Phase diagram of the Holstein-Hubbard two-leg ladder using a functional renormalization-group method

Using a functional renormalization-group method, we obtain the phase diagram of the two-leg ladder system within the Holstein-Hubbard model, which includes both electron-electron and electron-phonon interactions. Our renormalization-group technique allows us to analyze the problem for both weak and strong electron-phonon couplings. We show that, in contrast to results from conventional weak-coupling studies, electron-phonon interactions can dominate electron-electron interactions, and retardation effects play an important role on the phase diagram.

Figure 1

Phase diagram of Holstein-Hubbard ladder as a function of electron-phonon coupling $g_{\mathrm{ep}}$ and doping ($t=1$ and $U=2$). Filled circles, ${\omega }_0=1$; open circles, without retardation (antiadiabatic limit). Zero doping corresponds to half-filling.

Figure 2

Energy dispersion for free electrons in a two-leg ladder.

Figure 3

The flows of $s$-wave SC (solid lines) and $d$-wave SC (dotted lines) susceptibilities as a function of $l=\ln ({\Lambda }_0∕\Lambda )$ for $U=2$, ${\omega }_0=1$, and doping of 0.2.