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We propose a scaling theory for the many-body localization (MBL) phasetransition in one dimension, building on the idea that it proceeds via a'quantum avalanche'. We argue that the critical properties can be captured at acoarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG)flow. On phenomenological grounds, we identify the scaling variables as thedensity of thermal regions and the lengthscale that controls the decay oftypical matrix elements. Within this KT picture, the MBL phase is a line offixed points that terminates at the delocalization transition. We discuss twopossible scenarios distinguished by the distribution of rare, fractal thermalinclusions within the MBL phase. In the first scenario, these regions have astretched exponential distribution in the MBL phase. In the second scenario,the near-critical MBL phase hosts rare thermal regions that are power-lawdistributed in size. This points to the existence of a second transition withinthe MBL phase, at which these power-laws change to the stretched exponentialform expected at strong disorder. We numerically simulate two differentphenomenological RGs previously proposed to describe the MBL transition. BothRGs display a universal power-law length distribution of thermal regions at thetransition with a critical exponent $\alpha_c=2$, and continuously varyingexponents in the MBL phase consistent with the KT picture.

Kosterlitz-Thouless scaling at many-body localization phase transitions