The connection between nonzero density and spontaneous symmetry breaking for interacting scalars

We consider ${\rm U}(1)$-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in $d>2$ spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of $\mu$ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincar\'e invariant vacuum of the theory. The same conclusions hold non-perturbatively for an ${\rm O}(N)$ theory with quartic interactions in $d=3$ and $4$, at leading order in the $1/N$ expansion. We derive these results by computing analytically the zero-temperature, finite-$\mu$ one-loop effective potential. We check our results against the one-loop low-energy effective action for the superfluid phonons in $\lambda \phi^4$ theory in $d=4$ previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-$n$ operator for the $\lambda \phi^6$ conformal superfluid in $d=3$, at leading order in $1/n$, reproducing a numerical result of Badel et al. For a $\lambda \phi^4$ superfluid in $d=4$, we also reproduce the Lee--Huang--Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.