Mean-field theory of competing orders in metals with antiferromagnetic exchange interactions

It has long been known that two-dimensional metals with antiferromagnetic exchange interactions have a weak-coupling instability to the superconductivity of spin-singlet, d-wave electron pairs. We examine additional possible instabilities in the spin-singlet particle-hole channel and study their interplay with superconductivity. We perform an unrestricted Hartree-Fock-BCS analysis of bond order parameters in a single-band model on the square lattice with nearest-neighbor exchange and repulsion, while neglecting on-site interactions. The dominant particle-hole instability is found to be an incommensurate, bidirectional, bond density wave with wave vectors along the (1,1) and (1,-1) directions and an internal d-wave symmetry. The magnitude of the ordering wave vector is close to the separation between points on the Fermi surface, which intersect the antiferromagnetic Brillouin zone boundary. The temperature dependence of the superconducting and bond order parameters demonstrates their mutual competition. We also obtain the spatial dependence of the two orders in a vortex lattice induced by an applied magnetic field: halos of the bond order appear around the cores of the vortices.