Superconductivity and charge density wave order in the 2D 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 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 (KT) universality class, and is much less well characterized. In this paper we determine Tsc for the square lattice, for several values of the density ρ and phonon frequency ω0. We find that quasilong range order sets in at Tsc 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.

Quantum Monte Carlo study of an anharmonic Holstein model

“…when the system is doped away from a CDW phase. 51 Such a transition was recently observed in variational Monte Carlo simulations 22 and was also reported in [52].…”

“…This first order transition is present, though not widely studied previously, in the harmonic Holstein model. 22,52 In the homogeneous phase below half-filling, for ω = 0.5, we observed a clear enhancement of the superconducting susceptibility at finite temperature as ω 4 is increased. However, with the limited range of accessible temperatures, we were not able to observe the superconducting transition.…”

“…Using two values of ω yields the evolution of the anharmonic effects as a function of ω and also allows comparison with previous studies. 40,52 The average value of x i on a doubly occupied site can be roughly estimated as −2λ/ω 2 (see Appendix A). With this expression, the ratio η of the anharmonic to harmonic terms is given by…”

“…However, in general, the transition temperatures appear to be low. 31,52 For the ω = 1 harmonic Holstein model, the transition towards a SC state happens for β ≃ 28 52 and even larger inverse temperature for ω = 0.5. This makes it difficult to observe the effects of anharmonicity on the critical temperature itself, especially since ω 4 is expected to reduce T c even further.…”

Charge Singlets and Orbital Selective Charge Density Wave Transitions

“…Quantum Monte Carlo (QMC) simulations have already been applied to the analysis of charge-density wave (CDW) and superconducting (SC) transitions in the single band Holstein model [28]. However, thus far work has focused on mostly on two-dimensional or three-dimensional models with a single kinetic energy scale [29][30][31][32][33][34][35][36][37][38][39][40].…”