Exact Gaussian pinning of a relative-phase mode and interaction-driven detuning from the pinned frequency

Abstract

Hybridization between two identical fermionic layers or components produces a rigid bonding--antibonding splitting $2t_h$. We show that in a paired state this scale enters the collective dynamics exactly within Gaussian fluctuation theory: the copy-odd relative-phase kernel vanishes at $ω=2t_h$, with the underlying parity-pair identity holding pointwise in momentum. Within a broad class of conventionally paired systems, the result is independent of chemical potential, band structure, orbital content, and dimensionality. It follows from copy-pseudospin Larmor precession generated by intercopy hopping together with the Goldstone--Ward condition imposed by superconducting self-consistency. Pairing also isolates $2t_h$ from quasiparticle excitations, with thermal scattering below and pair creation above it whenever the gap is finite. At particle-hole symmetry, where the copy-odd amplitude decouples, the kernel zero becomes an isolated relative-phase pole at $Ω_L=2t_h$, and the layer-imbalance response is a single Gaussian mode exhausting its Gaussian first moment; the corresponding full-Hamiltonian first moment obeys an exact operator sum rule. This pinning is not protected beyond Gaussian order. An exact interaction torque destroys the Larmor closure, while an exactly solvable rung and exact diagonalization of the attractive Hubbard model on a periodic $N_s=12$, $z=3$ bilayer honeycomb cluster show strong detuning. At $|U|=4t$, the dominant layer-imbalance excitation occurs at $0.467(2t_h)$ while retaining $98.6\%$ of the zeroth-moment spectral weight but only $92.4\%$ of the exact first moment. Thus the Gaussian result provides an exact reference frequency whose many-body renormalization can be substantial while a dominant layer-odd excitation survives.