We determine the finite-temperature phase diagram of a one-dimensional disordered Bose gas using bosonization and the nonperturbative functional renormalization group (RG). We discuss two different scenarios, based on distinct truncations of the effective action. In the first scenario, the Bose glass is destabilized at any finite temperature, giving rise to a normal fluid. Nevertheless, one can distinguish a low-temperature glassy regime, where disorder plays an important role on intermediate length and time scales, from a high-temperature regime, where disorder becomes irrelevant. In the second scenario, below a temperature $T_c$, the RG flow exhibits a singularity at a finite value of the RG momentum scale. We propose that this singularity signals a lack of thermalization and the existence of a localized phase for $T<T_c$. We provide a description of this low-temperature localized phase within a droplet picture and highlight a number of possible similarities with characteristics of many-body localized phases, including nonthermal behavior, avalanche instabilities and many-body resonances, the structure of the many-body spectrum, and slow dynamics in the ergodic phase. The normal fluid above $T_c$, and below a crossover temperature $T_g$, exhibits glassy properties on intermediate scales.

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• Many-body localization
• 1-dimensional systems
• Bose gases
• Disordered systems
• Bosonization
• Functional renormalization group