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Tunneling and Fluctuating Electron-Hole Cooper Pairs in Double Bilayer Graphene

A strong low-temperature enhancement of the tunneling conductance between graphene bilayers has been reported recently, and interpreted as a signature of equilibrium electron-hole pairing, first predicted in bilayers more than forty years ago but previously unobserved. Here we provide a detailed theory of conductance enhanced by fluctuating electron-hole Cooper pairs, which are a precursor to equilibrium pairing, that accounts for specific details of the multi-band double graphene bilayer system which supports several different pairing channels. Above the equilibrium condensation temperature, pairs have finite temporal coherence and do not support dissipationless tunneling. Instead, they strongly boost the tunneling conductivity via a fluctuational internal Josephson effect. Our theory makes predictions for the dependence of the zero bias peak in the differential tunneling conductance on temperature and electron-hole density imbalance that capture important aspects of the experimental observations. In our interpretation of the observations, cleaner samples with longer disorder scattering times would condense at temperatures $T_c$ up to $\sim 50 {\rm K}$, compared to the record $T_c \sim 1.5 $K achieved to date in the experiment.