  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.
