  The electron-phonon (e-ph) interaction remains of great interest in condensed
matter physics and plays a vital role in realizing superconductors,
charge-density-waves (CDW), and polarons. We study the two-dimensional Holstein
model for e-ph coupling using determinant quantum Monte Carlo across a wide
range of its phase diagram as a function of temperature, electron density,
dimensionless e-ph coupling strength, and the adiabatic ratio of the phonon
frequency to the Fermi energy. We describe the behavior of the CDW
correlations, the competition between superconducting and CDW orders and
polaron formation, the optimal conditions for superconductivity, and the
transition from the weak-coupling regime to the strong-coupling regime.
Superconductivity is optimized at intermediate e-ph coupling strength and
intermediate electron density, and the superconducting correlations increase
monotonically with phonon frequency. The global maximum for superconductivity
in the Holstein model occurs at large phonon frequency, the limit where an
attractive Hubbard model effectively describes the physics.
