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.
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 present a theoretical and numerical study of the many-body localization (MBL) transition and entanglement dynamics in disordered spin-fermion chains with long-range interactions. Moving beyond standard single-particle pictures, we investigate the breakdown of ergodicity and thermalization by constructing an effective Hamiltonian via iterative Schrieffer-Wolff transformations and utilizing the local integrals of motion (l-bits) formalism. Our analysis examines entanglement entropy growth, level spacing statistics, and the transition from Wigner-Dyson to Poisson statistics. Furthermore, we derive analytical estimates for the dephasing rate scaling as a function of long-range potential parameters and validate the theoretical findings using exact diagonalization (ED) numerical protocols with finite-size scaling.
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.
Understanding how closed quantum systems dynamically approach thermal equilibrium presents a major unresolved problem in statistical physics. Generically, non-integrable quantum systems are expected to thermalize as they comply with the Eigenstate Thermalization Hypothesis. However, in the presence of strong disorder, the dynamics can possibly slow down to a degree that systems fail to thermalize on experimentally accessible timescales, as in spin glasses or many-body localized systems. In general, particularly in long-range interacting quantum systems, the specific nature of the disorder necessary for the emergence of a prethermal, metastable state--distinctly separating the timescales of initial relaxation and subsequent slow thermalization--remains an open question. We study an ensemble of Heisenberg spins with a tunable distribution of random coupling strengths realized by a Rydberg quantum simulator. We observe a drastic change in the late-time magnetization when increasing disorder strength. The data is well described by models based on pairs of strongly interacting spins, which are treated as thermal for weak disorder and isolated for strong disorder. Our results indicate a crossover into a pair-localized prethermal regime in a closed quantum system of thousands of spins in the critical case where the exponent of the power law interaction matches the spatial dimension.