Second order Josephson effect in excitonic insulators

We show that in electron-hole bilayers with excitonic order arising from conduction and valence bands formed by atomic orbitals that transform differently under inversion, nonzero interlayer tunneling leads to a second order Josephson effect. This means the interlayer electrical current is related to the phase of the excitonic order parameter as $J = J_c sin2theta$ instead of $J = J_c sin theta$, and that the system has two degenerate ground states that can be switched by an interlayer voltage. In a three dimensional stack of alternating electron-hole planes or a two dimensional stack of chains, the second order Josephson coupling can lead to a Weyl semimetal or a quantum anomalous hall insulator, respectively. A generic order parameter steering effect is demonstrated, whereby electric field pulses perpendicular to the layers and chains can steer the order parameter phase between the two degenerate ground states. The steering is also applicable to the excitonic insulator candidate Ta$_2$NiSe$_5$.