Pseudogap crossover in the electron-phonon system

Thermodynamic properties of the square-lattice Holstein model of the electron-phonon problem with phonon frequencies small compared to the bare Fermi energy are obtained using Monte Carlo methods, a strong-coupling (bipolaronic) expansion, and a weak coupling Migdal-Eliashberg approach. Already at elevated temperatures where the charge-density wave (CDW) and superconducting (SC) correlations are very short-range, a crossover occurs as a function of increasing electron-phonon coupling, $\lambda_0$, from a normal metallic regime to a pseudogap regime. At sufficiently low $T$, a SC phase is found for small $\lambda_0$ and a commensurate insulating CDW phase for large $\lambda_0$.