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arXiv:cond-mat/0303092 · Physical Review B

Monitoring the localization-delocalization transition within a one-dimensional model with nonrandom long-range interaction

А В Малышев, Victor A Malyshev, Francisco Domínguez-Adame

Published Nov 9, 2004• 43 citations
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Abstract

We consider a two-parameter one-dimensional Hamiltonian with uncorrelated diagonal disorder and nonrandom long-range intersite interaction ${J}_{mn}=J∕{\ensuremath{\mid}m\ensuremath{-}n\ensuremath{\mid}}^{\ensuremath{\mu}}$. The model is critical at $1<\ensuremath{\mu}<3∕2$ and reveals the localization-delocalization transition with respect to the disorder magnitude. To detect the transition we analyze level and wave function statistics. It is demonstrated also that in the marginal case $(\ensuremath{\mu}=3∕2)$ all states are localized.

DOI
10.1103/physrevb.70.172202

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We consider a two-parameter one-dimensional Hamiltonian with uncorrelated diagonal disorder and nonrandom long-range intersite interaction Jmn=J∕\ensuremath∣m\ensuremath−n\ensuremath∣\ensuremathμ{J}_{mn}=J∕{\ensuremath{\mid}m\ensuremath{-}n\ensuremath{\mid}}^{\ensuremath{\mu}}Jmn​=J∕\ensuremath∣m\ensuremath−n\ensuremath∣\ensuremathμ. The model is critical at 1<\ensuremathμ<3∕21<\ensuremath{\mu}<3∕21<\ensuremathμ<3∕2 and reveals the localization-delocalization transition with respect to the disorder magnitude. To detect the transition we analyze level and wave function statistics. It is demonstrated also that in the marginal case (\ensuremathμ=3∕2)(\ensuremath{\mu}=3∕2)(\ensuremathμ=3∕2) all states are localized.
Abstract We focus on tight‐binding Hamiltonians on a regular one‐dimensional lattice with non‐random long‐range inter‐site coupling Jmn = J/|m−n|μ and random uncorrelated site energies. Within the model the localization‐delocalization transition occurs at one of the energy band edges provided 1 < μ < 3/2. Using the model we demonstrate that the ratio of the first two momenta of the participation number distribution for the critical states is a size invariant parameter at some value of the disorder magnitude Δc. We claim that the invariance manifests the transition. We find that Δc ≠ 0 at 1 < μ < 3/2, suggesting that the system undergoes the localization‐delocalization transition with respect to disorder magnitude. At μ ≥ 3/2, all states are localized. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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 consider long-range correlated disorder and mutual interacting particles according to a dipole-dipole coupling as modifications to the one-dimensional Anderson model. Technically, we rely on the numerical exact diagonalization of the system's Hamilitonian. From the perspective of different localization measures, we confirm and extend the picture of the emergence of delocalized states with increasing correlations. Besides these studies, a definition for multiparticle localization is proposed. In the case of two interacting bosons, we observe a sensitivity of localization with respect to the range of the particle-particle interaction and insensitivity to the coupling's sign, which should stimulate new theoretical approaches and experimental investigations with, e.g., dipolar cold quantum gases.
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 describe the critical behavior of the Anderson-like transition predicted to occur in the 2D tight-binding model with isotropic scale-free long-range correlated disorder characterized by a power-law spectral density. We explore the scale invariance of the participation function relative fluctuation at the critical point to locate the mobility edge as a function of the power-law spectral density exponent α 2D . The states near the band center, which exhibit power-law localization for uncorrelated disorder, become delocalized for α 2D >2. In addition, we consider the finite-size scaling hypothesis to estimate the correlation length critical exponent. We find that the critical exponent ν depends on α 2D , thus indicating that correlations in the disorder distribution are indeed relevant in this regime, in agreement with the extended Harris criterion.