Gap Formation and Soft Phonon Mode in the Holstein Model

We investigate electron-phonon coupling in many-electron systems using the dynamical mean-field theory in combination with the numerical renormalization group. This nonperturbative method reveals significant precursor effects to the gap formation at intermediate coupling strengths. The emergence of a soft phonon mode and very strong lattice fluctuations can be understood in terms of Kondo-like physics due to the development of a double-well structure in the effective potential for the ions.

Figure 1

Electronic spectral density $\rho (\omega )=-\frac{1}{\pi }\mathrm{I}\mathrm{m}\mathbf{⟨}⟨c_{i\sigma };c_{i\sigma }^{†}⟩\mathbf{⟩}$ from the DMFT-NRG calculation for ${\omega }_0=0.05$ and various coupling strengths $g$.

Figure 2

Quasiparticle weight $Z=(1-\frac{\partial \Sigma (\omega )}{\partial \omega }{|}_{\omega =0}{)}^{-1}$ as a function of $g$ from the NRG and the ME calculation.

Figure 3

Electronic spectral function as in Fig. 1, but for larger ${\omega }_0=1$ and $3$. The values of $g$ are given in units of $g_c$ which is $g_c=0.69$ ($1.48$) for ${\omega }_0=1$ ($3$).

Figure 4

Lattice fluctuations $\mathbf{⟨}{\widehat{x}}^2-⟨\widehat{x}{⟩}^2\mathbf{⟩}$ for the NRG and the ME calculation. Additionally the value of $x_{\mathrm{m}\mathrm{f}}^2$ as obtained within the mean-field calculation is plotted using a thick dotted line. The thin dotted line shows the limiting behavior of $x_{\mathrm{m}\mathrm{f}}^2$ for large $g$, $2\frac{g^2}{{\omega }_0^3}$.

Figure 5

Phonon propagator $\sigma (\omega )=-\frac{1}{\pi }\mathrm{I}\mathrm{m}d(\omega )$ for ${\omega }_0=0.05$ and various coupling strengths $g$. The upper panel shows the Migdal-Eliashberg, the lower panel the NRG results.