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

Abstract

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 ${T}_{\mathrm{cdw}}$ in two dimensions. Quantum Monte Carlo calculations have determined ${T}_{\mathrm{cdw}}$ 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 ${T}_{\mathrm{sc}}$ for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, 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.