Dependence of incommesurability on superconducting and charge density wave phases in multiband intermetallic systems

The interplay between superconductivity (SC) and charge density wave (CDW) phases is an issue under intense debate in many quasi-2D compounds - from cuprates to transition-metal dichalcogenides. In particular, it is worth investigating the effects of incommensurable CDW states, and how such incommensurability affects the pairing response of the system. As a step towards this end, we study a two-band model in the square lattice, with the main goal to describe intermetallic systems with a d-band of moderately correlated electrons coexisting with a large sp band.
For simplicity, here we assume all interactions and the hybridizations as site-independent. We examine this problem by implementing an unrestricted Hartree-Fock approximation, and obtaining the mean-field variables by minimizing the free energy density. As our main results, at half-filling, and for commensurate values of
wavevectors, we obtain a staggered CDW phase as the most stable state, therefore reproducing previous results [1].
However, away from half-filling, we find an incommensurate CDW phase at the ground state, which is sensitive to the parameters introduced in the model. Then, we provide phase diagrams that emphasize the competition between these phases, and the role of incommensurability in their interplay.

[1] Nei Lopes, Daniel Reyes, Mucio A. Continentino, Christopher Thomas, Phys. Rev. B 103, 195150 (2021)