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.
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