Magic transition in monitored free fermion dynamics

TL;DR

This paper studies the behavior of quantum magic and entanglement in 1+1D free fermion circuits, revealing a delocalization transition of magic and a critical phase with unique relaxation dynamics.

Contribution

It introduces a novel analysis of magic in free fermion circuits, demonstrating a phase transition in magic structure and characterizing its dynamics and scaling behavior.

Findings

Magic remains extensive across the phase transition.

The structure of magic undergoes a delocalization transition.

Relaxation time in the critical phase scales linearly with system size.

Abstract

We investigate magic and its connection to entanglement in 1+1 dimensional random free fermion circuits, with a focus on hybrid free fermion dynamics that can exhibit an entanglement phase transition. To quantify magic, we use the Stabilizer R\'enyi Entropy (SRE), which we compute numerically via a perfect sampling algorithm. We show that although the SRE remains extensive as the system transitions from a critical phase to an area-law (disentangled) phase, the structure of magic itself undergoes a delocalization phase transition. This transition is characterized using the bipartite stabilizer mutual information, which exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in $O(1)$ time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.