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Communications Physics

Emergence of non-ergodic multifractal quantum states in geometric fractals

F. Salvati, Mikhail I. Katsnelson, Andrey A. Bagrov

Published Jan 8, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.1038/s42005-025-02455-w

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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.
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.
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.
The fate of ergodicity and localization in disordered, interacting quantum systems in two spatial dimensions remains an outstanding open problem in non-equilibrium physics. Theoretical and numerical approaches are severely constrained by exponential Hilbert space growth. Here, we leverage the high data-acquisition rates of superconducting quantum processors to extensively sample Hilbert space configurations, effectively imaging the state as a wavefunction network (WFN). Tracking the bitstring return probability across varied disorder strengths, we find heavy-tailed distributions and power-law typical decay characteristic of glassy relaxation. Furthermore, the distribution of bitstring probabilities transitions from a Porter-Thomas form at low disorder to an eight-decade power law at stronger disorder. In the reconstructed configuration space, WFN connectivity is compatible with a random geometric network at low disorder and becomes scale-free at stronger disorder. Finally, a multifractal analysis based on the binary intrinsic dimension (BID) reveals a non-ergodic regime where the wavefunction support saturates to a finite fraction of the total Hilbert space. Together, these results rule out a direct ergodic-to-localized transition, pointing instead to an extended multifractal regime. By visualizing wavefunction dynamics directly in configuration space, this work complements traditional real-space studies, showing the potential of quantum processors to resolve long-standing questions in non-equilibrium physics.
Periodically driven, or Floquet, disordered quantum systems have generated many unexpected discoveries of late, such as the anomalous Floquet Anderson insulator and the discrete time crystal. Here, we report the emergence of an entire band of multifractal wavefunctions in a periodically driven chain of non-interacting particles subject to spatially quasiperiodic disorder. Remarkably, this multifractality is robust in that it does not require any fine-tuning of the model parameters, which sets it apart from the known multifractality of critical wavefunctions. The multifractality arises as the periodic drive hybridises the localised and delocalised sectors of the undriven spectrum. We account for this phenomenon in a simple random matrix based theory. Finally, we discuss dynamical signatures of the multifractal states, which should betray their presence in cold atom experiments. Such a simple yet robust realisation of multifractality could advance this so far elusive phenomenon towards applications, such as the proposed disorder-induced enhancement of a superfluid transition.
Non-ergodic quantum many-body dynamics offers a route to persistent quantum coherence far from equilibrium, beyond the conventional expectations of thermalizing statistical mechanics. Most known examples have been identified through intuition, analogy, or numerical search, rather than by systematic Hamiltonian design. This leaves few general methods for engineering robust non-ergodic dynamics in disorder-free interacting quantum many-body systems, especially in higher dimensions where enhanced connectivity generally disfavors ergodicity breaking. Here, we use an eigenstate-to-Hamiltonian construction approach to systematically engineer two-dimensional quantum spin Hamiltonians with tunable non-ergodic properties. Starting from a common parent model and input eigenstate, we construct two target Hamiltonians on the square lattice that differ in their interaction geometry: axial versus diagonal. We show that the resulting interaction geometry and coupling pattern control the Hilbert-space structure and dynamics. The axial model forms a connected Hilbert-space network and exhibits an anisotropy-driven crossover from ergodic to non-ergodic behavior through the suppression of resonances. In contrast, the diagonal model exhibits Hilbert-space fragmentation and, within its largest irreducible sector, signatures of quantum many-body scar dynamics including towers of highly localized, low-entanglement eigenstates, and long-lived coherent revivals. These results establish a direct link between Hamiltonian design, Hilbert-space network structure, and emergent non-ergodic dynamics.
Periodically driven, or Floquet, disordered quantum systems have generated many unexpected discoveries of late, such as the anomalous Floquet Anderson insulator and the discrete time crystal. Here, we report the emergence of an entire band of multifractal wavefunctions in a periodically driven chain of non-interacting particles subject to spatially quasiperiodic disorder. Remarkably, this multifractality is robust in that it does not require any fine-tuning of the model parameters, which sets it apart from the known multifractality of critical wavefunctions. The multifractality arises as the periodic drive hybridises the localised and delocalised sectors of the undriven spectrum. We account for this phenomenon in a simple random matrix based theory. Finally, we discuss dynamical signatures of the multifractal states, which should betray their presence in cold atom experiments. Such a simple yet robust realisation of multifractality could advance this so far elusive phenomenon towards applications, such as the proposed disorder-induced enhancement of a superfluid transition.