Study of the Two Dimensional Holstein Model using Determinant Quantum Monte Carlo and Migdal-Eliashberg Theory.

The simplest model for studying electron-phonon coupling in many body systems is the singleband Holstein model. Most studies that utilize this model examine the nature of charge-density and pairing correlations at or around half-filling, but no comprehensive calculation for the dependence of the critical temperatures T_c ^SC , T_c ^CDW on the electron occupancy 〈n〉 far away from half-filling seems to exist in the literature. Furthermore, many of these studies compare results from quantum Monte-Carlo (QMC) to Migdal-Eliashberg (ME) theory in an attempt to shed light on the importance of the vertex corrections neglected in ME theory. In this presentation, we will show results highlighting the ME theory phase diagram with respect to 〈n〉over multiple phonon frequencies Ω_ph using relatively large lattice sizes. Moreover, we will also compare DQMC against ME theory on smaller lattices across a wide range of 〈n〉 and for multiple phonon frequencies. The corresponding ME theory results are found to be comparable to the numerically exact DQMC for lower occupancies (〈n〉≤0.4 ). Lastly, we will discuss the implications of these findings for superconductors with low numbers of charge carriers.