= Solution
A neutral <superfluid> has a complex <order parameter> whose phase describes coherent particle transport. For a Bose condensate it is represented by a field amplitude; for a paired Fermi superfluid it is a pair amplitude. In a symmetry-breaking description, selecting its phase breaks the global particle-number $U(1)$ symmetry. A common <phase shift> leaves the <energy> unchanged, while a phase gradient costs <energy> and produces <superfluid velocity>. Long-wavelength phase oscillations give a gapless <Goldstone mode>, generally a sound wave when density fluctuations provide the conjugate dynamics. A finite isolated state with exact particle number need not have a nonzero field expectation: the phase-selected language describes the ordered limit and relative-phase coherence.
In a <superconductor>, the paired <order parameter> carries charge of magnitude $2e$ and couples to the <electromagnetic field>. The meaningful spatial phase combination is $\hbar\nabla\theta-q\mathbf A$, with signed pair charge $q$. <Gauge symmetry> is a redundancy, so “breaking local gauge symmetry” is not a literal breaking of a measurable physical symmetry. The gauge-invariant electromagnetic response includes the <Meissner effect> and a finite <London penetration depth>. Long-range Coulomb <forces> move the would-be neutral phase mode to a plasma scale through the <Anderson-Higgs mechanism in a superconductor>.
Additional symmetries can be broken by particular pairing states. Conventional isotropic spin-singlet pairing need not break spatial rotations or time reversal. A <d-wave superconductor> has a gap transforming nontrivially under crystal symmetries, and some triplet or multicomponent states can also break spin-rotation or time-reversal symmetries. These additional choices are not required by superconductivity itself. A nonzero pairing amplitude, phase coherence and electromagnetic screening describe related but distinct aspects of the ordered state.
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