Nonstabilizerness dynamics in many-body localized systems

Nonstabilizerness, also known as ``magic'', quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this study, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer R\'enyi entropy (SRE). Leveraging a phenomenological description based on the $\ell$-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.