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arXiv:cond-mat/0505410 · Journal of Statistical Mechanics Theory and Experiment

Perturbation theory for optical excitations in the one-dimensional extended Peierls–Hubbard model

Anja Grage, Florian Gebhard, Jörg Rissler

Published Aug 17, 2005• 9 citations
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Abstract

For the one-dimensional, extended Peierls–Hubbard model we calculate analytically the ground-state energy and the single-particle gap to second order in the Coulomb interaction for a given lattice dimerization. The comparison with numerically exact data from the density-matrix renormalization group shows that the ground-state energy is quantitatively reliable for Coulomb parameters as large as the bandwidth. The single-particle gap can almost triple from its bare Peierls value before substantial deviations appear. For the calculation of the dominant optical excitations, we follow two approaches. In Wannier theory, we perturb the Wannier exciton states to second order. In two-step perturbation theory, similar in spirit to the GW -BSE approach (where the Bethe–Salpeter equation is solved for quasi-particles which are calculated in the GW -approximation), we form excitons from dressed electron–hole excitations. We find the Wannier approach to be superior to the two-step perturbation theory. For singlet excitons, Wannier theory is applicable up to Coulomb parameters as large as half the bandwidth. For triplet excitons, second-order perturbation theory quickly fails completely.

DOI
10.1088/1742-5468/2005/08/p08009

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For the one-dimensional, extended Peierls–Hubbard model we calculate analytically the ground-state energy and the single-particle gap to second order in the Coulomb interaction for a given lattice dimerization. The comparison with numerically exact data from the density-matrix renormalization group shows that the ground-state energy is quantitatively reliable for Coulomb parameters as large as the bandwidth. The single-particle gap can almost triple from its bare Peierls value before substantial deviations appear. For the calculation of the dominant optical excitations, we follow two approaches. In Wannier theory, we perturb the Wannier exciton states to second order. In two-step perturbation theory, similar in spirit to the GW -BSE approach (where the Bethe–Salpeter equation is solved for quasi-particles which are calculated in the GW -approximation), we form excitons from dressed electron–hole excitations. We find the Wannier approach to be superior to the two-step perturbation theory. For singlet excitons, Wannier theory is applicable up to Coulomb parameters as large as half the bandwidth. For triplet excitons, second-order perturbation theory quickly fails completely.
In this work the optical excitations in the one-dimensional extended Peierls-Hubbard model are studied. There are two competing interactions in this model, namely the Coulomb interaction and the Peierls interaction. The latter causes a dimerised lattice structure. To calculate the optical excitations perturbatively, one needs to identify the dominant interaction of the model. In the field-theoretical limit using different methods, such as a semiclassical approach, mean-field theory and the renormalisation group, one finds that it is the Peierls-interaction which dominates the system and therefore defines the starting point for perturbation theory. Rayleigh-Schrödinger perturbation theory is carried out up to second order in the weak Coulomb interaction for the ground-state energy and the one-particle gap. The singlet and triplet exciton gaps, i.e. charge and spin gap, are calculated using Wannier perturbation theory, two-step perturbation theory and Downfolding perturbation theory. All results are compared to results obtained by using the density-matrix renormalisation group approach (DMRG).
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.
We study the one-dimensional extended Hubbard model with alternating size of the hopping integrals using the density-matrix renormalization group method. We calculate the spin gap, the Tomonaga-Luttinger parameter, and the charge-density-wave order parameter for various dimerizations, interaction strengths, and band fillings. At half band-filling the spin and charge excitations are gapped but these gaps disappear for infinitesimal hole doping. At quarter filling, the umklapp scattering in the half-filled lower Peierls band generates a gap for the charge excitations but the gapless spin excitations can be described in terms of an effective antiferromagnetic Heisenberg model. Beyond a critical strength for the nearest-neighbor interaction, the dimerized extended Hubbard model at quarter filling develops a charge-density-wave ground state. The dimerization and the nearest-neighbor Coulomb interaction strongly reduce the Tomonaga-Luttinger parameter from its value for the bare Hubbard model. We discuss the relevance of our findings for the Bechgaard salts.
We study the elementary excitations of a model Hamiltonian for the $\ensuremath{\pi}$ electrons in polydiacetylene chains. In these materials, the bare band gap is only half the size of the observed single-particle gap and the binding energy of the exciton of 0.5 eV amounts to 20% of the single-particle gap. Therefore, exchange and correlations due to the long-range Coulomb interaction require a numerically exact treatment which we carry out using the density-matrix renormalization group method. Employing both the Hubbard-Ohno potential and the screened potential in one dimension, we reproduce the experimental results for the binding energy of the singlet exciton and its polarizability. Our results indicate that there are optically dark states below the singlet exciton, in agreement with experiment. In addition, we find a weakly bound second exciton with a binding energy of 0.1 eV. The energies in the triplet sector do not match the experimental data quantitatively, probably because we do not include polaronic relaxation effects.
Using a symmetrized density-matrix renormalization-group formulation, we have investigated the electronic binding energy of the lowest optically allowed exciton (1Bu{1\mathrm{B}}_{\mathrm{u}}1Bu​) within an extended Hubbard-Peierls model (with parameters U, V, and \ensuremath{\delta}) for conjugated chains with N=80 sites. Three symmetries, the C2{\mathrm{C}}_{2}C2​, spin parity, and electron-hole symmetries, have been applied to construct the projector operator. Analysis of our results on the ratios between exciton binding energies and exciton energies, sheds light on the current experimental and theoretical controversy on exciton binding for polyacetylene and polyparaphenylene vinylene. We show that in the absence of dimerization, the exciton binding energy is vanishingly small for U/t up to 5 and V/t up to 2.3; for a finite dimerization, it is only when V is large enough that the exciton gets to be significantly bound. This result questions the applicability to conjugated polymers of the strong correlation picture, in which the exciton binding energy is equal to V.
We study theoretically polydiacetylene chains diluted in their monomer matrix. We employ the density matrix renormalization group method on finite chains to calculate the ground state and low-lying excitations of the corresponding Peierls--Hubbard-Ohno Hamiltonian which is characterized by the electron transfer amplitude t0{t}_{0}t0​ between nearest neighbors, by the electron-phonon coupling constant $\ensuremath{\alpha}$, by the Hubbard interaction UUU, and by the long-range interaction VVV. We treat the lattice relaxation in the adiabatic limit, i.e., we calculate the polaronic lattice distortions for each excited state. Using chains with up to 102 lattice sites, we can safely perform the extrapolation to the thermodynamic limit for the ground-state energy and conformation, the single-particle gap, and the energies of the singlet exciton, the triplet ground state, and the optical excitation of the triplet ground state. The corresponding gaps are known with high precision from experiments. We determine a coherent parameter set $({t}_{0}^{*}=2.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{\ensuremath{\alpha}}^{*}=3.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV}/\text{\AA{}},{U}^{*}=6\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{V}^{*}=3\phantom{\rule{0.16em}{0ex}}\mathrm{eV})$ from a fit of the experimental gap energies to the theoretical values which we obtain for 81 parameter points in the four-dimensional search space $({t}_{0},\ensuremath{\alpha},U,V)$. We identify dark in-gap states in the singlet and triplet sectors as seen in experiments. Using a fairly stiff spring constant, the length of our unit cell is about 1% larger than its experimental value.