Charge-density-wave and superconducting states in the Holstein model on a square lattice

Abstract

The physics of the charge-density-wave (CDW) and superconducting states (and the competition between them) in the two-dimensional Holstein model is studied by means of a unitary transformation method. The nonadiabatic effect due to a finite phonon frequency ${\mathrm{\ensuremath{\omega}}}_{0}$>0 is treated through two energy-dependent electron-phonon scattering functions ${\mathrm{\ensuremath{\delta}}}_{1}$(${\mathrm{k}}^{|\mathrm{I}\mathrm{H}}$,k) and ${\mathrm{\ensuremath{\delta}}}_{2}$(${\mathrm{k}}^{|\mathrm{I}\mathrm{H}}$,k) introduced in the transformation. This leads to a weakening of the effective potential stabilizing the CDW state and results in the four-fermion terms, which lead to umklapp scattering terms in the CDW state and to Cooper pairing terms in the superconducting state. For the CDW state our calculated transition temperature ${\mathrm{T}}_{\mathrm{p}}$ is much lower than the adiabatic one. In particular, when ${\mathrm{\ensuremath{\omega}}}_{0}$/t\ensuremath{\ll}1 the ratio 2\ensuremath{\lambda}${\mathrm{m}}_{\mathrm{p}}$/${\mathrm{T}}_{\mathrm{p}}$ is much larger than the BCS value 3.53, because ${\mathrm{T}}_{\mathrm{p}}$ is suppressed by the thermal lattice fluctuations in the finite temperature case. For larger ${\mathrm{\ensuremath{\omega}}}_{0}$, however, the calculated ${\mathrm{T}}_{\mathrm{p}}$ is higher than that of Monte Carlo simulations. The calculated density of states (DOS) of electrons has a gap. When ${\mathrm{\ensuremath{\omega}}}_{0}$ is small the DOS is peaked above the true gap edge. For the superconducting state the gap equation is solved by a numerical method. The critical temperature ${\mathrm{T}}_{\mathrm{c}}$ and the DOS of electrons are calculated from small to large phonon frequencies. Finally, we discuss the competition between the CDW and superconducting correlations by comparing ${\mathrm{T}}_{\mathrm{p}}$ and ${\mathrm{T}}_{\mathrm{c}}$ (as functions of ${\mathrm{\ensuremath{\omega}}}_{0}$) for different band fillings n.