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Physical review. B, Solid state

Intraband absorption in a low-density Hubbard chain

S. K. Lyo, Theodore D. Holstein

Published Dec 15, 1976• 4 citations
Publisher

Abstract

We have investigated the intraband absorption in a single-band Hubbard model at a low concentration $c$ ($c$ is unity for a filled band) of electrons with a particular emphasis on one-dimensional nearest-neighbor electron transfer. Using a two-particle approximation, we find, for the latter in the frequency range $c\ensuremath{\ll}\ensuremath{\hbar}\ensuremath{\omega}$ (the bandwidth is unity) that the absorption is independent of frequency and vanishes as ${U}^{\ensuremath{-}2}$ for a large on-site Coulomb repulsion $U$. The absorption occurs when an electron is accelerated by another electron in the presence of an oscillating field (i.e., inverse bremsstrahlung). The transport relaxation rate is given by ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\propto}c{\ensuremath{\omega}}^{2}{U}^{\ensuremath{-}2}$. In the limit $U=\ensuremath{\infty}$ the present result supports the spinless fermion model, according to which the intraband absorption vanished for $\ensuremath{\omega}\ensuremath{\ne}0$. In the presence of second-nearest-neighbor electron transfer (${t}_{2}$) such that $c\ensuremath{\ll}|{t}_{2}|$, the ${\ensuremath{\omega}}^{\ensuremath{-}2}$-frequency dependence of the absorption obtains as in two or three dimensions.

DOI
10.1103/physrevb.14.5137

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A theory of absorption is presented for a half-filled Hubbard band with nearest-neighbour electron transfer (t) and strong intra-site Coulomb repulsion (U). The lineshape is calculated at low temperature (i.e. k B T<<U) to lowest order in t/U for antiferromagnetic, random and ferromagnetic spin configurations in a Cayley tree lattice with a particular emphasis on one dimension. The authors find that the absorption diverges logarithmically at the centre of the absorption band (i.e. at h(cross) omega =U). In a one-dimensional lattice, the absorption width equals twice the bandwidth of a single hole (or an extra electron) in an otherwise half-filled Hubbard band.
In this last of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain we address the dimerized extended Hubbard model in the limit of a large on-site interaction U. When the Hubbard interaction is large both compared to the band width W and the nearest-neighbor interaction V the charge dynamics is properly described by the Harris-Lange model. This model can be exactly mapped onto a model of free spinless Fermions in parallel (Hubbard-)bands of width W which are eventually Peierls-split. To determine the coherent absorption features at low temperatures we design and employ the “no-recoil approximation ” in which we assume that the momentum transfer to the spin degrees of freedom can only be ∆qS = 0 or ∆qS = π/a during an optical excitation. We present explicit analytical results for the optical absorption in the presence of a lattice dimerization δ and a nearest-neighbor interaction V for the Néel and dimer state. We find
We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio Ut\frac{U}{t}tU​, where UUU represents the on-site Coulomb repulsion and ttt is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.
In this, the last of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain, we address the dimerized extended Hubbard model in the limit of a large on-site interaction U. When the Hubbard interaction is large compared with both the bandwidth W and the nearest-neighbour interaction V, the charge dynamics are properly described by the Harris-Lange model. This model can be exactly mapped onto a model of free spinless fermions in parallel (Hubbard) bands of width W which are eventually Peierls split. To determine the coherent absorption features at low temperatures, we design and employ the ‘no-recoil approximation’ in which we assume that the momentum transfer to the spin degrees of freedom can only be Δqs = 0 or Δqs = π/a during an optical excitation. We present explicit analytical results for the optical absorption in the presence of a lattice dimerization δ and a nearest-neighbour interaction V for the Néel and dimer state. We find that the coherent part of the optical absorption for and the spin dimerization of the ground state.
In this, the second of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain, we present exact results for strongly correlated tight-binding electrons. In the limit of a strong on-site interaction U, we map the Hubbard model onto the Harris-Lange model which can be solved exactly in one dimension in terms of spinless fermions for the charge excitations. The exact solution allows for an interpretation of the charge dynamics in terms of parallel Hubbard bands with a free-electron dispersion of bandwidth W, separated by the Hubbard interaction U. The spin degrees of freedom enter the expressions for the optical absorption only via a momentum-dependent but static ground-state expectation value. The remaining spin problem can be traced out exactly since the eigenstates of the Harris-Lange model are spin degenerate. This corresponds to the Hubbard model at temperatures large compared with the spin exchange energy. Explicit results are given for the optical absorption in the presence of a lattice distortion δ and a nearest-neighbour interaction V. We find that the optical absorption for V = 0 is dominated by a peak at ω = U and broad but weak absorption bands for |ω - U| ≤ W. For an appreciable nearest-neighbour interaction V < W/2, almost all the spectral weight is transferred to Simpson's exciton band which is eventually Peierls split.
We consider a narrow one-dimensional tight-binding chain with intra-atomic Coulomb repulsions described in the Hubbard model. The electrons at each site are furthermore assumed to be coupled to a high-frequency local oscillator. In the limit where the oscillator frequency is large compared to the bandwidth, and in the extreme-band-narrowing regime, for various electron densities, we find that: the half-filled-band ground state is an antiferromagnetic insulator; the one-third-filled-band ground state is a diamagnetic insulator; and the one-quarter-filled-band ground state is a weakly coupled system of bound singlet pairs somewhat reminiscent of superconductivity.