Interplay Between $d$-wave Superconductivity and a Bond Density Wave in the one-band Hubbard model

It is now well established that superconducting cuprates support a charge density wave state in the so-called underdoped region of their phase diagram. We investigate the possibility of charge order in the square-lattice Hubbard model, both alone and in coexistence with $d$-wave superconductivity. The charge order has a period four in one direction, is centered on bonds and has a $d$ form factor. We use the variational cluster approximation, an approach based on a rigorous variational principle that treats short-range correlations exactly, with two clusters of size $2\times6$ that together tile the infinite lattice and provide a non-biased unit for a period-four bond density wave (BDW). We find that the BDW exists in a finite range of hole doping and increases in strength from $U=5$ to $U=8$. Its location and intensity depends strongly on the band dispersion. When probed simultaneously with $d$-wave superconductivity, the energy is sometimes lowered by the presence of both phases, depending on the interaction strength. Whenever they coexist, a pair-density wave (a modulation of superconducting pairing with the same period and form factor as the BDW) also exists.