Phonons in Hubbard ladders studied within the framework of the one-loop   renormalization group
  We study the effects of phonons in $N$-leg Hubbard ladders within the framework of a one-loop renormalization group. In particular, we explicitly demonstrate that the role of phonons changes {\em qualitatively} even in the simplest two-leg ladder, as compared to the single-chain system where phonons always dominate. Our numerical results suggest that in the spin-gapped phase of the two-leg ladder, the opening of the spin gap by electron-electron interaction also drives the electron-phonon interaction to strong coupling, but in a {\em subdominant} fashion. Therefore, even though the inclusion of phonons does not alter the phase, their subdominant relevance strongly renormalizes some physical properties below the energy scale of the spin gap. This might shine some light on the recent experiments showing an anomalous isotope effects in high-temperature superconductors. 
Alexander Seidel, Hsiu-Hau Lin, Dung-Hai Lee