Real space Eliashberg approach to charge order of electrons coupled to dynamic antiferromagnetic fluctuations

We study charge ordered solutions for fermions on a square lattice interacting with dynamic antiferromagnetic fluctuations. Our approach is based on real space Eliashberg equations which are solved self-consistently. We first show that the antiferromagnetic fluctuations can induce arc features in the spectral functions, as spectral weight is suppressed at the hot spots; however, no real pseudogap is generated. At low temperature spontaneous charge order with a $d$-form factor can be stabilized for certain parameters. As long as the interacting Fermi surfaces possesses hot spots, the ordering wave vector corresponds to the diagonal connection of the hot spots, similar to the non-self-consistent case. Tendencies towards observed axial order only appear in situations without hot spots.