Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results

TL;DR

This work links the spectral form factor $S(t)$ to a planar random-walk problem and proves that, under spectrum-independence and a Lyapunov-type condition, the corresponding walk converges to a 2D Wiener process in the thermodynamic limit, yielding Gaussian statistics for chaotic models and log-normal statistics for integrable ones. It introduces a fractal-geometry framework, showing that the Wiener frontier has Hausdorff dimension $d_F=4/3$, while integrable/quasi-free walks have $d_F$ near 1, and provides exact, non-Gaussian expressions for all SFF moments via a recursive construction. The paper derives exact moments $I_m$ with a generating function and a recursion, clarifying how Gaussian terms dominate for generic weights and how degeneracies or low-temperature regimes produce corrections. Numerical experiments on XXZ+NNN chains, XY models, Bethe-Ansatz cases, and SYK variants corroborate the theoretical predictions, illustrating exponential versus log-normal SFF distributions and the fractal-frontier behavior across chaotic and integrable regimes. The results offer a robust diagnostic of spectral independence and a toolkit for beyond-Gaussian analysis of quantum chaos.

Abstract

We establish the mathematical equivalence between the spectral form factor, a quantity used to identify the onset of quantum chaos and scrambling in quantum many-body systems, and the classical problem of statistical characterization of planar random walks. We thus associate to any quantum Hamiltonian a random process on the plane. We set down rigorously the conditions under which such random process becomes a Wiener process in the thermodynamic limit and the associated distribution of the distance from the origin becomes Gaussian. This leads to the well known Gaussian behavior of the spectral form factor for quantum chaotic (non-integrable) models, which we show to be violated at low temperature. For systems with quasi-free spectrum (integrable), instead, the distribution of the SFF is Log-Normal. We compute all the moments of the spectral form factor exactly without resorting to the Gaussian approximation. Assuming degeneracies in the quantum chaotic spectrum we solve the classical problem of random walker taking steps of unequal lengths. Furthermore, we demonstrate that the Hausdorff dimension of the frontier of the random walk, defined as the boundary of the unbounded component of the complement, approaches 1 for the integrable Brownian motion, while the non-integrable walk approaches that obtained by the Schramm-Loewner Evolution (SLE) with the fractal dimension $4/3$. Additionally, we numerically show that Bethe Ansatz walkers fall into a category similar to the non-integrable walkers.

Figures (8)

• Figure 1: Fractals and their frontiers, in black, corresponding to physical models: non-integrable (left) vs integrable (right). Color from blue to pink corresponds to increasing time-steps of the random walks. The frontier is essentially the boundary of the fractal without the inner islands, see text for details. The non-integrable Hamiltonian is the XXZ model with next nearest neighbor interactions (see Eq. (\ref{['XXZNNNHamiltonian']})), with $(\Delta,\alpha) = (0.4,0.5)$. The integrable Hamiltonian is the XY model with parameters $(h,\gamma)=(0.2,0.3)$; see Appendix \ref{['sec:appfractral']}.
• Figure 2: Numerical check of \ref{['eq:generic_weights']} for the paradigmatic example of a physical (local) Hamiltonian given by XXX+NNN chain in \ref{['XXZNNNHamiltonian']} at high temperature. Upper panels: Decay of the ratios $R_q^{N_B}$ for various system sizes $N$, where $N_B$ denotes the total number of blocks in the Hamiltonians (corresponding to the number of $d_j$'s). Dashed lines are included as guides for the eye. Lower panels: Decay of the ratios $R_q^{n}$ for fixed system size $N=12$, the largest size for which we computed the full spectrum. Solid lines are fits to the power-law function $f(n) = b/n^{a-1}$, with the extracted exponent satisfying $a \approx q$ up to small deviations. Panels on the left correspond to the Bethe Ansatz integrable case with $\alpha=0$ and $\Delta=0.1$, while panels on the right show the non-integrable case with $\alpha=0.1$ and $\Delta=0.1$.
• Figure 3: Numerical check of \ref{['eq:Lyapunov_wiener']} for the paradigmatic example of a physical (local) Hamiltonian given by XXX+NNN chain in \ref{['XXZNNNHamiltonian']}. We fix the system size $N=12$, show the result of the quantity $sR_{1}^{N}(h,s)$ which is expected to be $sh$. Upper panel: Bethe Ansatz integrable case with $\alpha=0$ and $\Delta=0.1$. Lower panel: non-integrable case with parameters $\alpha=0.1$ and $\Delta=0.1$. Infinite temperature $T \rightarrow \infty$ regime.
• Figure 4: Numerical check of \ref{['eq:generic_weights']} for the paradigmatic example of a physical (local) Hamiltonian given by XXX+NNN chain in \ref{['XXZNNNHamiltonian']} at low and different temperatures $T$ and breakdown of the Lyapunov conditions. Decay of the ratios $R_q^{N_B}$ for various system sizes $N$, where $N_B$ denotes the total number of blocks in the Hamiltonians (corresponding to the number of $d_j$'s). Dashed lines are included as guides for the eye. Significant deviation from an exponential decay of the ratios signals the breakdown of the generic condition. Decrease of the temperature from the top to the bottom panel shows how exactly the condition is not longer satisfied in the low temperature limit.
• Figure 5: Probability distribution of the normalized Spectral Form Factor (SFF) for the XXZ + NNN chain spectrum at infinite temperature ($\rho={\mathrm{1\mkern4.8mu I}}$) for system size $N = 8$. The distribution is obtained sampling in the time domain $[10^5,2 \cdot 10^5]$. The prediction of Theorem \ref{['thm:CLT_lyapunovplus']}, $e^{-x}$, is shown in red. Upper panel: non-integrable spectrum with $\Delta = 0.1$ and $\alpha = 0.1$. Lower panel: Bethe Ansatz integrable spectrum with $\Delta = 0.1$ and $\alpha = 0.0$. Notice the small system sizes used to perform the checks, and how well the exponential function captures the behavior.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 1
• Lemma
• proof