Retardation effects in the Holstein-Hubbard chain at half filling

The ground-state phase diagram of the half filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon $\left(e\text{−}ph\right)$ coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron $\left(e\text{−}e\right)$ repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also $e\text{−}ph$ couplings. Our method [Tsai et al., Phys. Rev. B 72, 054531 (2005); Philos. Mag. B 86, 2631 (2006)] treats $e\text{−}e$ and $e\text{−}ph$ interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the SDW and CDW states. We study the effects of retardation, which are particularly important near the transition, and find that umklapp processes at finite frequencies drive the CDW instability close to the transition.

Figure 1

Discretization of the momenta in the Brillouin zone and frequencies in the frequencies axis. This figure shows the case $N_k=2$, $N_w=15$.

Figure 2

(Color online) Left: flows of SC, SDW, and CDW susceptibilities for $U=0.5$ and ${\omega }_0=1.0$. Right: flows of umklapp $g_3$ and backscattering $g_1$ at zero frequencies. Top: $g_{ep}=0.2$ $(U_{\mathrm{eff}}>0)$. Bottom: $g_{ep}=0.8$ $(U_{\mathrm{eff}}<0)$.

Figure 3

(Color online) Left: flow of susceptibilities for $U=0.5$, ${\omega }_0=1.0$, $g_{\mathrm{ep}}=0.55$ $(U_{\mathrm{eff}}<0)$. Right: flows of the umklapp scattering $g_3$ and backscattering $g_1$ at zero frequency.

Figure 4

Plots of the umklapp scattering $g_3({\omega }_1,{\omega }_2,{\omega }_2,{\omega }_1)$ for $U=0.5$ and ${\omega }_0=1.0$. Left: $g_{\mathrm{ep}}=0.2$. Right: $g_{\mathrm{ep}}=0.8$.

Figure 5

Plot of the umklapp scattering $g_3({\omega }_1,{\omega }_2,{\omega }_2,{\omega }_1)$ for $U=0.5$, ${\omega }_0=1.0$, and $g_3=0.55$. Note that $g_3(0,0,0,0)$ is flowing towards zero.