The layer-antisymmetric pair-phase resonance sits at the bonding-antibonding splitting 2t_h because the antisymmetric bubble reduces pointwise to the static symmetric bubble at that frequency.
Each term inP †,un − carries one bonding and one antibonding creation operator, so the contributions ofH th cancel pairwise and one finds [H th , P †,un − ] = 0
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Layer-antisymmetric pair-phase resonance at the bonding-antibonding splitting in the AA-stacked bilayer attractive Hubbard model
The layer-antisymmetric pair-phase resonance sits at the bonding-antibonding splitting 2t_h because the antisymmetric bubble reduces pointwise to the static symmetric bubble at that frequency.