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arXiv:cond-mat/9609291

Complete Pseudoparticle Representation and the Conductivity Spectrum of the Hubbard Chain

J. M. P. Carmelo, Nuno M. R. Peres, David K Campbell

Published Sep 30, 1996• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/9609291

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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 introduce the pseudohole and heavy-pseudoparticle operator algebra that generates all Hubbard-chain eigenstates from a single reference vacuum. In addition to the pseudoholes already introduced for the description of the low-energy physics, this involves the heavy pseudoparticles associated with Hamiltonian eigenstates whose energy spectrum has a gap relative to the many-electron ground state. We introduce a generalized pseudoparticle perturbation theory that describes the relevant finite-energy ground-state transitions. In the present basis these excitations refer to a small density of excited pseudoparticles. Our operator basis goes beyond the Bethe-ansatz solution and it is the suitable and correct starting point for the study of the finite-frequency properties, which are of great relevance for the understanding of the unusual spectral properties detected in low-dimensional novel materials.
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 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.
We investigate the optical conductivity of a Hubbard ring in presence of an impurity by means of exact diagonalization using the Lanczos algorithm. We concentrate thereby on the first excited, open shell state, i.e. on twisted boundary conditions. In the metallic phase a substantial part of the spectral weight lies in the Drude peak, σ(ω)=Dδ(ω)+σregσ(ω) = Dδ(ω)+ σ_{\rm reg}σ(ω)=Dδ(ω)+σreg​. In the non-interacting system, the Drude peak can be visualized in our calculations at ω=0ω= 0ω=0 even for finite chain lengths. Adding the impurity, the main peak is shifted to finite frequencies proportional to the impurity strength. The shift indicates the energy gap of the disturbed finite size system, also in the interacting system. Thus, we can pursue in the optical conductivity for finite metallic systems the energy gap. However, due to level crossing, the impurity-induced peak arises in the interacting system first when a certain impurity strength is exceeded. In the Mott insulating phase, the impurity leads to an impurity state within the gap.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in the two-dimensional Hubbard model is examined by applying the exact diagonalization technique to small square clusters with periodic boundary conditions up to $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ sites. Spectral-weight distributions at half-filling and their doping dependence in the 20-site cluster are found to be similar to those in a $\sqrt{18}\ifmmode\times\else\texttimes\fi{}\sqrt{18}$ cluster, but different from $4\ifmmode\times\else\texttimes\fi{}4$ results. The results for the 20-site cluster enable us to perform a systematic study of the doping dependence of the spectral-weight transfer from the region of the Mott-gap excitation to lower-energy regions. We discuss the dependence of the Drude weight and the effective carrier number on the electron density at a large on-site Coulomb interaction.