Spectral Form Factor (SFF) in Quantum Chaos

• Spectral Form Factor (SFF) is a two-point correlator that captures universal spectral correlations and the dip–ramp–plateau signature of quantum chaos.
• It is computed via the Fourier transform of the two-level correlation function and generalized to both Hermitian and non-Hermitian systems.
• Recent advances offer exact results in solvable models, providing clear insights into level repulsion, ergodicity, and experimental measurement techniques.

The spectral form factor (SFF) is a central object in quantum chaos, random matrix theory, and the paper of many-body quantum dynamics. It provides a direct probe of universal spectral correlations, level repulsion, and the onset of ergodicity in closed and open quantum systems, as well as in generalized settings including non-Hermitian and non-interacting models. The SFF admits rigorous characterization and exact computation in several contexts, with its prototypical “dip–ramp–plateau” structure forming a diagnostic hallmark for quantum chaoticity. This entry surveys the definition, mathematical structure, universal dynamical signatures, experimental measurement, exact solutions for solvable models, and modern generalizations of the SFF, referencing especially the recent exact results on non-interacting fermions with correlated random circuits (Flynn et al., 9 Oct 2024, Ikeda et al., 10 Oct 2024).

1. Definitions and Mathematical Formulation

The spectral form factor is a two-point correlator in the energy or quasienergy domain, quantifying correlations in the spectrum of a quantum system. The standard definition for a system of dimension $N$ with eigenvalues $\{E_n\}$ is

$$K(t) = \left\langle \left| \operatorname{Tr} e^{-iHt} \right|^2 \right\rangle = \left\langle \sum_{m,n} e^{-i(E_m-E_n)t} \right\rangle,$$

where $\langle \cdot \rangle$ denotes an ensemble or disorder average, or a time average in the absence of ensemble averaging (Dong et al., 25 Mar 2024, Okuyama et al., 2023). For Floquet or circuit systems, $H$ is replaced by a unitary $U$, so

$$K_U(L, t) = \left\langle | \operatorname{Tr} U^t |^2 \right\rangle_U,$$

where $L$ is the single-particle Hilbert space dimension (Flynn et al., 9 Oct 2024, Ikeda et al., 10 Oct 2024).

The SFF can equivalently be expressed as the Fourier transform of the two-level correlation function $R_2(E_1, E_2)$,

$$K(t) = \int dE_1 dE_2\, R_2(E_1, E_2)\, e^{i(E_1 - E_2)t},$$

with $$R_2(E_1, E_2) = \left\langle \sum_{i \ne j} \delta(E_1 - E_i)\delta(E_2 - E_j) \right\rangle$$ (Wei et al., 4 Jan 2024, Matsoukas-Roubeas et al., 2023, Cipolloni et al., 2021).

For mixed or open systems, the SFF generalizes to sums over complex (non-Hermitian) spectra or Liouvillian eigenvalues (Charamis et al., 17 Sep 2025, Zhou et al., 2023).

2. Universal Structure: Dip–Ramp–Plateau and Spectral Diagnostics

In chaotic quantum systems, the time evolution of the SFF displays a characteristic sequence of regimes (Dong et al., 25 Mar 2024, Ageev et al., 27 Dec 2024, Talib et al., 30 Nov 2025, Okuyama et al., 2023):

• Slope/Dip: At short times, interference is suppressed, and the SFF decays due to the envelope of the mean density of states.
• Ramp: At intermediate times, the SFF displays a linear rise, $K(t) \propto t$, called the "ramp," reflecting universal long-range correlations and spectral rigidity as predicted by random matrix theory (RMT). The ramp persists up to the Heisenberg time, $t_H \sim N$, the inverse mean level spacing.
• Plateau: For $t \gtrsim t_H$, only diagonal contributions survive, and the SFF saturates to a plateau, $K(t) \sim 1/N$.

This "dip–ramp–plateau" (DRP) structure is a signature of Wigner–Dyson statistics and quantum chaos. In non-chaotic or many-body localized phases, the ramp is absent; the SFF displays only a dip followed by a rapid plateau (Dong et al., 25 Mar 2024, Talib et al., 30 Nov 2025, Buijsman et al., 2020).

Higher-order SFFs, such as the generalized or multi-level spectral form factor $g^{(k)}(t)$, extend this paradigm to probe $k$-body correlations, with distinct plateau heights and more complex transient features (Wei et al., 4 Jan 2024, Sohail et al., 1 Dec 2025).

3. Exact Results and Solvable Models

Recent progress has achieved closed-form analytic expressions for the SFF in several exactly solvable ensembles:

• Non-Interacting Fermions with CUE Disorder: For non-interacting fermions on $L$ sites with single-particle phases $\{\theta_j\}$ drawn from the CUE, the many-body Floquet operator is $U = \exp(-i \sum_{i=1}^L \theta_i n_i)$, and the SFF shows exact piecewise-exponential growth:

$$K_U(L,t) = (N+1)^t \left( \frac{N+2}{N+1} \right)^{L-N t}, \quad N = \left\lfloor \frac{L}{t} \right\rfloor,\, r = L - N t,$$

with explicit forms for special divisors of $L$, scaling collapse at the single-particle Heisenberg time $t_H^{(1)} = L$, and an exponential "ramp" distinct from the linear ramp of interacting many-body RMT (Flynn et al., 9 Oct 2024, Ikeda et al., 10 Oct 2024).

• Circular Ensembles and Polygamma Structure: For the CUE of size $N$, the two-level SFF admits an exact analytic form in terms of the trigamma function $\psi^{(1)}$,

$$K_2(t;N) = N - \psi^{(1)}(N+1 - t/2\pi) + \psi^{(1)}(1 - t/2\pi),$$

reproducing the linear ramp and plateau in the large-$N$ limit, with logarithmic growth at finite $N$ for times $t = O(1)$ (Sohail et al., 1 Dec 2025).

• General β-Ensembles: Interpolating formulas for the SFF exist for all classical Gaussian ensembles (GOE, GUE, GSE), with a universal ramp slope $(2/\beta)\tau$ for rescaled time $\tau = t/2\pi$, and plateau at $\tau = 1$ (GUE), $\tau = 2$ (GSE) (Bianchi et al., 1 Mar 2024).
• Non-Gaussian Random Matrices: In polynomial potential ensembles, the universal late-time SFF is governed by the sine-kernel regardless of potential, while multi-criticalities manifest in the short-time decay laws of the disconnected part (Gaikwad et al., 2017).
• Open Quantum Systems: For Lindblad evolution, the SFF decays exponentially at early times, then displays a ramp and saturates to a plateau set by the number of steady states, with the early-time decay rate only dependent on Lindblad operators (Zhou et al., 2023).

4. SFF as a Probe of Quantum Chaos, Localization, and Transitions

The SFF serves as a robust diagnostic for distinguishing quantum-chaotic (ergodic) regimes from integrable or many-body localized (MBL) phases (Dong et al., 25 Mar 2024, Talib et al., 30 Nov 2025, Ageev et al., 27 Dec 2024, Buijsman et al., 2020):

• Quantum Many-Body Chaos: In interacting systems, a linear ramp up to $t_H$ is observed; SFF plateaus and ramp time grow exponentially with system size. Floquet and static models with random onsite fields, probed on superconducting quantum processors, confirm RMT predictions (Dong et al., 25 Mar 2024).
• Many-Body Localization: In MBL, the SFF shows a rapid dip and plateau, with ramp time not scaling with size—an immediate diagnostic of localization (Dong et al., 25 Mar 2024, Buijsman et al., 2020).
• Singling Out Single-Particle vs Many-Body Chaos: In non-interacting limits (e.g., quadratic circuits), the SFF exhibits an exponential ramp determined by single-particle dynamics, saturating at much earlier times ($t \sim L$) than full many-body “Heisenberg time” ($t_H \sim 2^L$) (Flynn et al., 9 Oct 2024, Ikeda et al., 10 Oct 2024). This provides a "single-particle chaos baseline" for detecting interaction-induced many-body chaos.
• Partial SFF and Subsystem Purity: Partial SFFs restricted to a subsystem probe eigenstate thermalization and entanglement properties, with plateau values linked to reduced density matrix purities (Dong et al., 25 Mar 2024).
• Beyond Chaos: Cautionary Examples: Systems with brick-wall normal modes (e.g., BTZ black hole, de Sitter) display DRP SFF but lack higher-point RMT signatures—demonstrating that DRP alone does not guarantee quantum chaos (Ageev et al., 27 Dec 2024).

5. Experimental Measurement and Self-Averaging Properties

Direct measurement of the SFF in many-body systems is challenging due to exponentially small level spacings in large Hilbert spaces. Experimental advances utilize randomized measurement protocols to extract SFF and partial SFF on quantum processors, thereby circumventing level resolution limitations (Dong et al., 25 Mar 2024).

Averaging procedures—over disorder, time, or frequency—are required to reveal universal SFF structure, as single-instance SFF is strongly non-self-averaging due to quantum fluctuations ("quantum noise") (Matsoukas-Roubeas et al., 2023, Charamis et al., 17 Sep 2025). Frequency and energy filtering can restore self-averaging, described mathematically as mixed-unitary or non-Hermitian quantum channels which suppress the quantum noise while retaining universal SFF features (Matsoukas-Roubeas et al., 2023, Charamis et al., 17 Sep 2025). At long times, the logarithm of the SFF can have Gumbel-distributed fluctuations due to rare close approaches to Fisher zeros (Charamis et al., 17 Sep 2025).

Recent experiments on interacting ultracold bosons confirm the SFF’s sensitivity for diagnosing the crossover from integrability (no ramp), through pseudo-integrability (weak ramp), to chaos (strong ramp)—with results corroborated by $1/f^\alpha$ noise exponents in spectral power diagnostics (Talib et al., 30 Nov 2025).

6. Advanced Theoretical Developments and Generalizations

Several further directions illustrate the depth and flexibility of the SFF framework:

• Higher-Order and Generalized SFFs: The $k$-th order generalized SFFs probe multi-level correlations, offering refined diagnostics of spectral rigidity and long-range order beyond the conventional two-level form factor (Wei et al., 4 Jan 2024, Sohail et al., 1 Dec 2025).
• Open and Non-Hermitian Systems: The SFF generalizes to dissipative/Hermitian and non-Hermitian spectra (DSFF), retaining sensitivity as a chaos diagnostic, with their quenched counterparts inheriting self-averaging properties (Charamis et al., 17 Sep 2025, Zhou et al., 2023).
• Effective Field Theory and Hydrodynamics: In hydrodynamic systems with conserved quantities, the SFF is governed by slow-mode dynamics and displays enhancements or anomalous fluctuations; the logarithm of the SFF can be mapped to single-particle SFF in the same cavity (Winer et al., 2022).
• SFF in Gravity and Holography: In supersymmetric and holographic theories, SFFs encode black hole and saddle-point physics, with late-time "ramp" phases arising from subdominant orbifolded saddles corresponding to physical Euclidean black holes in AdS (Choi et al., 2022).
• Representation Theory and Topological Matrix Models: Exact SFF results in Chern–Simons matrix models and related systems provide analytic control of intermediate regimes between Poisson, WD, and gapped spectral statistics (Vleeshouwers et al., 2022).
• Thermalization Echo: Late-time SFF displays a "second break" from RMT predictions, encoding a quantitative memory of early-time thermalization dynamics—revealed via the Riemann–Siegel lookalike formula (Winer et al., 2023).

7. Tables: SFF Regimes and Key Scaling Laws

Regime	SFF Behavior	Physical Interpretation
Slope/Dip	Model-specific decay	Mean density envelope, pre-ergodic regime
Linear Ramp	$K(t) \sim (2/\beta)\tau$	Spectral rigidity, RMT universality
Plateau	$K(t) \sim 1/N$	Diagonal terms only, finite Hilbert space
Exponential Ramp	$K(t) \propto \exp(\alpha t)$	Single-particle chaos, non-interacting baseline
Rapid Plateau	$K(t)$ flat post-dip	Localized or integrable spectrum

Examining the growth law and timescale for the SFF ramp and plateau yields diagnostic power for distinguishing ergodic, localized, and pseudo-integrable behavior.

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In summary, the SFF provides an analytically and experimentally tractable window into the ergodic and chaotic properties of quantum spectra, with recent progress yielding exact solutions, rigorous universality results, and a wealth of physical diagnostics across domains. Current research continues to expand its scope to higher-order forms, open systems, and experimental platforms (Flynn et al., 9 Oct 2024, Ikeda et al., 10 Oct 2024, Dong et al., 25 Mar 2024, Wei et al., 4 Jan 2024, Sohail et al., 1 Dec 2025).