System-environmental entanglement in critical spin systems under $ZZ$-decoherence and its relation to strong and weak symmetries

Open quantum many-body systems exhibit nontrivial behavior under decoherence. In particular, system-environmental entanglement (SEE) is one of the efficient quantities for classifying mixed states subject to decoherence. In this work, we investigate the SEE of critical spin chains under nearest-neighbor $ZZ$-decoherence. We numerically show that the SEE exhibits a specific scaling law, in particular, its system-size-independent term (``$g$-function'') changes drastically its behavior in the vicinity of phase transition caused by decoherence. For the XXZ model in its gapless regime, a transition diagnosed by strong R\'{e}nyi-2 correlations occurs as the strength of the decoherence increases. We determine the location of the phase transition by investigating the $g$-function that exhibits a sharp change in the critical region of the transition. Furthermore, we find that the value of the SEE is twice that of the system under single-site $Z$-decoherence, which was recently studied by conformal field theory. From the viewpoint of R\'{e}nyi-2 Shannon entropy}, which is closely related to the SEE at the maximal decoherence, we clarify the origin of this $g$-function behavior.