Precise correspondence between the p-wave chiral superfluid and the spinless bosonic superfluid on the lowest Landau level

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 accounting for the correct superfluid vorticity in the effective theory of the former. In literature, the vorticity manifests itself incompletely as the magnetic field. This paper demonstrates this substitution can be understood as a truncation in the relevant expansion scheme, together with field redefinitions. Those discarded components are critical to restoring the Berry phase term in the effective theory, and the restoration incapsulates these two systems in the same master Lagrangian. In addition to clarifying the structure of the Berry phase, this formalism permits solving the bosonic system on the lowest Landau level (LLL) based on the method of analogue. In particular, we show that in the linear regime, the Maxwell equations governing these systems are identical by reformulating the vortex crystal with an auxiliary electromagnetic field. This approach offers a unified view for these systems and yields solutions that are rotationally covariant, gauge invariants, and physically interpretable.