Momentum-Space AC Josephson Effect and Intervalley Coherence in Multilayer Graphene

Electron transport driven by the phase coherence and interference of quantum many-body wavefunctions is a fascinating phenomenon with potential technological significance. Superconductivity, for example, enables dissipationless transport through macroscopic phase twisting. Similarly, in charge-density waves, once the phase degree of freedom—representing the collective position of electrons relative to the lattice—is depinned, it generates characteristic broadband noise and intriguing AC-DC interference patterns. In this work, we point out a phase-coherent dynamics in the intervalley coherent (IVC) state, also known as the bond-ordered or Kekul\'e distorted state, frequently reported in rhombohedral multilayer graphene. Under a static magnetic field, the IVC state responds with an oscillating intervalley current, which in turn causes oscillating orbital magnetization, thereby inducing a detectable AC Hall effect. This mechanism mirrors the AC Josephson effect observed in superconductors but now happening in momentum space. In this analogy, the static magnetic field acts as the DC voltage, while the oscillating intervalley current assumes the role of the AC Josephson current. We present detailed microscopic calculations for all the parameters of the phase-number free-energy in rhombohedral trilayer graphene, predicting an orbital magnetization oscillation frequency of approximately 12 GHz at 0.1 Tesla. We comment on this phase-coherent dynamics in other 2D materials like twisted homobilayer transition metal dichalcogenides.