Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors

Abstract: Under unitary evolution, chaotic quantum systems initialized in simple states rapidly develop high complexity, precluding any efficient classical description. Quantum chaos is traditionally characterized by spectral properties of the Hamiltonian, most notably through the spectral form factor, while the hardness of classical simulation within the stabilizer formalism, commonly referred to as quantum magic, can be quantified by the stabilizer Rényi entropy. In this Letter, we propose a relation between the dynamics of the stabilizer Rényi entropy for thermofield double states and the spectral form factor, based on general arguments for chaotic systems with all-to-all interactions. This relation implies that the saturation of the stabilizer Rényi entropy is governed by a first-order dynamical transition. We then demonstrate this relation explicitly in the Sachdev-Ye-Kitaev model, using an auxiliary-spin representation of the stabilizer Rényi entropy that exhibits an emergent $Z_2$ symmetry. We further find that, in the high-temperature regime of the SYK model, the transition occurs at a finite time, with the long-time phase marked by spontaneous $Z_2$ symmetry breaking. In contrast, at low temperatures, the transition is pushed to times exponentially long in the system size. Our results reveal an intriguing interplay between quantum chaos and quantum magic.