Coexistence of Charge Density Wave and Superconductivity in a correlated regime

Many different systems present the common feature of the coexistence between superconductivity and charge density wave [1,2,3,4]. As an example of such systems, we can highlight the cuprate superconductors [4], in which, it seems to be essential to understand the mechanisms behind the competition of these ordered states, in order to get a clear comprehension about the whole phase diagram, including the pseudogap phenomena. With the intention of studying both superconductivity and charge density wave (CDW) and its coexistence in a correlated regime, we combine the Green's functions formalism and the Hubbard-I approximation, as a way to take into account the correlations due to the repulsive Coulomb interaction $U$. The superconductivity and the CDW have been considered starting from a BCS like model [5], while the Coulomb interaction is included through a two-dimensional one band Hubbard model [6].
The obtained results show that the presence of superconductivity tends to suppress the charge density wave and vice versa. In addition, the careful analysis of the results obtained in this work showed that the Coulomb interaction $U$ acts in the sense of suppressing both superconductor and CDW states. As a consequence, the region of coexistence of both phases, in the parameters space, is also reduced by the effects of $U$.
For the CDW gap, it has been considered a $s$-wave symmetry. On the other hand, the superconductor gap can assumes $s$-wave symmetry or $d_{x^2-y^2}$-wave symmetry. The results show that in the coexistence region, the superconductivity with $d_{x^2-y^2}$-wave symmetry is more robust when compared with a superconductor state with $s$-wave symmetry.
The effect of the second second nearest neighbors hopping $t_1$ on the coexistence regime, is also investigated.
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