Contrary to canonical expectations we show that lattice translational symmetry breaking often accompanies uniformly ordered flux phases. We demonstrate this phenomena by studying a spinless-fermion model on a square latttice with nearest-neighbor repulsion. We find an array of flux patterns, as a function interaction strength and filling factor, that break time-reversal symmetry but may or may not preserve translational symmetry. A key finding is that the pattern of flux ordering does not uniquely determine the electronic properties. As such a class of new phases of matter are introduced that expand the candidate ground states in many body interacting systems.
A two-dimensional (2D) Wigner-Jordon transformation which maps spin variables into spinless fermion variables is found and is used to study the 2D spin-1/2 antiferromagnetic Heisenberg model. The transformation generates a fictitious gauge field in the XY component of the Heisenberg model, and hence induces an in-phase orbital current in the Ising component flowing around each elementary plaquette of the underlying lattice. The ground state of the Heisenberg model in a 2D square lattice is found to be an in-phase N\'eel flux phase, i.e., a coexisting state of the flux phase with in-phase orbital currents and a long-range antiferromagnetic spin order. The zero-temperature mean-field energy of the in-phase N\'eel flux state,
E0=-0.33J per bond, is only 1% higher than the best-estimated ground-state energy, -0.334J per bond.