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arXiv:cond-mat/0508712 · Journal of Physics Condensed Matter

Spectral functions of the spinless Holstein model

Jan Loos, Martin Hohenadler, Holger Fehske

Published Feb 10, 2006• 20 citations
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Abstract

An analytical approach to the one-dimensional spinless Holstein model is proposed, which is valid at finite charge-carrier concentrations. Spectral functions of charge carriers are computed on the basis of self-energy calculations. A generalization of the Lang–Firsov canonical transformation method is shown to provide an interpolation scheme between the extreme weak-coupling and strong-coupling cases. The transformation depends on a variationally determined parameter that characterizes the charge distribution across the polaron volume. The relation between the spectral functions of the polaron and electron, the latter corresponding to the photoemission spectrum, is derived. Particular attention is paid to the distinction between the coherent and incoherent parts of the spectra, and their evolution as a function of band filling and model parameters. Results are discussed and compared with recent numerical calculations for the many-polaron problem.

DOI
10.1088/0953-8984/18/8/011

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An analytical approach to the one-dimensional spinless Holstein model is proposed, which is valid at finite charge-carrier concentrations. Spectral functions of charge carriers are computed on the basis of self-energy calculations. A generalization of the Lang–Firsov canonical transformation method is shown to provide an interpolation scheme between the extreme weak-coupling and strong-coupling cases. The transformation depends on a variationally determined parameter that characterizes the charge distribution across the polaron volume. The relation between the spectral functions of the polaron and electron, the latter corresponding to the photoemission spectrum, is derived. Particular attention is paid to the distinction between the coherent and incoherent parts of the spectra, and their evolution as a function of band filling and model parameters. Results are discussed and compared with recent numerical calculations for the many-polaron problem.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
Cluster perturbation theory in combination with the Lanczos method is used to compute the one-electron spectral function of the Holstein polaron in one and two dimensions. It is shown that the method allows reliable calculations using relatively small clusters, and at the same time significantly reduces finite-size effects. Results are compared with exact data and the relation to existing work is discussed. We also use a strong-coupling perturbation theory---equivalent to the Hubbard I approximation---to calculate the spectral function of the quarter-filled Holstein model of spinless fermions, starting from the exact atomic-limit Green function. The results agree well with previous calculations within the many-body coherent potential approximation.
A recently proposed variational approach based on an expansion in coherent states is used to investigate the single polaron features in the one-dimensional Holstein model. The method provides highly accurate results, both for static and dynamic properties, over a wide range of the polaron parameter space, from weak to strong coupling regimes, with a little numerical effort when compared with standard numerical techniques. The results for the spectral weight function and the real part of the optical conductivity are reported in the moderate adiabatic regime. We show that the spectral function exhibits a strong dependence on the polaron wave number kkk that is not present in the standard analytical approaches. The optical conductivity, in the strong coupling regime, displays a high-energy peak due to the photoionization of the small polaron, while, at lower energies, there is evidence of electronic transitions between localized states. These results show that the coherent-state basis is a very powerful basis to study interacting electron-phonon systems.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.