Hohenberg-Mermin-Wagner-type theorems and dipole symmetry

We study the possibility of spontaneous symmetry breaking in systems with both charge and dipole symmetries. For $d$-dimensional systems at a positive temperature, we show that charge symmetry cannot be spontaneously broken for $d\leq 4$, while dipole symmetry cannot be spontaneously broken for $d\leq 2$. For $T=0$, we show that charge symmetry cannot be spontaneously broken for $d\leq 2$ if the compressibility is finite. We also show that continuum systems with a dipole symmetry have infinite inertial mass density.