Spectral Form Factor of Gapped Random Matrix Systems

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. 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.

💡 Research Summary

The paper investigates the spectral form factor (SFF) of random matrix ensembles that feature a macroscopic gap and an exponentially large number of degenerate ground states. Such spectra arise naturally in supersymmetric theories, for example in $\mathcal{N}=2$ JT supergravity where a huge number of BPS states sit at zero energy while non‑BPS excitations are separated by a gap $E_{\rm gap}$. The authors ask how the standard picture of the SFF—early‑time dip, linear ramp, and late‑time plateau—gets altered when a sizable degenerate sector is present.

They begin by reviewing the conventional decomposition of the SFF into a disconnected part $\langle Z(\beta+it)\rangle\langle Z(\beta-it)\rangle$ and a connected part $\langle Z(\beta+it)Z(\beta-it)\rangle_{\rm con}$. In ordinary ensembles (GOE, GUE, etc.) the disconnected contribution decays to zero at late times, leaving the connected piece to generate the ramp and plateau. The authors then consider a mixed spectrum consisting of $N$ random eigenvalues and $\Gamma\sim e^{S_0}$ non‑random, degenerate eigenvalues. By explicit counting they show that at low temperature ($\beta\gg\hbar/E_{\rm gap}$) the disconnected term is dominated by the degenerate ground states and never vanishes, even as $t\to\infty$. Consequently the usual dominance of the connected part at late times is overturned: the SFF remains essentially constant (up to small oscillations) because of the huge degeneracy.

The connected part, however, receives contributions only from the non‑degenerate sector. To compute it the authors employ the Christoffel‑Darboux kernel $K_N(x,y)$ associated with the underlying orthogonal polynomial ensemble. By taking the semiclassical limit $\hbar\to0$, they truncate the full kernel to the universal sine kernel \