We investigate the stationary and dynamical behavior of an Anderson localized chain coupled to a single central bound state. Although this coupling partially dilutes the Anderson localized peaks towards nearly resonant sites, the most weight of the original peaks remains unchanged. This leads to multifractal wave functions with a frozen spectrum of fractal dimensions, which is characteristic for localized phases in models with power-law hopping. Using a perturbative approach we identify two different dynamical regimes. At weak couplings to the central site, the transport of particles and information is logarithmic in time, a feature usually attributed to many-body localization. We connect such transport to the persistence of the Poisson statistics of level spacings in parts of the spectrum. In contrast, at stronger couplings the level repulsion is established in the entire spectrum, the problem can be mapped to the Fano resonance, and the transport is ballistic.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum
f(0≤α≤2)=2α, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions
d=2,3, we study numerically the statistical properties of the renormalized on-site energies
ϵ and of the renormalized hoppings
V as a function of the linear size
L. We find that the renormalized on-site energies
ϵ remain finite in the localized phase in
d=2,3 and at criticality
(d=3), with a finite density at
ϵ=0 and a power-law decay
1/ϵ2 at large
∣ϵ∣. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and
u a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality
(d=3), the statistics of renormalized hoppings
V is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.
Quantum many-body Hamiltonians with two-body interactions can be represented by graphs, with sites as nodes and two-site couplings as edges. We investigate how the geometry of these graphs affects transport and entanglement growth. To do so, we study dynamics for quantum hopping on a random graph model with an effective range parameter. Tuning it, we interpolate between nearest-neighbor and all-to-all graphs, identifying a new localization regime that is robust in both single-particle and finite-density cases. This provides a model of Anderson localization induced by structural disorder, with implications for amorphous materials and tunable-range analog quantum simulators.
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.