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arXiv:2601.20954

Spectral Form Factor of Gapped Random Matrix Systems

Krishan Saraswat

hep-thgr-qcquant-phSubmitted Jan 28, 2026• Announced Jan 30, 2026• Revised Apr 2, 2026 (v2)
v1 · Jan 28, 2026v2 · Apr 2, 2026
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Abstract

In this work, we study the spectral form factor of random matrix models which exhibit a large number of degenerate ground states accompanied by a macroscopic gap in the spectrum. The central aim of this work is to understand how the standard narrative about the behavior of the spectral form factor is modified in the presence of these parametrically large number of ground states. We show that, at sufficiently low temperatures, the spectral form factor is dominated by the disconnected contribution, even at arbitrarily late times. Moreover, we demonstrate that the connected form factor only depends on the eigenvalues of the non-degenerate sector, implying that BPS states do not contribute to wormhole calculations in the gravity context. Using the Christoffel-Darboux kernel, we analyze a number of examples including the Bessel model and $\mathcal{N}=2$ Jackiw-Teitelboim supergravity. In these examples, we find damped oscillations in the disconnected form factor, with a period set by the inverse size of the gap. Furthermore, we demonstrate that the slope of the ramp in the connected form factor arises from a universal sine-kernel, which emerges from a truncation of the full non-perturbative kernel in the $\hbar \to 0$ limit, and find agreement with the leading double trumpet result. Finally, we present predictions for how the ramp will transition to a plateau in the connected form factor and demonstrate how the transition depends on the details of the leading spectral density of states.

Comments
44 pages, 12 figures, and appendices. Added discussion (in appendix A) of BPS states and wormholes as well as updated references in V2

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In this work, we study the spectral form factor of random matrix models which exhibit a large number of degenerate ground states accompanied by a macroscopic gap in the spectrum. The central aim of this work is to understand how the standard narrative about the behavior of the spectral form factor is modified in the presence of these parametrically large number of ground states. We show that, at sufficiently low temperatures, the spectral form factor is dominated by the disconnected contribution, even at arbitrarily late times. Moreover, we demonstrate that the connected form factor only depends on the eigenvalues of the non-degenerate sector, implying that BPS states do not contribute to wormhole calculations in the gravity context. Using the Christoffel-Darboux kernel, we analyze a number of examples including the Bessel model and N=2\mathcal{N}=2N=2 Jackiw-Teitelboim supergravity. In these examples, we find damped oscillations in the disconnected form factor, with a period set by the inverse size of the gap. Furthermore, we demonstrate that the slope of the ramp in the connected form factor arises from a universal sine-kernel, which emerges from a truncation of the full non-perturbative kernel in the ℏ→0\hbar \to 0ℏ→0 limit, and find agreement with the leading double trumpet result. Finally, we present predictions for how the ramp will transition to a plateau in the connected form factor and demonstrate how the transition depends on the details of the leading spectral density of states.
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One manifestation of quantum chaos is a random-matrix-like fine-grained energy spectrum.Prior to the inverse level spacing time, random matrix theory predicts a "ramp" of increasing variance in the connected part of the spectral form factor.However, in realistic quantum chaotic systems, the finite-time dynamics of the spectral form factor is much richer, with the pure random matrix ramp appearing only at sufficiently late time.In this article, we present a hydrodynamic theory of the connected spectral form factor prior to the inverse level spacing time.We first derive a general formula for the spectral form factor of a system with almost-conserved sectors in terms of return probabilities and spectral form factors within each sector.Next we argue that the theory of fluctuating hydrodynamics can be adapted from the usual Schwinger-Keldysh contour to the periodic time setting needed for the spectral form factor, and we show explicitly that the general formula is recovered in the case of energy diffusion.We also initiate a study of interaction effects in this modified hydrodynamic framework and show how the Thouless time defined as the time required for the spectral form factor to approach the pure random matrix result is controlled by the slow hydrodynamics modes.We then extend the formalism to Floquet systems, where a ramp is expected but with a different coefficient, and we derive a crossover formula from the Hamiltonian ramp to the Floquet ramp when the Floquet drive is weak.Taken together, these results establish an effective field theory of chaotic spectral correlations which predicts the random matrix ramp at late time and computes corrections to it at earlier times.
Universal hallmarks of quantum chaos---such as the ramp and plateau in the spectral form factor---and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models---including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons---can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-NNN limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in 1/N21/N^21/N2. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects---such as baby-universe and wormhole condensates---which yield double exponential effects in GNG_NGN​. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.
We study the behaviour of two-time correlation functions at late times for finite system sizes considering observables whose (one-point) average value does not depend on energy. In the long time limit, we show that such correlation functions display a ramp and a plateau determined by the correlations of energy levels, similar to what is already known for the spectral form factor. The plateau value is determined, in absence of degenerate energy levels, by the fluctuations of diagonal matrix elements, which highlights differences between different symmetry classes. We show this behaviour analytically by employing results from Random Matrix Theory and the Eigenstate Thermalisation Hypothesis, and numerically by exact diagonalization in the toy example of a Hamiltonian drawn from a Random Matrix ensemble and in a more realistic example of disordered spin glasses at high temperature. Importantly, correlation functions in the ramp regime do not show self-averaging behaviour, and, at difference with the spectral form factor the time average does not coincide with the ensemble average.
We investigate crystalline-like behavior of the spectral form factor in unitary quantum systems with extremely strong eigenvalue repulsion. Using a low-temperature Coulomb gas as a model of repulsive eigenvalues, we derive the Debye-Waller factor suppressing periodic oscillations of the spectral form factor and estimate the order of its singularities at multiples of the Heisenberg time. We also reproduce this crystalline-like behavior using perturbed permutation circuits and random matrix ensembles associated with Lax matrices. Our results lay a foundation for future studies of quantum systems that exhibit intermediate level statistics between standard random matrix ensembles and permutation circuits.
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.