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We discover a novel localization transition that alters the dynamics of coherence in disordered many-body spin systems subject to Markovian dissipation.
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In addition to the exponential decay whose rate is proportional to γ \gamma , the coherence may have a polynomial decay factor due to interactions as in Hermitian cases Serbyn and Abanin 2017 ; Abanin et al. 2019 . However, this factor is negligible compared with the exponential decay
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While the analytical formula for ensembles different from a complex Ginibre ensemble (e.g., a real Ginibre ensemble and an ensemble with transposition symmetry Hamazaki et al. 2020 ) is not known, we have numerically confirmed this distribution. If λ α \lambda_{\alpha} is real, then r α G r_{\alpha}^{G} and r α G ′ r^{G^{\prime}}_{\alpha} can be taken to be real, and their distributions may be different from the case of complex eigenvalues. We here do not consider real eigenvalues because their fractions vanish in the limit of large system size
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