We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as $1/{r}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ satisfying $d\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2d$, where $d$ is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where $W$ is the disorder strength and $L$ the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder $W$ with the system size $L$ and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.
We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as 1/r\ensuremathα with an exponent \ensuremathα satisfying d\ensuremath≤\ensuremathα\ensuremath≤2d, where d is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder W∗=W/L2d\ensuremath−\ensuremathαlnL, where W is the disorder strength and L the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder W with the system size L and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.
We investigate the transition from the many-body localized phase to the ergodic thermalized phase at an infinite temperature in an XY spin chain with L spins, which experiences power-law decaying interactions in the form of ${V}_{ij}\ensuremath{\propto}1/{\left|i\ensuremath{-}j\right|}^{\ensuremath{\alpha}}$ (i,j=1,...,L) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to L=18, we systematically analyze the energy gap statistics, the half-chain entanglement entropy, and the uncertainty of the entanglement entropy of the system at different interaction exponents $\ensuremath{\alpha}$. The finite-size critical scaling allows us to determine the critical disorder strength Wc and the critical exponent $\ensuremath{\nu}$ at the many-body-localization phase transition, as a function of the interaction exponent $\ensuremath{\alpha}$ in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$. We find that both Wc and $\ensuremath{\nu}$ diverge when $\ensuremath{\alpha}$ decreases to a critical power of ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.16\ifmmode\pm\else\textpm\fi{}0.17$, indicating the absence of many-body localization for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. Our result differs from the analytical prediction of ${\ensuremath{\alpha}}_{c}=3/2$ found in a previous work based on a scaling argument [A. L. Burin, Phys. Rev. B 92, 104428 (2015)], but is close to the recent numerical result ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ obtained from quantum dynamics simulation [A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Phys. Rev. A 99, 033610 (2019)].
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 perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder WE which is strictly less than the Anderson localization critical disorder WC, and that of other works which suggest that there is no such region. We find that, although no separate WE exists from WC, the length scale at which fully developed ergodicity is found diverges like ∣W−WC∣−1, while the critical length over which delocalization develops is ∼∣W−WC∣−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.
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.
Anderson localization on random regular graphs (RRG) serves as a toy-model of many-body localization (MBL). We explore the transition from ergodicity to localization on RRG with large connectivity m. In the analytical part, we focus on the inverse participation ratio of eigenstates and identify several regimes on the ergodic side of the transition---self-consistent golden rule, golden rule, precritical, and critical---that the system consecutively goes through when the disorder increases toward the point Wc of the localization transition. We also perform exact-diagonalization numerics as well as a population-dynamics analysis combining analytical and numerical techniques. Results of all the approaches are in excellent mutual agreement. We further explore the evolution from ergodicity to localization in two models of Fock-space MBL: fermionic and spin quantum dots. Large-connectivity RRG models serve as an approximation for this class of many-body problems; one of our central goals is to better understand the status of this approximation. Our numerical simulations support the conjecture that, in the limit of a large system, there is a sharp ergodicity-to-localization transition in the quantum-dot models. Furthermore, our results are consistent with the RRG-like scaling of the critical disorder, ${W}_{c}\ensuremath{\sim}mlnm$. While in the golden-rule range the behavior of quantum-dot models is in agreement with analytical predictions based on the RRG model, substantial deviations occur in the precritical regime. Our results indicate that the precritical and critical behavior (as well as the numerical coefficient in the formula for Wc) in the quantum-dot models may be different from what one would expect from the RRG-like approximation.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder W_{E} W E which is strictly less than the Anderson localization critical disorder W_C W C , and that of other works which suggest that there is no such region. We find that, although no separate W_{E} W E exists from W_C W C , the length scale at which fully developed ergodicity is found diverges like |W-W_C|^{-1} | W − W C | − 1 , while the critical length over which delocalization develops is \sim |W-W_C|^{-1/2} ∼ | W − W C | − 1 / 2 . The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.