Superconducting circuits provide a very attractive platform to study the transition from chaotic ergodic to full localized behavior expected in closed quantum systems. The present work reports the emergence of a chaotic, yet nonergodic regime that preempts the transition into a fully localized state in Josephson junction chains. The lack of ergodicity is reflected in multifractal eigenstates in the many-body space. Furthermore, the multifractality in the phase space is accompanied by the fractal properties of the local energy spectrum that translates into anomalous power-law behavior of the physical correlators. The observation of a fractal structure in the local energy spectrum of Josephson junction arrays may provide a bridge between nonergodic dynamics of quantum systems and the hierarchical structure of the local minima in the theory of classical spin glasses.
Superconducting circuits provide a very attractive platform to study the transition from chaotic ergodic to full localized behavior expected in closed quantum systems. The present work reports the emergence of a chaotic, yet nonergodic regime that preempts the transition into a fully localized state in Josephson junction chains. The lack of ergodicity is reflected in multifractal eigenstates in the many-body space. Furthermore, the multifractality in the phase space is accompanied by the fractal properties of the local energy spectrum that translates into anomalous power-law behavior of the physical correlators. The observation of a fractal structure in the local energy spectrum of Josephson junction arrays may provide a bridge between nonergodic dynamics of quantum systems and the hierarchical structure of the local minima in the theory of classical spin glasses.
Eigenstate multifractality, a hallmark of non-interacting disordered metals, which may also be observed in many-body localized states, is characterized by anomalous slow dynamics and appears relevant for many areas of quantum physics, from measurement-driven systems to superconductivity. We propose a novel approach to achieve non-ergodic multifractal states (NEMs) without disorder by iteratively introducing defects into a crystal lattice, reshaping it from a plain structure into a fractal geometry. By comprehensive analysis of the Sierpiński gasket case, we find robust evidence of the emergence of NEMs that go beyond the conventional classification of quantum states and designate new pathways for quantum transport studies. We discuss potential experimental signatures of these states.
Eigenstate multifractality, a hallmark of non-interacting disordered metals, features anomalously slow dynamics and is relevant across quantum platforms ranging from measurement-driven systems to superconductivity. Despite extensive work on multifractality in disordered and quasiperiodic systems, deterministic, disorder-free lattice routes remain underexplored. Here, we demonstrate non-ergodic multifractal states (NEMs) by iteratively placing defects to shape a crystal lattice into a fractal geometry. Through a comprehensive analysis of the Sierpiński gasket, we find consistent evidence for the emergence of NEMs that go beyond the conventional classification of quantum states and open new pathways for quantum transport studies. We further outline experimental signatures of these states. Quantum fractals offer a clean route to unconventional states between a metal and an insulator. In a disorder-free Sierpiński gasket lattice, the authors characterise the emergence of non-ergodic multifractal states, predict experimental signatures, and suggest pathways toward precise quantum control.
The disordered XXZ model is a prototype model of the many-body localization (MBL) transition. Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due to severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.
The disordered XXZ model is a prototype model of the many-body localization transition (MBL). Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.
Fractal dimensions have been used as a quantitative measure for structure of eigenstates of quantum many-body systems, useful for comparison to random matrix theory predictions or to distinguish many-body localized systems from chaotic ones. For chaotic systems at midspectrum the states are expected to be ``ergodic'', infinite temperature states with all fractal dimensions approaching 1 in the thermodynamic limit. However, when moving away from midspectrum, the states develop structure, as they are expected to follow the eigenstate thermalization hypothesis, with few-body observables predicted by a finite-temperature ensemble. We discuss how this structure of the observables can be used to bound the fractal dimensions from above, thus explaining their typical arc-shape over the energy spectrum. We then consider how such upper bounds act as a proxy for the fractal dimension over the many-body localization transition, thus formally connecting the single-particle and Fock space pictures discussed in the literature.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.