Electron-phonon-coupling-driven pairing symmetry transition in a ladder

We address the effects of electron-phonon coupling in the electron-electron interacting ladder using the recently developed functional renormalization group method, in which the full retardation effects can be taken into account impartially\footnote{S.-W. Tsai, A.H. Castro Neto, R. Shankar, D.K. Campbell, Phys. Rev. B 72, 054531 (2005)}. We study the doped Holstein-Hubbard ladder as a typical example and show that there is a transition between s-wave and d-wave pairing as a function of electron-phonon coupling and doping level. This contrasts with recent results from a two-step renormalization group, which suggest that the electron-phonon coupling only contributes in a subdominant fashion and that the spin-gapped pairing phase always has d-wave symmetry, unchanged from the doped Hubbard ladder without electron-phonon interaction\footnote{Alexander Seidel, Hsiu-Hau Lin, Dung-Hai Lee, Phys. Rev. B 71, 220501 (2005)}.