Dissipative Spectral Form Factor (DSFF)

• DSFF is a two-point diagnostic that extends the spectral form factor to non-Hermitian settings, capturing complex-eigenvalue correlations in open quantum systems.
• Its methodology employs a two-dimensional Fourier transform and ensemble averaging, revealing dip, quadratic ramp, and plateau regimes that distinguish chaotic from integrable dynamics.
• DSFF has practical applications in analyzing dissipative systems such as random Kraus circuits, SYK-Lindbladians, and spin chains, bridging theoretical predictions with experimental non-unitary dynamics.

The Dissipative Spectral Form Factor (DSFF) is a two-point spectral diagnostic that generalizes the traditional spectral form factor (SFF) to non-Hermitian or non-unitary settings. The DSFF provides a direct probe of complex-eigenvalue correlations in open quantum systems, non-unitary quantum channels, and dissipative random matrix models, revealing universal characteristics of quantum chaos and integrability beyond the constraints of Hermiticity. Its mathematical structure enables quantitative distinction between chaotic (random-matrix-type) and integrable (Poissonian) dissipative dynamics, and establishes rigorous connections with classical and quantum statistical mechanics, random matrix theory (RMT), and open-system spectral statistics (Li et al., 2021, Sen et al., 24 Jul 2024, Cipolloni et al., 2023, Li et al., 2 May 2024).

1. Definition and Mathematical Structure

For an $N\times N$ non-Hermitian matrix with complex eigenvalues $z_i = x_i + i y_i$, the DSFF is constructed by introducing a complex “time” parameter $\tau = t + i s$, where $t, s \in \mathbb{R}$. The DSFF is defined as

$$K(\tau,\tau^*) = \left\langle \frac{1}{N}\sum_{m,n = 1}^N \exp\bigl[i(x_m - x_n)t + i(y_m - y_n)s\bigr] \right\rangle$$

or equivalently,

$$K(\tau,\tau^*) = \left\langle \frac{1}{N} \sum_{m,n} e^{i(z_m\tau^* + z_m^*\tau)/2} e^{-i(z_n\tau^* + z_n^*\tau)/2} \right\rangle$$

where angle brackets denote the ensemble or disorder average (Li et al., 2021, Sen et al., 24 Jul 2024, Cipolloni et al., 2023). The DSFF can be recast as a two-dimensional Fourier transform of the two-point spectral density, providing rotationally invariant diagnostics in the complex plane for ensembles such as Ginibre.

In the Hermitian limit (real eigenvalues), the DSFF reduces to the standard SFF,

$$\mathrm{SFF}(t) = \left\langle \left| \sum_{n = 1}^N e^{i t E_n} \right|^2 \right\rangle$$

so that $\mathcal{F}(t,s)$ is independent of $s$ (Sen et al., 24 Jul 2024).

2. DSFF in Random Matrix Theory: Ginibre and Beyond

For the complex Ginibre ensemble (GinUE), where eigenvalues densely populate the complex plane, the DSFF admits an exact determinantal formula and large-$N$ asymptotics: $$K_{\rm GinUE}(\tau,\tau^*) \approx N + N^2 \frac{4 J_1(|\tau|)^2}{|\tau|^2} - N e^{-|\tau|^2/(4N)}$$ with $J_1$ the Bessel function of the first kind. The quadratic ramp and plateau regime emerge in the large-$N$ limit: $$K_{\rm GinUE}(\tau) \rightarrow \begin{cases} |\tau|^2/4, & |\tau| \ll \sqrt{N} \ N, & |\tau| \gg \sqrt{N} \end{cases}$$ In contrast, for Poissonian (uncorrelated) spectra,

$K_{\rm Poisson}(\tau) = N + N(N-1) e^{-|\tau|^2}$

which exhibits only a trivial plateau after a small dip region (Li et al., 2021).

For the elliptic Ginibre ensemble (eGinUE), interpolating between fully non-Hermitian GinUE and Hermitian GUE via a parameter $\tau$, DSFF exhibits a dip–ramp–plateau structure with crossover scaling: $$\mathcal{F}^{\rm eGinUE}(t, s) = \exp \left[ - \frac{(1-\tau^2)(t^2 + s^2)}{4D} \right] \, \mathcal{F}^{\rm GUE}(\sqrt{D\,T})$$ with $T$ a quadratic form in $t$ and $s$, providing a direct mapping between Hermitian and non-Hermitian universality classes (Sen et al., 24 Jul 2024).

For i.i.d. random matrices, the DSFF is governed at short times by the fourth cumulant $\kappa_4$ of the entry distribution, with universal Ginibre behavior recovered at longer timescales. The leading terms are: $$K_{\mathbf{F}}(\tau, \bar{\tau}) = \left[ 4 \frac{J_1(|\tau|)^2}{|\tau|^2} \right] + \frac{1}{N^2} \left( \frac{|\tau|^2}{4} + \cdots \right) + o(1)$$ with explicit nonuniversal corrections containing $\kappa_4$ at $|\tau| \sim \mathcal{O}(1)$ (Cipolloni et al., 2023).

3. Universal Dynamical Structure: Dip, Ramp, and Plateau

The DSFF displays universal "dip–ramp–plateau" behavior that characterizes chaotic (RMT-type) and integrable (Poisson-type) open quantum systems:

• Dip: At early times ($|\tau| \lesssim 1$), the DSFF decreases from its maximal initial value due to the disconnected (one-point) component.
• Quadratic ramp: For $1 \ll |\tau| \ll \sqrt{N}$, the DSFF increases quadratically with $|\tau|$; this distinguishes non-Hermitian chaos from the linear ramp in Hermitian random matrices.
• Plateau: For $|\tau| \gg \sqrt{N}$ (Heisenberg time, $\tau_{\mathrm{H}} \sim \sqrt{N}$), the DSFF saturates at a constant plateau determined by system size $N$ (Li et al., 2021, Sen et al., 24 Jul 2024, Li et al., 2 May 2024).

Integrable and many-body localized (MBL) systems do not exhibit a ramp: after a brief transient, the DSFF reaches a plateau without universal scaling, providing a sharp diagnostic for the breakdown of random-matrix universality in dissipative dynamics (Li et al., 2 May 2024).

The following table summarizes the DSFF key regimes in chaotic and integrable/dissipative systems:

Regime	Chaotic (GinUE) DSFF	Integrable (Poisson) DSFF
Early (Dip)	$K \downarrow$ from $N^2$	Dip (Gaussian width)
Intermediate (Ramp)	$K \propto |\tau|^2/4$	No ramp
Late (Heisenberg)	Plateau at $N$	Plateau at $N$

4. DSFF in Open Quantum Many-Body Systems

Open quantum systems, governed by quantum channels (Kraus maps) or Lindbladian superoperators, have spectra comprising complex eigenvalues. The DSFF, as generalized to such spectra, requires careful treatment to reveal the expected RMT universality: $$K(t, s) = \left\langle \left| \sum_{n = 1}^N e^{i x_n t + i y_n s} \right|^2 \right\rangle - \left| \left\langle \sum_{n=1}^N e^{i x_n t + i y_n s} \right\rangle \right|^2$$ Typical analyses employ "unfolding" (conformal flattening of the spectral density) and "filtering" (cutoff functions) to remove coarse-grained density of states (DOS) variations that would otherwise obscure the universal quadratic ramp of the DSFF (Li et al., 2 May 2024).

In all surveyed chaotic open many-body models—channels, random Kraus circuits, SYK-Lindbladians, dissipative spin chains—filtered and unfolded DSFF displays an identical quadratic ramp and plateau as the Ginibre ensemble. The many-body Thouless time ($\tau_{\mathrm{dev}}$) marks the onset of this universal behavior and scales with system size (Li et al., 2 May 2024).

5. Analytical Scaling Relations and Crossover Universality

The DSFF in the elliptic Ginibre ensemble exhibits a scaling relationship with the Hermitian SFF: $$\mathcal{F}_{\rm dis,conn}^{\rm eGinUE}(t, s) = \exp \left[ -\frac{(1-\tau^2)(t^2 + s^2)}{4D} \right] \mathcal{F}_{\rm dis,conn}^{\rm GUE}(\sqrt{D\,T})$$ This allows direct transplantation of GUE SFF results to the non-Hermitian domain, establishing a quantitative bridge between Hermitian and non-Hermitian quantum chaos (Sen et al., 24 Jul 2024).

As the symmetry parameter $\tau$ interpolates between GinUE and GUE, the DSFF smoothly transitions from a quadratic to a linear ramp, with explicit dependence on Thouless and Heisenberg times:

• In the non-Hermitian regime ($\tau$ small): $T_\mathrm{Th}\propto D^{2/5}$; $T_\mathrm{H} \propto \sqrt{D}$.
• In near-Hermitian regime ($\tau \to 1$): $T_\mathrm{Th} \propto D^{1/2}$; $T_\mathrm{H} \propto D$.

Monte Carlo simulations validate these scaling predictions across finite sizes, confirming the regime-dependent transition structure and exact asymptotics (Sen et al., 24 Jul 2024).

6. Physical Interpretation and Applications

The DSFF serves as a spectral probe for quantum chaos in open and dissipative systems:

• It generalizes spectral rigidity measures from closed (Hermitian) to open (non-Hermitian) dynamics, encoding long-range level repulsion in two dimensions.
• The "quadratic ramp" is a diagnostic marker for non-Hermitian random-matrix universality and thus for dissipative quantum chaos, in contrast to the linear ramp of closed systems (Li et al., 2021, Cipolloni et al., 2023, Li et al., 2 May 2024).
• The DSFF can distinguish between dissipative systems that are chaotic (Ginibre universality class) and those that are integrable or localized (Poisson statistics).
• It connects to field-theoretical frameworks (nonlinear $\sigma$-models), where dissipation dynamically gaps the spectral soft mode, suppressing the ramp and setting a plateau value (Zhou et al., 2023).

Physical examples include dissipative quantum maps (e.g., kicked top), Lindbladian many-body chains (e.g., SYK, XXZ, Ising), and classical stochastic networks, where the DSFF quantifies transitions between dissipative chaos and integrability (Li et al., 2021, Li et al., 2 May 2024).

7. Extensions, Filtering, and Self-Averaging

Application of frequency or eigenvalue filters in otherwise unitary systems induces effective dissipation and leads to a DSFF interpretation as quantum-channel fidelity: $$K_{\rm DSFF}(t) = \operatorname{Tr}\left[ \rho_0 \Phi_t(\rho_0) \right]$$ A Gaussian frequency filter produces a Lindblad energy-dephasing master equation

$$\frac{d\rho}{dt} = -i[H, \rho] - \kappa [H, [H, \rho]]$$

where the long-time DSFF plateau becomes the thermal purity $Z(2\beta)/Z(\beta)^2$, self-averaging in the large dimension limit (Matsoukas-Roubeas et al., 2023).

This channel perspective reveals that disorder, time, and frequency filtering all generically induce unitarity breaking, resulting in dissipative spectral diagnostics essentially equivalent to the DSFF. Non-Hermitian evolution without quantum jumps corresponds to eigenvalue filtering and requires post-selection; both approaches ultimately stabilize the long-time DSFF at a plateau determined by diagonal spectral weights (Matsoukas-Roubeas et al., 2023).

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For comprehensive references and additional technical discussion, see (Li et al., 2021, Sen et al., 24 Jul 2024, Cipolloni et al., 2023, Li et al., 2 May 2024, Zhou et al., 2023), and (Matsoukas-Roubeas et al., 2023).