 Discrete remnant of a circle symmetry

Suppose fields with integer U(1) charges $Q_i$ acquire nonzero vacuum expectation values, with at least one $Q_i\ne 0$. The stabilizer in the circle group is exactly ${\mathbb{Z}}_d$, where $d=\gcd (|Q_1|,\dots ,|Q_r|)$. Indeed, $e^{i\theta }$ preserves every condensate if and only if $e^{i{Q}_i\theta }=1$ for every $i$. The Bezout identity implies $e^{id\theta }=1$, and conversely $d\mid Q_i$ proves every $d$th root of unity preserves every condensate. Neutral fields impose no constraint; if all condensing fields are neutral, the whole circle group survives.