical Absorption in 1D Conductors and Mott-Hubbard Insulators

J. M. P. Carmelo, Nuno M. R. Peres, Pedro D Sacramento

Published May 15, 2000• 29 citations
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Abstract

The frequency-dependent conductivity is studied for the one-dimensional Hubbard model, using a selection rule, the Bethe ansatz, and symmetries associated with conservation laws. For densities where the system is metallic the absorption spectrum has two contributions, a Drude peak at omega = 0 separated by a pseudogap from a broad absorption band whose lower edge is characterized by a nonclassical critical exponent. Our findings shed new light on the "far infrared puzzle" and other optical properties of metallic organic chain compounds.

DOI
10.1103/physrevlett.84.4673

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The frequency-dependent conductivity is studied for the one-dimensional Hubbard model, using a selection rule, the Bethe ansatz, and symmetries associated with conservation laws. For densities where the system is metallic the absorption spectrum has two contributions, a Drude peak at omega = 0 separated by a pseudogap from a broad absorption band whose lower edge is characterized by a nonclassical critical exponent. Our findings shed new light on the "far infrared puzzle" and other optical properties of metallic organic chain compounds.
The frequency-dependent conductivity is studied for both the one-dimensional Hubbard model and a model of spinless fermions, using a selection rule, the Bethe ansatz energy eigenstates, and conformal invariance. For densities where the system is metallic the absorption spectrum has two contributions, a Drude peak at ω= 0 separated by a pseudo-gap from a broad absorption band whose lower edge is characterized by a non-classical critical exponent. Our findings are expected to shed new light on the ``far infrared puzzle'' of metallic organic chain compounds.
We consider the optical conductivity in the one-dimensional Hubbard model in the metallic phase close to half-filling. In this regime, most of the spectral weight is located at frequencies above an energy scale Eopt{E}_{\mathrm{opt}}Eopt​ that tends towards the optical gap in the Mott insulating phase for vanishing doping. Using the Bethe ansatz, we relate Eopt{E}_{\mathrm{opt}}Eopt​ to thresholds of particular kinds of excitations in the Hubbard model. We then employ a mobile impurity model to analyze the optical conductivity for frequencies slightly above these thresholds. This entails generalizing mobile impurity models to excited states that are not the highest weight with regards to the SU(2) symmetries of the Hubbard chain, and that occur at a maximum of the impurity dispersion.
We use a complete pseudoparticle operator representation to study the explicit form of the finite-frequency conductivity for the Hubbard chain. Our study reveals that the spectral weight is mostly concentrated at the ω=0 Drude peak (except at density n=1) and at an absorption starting just below twice the value of the chemical potential whose ω-dependence we evaluate exactly. We also obtain the exact ωdependence of the higher-energy, less-pronounced absorption edges, whose weights vanish for large on-site interaction U.
We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}t2{\mathit{t}}^{2}t2/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.
The optical conductivity for the Hubbard and t-J models is calculated numerically by use of a Lanczos algorithm. One- and two-dimensional systems of up to 16 sites with periodic and open boundary conditions are examined for the t-J model, as well as Hubbard-model systems of up to 9 sites. In the one-dimensional case, the low-frequency part of the conductivity away from half filling is shown to consist almost entirely of a single \ensuremath{\delta} function (Drude peak) even for J of the order of t. This is a direct consequence of the decoupling of charge and spin in one dimension. In two dimensions there is an additional broad absorption band above the Drude peak, consistent with the experimentally observed midinfrared band in the copper oxide superconductors. The importance of vertex corrections is demonstrated by comparing the exact conductivity with the convolution integral of the exact single-particle Green's function.