Spectral Form Factor in Quantum Chaos

• Spectral Form Factor is a measure of energy-level correlations in quantum systems, defined as the squared modulus of the Fourier-transformed density of states.
• It employs Fourier analysis of deformed eigenvalue densities from polynomial potentials to reveal multi-critical behavior and model-specific early-time decay.
• Scaling studies show that while early-time dynamics are tuned by non-Gaussian deformations, the universal sine kernel governs the constant late-time ramp and plateau.

The spectral form factor (SFF) is a central object in random matrix theory (RMT) and quantum chaos, providing a dynamic diagnostic of spectral correlations and ergodicity in quantum many-body and disordered systems. It is defined as the squared modulus of the Fourier-transformed density of states or, more precisely, the analytic continuation of the partition function to complex time, and encodes the transition from short-time, model-dependent behavior to universal long-time correlations characterized by the “ramp” and “plateau.” Deformations of the RMT ensemble by non-Gaussian (higher-order polynomial) potentials induce rich phase structures and multi-critical phenomena that alter early-time SFF dynamics, yet robustly leave the universal long-time structure intact. This property establishes the SFF as a powerful and versatile tool for diagnosing quantum chaotic behavior in both fundamental models and real quantum systems.

1. Definition and Structure of the Spectral Form Factor

For a quantum system with a discrete spectrum $\{E_n\}$, the spectral form factor is constructed from the analytically continued partition function $Z(\beta + it)$,

$$|Z(\beta + it)|^2 = \sum_{mn} e^{-\beta(E_m + E_n)} e^{-it(E_m - E_n)} \,,$$

where typically $\beta = 0$ for pure quantum dynamics. The SFF is naturally decomposed into disconnected and connected pieces. The disconnected part measures the product of one-point spectral densities (mean-level behavior), while the connected part encodes genuine energy-level correlations: $$\mathrm{SFF}(t) \equiv \left|\int d\lambda~e^{-it\lambda}\,\rho(\lambda)\right|^2\,.$$ For matrix ensembles, the eigenvalue density $\rho(\lambda)$ is model and potential dependent.

The SFF generically exhibits distinct time regimes:

• Early time (disconnected): Power-law decay determined by global density and edge singularities.
• Intermediate (“ramp”): Universal linear growth governed by short-distance spectral correlations.
• Late time (“plateau”): Saturation to a system-size-dependent constant—a haLLMark of spectral rigidity.

2. Non-Gaussian Deformations and Model Construction

Standard RMT considers Gaussian potentials $V(M) \propto M^2$ (Wigner-Dyson ensembles). The focus of (Gaikwad et al., 2017) is to generalize to non-Gaussian, higher-order polynomial potentials:

• Quartic: $V(M) = \frac{1}{2} M^2 + g N M^4$
• Sextic: $V(M) = 12 M^2 + g N M^4 + h N^2 M^6$

The eigenvalue density $\rho(\lambda)$ changes according to the modified saddle-point/resolvent equation:

$$\rho_{\text{quartic}}(\lambda) = \frac{1}{\pi}\left[12 + 4g a^2 + 2g\lambda^2\right]\sqrt{4a^2 - \lambda^2}$$

with $a^2$ subject to the constraint $12 g a^4 + a^2 - 1 = 0$.

For the sextic model,

$$\rho_{\text{sextic}}(\lambda) = \frac{1}{\pi}\left[3h\lambda^4 + (2g + 6h a^2)\lambda^2 + (12 + 4g a^2 + 18 h a^4)\right]\sqrt{4a^2 - \lambda^2}$$

with $60 h a^6 + 12g a^4 + a^2 - 1 = 0$.

The SFF for these models is computed via the Fourier transform of $\rho(\lambda)$, utilizing explicit Bessel function expressions for $Z(it)$.

3. Phase Structure and Multi-Criticality

Deformations by quartic and sextic (and higher) terms induce nontrivial phase diagrams in the space of coupling constants:

• The quartic model admits a critical point at $g_c = -1/48$; below this, a single support (one-cut) solution is lost.
• The sextic model supports a richer phase diagram, including a critical line and tri-critical points (e.g., $g_{\text{tri}} = -1/36$, $h_{\text{tri}} = 1/1620$ with $a^2 = 3$).

At these criticalities, the edge behavior of $\rho(\lambda)$ is altered:

• Generic: $\rho(\lambda) \sim (2a - \lambda)^{1/2}$
• Quartic critical: $(2a - \lambda)^{3/2}$
• Sextic tri-critical: $(2a - \lambda)^{5/2}$

This directly impacts the power-law decay in the early-time SFF. Away from criticality, the disconnected SFF decays as $\tau^{-3/2}$; at the quartic critical point, as $\tau^{-5/2}$; and at the tri-critical sextic point, as $\tau^{-7/2}$.

4. Dip Time and Scaling Analysis

The “dip-time”—at which the initially decaying disconnected SFF crosses over to the ramp set by the connected part—is a key diagnostic of quantum chaos onset. It is estimated by equating the decaying disconnected contribution to the ramping connected contribution:

• Gaussian: $\tau^{-3} \sim \tau N^2 \implies \tau \sim N^{1/2}$
• Quartic critical: $\tau^{-5} \sim \tau N^2 \implies \tau \sim N^{1/3}$
• Sextic tri-critical: $t \sim N^{-1/4}$

A higher decay exponent at criticality signals an earlier ramp onset when expressed in scaled time; hence, multi-critical deformations permit tuning of the ramp time and chaos diagnostic.

5. Universality of Late-Time Behavior

While the early/disconnected SFF (and thus the dip and ramp onset) demonstrates sharp sensitivity to the details of the global spectral density and edge behavior (hence, to non-Gaussian deformations), the late-time SFF is determined solely by the universal sine kernel: $$K(\lambda, \mu) \sim \frac{\sin^2(N(\lambda - \mu))}{(\pi N (\lambda - \mu))^2}$$ The ramp (linear in $t$) and plateau values thus do not depend on the specifics of the global eigenvalue density or the details of the potential, and emerge identically in all polynomially-deformed ensembles. This universality mechanism is demonstrated through explicit computation and analytical continuation of the kernel.

6. Explicit SFF Expressions and Asymptotics

For the quartic potential, the explicit disconnected SFF is given by

$$Z(it) = \frac{1}{\tau^2} \left[a\tau(1 + 24 a^2 g) J_1(2a\tau) - 24 a^2 g J_2(2a\tau)\right]$$

with $J_n$ a Bessel function. Leading asymptotics at large $\tau$ show the dominant decay shifts to higher negative powers at multicritical points as outlined above.

For the sextic model, the analogous Fourier transform structures emerge from the derived density.

7. Implications for Quantum Chaos and Chaotic Many-Body Systems

The analysis establishes that, despite strong variations in short-time (dip/ramp onset) SFF behavior sensitive to eigenvalue density edges and phase structure, all polynomial deformations of RMT display universal ramp/plateau behavior at late times. This underpins the observed RMT universality in quantum chaotic systems—including models such as the SYK model—even when the averaging ensemble is non-Gaussian.

Hence, tuning multi-criticality in random matrix models permits exploration of the temporal crossover to chaos, while the robust long-time spectral rigidity is a stringent diagnostic of quantum chaos insensitive to non-Gaussian details. This provides a dynamical mechanism for the emergence of universality observed in quantum chaotic dynamics.

8. Table: Scaling of Early-Time SFF Decay and Dip Time

Potential	Edge Density Exponent	Decay Power $\sim \tau^{-\alpha}$	Dip Time Scaling $t_\text{dip}$
Gaussian	$1/2$	$3/2$	$N^0$ (in $t$ units)
Quartic crit.	$3/2$	$5/2$	$N^{-1/6}$
Sextic tri-crit	$5/2$	$7/2$	$N^{-1/4}$

Increasing the edge exponent at multi-criticality increases the decay power and decreases the physical dip time, enabling model-dependent control of quantum chaos diagnostics while preserving late-time universal behavior.

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In summary, non-Gaussian random matrix theories display a rich phenomenology in their spectral form factor, demonstrating strong multi-critical effects in short-time behavior while preserving the ramp and plateau universality at late times. This framework provides a model for understanding the interplay between ensemble potential details and the emergence of universal quantum chaotic dynamics.