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Scrambling Without Chaos in Random Free-Fermionic Systems

Ali Mollabashi, Mohammad-Javad Vasli

TL;DR

This work demonstrates that randomness can induce information scrambling in integrable free-fermionic systems by analyzing quadratic Hamiltonians with varying randomness using entanglement-based diagnostics. Through the correlator method for fermionic Gaussian states, the authors show that local randomness eliminates the memory effect, while the tripartite mutual information saturates at a negative value analogous to chaotic systems, albeit weaker. Spectral diagnostics reveal a nontrivial ramp in the spectral form factor and, for the single-particle sector, a partial Poisson-to-Wigner-Dyson crossover as the randomness range is tuned, highlighting a link between spectral statistics and scrambling in Gaussian models. Overall, randomness acts as a minimal ingredient for scrambling in integrable quadratic fermionic models, connecting scrambling behavior to spectral correlations beyond conventional chaos indicators.

Abstract

We study the role of randomness in the scrambling of quantum information within integrable free-fermionic systems. Considering quadratic Hamiltonians with varying degrees of randomness, we analyze entanglement-based measures to characterize the scrambling structure. We show that the memory effect in the entanglement of disjoint subsystems of Gaussian states vanishes when the local couplings are random, indicating information delocalization. The tripartite mutual information exhibits negative saturation values similar to those in chaotic systems, albeit with a smaller magnitude, revealing weaker scrambling under integrable quadratic dynamics. Despite integrability, spectral analyses reveal that local random models display a spectral-form-factor ramp and a partial crossover in the single-particle level-spacing ratio from Poisson-like to Wigner--Dyson-like behavior within a certain range of random couplings. These results demonstrate that randomness can act as a minimal ingredient for inducing information scrambling in integrable quadratic fermionic models.

Scrambling Without Chaos in Random Free-Fermionic Systems

TL;DR

This work demonstrates that randomness can induce information scrambling in integrable free-fermionic systems by analyzing quadratic Hamiltonians with varying randomness using entanglement-based diagnostics. Through the correlator method for fermionic Gaussian states, the authors show that local randomness eliminates the memory effect, while the tripartite mutual information saturates at a negative value analogous to chaotic systems, albeit weaker. Spectral diagnostics reveal a nontrivial ramp in the spectral form factor and, for the single-particle sector, a partial Poisson-to-Wigner-Dyson crossover as the randomness range is tuned, highlighting a link between spectral statistics and scrambling in Gaussian models. Overall, randomness acts as a minimal ingredient for scrambling in integrable quadratic fermionic models, connecting scrambling behavior to spectral correlations beyond conventional chaos indicators.

Abstract

We study the role of randomness in the scrambling of quantum information within integrable free-fermionic systems. Considering quadratic Hamiltonians with varying degrees of randomness, we analyze entanglement-based measures to characterize the scrambling structure. We show that the memory effect in the entanglement of disjoint subsystems of Gaussian states vanishes when the local couplings are random, indicating information delocalization. The tripartite mutual information exhibits negative saturation values similar to those in chaotic systems, albeit with a smaller magnitude, revealing weaker scrambling under integrable quadratic dynamics. Despite integrability, spectral analyses reveal that local random models display a spectral-form-factor ramp and a partial crossover in the single-particle level-spacing ratio from Poisson-like to Wigner--Dyson-like behavior within a certain range of random couplings. These results demonstrate that randomness can act as a minimal ingredient for inducing information scrambling in integrable quadratic fermionic models.
Paper Structure (9 sections, 23 equations, 5 figures)

This paper contains 9 sections, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our Models
  3. Diagonalization
  4. Review of Correlator Method
  5. Memory Effect
  6. Tripartite Mutual Information
  7. Spectral Form Factor
  8. Spectral Statistics
  9. conclusions and discussions

Figures (5)

  • Figure 1: Left: Time evolution of $S_{A_1\cup A_2}$ in Ising model, local disordered Ising model, and non-local models. The subregion configuration in specified in the legend where $L$ denotes the size of the total system. $J_i$ is chosen from uniform random distribution specified in the plot legend. Middle: Time evolution of mutual information corresponding to the left panel. Right: Time evolution of $S_{A_1\cup A_2}$ in non-local random models. In all plots corresponding to random models, the results are averaged over 20 samples.
  • Figure 2: Upper: Time evolution of Tripartite Mutual Information in Ising model versus local disordered Ising model. The configuration is specified in the legend. $J_i$ is chosen from a uniform random distribution specified in the plot legend. Lower: Time evolution of Tripartite Mutual Information in non-local random models. The dashed line indicates the lower bound of the TMI for Gaussian states evolving under quadratic Hamiltonians. In all plots, the results are averaged over 20 samples, except for the Ising model, which does not contain any random parameters.
  • Figure 3: This figure shows the spectral form factor for two cases: (i) the full spectrum of the model and (ii) the single-particle sector of the spectrum. The two plots on the left correspond to the local disordered Ising model. In the inset of the left plot, we have magnified the main plot to show the ramp clearly. The right plot corresponds to GSYK$_2$ versus SYK$_2$. In all plots, the results are averaged over 2000 samples.
  • Figure 4: The level-statistics ratio for the local random model. The rightmost panel shows the level-statistics ratio for the full spectrum, which follows a Poisson $r$-parameter distribution. Moving from right to left, starting from the second panel, the remaining panels display the level-statistics ratio for the single-particle sector, arranged in order of decreasing range of the random coupling. The corresponding coupling ranges are indicated in the legends of each panel. From left to write $\langle \tilde{r} \rangle= 0.386, 0.397, 0.431, 0.499, 0.373$. For reference, the Poisson and Wigner–Dyson distributions of the $r$-parameter are also shown for comparison. All results are averaged over 300 samples.
  • Figure 5: The level-statistics ratio for GSKY$_2$ model. The left panel following a Poisson $r$-parametr distribution corresponds to the full spectrum, and the right panel following a Wigner-Dyson $r$-parametr distribution corresponds to the single-particle sector. For the left and right panels we find $\langle \tilde{r} \rangle \approx 0.384, 0.598$, respectively. The results are averaged over 300 samples.