  In this note, we perform the large-charge expansion for non-relativistic
systems with a global $U(1)$ symmetry in $3+1$ and $2+1$ space-time dimensions,
motivated by applications to the unitary Fermi gas and anyons. These systems do
not have full conformal invariance, but are invariant under the Schr\"odinger
group. Also here, the low-energy physics is encoded by a Goldstone boson which
is due to the breaking of the global symmetry when fixing the charge. We find
that in $2+1$ dimensions and higher, there is a large-charge expansion in which
quantum corrections are suppressed with respect to the next-to-leading order
terms in the Lagrangian. We give the next-to-leading-order expressions for the
ground state energy and the speed of sound.
