Effective model for a supercurrent in a pair-density wave

We extend the standard effective model of d-wave superconductivity of a single band tight-binding Hamiltonian with nearest-neighbor attraction to include finite range periodically modulated pair-hopping. The pair-hopping is characterized by a fixed wave number \(\pmb{\mathcal{Q}}=\mathcal{Q}\hat{x}\) breaking lattice rotational symmetry. Within self-consistent BCS theory we study the general variational state consisting of two incommensurate singlet pair amplitudes \(\Delta_{{\bf Q}_1}\) and \(\Delta_{{\bf Q}_2}\) and find two types of near degenerate ground states; of the Larkin-Ovchnnikov (LO) or pair-density wave (PDW) type with \(\Delta_{{\bf Q}_1}=\Delta_{{\bf Q}_2}\) and \({\bf Q}_1=-{\bf Q}_2\approx \mathcal{Q}\) or of the Fulde-Ferrell (FF) type with \(\Delta_{{\bf Q}_2}=0\) and \({\bf Q}_1\approx \pm \mathcal{Q}\). An anomalous term in the static current operator arising from the pair-hopping ensures that Bloch's theorem on zero current in the ground state is enforced also for the FF ground state, despite broken time-reversal symmetry without spin-population imbalance. We also consider a supercurrent by exploring the space of pair-momenta \({\bf Q}_1\) and \({\bf Q}_2\) and identify characteristics of a state with multiple finite momentum order-parameters. This includes the possibility of phase-separation of current densities and spontaneous mirror-symmetry breaking manifested in the directional dependence of the depairing current.