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The Holstein Hamiltonian describes fermions hopping on a lattice andinteracting locally with dispersionless phonon degrees of freedom. In the lowdensity limit, dressed quasiparticles, polarons and bipolarons, propagate withan effective mass. At higher densities, pairs can condense into a lowtemperature superconducting phase and, at or near commensurate filling on abipartite lattice, to charge density wave (CDW) order. CDW formation breaks adiscrete symmetry and hence occurs via a second order (Ising) transition, andtherefore at a finite $T_{\rm cdw}$ in two dimensions. Quantum Monte Carlocalculations have determined $T_{\rm cdw}$ 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 (KT) universalityclass, and is much less well characterized. In this paper we determine $T_{\rmsc}$ for the square lattice, for several values of the density $\rho$ andphonon frequency $\omega_0$. We find that quasi-long range order sets in at$T_{\rm sc} \lesssim t/20$, where $t$ is the near neighbor hopping amplitude,consistent with previous rough estimates from simulations which onlyextrapolated to the temperatures we reach from considerably higher $T$. We alsoshow evidence for a discontinuous evolution of the density as the CDWtransition is approached at half-filling.

Superconductivity and charge density wave order in the 2D Holstein model