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arXiv:cond-mat/9402029 · Physical review. B, Condensed matter

Conserving approximations for the attractive Holstein and Hubbard models

James K. Freericks

Published Jul 1, 1994• 61 citations
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Abstract

Conserving approximations are applied to the attractive Holstein and Hubbard models (on an infinite-dimensional hypercubic lattice). All effects of nonconstant density of states and vertex corrections are taken into account in the weak-coupling regime. Infinite summation of certain classes of diagrams turns out to be a quantitatively less accurate approximation than truncation of the conserving approximations to a finite order, but the infinite summation approximations do show the correct qualitative behavior of generating a peak in the transition temperature as the interaction strength increases.

DOI
10.1103/physrevb.50.403

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Conserving approximations are applied to the attractive Holstein and Hubbard models (on an infinite-dimensional hypercubic lattice). All effects of nonconstant density of states and vertex corrections are taken into account in the weak-coupling regime. Infinite summation of certain classes of diagrams turns out to be a quantitatively less accurate approximation than truncation of the conserving approximations to a finite order, but the infinite summation approximations do show the correct qualitative behavior of generating a peak in the transition temperature as the interaction strength increases.
A strictly truncated (weak-coupling) perturbation theory is applied to the attractive Holstein and Hubbard models in infinite dimensions. These results are qualified by comparison with essentially exact quantum Monte Carlo results. The second-order iterated perturbation theory is shown to be quite accurate in calculating transition temperatures for retarded interactions, but is not as accurate for the self-energy or the irreducible vertex functions themselves. Iterated perturbation theory is carried out through fourth order for the Hubbard model. The self-energy is quite accurately reproduced by the theory, but the vertex functions are not. Anomalous behavior occurs near half-filling because the iterated perturbation theory is not a conserving approximation.
We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in Tc{T}_{c}Tc​ as a function of U and a fluctuation-broadened specific heat peak. At n=1,n=1,n=1, competing pair fluctuations and charge-density fluctuations fail to drive Tc{T}_{c}Tc​ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because Tc{T}_{c}Tc​ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in 106.{10}^{6}.106. We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.
Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.
The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.