Second order Josephson effect in excitonic insulators

We show that in electron-hole bilayers with conduction and valence bands formed by atomic orbitals with different parities, nonzero interlayer tunneling leads to a second order Josephson effect in the excitonic insulating state, where the interlayer electrical current is related to the phase of the excitonic order parameter as J = J_c sin(2θ) instead of J = J_c sin(θ). An interlayer voltage rotates the phase in the same way as AC Josephson effect. Suitable voltage pulses can switch the system between the two degenerate ground states with θ=0,π, making it a potentially ultra fast memory device. We also discuss a three dimensional stack of alternating electron-hole layers and a two dimensional stack of electron-hole chains, where the excitonic insulating states exhibit nontrivial topology.