Low-frequency conductivity in many-body localized systems

Abstract

We argue that the a.c. conductivity σ(ω) in the many-body localized phase is a power law of frequency ω at low frequency: specifically, σ(ω) ∼ ω α with the exponent α approaching 1 at the phase transition to the thermal phase, and asymptoting to 2 deep in the localized phase. We identify two separate mechanisms giving rise to this power law: deep in the localized phase, the conductivity is dominated by rare resonant pairs of configurations; close to the transition, the dominant contributions are rare regions that are locally critical or in the thermal phase. We present numerical evidence supporting these claims, and discuss how these power laws can also be seen through polarization-decay measurements in ultracold atomic systems.

Rare‐region effects and dynamics near the many‐body localization transition

“…Because the dynamics in the MBL phase are exponentially sensitive to localization length, even inclusions that are slightly less (or more) localized than typical regions can dominate typical regions in response [41]. These "same-phase" rare regions-which may be dominant deep inside the MBL phase-are outside the scope of this review.…”

“…Specifically, the conductivity would go as σ (ω) ∼ ω 1+g near the MBL transition, while the spectral function would go as S(ω) ∼ ω g −1 . This Griffiths power-law competes with a separate continuously varying power-law due to many-body resonances [41], which may dominate it deep in the MBL phase. (Note, however, that generic spectral functions such as the structure factor also have a zero-frequency "Drude" contribution coming from typical regions.…”

“…These couplings are, by construction, the only processes that can flip the l-bits. Anticipating that the thermal region will be able to thermalize some spins in the initially MBL region, we introduce the following terminology: the thermal "core" is the microscopically rare region, whereas the "periphery" consists of those initially MBL spins that are thermalized by the inclusion [41].…”

“…On the other hand, in the MBL phase of a system with quenched randomness there may be rare regions that are less disordered and thus locally thermalizing [41]. Ultimately, at least in one-dimensional systems with short-range interactions, these rare regions get "localized" because they are effectively finite [41,58,63,64]: thus the spectral features of an inclusion become discrete on frequency scales small compared with the level spacing of the inclusion. In addition to rare regions of the disorder potential, the MBL phase has rare regions of the state [65], in which the conserved densities have atypical values (such "rare regions" would of course not be dynamically stable in the thermal phase).…”

“…The MBL phase resembles noninteracting Anderson insulators in some ways (e.g., spatial correlations decay exponentially, and eigenstates have area-law entanglement [35]). However, there are also important distinctions in entanglement dynamics [36,37], dephasing [38][39][40], linear [41] and nonlinear [42][43][44][45][46][47][48] response, and the entanglement spectrum [49,50]. These developments (reviewed in Refs.…”