Phonons in Hubbard ladders studied within the framework of the one-loop renormalization group

TL;DR

This paper investigates how phonons influence the physics of Hubbard ladders using one-loop renormalization group analysis, revealing that phonons play a qualitatively different and subdominant role compared to single chains, especially in the spin-gapped phase.

Contribution

It demonstrates the qualitative change in phonon effects in Hubbard ladders versus single chains and analyzes their subdominant but significant influence on physical properties.

Findings

Phonons behave differently in two-leg ladders compared to single chains.

Electron-phonon interaction becomes strongly coupled in the spin-gapped phase.

Phonons do not change the phase but significantly renormalize physical properties.

Abstract

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.