Dephasing Catastrophe in 4-ε Dimensions: A Toy Model for the Ergodic to Many-Body-Localized Phase Transition

In this work, we propose a strategy to investigate the two-dimensional ergodic to many-body-localized (MBL) phase transition as a dephasing catastrophe by approaching from the ergodic side. In a closed interacting fermion system with quenched disorder, the dephasing of weak localization corrections to conductivity is caused by inelastic electron-electron collisions, which can be interpreted as interactions between electrons and thermal fluctuations of the hydrodynamic mode. For system with short-range interactions, the dephasing problem does not admit a closed-form solution due to the diffusive and non-Markovian nature of the fluctuations. It is reformulated as a geometric statistical-mechanical problem of a self-interacting polymer loop whose characteristic length scale is determined by the dephasing length. In the renormalization group framework, we study the critical behavior through a controlled epsilon expansion from the upper critical dimensions d=4. We find a nontrivial fixed point corresponding to temperature T*>0 where the dephasing rate vanishes. This critical point is associated with the toy version of ergodic-MBL transition in d=2 if it survives to ε=2. The analytical results reported here could be tested with a lattice polymer simulation.