We present a finite-size scaling for both interaction and disorder strengths in the critical regime of the many-body localization (MBL) transition for a spin-1/
2XXZ spin chain with a random field by studying level statistics. We show how the dynamical transition from the thermal to MBL phase depends on interaction together with disorder by evaluating the ratio of adjacent level spacings, and thus, extend previous studies in which interaction coupling is fixed. We introduce an extra critical exponent in order to describe the nontrivial interaction dependence of the MBL transition. It is characterized by the ratio of the disorder strength to the power of the interaction coupling with respect to the extra critical exponent and not by the simple ratio between them.
We study a canonical many-body-localized (MBL) system with power-law-correlated disorders:
s=21 spin chain in a random magnetic field. The power-law-correlated disorder can control the critical regime between the MBL and thermal (ergodic) phases by varying its exponent, and it let us investigate the MBL transitions in detail. Static-eigenstate and dynamic properties of MBL are studied by numerical methods for systems with various long-range correlations. By using energy-resolved distribution of the participation ratio (PR) and calculating some physical quantities related to localization length, we show that the MBL transition exhibits certain universal behavior. We also investigate entanglement properties for the static and dynamics system. These studies elucidate the impact of power-law-correlated disorders in the canonical MBL system.
We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.
We analyze the finite-size scaling of the average gap ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of the Berezinskii-Kosterlitz-Thouless transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sublinear drift with the system size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.
The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.
We investigate many-body localization in the chain of interacting spins with a transverse power-law interaction, ${J}_{0}/{r}^{\ensuremath{\alpha}}$, and random on-site potentials, ${\ensuremath{\phi}}_{i}\ensuremath{\in}\left(\ensuremath{-}W/2,W/2\right)$, in the long-range limit, $\ensuremath{\alpha}<3/2$, which has been recently examined experimentally on trapped ions. The many-body localization threshold is characterized by the critical disordering,
Wc, which separates localized (
W>Wc) and chaotic (
W<Wc) phases. Using the analysis of the instability of localized states with respect to resonant interactions complemented by numerical finite-size scaling, we show that the critical disordering scales with the number of spins,
N, as ${W}_{c}\ensuremath{\approx}[1.37{J}_{0}/(4/3\ensuremath{-}\ensuremath{\alpha})]{N}^{4/3\ensuremath{-}\ensuremath{\alpha}}lnN$ for $0<\ensuremath{\alpha}\ensuremath{\le}1$, and as ${W}_{c}\ensuremath{\approx}[{J}_{0}/(1\ensuremath{-}2\ensuremath{\alpha}/3)]{N}^{1\ensuremath{-}2\ensuremath{\alpha}/3}{ln}^{2/3}N$ for $1<\ensuremath{\alpha}<3/2$, while the transition width scales as ${\ensuremath{\sigma}}_{W}\ensuremath{\propto}{W}_{c}/N$. We use this result to predict the spin long-term evolution for a very large number of spins (
N=50), inaccessible for exact diagonalization, and to suggest the rescaling of hopping interaction with the system size to attain the localization transition at finite disordering in the thermodynamic limit of infinite number of spins.