Complexity study of the Hartle-Hawking state in JT gravity

Abstract:The negativity of the Wigner function gives an operationally meaningful measure of the complexity of simulating a quantum state on a classical computer. We study its growth for the Hartle-Hawking state under time evolution in Jackiw-Teitelboim gravity, in the length basis. At the disk level, and to leading order in the semiclassical parameter $\beta$, the Hartle-Hawking state is a minimum-uncertainty Gaussian wavepacket at rest at the turning point of the Liouville wall, centred at $x=2\log(2\beta/\pi)$ with $\langle p^2\rangle=\pi^2/8\beta$. Its Wigner function is positive, so its negativity is $1+o(1)$ for all sub-exponential times; it is even in $t$ by time-reversal symmetry, and exactly frozen once the reflected packet decouples from the wall, because free evolution is a Clifford shear. At late times the negativity saturates close to its upper bound, at $\sqrt{2/\pi}\sqrt{d_{\rm eff}(\beta)}$ with $d_{\rm eff}(\beta)=Z(\beta)^2/Z(2\beta)$, a value that requires the discrete spectrum. Unlike the spectral form factor, the negativity displays no ramp: the two-boundary wormhole contribution is suppressed by $e^{-2S_0}$ relative to a disk term that, unlike that of the spectral form factor, does not decay. The exact survival amplitude $Z(\beta+it)/Z(\beta)$ gives the spread complexity growing as $\sigma_E^2t^2$, and the seed-normalised negativity is, at the disk level, the inverse of the exact survival probability. We take this as evidence that the length basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large $e^{S_0}$ at sub-exponential times.

arXiv:2610.00844 [hep-th]

(or arXiv:2610.00844v1 [hep-th] for this version)

https://doi.org/10.48550/arXiv.2610.00844 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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