Imaginary-spin-bond order induced by a magnetic field

We show that a 2D fermion system with well separated electron and hole pockets may develop a time-reversal symmetric directional imaginary-spin-bond order. It emerges when some interactions are repulsive and some are attractive. Moreover, we demonstrate that this order is necessarily triggered by a magnetic field, even if the order at zero field is a conventional spin-density-wave state. We show that a finite field linearly couples the bond order to the spin-density-wave order. We also discuss the ways to probe such an order in experiments.