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Modern Physics Letters B

GREEN FUNCTION CALCULATIONS FOR THE THREE-DIMENSIONAL HOLSTEIN MODEL

Xincheng Huang, Qiang Luo

Published Nov 10, 2008• 0 citations
Publisher

Abstract

The many-body Green functions for the three-dimensional electron–phonon Holstein model are calculated via the Feynman diagram technique. The lowest-order contributions to the one-electron self-energy are considered (the Hartree–Fock approximation). The chemical potential is adjusted to yield the desired band-filling. The temperature dependence of the chemical potential has the correct asymptotic form in the non-interacting limit. The behavior of the one-electron occupancy factor is examined for various parameter values of the model. Tian's theorem is satisfied at half-filling. Luttinger liquid behavior is observed in the strong electron–phonon coupling regime.

DOI
10.1142/s0217984908017266

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The many-body Green functions for the three-dimensional electron–phonon Holstein model are calculated via the Feynman diagram technique. The lowest-order contributions to the one-electron self-energy are considered (the Hartree–Fock approximation). The chemical potential is adjusted to yield the desired band-filling. The temperature dependence of the chemical potential has the correct asymptotic form in the non-interacting limit. The behavior of the one-electron occupancy factor is examined for various parameter values of the model. Tian's theorem is satisfied at half-filling. Luttinger liquid behavior is observed in the strong electron–phonon coupling regime.
We examine the behavior of the electron and phonon self-energies obtained from quantum Monte Carlo simulations of the two-dimensional Holstein model as a function of Matsubara frequency and momentum. For the half-filled system, there is a finite-temperature phase transition to an insulating charge-density-wave state, and the electron self-energy develops a distinctive divergence. The renormalized phonon frequency softens strongly with temperature at the commensurate wave vector. Away from half-filling, the electron self-energy behaves in a manner consistent with that of a Fermi liquid.
We consider a one-dimensional system consisting of electrons with short-ranged repulsive interactions and coupled to small-momentum transfer acoustic phonons. The interacting electrons are bosonized and described in terms of a Luttinger liquid which allows us to calculate exactly the one- and two-electron Green function. For non-interacting electrons, the coupling to phonons alone induces a singularity at the Fermi surface which is analogous to that encountered for electrons with an instantaneous attractive interaction. The exponents which determine the presence of singlet/triplet superconducting pairing fluctuations, and spin/charge density wave fluctuations are strongly affected by the presence of the Wentzel-Bardeen singularity, resulting in the favoring of superconducting fluctuations. For the Hubbard model the equivalent of a phase diagram is established, as a function of: the electron-phonon coupling, the electron filling factor, and the on-site repulsion between electrons. The Wentzel-Bardeen singularity can be reached for arbitrary values of the electron-phonon coupling constant by varying the filling factor. This provides an effective mechanism to push the system from the antiferromagnetic into the metallic phase, and finally into the superconducting phase as the electron filling factor is increased towards half-filling.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
The one-electron spectral function of the one-dimensional spin-1∕21∕21∕2 Holstein model at half filling is computed by use of the cluster perturbation theory. The cluster Green's function is obtained by the Lanczos exact diagonalization method within an optimized phonon approach. It is shown that the method allows reliable calculations using a relatively small size cluster and a few optimal phonon bases for the system from weak to strong electron-phonon coupling. In the strong-coupling limit, the spectral function shows the excitation behavior of a bipolaron state with a large gap at the Fermi surface. However, the obtained spectral function displays a metallic character in the weak-coupling regime, which is in accord with the suggestion that the Peierls gap is suppressed by quantum fluctuation of the phonons.
Phase diagram of the Hubbard-Holstein model in the coexistence of electron-electron and electron-phonon interactions has been theoretically obtained with the density-matrix renormalization group method for one-dimensional systems, where an improved warm-up (the recursive sweep) procedure has enabled us to calculate various correlation functions. We have examined the cases of (i) the systems half-filled by electrons for the full parameter space spanned by the electron-electron and electron-phonon coupling constants and the phonon frequency, (ii) non-half-filled system, and (iii) trestle lattice. For (i), we have detected a region where both the charge and on-site pairing correlations decay with power laws in real space, which suggests a metallic behavior. While pairing correlations are not dominant in (i), we have found that they become dominant as the system is doped in (ii) or as the electronic band structure is modified (with a broken electron-hole symmetry) in (iii) in certain parameter regions.
Electron- and phonon spectral functions of the one-dimensional, spinless-fermion Holstein model at half filling are calculated in the four distinct regimes of the phase diagram, corresponding to an attractive or repulsive Luttinger liquid at weak electron-phonon coupling, and a band- or polaronic insulator at strong coupling. The results obtained by means of kernel polynomial and systematic cluster approaches reveal substantially different physics in these regimes and further indicate that the size of the phonon frequency significantly affects the nature of the quantum Peierls phase transition, the latter being either of the soft-mode or central-peak type. The generic features observed are relevant to several classes of low-dimensional materials.