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Abstract

We establish a precise formal correspondence between a spinless p-wave chiral fermionic superfluid and a spinless bosonic superfluid in a strong magnetic field by correctly accounting for superfluid vorticity in the effective theory of the former. In the literature, this vorticity term incompletely manifests as the magnetic field. This paper demonstrates this substitution can be understood as a truncation within the relevant expansion scheme, accompanied by field redefinitions. The components discarded in this truncation are critical for restoring the Berry phase term in the effective theory, encapsulating both systems in the same master Lagrangian. Beyond clarifying the structure of the Berry phase, this formalism allows for solving the bosonic system in the lowest Landau level (LLL) by analogy. Specifically, we show that, in the linear regime, the Maxwell equations governing these systems are identical when the vortex crystal is reformulated using an auxiliary electromagnetic field. This approach offers a unified perspective on these systems and yields solutions that are rotationally covariant, gauge invariants, and physically interpretable.