Superconductivity and charge density wave order in the 2D 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 $T_{\rm cdw}$ in two dimensions. Quantum Monte Carlo calculations 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) universality class, and is much less well characterized. In this paper we determine $T_{\rm sc}$ for the square lattice, for several values of the density $\rho$ and phonon 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 only extrapolated to the temperatures we reach from considerably higher $T$. We also show evidence for a discontinuous evolution of the density as the CDW transition is approached at half-filling.