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arXiv:cond-mat/9806275 · Physical review. B, Condensed matter

Vertex-corrected perturbation theory for the electron-phonon problem with nonconstant density of states

James K. Freericks, V. Zlatić, Woonki Chung, Mark Jarrell

Published Nov 1, 1998• 12 citations
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Abstract

A series of weak-coupling perturbation theories that include the lowest-order vertex corrections are applied to the attractive Holstein model in infinite dimensions. The approximations are chosen to reproduce the iterated perturbation theory in the limit of half-filling and large phonon frequency (where the Holstein model maps onto the Hubbard model). Comparison is made with quantum Monte Carlo solutions to test the accuracy of different approximation schemes.

DOI
10.1103/physrevb.58.11613

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A series of weak-coupling perturbation theories that include the lowest-order vertex corrections are applied to the attractive Holstein model in infinite dimensions. The approximations are chosen to reproduce the iterated perturbation theory in the limit of half-filling and large phonon frequency (where the Holstein model maps onto the Hubbard model). Comparison is made with quantum Monte Carlo solutions to test the accuracy of different approximation schemes.
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.
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.
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.
I apply the newly developed dynamical cluster approximation (DCA) to the calculation of the electron and phonon dispersions in the two-dimensional Holstein model. In contrast to previous work, the DCA enables the effects of spatial fluctuations (non-local corrections) to be examined. Approximations neglecting and incorporating lowest-order vertex corrections are investigated. I calculate the phonon density of states, the renormalized phonon dispersion, the electron dispersion and electron spectral functions. I demonstrate how vertex corrections stabilize the solution, stopping a catastrophic softening of the (π, π) phonon mode. A kink in the electron dispersion is found in the normal state along the (ζ, ζ) symmetry direction in both the vertex- and non-vertex-corrected theories for low phonon frequencies, corresponding directly to the renormalized phonon frequency at the (π, 0) point. This kink is accompanied by a sudden drop in the quasi-particle lifetime. Vertex and non-local corrections enhance the effects at large bare phonon frequencies.