Topological fluctuating electron-hole Cooper pairs in graphene-GaAs heterostructures

Fluctuating Cooper pairs formed by spatially separated electrons and holes are precursors of their equilibrium condensation. Their presence strongly impacts transport phenomena and interlayer tunneling in double-layer systems above the transition temperature. Here, we consider a hybrid graphene/quantum-well double-layer system and focus on the dynamics of fluctuating Cooper pairs formed by conventional electrons and Dirac holes. We demonstrate that the chiral nature of Dirac holes is manifested in the presence of two (almost) degenerate competing pairing channels, which are intertwined by effective pseudospin-orbit interactions. We argue that the spectrum of the Ginzburg-Landau Hamiltonian describing the energetics of fluctuating Cooper pairs is geometrically nontrivial and can be characterized by the half-integer topological Chern number. We derive a kinetic equation for fluctuating Cooper pairs and demonstrate that their nontrivial geometries generate two anomalous velocities of distinct geometrical origins. These velocities are intricately connected with the Berry curvature and the quantum metric for the Ginzburg-Landau Hamiltonian, respectively. The resulting anomalous contributions to conductivity are singular at the transition temperature, and we discuss possible setups for their experimental observation.