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Symmetry classification of bond order parameters in cuprates

We study bond-order parameters for generalized $t$-$J$ models on a square lattice. Using the plane-wave limit the considered order parameters form basis functions for irreducible representations of the symmetry transformations of the point group and of time reversal. We show that for instability wave vectors along the diagonals all possible basis functions are either fine-tuned (i.e., obey restrictions beyond the requirements of symmetry) or break time reversal symmetry and thus describe flux states. For instability wave vectors along the crystalline axes, corresponding to the observed case in underdoped cuprates, there are only three representations with $A_1$, $B_1$, and $E$ symmetry which do not break time reversal symmetry in the general case. We suggest that one of them has recently been observed in resonant elastic X-ray scattering.