  Atomic Parity Violation (APV) is usually quantified in terms of the weak
nuclear charge $Q_W$ of a nucleus, which depends on the coupling strength
between the atomic electrons and quarks. In this work, we review the importance
of APV to probing new physics using effective field theory. Furthermore, using
$SU(2)$ invariance, we correlate our findings with those from neutrino-nucleus
coherent scattering. Moreover, we investigate signs of parity violation in
polarized electron scattering and show how precise measurements on the Weinberg
angle, $\sin \theta_W$, will give rise to competitive bounds on light mediators
over a wide range of masses and interactions strength. Lastly, apply our bounds
to several models namely, Dark Z, Two Higgs Doublet Model-$U(1)_X$ and 3-3-1,
considering both light and heavy mediator regimes.
