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arXiv:cond-mat/9609090 · Philosophical Magazine B

Exact results for the optical absorption of strongly correlated electrons in a half-filled Peierls-distorted chain

F. Gebhard, K. Born, M. Scheidler, P. Thomas, S. W. Koch

Published Jan 1, 1997• 36 citations
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Abstract

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.

DOI
10.1080/13642819708205701

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Journal of Physics C Solid State Physics•May 28, 1977•27 citations
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.
In this second of three articles on the optical absorption of electrons in a halffilled 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 band-width 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 spindegenerate. This corresponds to the Hubbard model at temperatures large compared to the spin exchange energy.
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 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
It is shown that the ground-state optical absorption spectrum of a dimerised strongly correlated half-filled Hubbard chain can be expressed in terms of local densities of states of a corresponding cleaved one-electron tight-binding chain with appropriate alternating hopping matrix elements. A general continued-fraction expression is presented which gives, in principle, the local density of states for any arbitrary distribution of hopping elements in a nearest-neighbour one-electron tight-binding chain, and from which, as an application, the absorption spectrum of the dimerised Hubbard chain can be calculated.
We show that charge excitations within one-dimensional extended Hubbard models in the limit of large on-site Coulomb interaction can be mapped onto pairs of spin reversals from the ferromagnetic Ising-Heisenberg spin Hamiltonian. The exciton in the extended Hubbard models is equivalent to the bound two-magnon state of the spin Hamiltonian. From exact results for the spin Hamiltonian, for the intersite Coulomb interactions limited between nearest neighbors, the lowest optical exciton is separated in energy from the electron-hole continuum only for the intersite interaction larger than half the one-electron bandwidth. The commutation relationship between spin operators allows us to numerically investigate exciton binding energies and optical absorptions from the ground state in the limit of large on-site Coulomb interaction even for cases where analytic results are not available. For intersite Coulomb interaction limited between nearest neighbors, the binding energy of the lowest optical exciton changes very weakly for realistic Peierls bond dimerization, although there is a profound change in the optical absorption. While in the undimerized chain a single optical exciton can occur, two additional excitons that occur above the threshold of the electron-hole continuum become optically visible in the dimerized case. For long-range intersite Coulomb interactions, multiple excitons can occur even for the undimerized chain, and additional exciton states become optically visible upon dimerization. With the 1/r1/r1/r form for the intersite Coulomb interaction, our numerical calculations suggest that the formation of at least one optical exciton is unconditional, unlike the case of nearest-neighbor interaction. The binding energy of the lowest optical exciton again depends weakly on the dimerization for long-range interaction. Our theoretical results can explain the occurrence of two charge-transfer absorption bands in half-filled band segregated stack charge transfer solids below the spin-Peierls dimerization temperature. Our results also suggest that the lowest optical state of \ensuremath{\pi}-conjugated polymers, in which Coulomb interactions are long ranged, is necessarily an exciton.
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.