Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.11662

Emergent quantum chaos from correlations on a random graph

Mrinal Sarkar, Valerio Pagni, Tilman Enss, Nicolò Defenu

Published Jul 13, 2026• 0 citations
arXivHTMLPDF

Abstract

This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability $p_{ij}= d_{ij}^{-(1+σ)}$. Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small $σ$ and localized eigenstates with Poisson statistics at larger $σ$. The transition occurs in the range $ 0.80 \lesssim σ_c \lesssim 0.85$, far above the summability threshold of the mean hopping profile ($σ=0$). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at $σ=1$, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.

DOI
10.48550/arxiv.2607.11662

Related papers

Quantum chaos in spin-fermion models

Gilles Montambaux, Didier Poilblanc, Jean Bellissard +1 more

The full spectrum of a single hole in a quantum antiferromagnetic background (t-${\mathit{J}}_{\mathit{z}}$-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model) is obtained by complete exact diagonalization of small two-dimensional clusters. Various statistical properties of the spectrum are investigated. On a very wide range of the parameters the level-spacing distribution follows Gaussian-orthogonal-ensemble Wigner law characteristic of chaotic spectra. At small separation, the spectral rigidity follows the universal behavior described by random-matrix theory and presents deviations at higher energies. We argue that quantum chaos is a generic feature of complex (i.e., nonintegrable) strongly correlated fermion systems. Our results suggest that random-matrix theory might be useful to investigate the incoherent part of dynamical correlation functions (such as the hole spectral density or the spin structure factor).

Physical Review Letters•Jan 25, 1993•142 citations

Localization challenges quantum chaos in the finite two-dimensional Anderson model

Jan Šuntajs, Tomaž Prosen, Lev Vidmar

It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy, and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter $u={(WlnV)}^{\ensuremath{-}1}$ where $W$ is the disorder strength and $V$ is the number of lattice sites.

arXiv:2212.10625•Physical review. B./Physical review. B•Feb 21, 2023•21 citations

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

Pavel Kos, Marko Ljubotina, Tomaž Prosen

A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.

arXiv:1712.02665•Physical Review X•Jun 8, 2018•183 citations

Topological Genesis of Quantum Chaos

Andres Sebaatian Pirolo

Macroscopic Quantum Chaos from Topological Degradation We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it arises as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. ◆ Key Features • 2D toroidal mesh: N = L × L sites (L = 20–50). • Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter). • Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss. • Exact diagonalization on mobile ARM (Snapdragon 8 Gen 2) hardware without spectral unfolding: ~25–75 s per 50-realization ensemble, achieving ~25× speedup over conventional HPC workflows. ◆ Main Results • Perfect torus exhibits symmetry-induced spectral degeneracy, numerically close to Poisson statistics. • Moderate disorder (W ≈ 1–4) yields ⟨r⟩ = 0.530–0.534, matching GOE Wigner–Dyson statistics. • Strong disorder (W ≳ 8) restores genuine Poisson statistics (⟨r⟩ ≈ 0.39). • Critical regime (W ≈ 4–6): strongly multifractal eigenstates, monotonic decay of generalized dimensions Dq, and f(α) peaking near α ≈ 2.2–2.3. • Finite-size scaling confirms Anderson-transition-like criticality. ◆ Implications These results show that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. The critical threshold (W ≈ 4–6) defines a quantitative failure boundary for distributed tensor-processing interconnects, while the unfolding-free methodology enables rapid, on-device spectral diagnostics for next-generation fault-tolerant AI accelerators.

•Aug 5, 2026•0 citations

Signatures of nonergodicity in sparse random matrices

Anonymous, Adway Kumar Das, Anandamohan Ghosh

The prevalence of sparsity in the Fock space graph of interacting many-body systems motivates an investigation into the spectral statistics of sparse random matrices with on-site disorder. We numerically determine the delocalization-localization transition in the ground state as a function of the sparsity. The short-range energy correlation in the bulk indicates that the Anderson transition at infinite temperature occurs at the critical percolation limit of the sparse graph. By analytically deriving the energy moments and calculating the shifted kurtosis, we show that the critical sparsity threshold matches the Anderson transition. Furthermore, long-range energy correlations in the bulk spectrum reveal a Thouless energy scale, suggesting a broad nonergodic regime within the delocalized phase.

arXiv:2603.21369•Physical review. B./Physical review. B•Aug 20, 2026•0 citations

Critical dynamics of long-range quantum disordered systems

Weitao Chen, Gabriel Lemarié, Jiangbin Gong

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.

arXiv:2307.00999•Physical review. E•Nov 17, 2023•5 citations
This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability pij=dij−(1+σ)p_{ij}= d_{ij}^{-(1+σ)}pij​=dij−(1+σ)​. Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small σσσ and localized eigenstates with Poisson statistics at larger σσσ. The transition occurs in the range 0.80≲σc≲0.85 0.80 \lesssim σ_c \lesssim 0.850.80≲σc​≲0.85, far above the summability threshold of the mean hopping profile (σ=0σ=0σ=0). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at σ=1σ=1σ=1, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.
The full spectrum of a single hole in a quantum antiferromagnetic background (t-Jz{\mathit{J}}_{\mathit{z}}Jz​-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model) is obtained by complete exact diagonalization of small two-dimensional clusters. Various statistical properties of the spectrum are investigated. On a very wide range of the parameters the level-spacing distribution follows Gaussian-orthogonal-ensemble Wigner law characteristic of chaotic spectra. At small separation, the spectral rigidity follows the universal behavior described by random-matrix theory and presents deviations at higher energies. We argue that quantum chaos is a generic feature of complex (i.e., nonintegrable) strongly correlated fermion systems. Our results suggest that random-matrix theory might be useful to investigate the incoherent part of dynamical correlation functions (such as the hole spectral density or the spin structure factor).
It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy, and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter $u={(WlnV)}^{\ensuremath{-}1}$ where WWW is the disorder strength and VVV is the number of lattice sites.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
Macroscopic Quantum Chaos from Topological Degradation We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it arises as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. ◆ Key Features • 2D toroidal mesh: N = L × L sites (L = 20–50). • Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter). • Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss. • Exact diagonalization on mobile ARM (Snapdragon 8 Gen 2) hardware without spectral unfolding: ~25–75 s per 50-realization ensemble, achieving ~25× speedup over conventional HPC workflows. ◆ Main Results • Perfect torus exhibits symmetry-induced spectral degeneracy, numerically close to Poisson statistics. • Moderate disorder (W ≈ 1–4) yields ⟨r⟩ = 0.530–0.534, matching GOE Wigner–Dyson statistics. • Strong disorder (W ≳ 8) restores genuine Poisson statistics (⟨r⟩ ≈ 0.39). • Critical regime (W ≈ 4–6): strongly multifractal eigenstates, monotonic decay of generalized dimensions Dq, and f(α) peaking near α ≈ 2.2–2.3. • Finite-size scaling confirms Anderson-transition-like criticality. ◆ Implications These results show that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. The critical threshold (W ≈ 4–6) defines a quantitative failure boundary for distributed tensor-processing interconnects, while the unfolding-free methodology enables rapid, on-device spectral diagnostics for next-generation fault-tolerant AI accelerators.
The prevalence of sparsity in the Fock space graph of interacting many-body systems motivates an investigation into the spectral statistics of sparse random matrices with on-site disorder. We numerically determine the delocalization-localization transition in the ground state as a function of the sparsity. The short-range energy correlation in the bulk indicates that the Anderson transition at infinite temperature occurs at the critical percolation limit of the sparse graph. By analytically deriving the energy moments and calculating the shifted kurtosis, we show that the critical sparsity threshold matches the Anderson transition. Furthermore, long-range energy correlations in the bulk spectrum reveal a Thouless energy scale, suggesting a broad nonergodic regime within the delocalized phase.
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.