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Physical review. B, Solid state

Optical spectrum of a Hubbard chain

Pierre F. Maldague

Published Sep 15, 1977• 235 citations
Publisher

Abstract

The optical absorption of the one-dimensional Hubbard model is calculated by three different methods. Linear chains of five atoms and rings of five and seven atoms containing four electrons are treated numerically. The infinite-chain problem is solved first in the $t$-matrix approximation of Lyo and Holstein. It is shown that in this approximation, most of the high-frequency absorption is due to a bound state which lies above the band continuum. Finally, the absorption is evaluated in the memory-function formalism of G\"otze and W\"olfle, which reduces to ordinary perturbation theory at high frequency. The three approaches are in qualitative agreement, and the differences between them can be explained by the nature of the approximations involved. Applications to tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) and the platinum salt ${\mathrm{K}}_{2}$Pt${(\mathrm{CN})}_{4}$${\mathrm{Br}}_{0.3}$ \ifmmode\cdot\else\textperiodcentered\fi{} 3${\mathrm{H}}_{2}$O (KCP) are discussed.

DOI
10.1103/physrevb.16.2437

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The optical absorption of the one-dimensional Hubbard model is calculated by three different methods. Linear chains of five atoms and rings of five and seven atoms containing four electrons are treated numerically. The infinite-chain problem is solved first in the ttt-matrix approximation of Lyo and Holstein. It is shown that in this approximation, most of the high-frequency absorption is due to a bound state which lies above the band continuum. Finally, the absorption is evaluated in the memory-function formalism of G"otze and W"olfle, which reduces to ordinary perturbation theory at high frequency. The three approaches are in qualitative agreement, and the differences between them can be explained by the nature of the approximations involved. Applications to tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) and the platinum salt K2{\mathrm{K}}_{2}K2​Pt{(\mathrm{CN})}_{4}$${\mathrm{Br}}_{0.3} \ifmmode\cdot\else\textperiodcentered\fi{} 3H2{\mathrm{H}}_{2}H2​O (KCP) are discussed.
Exact results for the one-hole excitation spectra of the one-dimensional half-filled Hubbard Hamiltonian are presented. We consider short chains of N=2, 4, 6 atoms and introduce modified periodic boundary conditions which allow us to make a comparison between different N and to extrapolate the results up to infinite chains. By the increasing weight of the scattering states in the one-hole spectrum, we demonstrate a gradual disappearance of the free-electron (band) picture with increasing local electron-electron interaction U. The exact spectra of the N=6 chain verify the results of second-order perturbation theory for U/W1, where W is the bandwidth. The extrapolation to the infinite system, made with the finite-cell-scaling method, gives the gap which agrees rather well with the exact result for U/W>0.2. The average energies and weights of the band and scattering states obtained with this method for an infinite chain do not differ significantly from those of the N=6 chain.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
A new, simple and efficient method is presented for calculation of the ground and a few excited states of Hubbard chain nanostructures. By using this method, the photoemission spectral function for organic charge transfer salt TTF-TCNQ, is calculated. For a chain with maximum 70 sites the result is in good agreement with the previous works but there is a difference for further number of sites in the chain, which is discussed in the text with all the specifics. We also show that a source of errors in density matrix renormalization group method for a one dimensional chain is the deficiency of the matrix product scheme for generating the desired states of the linear chain.
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
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.