Critical exponent gamma_c at the ergodic–MBL transition and its constancy

Determine the value of the critical exponent gamma_c that governs the power-law decay of the typical minimal deviation of local magnetization, delta_min^typ ~ A L^{-gamma}, at the ergodic-to-many-body localized transition in the random-field XXZ spin chain, and ascertain whether gamma_c is constant (i.e., universal) along the transition line.

Background

The paper analyzes diagnostics of ergodicity versus many-body localization (MBL) in the random-field XXZ chain using several observables, including extreme magnetization. For an eigenstate, the extreme magnetization is quantified by the minimal deviation of local magnetization from saturation, and its typical value scales with system size as delta_min^typ ~ A L^{-gamma} in the localized regime.

Near the transition out of the ergodic regime, the authors note that a jump of this exponent is expected: gamma takes a critical value gamma_c at the transition and vanishes in the ergodic phase. However, the precise value of gamma_c has not been established, and it is not known whether gamma_c remains constant across the transition for different parameters.