The large-charge expansion for Schr\"odinger systems

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.