Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors

TL;DR

This paper establishes a theoretical link between the dynamics of quantum magic, quantified by stabilizer Renyi entropy, and spectral form factors in chaotic systems, demonstrated explicitly in the Sachdev-Ye-Kitaev model, revealing a transition governed by system temperature.

Contribution

It introduces a novel relation between stabilizer Renyi entropy dynamics and spectral form factors in chaotic quantum systems, with explicit analysis in the SYK model.

Findings

Stabilizer Renyi entropy saturation is governed by a first-order dynamical transition.

In the SYK model, the transition occurs at finite time at high temperature.

At low temperature, the transition time grows exponentially with system size.

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 R\'enyi entropy. In this Letter, we propose a relation between the dynamics of the stabilizer R\'enyi 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 R\'enyi 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 R\'enyi 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.