Many-body-localization transition: strong multifractality spectrum for matrix elements of local operators

Abstract

For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f (0 ≤ α ≤ 2) = α 2 , well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.

Many-Body-Localization Transition in the Strong Disorder Limit: Entanglement Entropy from the Statistics of Rare Extensive Resonances

“…The fact that multifractality occurs also in the many-body localized phase is in agreement with the analysis of matrix elements of local operators and the statistics of hybridization ratios in nearest-neighbors many-body localized models [46].…”

“…For a quantum one-dimensional toy model, we have analyzed the statistical properties of the rare extensive resonances that are needed to destabilize the many-body localized phase. At criticality, we have found that the entanglement entropy can grow with an exponent 0 < α = 3/2 − a < 1 anywhere between the area law α = 0 and the volume law α = 1, as a function of the power law exponent a of the couplings (Equation (79)), while the entanglement spectrum follows the strong multifractality statistics of Equation (101), well-known as the "strong-multifractality" spectrum in the context of Anderson localization transition [47] and found recently for nearest-neighbor MBL models at criticality [46].…”

“…More recently, the strong multifractality spectrum of Equation (101) has been found to describe the statistics of matrix elements of local operators and the statistics of hybridization ratios in nearest-neighbors many-body localized models at criticality [46]. Our conclusion is thus that the present toy model is not so far from more realistic short-range models, since it is able to reproduce the same critical multifractal spectrum.…”

“…As in Anderson transitions where the critical point is characterized by multifractal eigenfunctions [47], one expects that the MBL transition is related to some multifractal properties [46,[74][75][76][77][78][79]. Besides the entanglement entropy described above, it is thus interesting to characterize the statistics of the whole entanglement spectrum of Equation (38), both in the many-body localized phase b > 1 and at the critical point b c = 1.…”

“…with the thermal coefficient fixed by Equation (3), then the continuous vanishing of the coefficient of the volume law of Equation (8) is possible, and the exponent α is not a priori fixed by the strong-subadditivity property (see [36] for more details). This scenario is suggested by the point of view that the MBL transition is somewhat similar to an Anderson localization transition in the Hilbert space of "infinite dimensionality" as a consequence of the exponential growth of the size of the Hilbert space with the volume [40][41][42][43][44][45][46].…”