Efficient Dynamical Mean Field Simulation of the Holstein-Hubbard Model

We present a method for solving impurity models with electron-phonon coupling, which treats the phonons efficiently and without approximations. The algorithm is applied to the Holstein-Hubbard model in the dynamical mean field approximation, where it allows access to strong interactions, very low temperatures, and arbitrary fillings. We show that a renormalized Migdal-Eliashberg theory provides a reasonable description of the phonon contribution to the electronic self-energy in strongly doped systems, but fails if the quasiparticle energy becomes of order of the phonon frequency.

Figure 1

Phase diagram for ${\omega }_0=0.2t$, $\beta t=50$, 400, and $U/t=6$ in the plane of chemical potential $\mu$ and phonon coupling strength $\lambda$. Half-filling corresponds to $\mu =U/2$. In the metallic phase, the filling increases with increasing $\mu$, $\lambda$.

Figure 2

Imaginary time dependence of the density-density correlation function plotted on a log-scale to highlight the intermediate-time exponential decay. If no exponential regime is visible, the phonon frequency is not renormalized; otherwise ${\omega }_0^{\mathrm{ren}}$ can be extracted from the slope. Upper panel: $U=0$ and $n=1$. Dashed lines: $\chi (\tau )$ from Eq. (7) for indicated phonon couplings. Solid lines with $1\mathrm{\text{−}}\sigma$ error bars show the QMC results for $\beta t=400$ and $\lambda /t=0.1$, 0.2, 0.3 (from bottom to top). Lower panel: correlation functions for the interacting model at half-filling, $U/t=4$ and $\lambda /t=0.6$, 0.65 and in the doped Mott insulator at $U/t=6$, $\lambda /t=0.4$, 0.6 (dopings per spin $\delta =0.043$, 0.138).

Figure 3

Phonon contribution to the electronic self-energy at $\beta t=400$. Black diamonds: renormalized phonon frequency ${\omega }_0^{\mathrm{ren}}$. Upper panel: $U/t=4$. Symbols: half-filling, $\lambda$ values indicated. Blue (or dark gray) lines: $\lambda /t=0.4$ and doping per spin $\delta =0.019$, 0.067, and 0.182 (from bottom to top). Black line: Migdal-Eliashberg result. Lower panel: $U=6t$ (above the critical value for Mott insulating behavior). Symbols: $\lambda$ and $\delta$ as shown. Solid lines: Migdal-Eliashberg calculations, Eq. (8), with renormalized electron-phonon couplings ${\lambda }_{\mathrm{ren}}$ and phonon frequencies ${\omega }_0^{\mathrm{ren}}$ as indicated.