Spectral features of a many-body localized system weakly coupled to a heat bath

We study many-body-localized (MBL) systems that are weakly coupled to thermalizing environments, focusing on the spectral functions of local operators. We argue that these spectral functions carry signatures of localization even away from the limit of perfectly isolated systems. We find that, in the limit of vanishing coupling to a bath, MBL systems come in two varieties, with either discrete or continuous local spectra. Both varieties of MBL systems exhibit a "soft gap" at zero frequency in the spatially-averaged spectral functions of local operators, which serves as a diagnostic for localization. We estimate the degree to which coupling to a bath broadens these spectral features, and find that characteristics of incipient localization survive as long as the system-bath coupling is much weaker than the characteristic energy scales of the system. Since perfect isolation is impossible, we expect the ideas discussed in this paper to be relevant for all experiments on many-body localization.