Aharonov-Bohm effect for the network of critical trajectories near Lifshitz transition leads to precursors of Brown-Zak oscillations.

Moiré quasi-crystals open previously inaccessible regimes of Fermi surface (FS) topologies. We consider the magnetotransport in the vicinity of Lifshitz transitions that happen when the size of FS compares to the size of mini-Brillouin zone and the FS touches with its copy in the next unit cell. As a result, FS forms an infinite periodic 2D network. In a perpendicular magnetic field, electrons drift in a fixed direction along the edges while having a choice (determined by the amplitude of magnetic breakdown) where to go at each vertex.

We show that interference of electronic paths on the network leads to conductance oscillations periodic in the inverse magnetic flux per unit cell that survive elevated temperatures and correspond to Brown-Zak oscillations. We develop an intuitive semiclassical approach explaining the magnitude of the oscillations. Surprisingly, for hexagonal systems with C₃ symmetry, the oscillations are maximal away from the Lifshitz transition, which is explained by the topology of interfering paths.

Our findings are supported by comparison with experimental data for twisted double-bilayer graphene.