Superconductivity and charge density wave order in the two-dimensional Holstein model

The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite T-cdw in two dimensions. Quantum Monte Carlo calculations have determined T-cdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d = 2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine T-sc for the square lattice for several values of the density rho and phonon frequency omega(0). We find that quasilong-range order sets in at T-sc less than or similar to t/20, where t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.