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In two dimensions (2D), dephasing by a bath cuts off Anderson localizationthat would otherwise occur at any energy density for fermions with disorder.For an isolated system with short-range interactions, the system can be its ownbath, exhibiting diffusive (non-Markovian) thermal density fluctuations. Werecast the dephasing of weak localization due to a diffusive bath as aself-interacting polymer loop. We investigate the critical behavior of the loopin $d=4-\epsilon$ dimensions, and find a nontrivial fixed point correspondingto a temperature $T^* \sim \epsilon >0$ where the dephasing time diverges.Assuming that this fixed point survives to $\epsilon=2$, we associate it to apossible instability of the ergodic phase. Our approach may open a new line ofattack against the problem of the ergodic to many-body-localized phasetransition in $d > 1$ spatial dimensions.

Dephasing catastrophe in $4 - \epsilon$ dimensions: A possible instability of the ergodic (many-body-delocalized) phase