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arXiv:1410.7858 · Physical Review B

Many-body delocalization in a strongly disordered system with long-range interactions: Finite-size scaling

Alexander L. Burin

Published Mar 4, 2015• 122 citations
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Abstract

Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/{R}^{\ensuremath{\alpha}}$ is investigated combining analytical theory considering resonant interactions and a finite-size scaling of exact numerical solutions with number of spins $N$. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a $d$-dimensional system including the absence of localization in the infinite system at $\ensuremath{\alpha}<2d$ and a universal scaling of a critical energy disordering ${W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}$.

DOI
10.1103/physrevb.91.094202

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Many-body localization in a disordered system of interacting spins coupled by the long-range interaction 1/R\ensuremathα1/{R}^{\ensuremath{\alpha}}1/R\ensuremathα is investigated combining analytical theory considering resonant interactions and a finite-size scaling of exact numerical solutions with number of spins NNN. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a ddd-dimensional system including the absence of localization in the infinite system at \ensuremathα<2d\ensuremath{\alpha}<2d\ensuremathα<2d and a universal scaling of a critical energy disordering Wc\ensuremath∝N2d\ensuremath−\ensuremathαd{W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}Wc​\ensuremath∝Nd2d\ensuremath−\ensuremathα​.
Many-body localization in a disordered system of interacting spins coupled by the long-range interaction 1/Rα1/R^α1/Rα is investigated combining analytical theory considering resonant interactions and a finite size scaling of exact numerical solutions with a number of spins NNN. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for ddd-dimensional system including the absence of localization in the infinite system at α<2dα<2dα<2d and a universal scaling of a critical energy disordering Wc∝N2d−αdW_{c} \propto N^{\frac{2d-α}{d}}Wc​∝Nd2d−α​. %The finite size effect on the interaction stimulated delocalization of energy in the ensemble of interacting two level systems in amorphous solids at low temperature is discussed.
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{W}_{c}Wc​, which separates localized (W>WcW>{W}_{c}W>Wc​) and chaotic (W<WcW<{W}_{c}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, NNN, 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=50N=50N=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.
Many-body localization in an XY model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power-law distance dependent hopping $V(R)\ensuremath{\propto}{R}^{\ensuremath{-}\ensuremath{\alpha}}$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\ensuremath{\alpha}<3d/2$ where ddd is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R)\ensuremath{\propto}{R}^{\ensuremath{-}2\ensuremath{\alpha}}$ of ``extended'' resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in an XY model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\ensuremath{\alpha}\ensuremath{\ge}2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.
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 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 ddd 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 WWW is the disorder strength and LLL 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 WWW with the system size LLL 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 XYXYXY spin chain with LLL 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,...,Li,j=1,...,Li,j=1,...,L) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to L=18L=18L=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{W}_{c}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{W}_{c}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)].