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Motivated by experiments on the carrier-doped bilayer iridate$(\text{Sr}_{1-x}\text{La}_x)_3\text{Ir}_2\text{O}_7$, we study the dynamics ofa single doped electron in a bilayer magnet in the presence of spin-orbitcoupling, taking into account the spatially staggered rotation of IrO$_6$octahedra. We employ an effective single-orbital bilayer $t$-$J$ model,concentrating on the quantum paramagnetic phase near the magnetic quantumcritical point. We determine the carrier dispersion using a combination ofself-consistent Born and bond-operator techniques. Extrapolating to finitesmall carrier density we find that, for experimentally relevant parameters, thecombination of octahedral rotation and spin-orbit coupling induces a bandfolding which results in a Fermi surface of small double electron pockets, instriking agreement with experimental observations. We also determine theinfluence of spin-orbit coupling on the location of the quantum critical pointin the undoped case, and discuss aspects of the global phase diagram of dopedbilayer Mott insulators.

Carrier dynamics in doped bilayer iridates near magnetic quantum criticality