Effective Description of Ajar Systems with a $U(1)$ Symmetry

We introduce the concept of "ajar systems" as an intermediate case between closed and open systems, where the time scale for charge exchange with the environment is parametrically larger than all other characteristic time scales. The Schwinger-Keldysh effective action for such finite-temperature systems exhibits a symmetry group $G_1 \times G_2$ weakly broken to its diagonal subgroup $G_{\text{diag}}$, which we systematically describe using spurion techniques. For $G = U(1)$, we calculate leading-order corrections to correlation functions in both diffusive and spontaneously broken phases. Unlike systems with approximate symmetries where both real and imaginary parts of dispersion relations receive corrections, ajar systems modify only the imaginary part -- preserving gapless modes while adding finite damping.