Layer-antisymmetric pair-phase resonance at the bonding-antibonding splitting in the AA-stacked bilayer attractive Hubbard model

Abstract: The relative phase between the two pair condensates of a bilayer s-wave superconductor is a collective degree of freedom distinct from the usual in-phase Anderson-Bogoliubov mode. Working at the Gaussian fluctuation level for the AA-stacked attractive-Hubbard honeycomb bilayer, we show analytically that the layer-antisymmetric pair-phase channel hosts an in-gap collective pole at twice the single-particle interlayer hopping, $2t_h$, precisely the bonding-antibonding band splitting. The mechanism is algebraic: at this frequency, the antisymmetric phase bubble reduces pointwise in momentum space to the static symmetric phase bubble that enforces the in-phase Goldstone pole. The resulting resonance scale is therefore fixed by the single-particle hybridization, rather than by the interaction-driven Josephson coupling that controls the canonical Leggett mode. The identity is verified numerically by direct Bogoliubov-de Gennes calculations. The diagonal antisymmetric phase-channel kernel zero is exact within Gaussian theory at any chemical potential; the full coupled amplitude-phase pole coincides with it at half filling and tracks it closely away from half filling. The excitation is Raman-forbidden by inversion, which motivates layer-odd probes. We find that a layer-imbalance drive has finite Gaussian-level overlap with the pair-phase sector, suggesting a possible cold-atom layer-bias response feature near the sub-kilohertz scale for typical optical-lattice parameters.