Eigenstate multifractality is of significant interest with potential applications in various fields of quantum physics. Most of the previous studies concentrated on fine-tuned quantum models to realize multifractality, which is generally believed to be a critical phenomenon and fragile to random perturbations. In this work, we propose a set of generic principles based on the power-law decay of the eigenstates that allow us to distinguish a fractal phase from a genuine multifractal phase. We demonstrate the above principles in a 1D tight-binding model with inhomogeneous nearest-neighbor hopping that can be mapped to the standard quantum harmonic oscillator via energy-coordinate duality. We analytically calculate the fractal dimensions and the spectrum of fractal dimensions, which are in agreement with numerical simulations.
Eigenstate multifractality is of significant interest with potential applications in various fields of quantum physics. Most of the previous studies concentrated on fine-tuned quantum models to realize multifractality, which is generally believed to be a critical phenomenon and fragile to random perturbations. In this work, we propose a set of generic principles based on the power-law decay of the eigenstates that allow us to distinguish a fractal phase from a genuine multifractal phase. We demonstrate the above principles in a 1D tight-binding model with inhomogeneous nearest-neighbor hopping that can be mapped to the standard quantum harmonic oscillator via energy-coordinate duality. We analytically calculate the fractal dimensions and the spectrum of fractal dimensions, which are in agreement with numerical simulations.
Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the Rosenzweig-Porter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies’ distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase.
The dynamical scaling for statistics of critical multifractal eigenstates proposed by Chalker is analytically verified for the critical random matrix ensemble in the limit of strong multifractality controlled by the small parameter b⪡1. The power-law behavior of the quantum return probability ${P}_{N}(\ensuremath{\tau})$ as a function of the matrix size N or time $\ensuremath{\tau}$ is confirmed in the limits $\ensuremath{\tau}/N\ensuremath{\rightarrow}\ensuremath{\infty}$ and $N/\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}$, respectively, and it is shown that the exponents characterizing these power laws are equal to each other up to the order b2. The corresponding analytical expression for the fractal dimension d2 is found.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.
Two established results have never been composed. The fidelity susceptibility of a quantum state to a Hamiltonian parameter equals one quarter of its quantum Fisher information, F_Q = 4·chi_f; and an exact calculation in the Rosenzweig-Porter (RP) ensemble shows that eigenfunction fidelity susceptibility is enhanced in the fractal non-ergodic-extended phase (Skvortsov, Amini and Kravtsov, PRB 106, 054208, 2022). Composed, they make the eigenvector fractal dimension D2 — until now a diagnostic of localization physics — an operational metrological figure of merit: along the RP fractal phase, at fixed distance from criticality and for field-like (local) couplings to the sensed parameter, the finite-size enhancement exponent of the quantum Fisher information over the ergodic reference is 1 − D2. We confirm this in-house under preregistered criteria on the generalized-RP ensemble with D2 measured from inverse-participation-ratio scaling (not assumed): enhancement = 1 − D2 within 0.15 at 4/4 interior points of the phase (deviations 0.009–0.078; correlation 0.996). Two further preregistered experiments fix the criterion's boundaries: geometric fractality alone — a disorder-free Sierpinski-gasket lattice with certified eigenpairs and degenerate-multiplet treatment — shows no effect (enhancement −0.12; the RP formula does not extend), while critical power-law banded random matrices extend the law at all three tested bandwidths (deviations 0.016–0.064). The criterion is therefore: multifractal eigenstates of random ensembles, sensed through local couplings — randomness, not fractal geometry, is the missing ingredient, and an operator-structure scope law (diagonal inherits; short-range inherits once its range fits the fractal support; collective and dense couplings are D2-blind) gives the design rule. On the platform side we construct — because it cannot be cited — the minimal model that carries the physics onto cold atoms: an interacting tilted+disordered lattice in which the tilt is only the sensed parameter (the bare Wannier-Stark ladder is integrable, so sourcing RP statistics from gravity would be immediately refutable) and the dense RP-like spectrum comes from interactions and disorder. A preregistered coexistence test on the interacting t–V chain finds a textbook ergodic-to-localized crossover whose typical susceptibility peaks at the crossover's ergodic edge — not mid-crossover — with the predicted heavy x^−2 tail (Hill index 1.96–2.04) in the chaotic regime and estimator-dependent rare-event gains inside the crossover. The platform statement is scoped accordingly, and one instrument caveat is stated up front: this attaches to lattice/Bloch gravimetry, not to free-fall light-pulse interferometry. Tier-2 throughout: a shared random-matrix description with preregistered numerical evidence, not a hardware claim.
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.