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arXiv:2512.16859

Slow growth of quantum magic in disorder-free Stark many-body localization

Han-Ze Li, Yi-Rui Zhang, Yu-Jun Zhao, Xuyang Huang, Jian-Xin Zhong

quant-phSubmitted Dec 18, 2025• Announced Dec 19, 2025• 0 citations• Revised Jan 6, 2026 (v2)
v1 · Dec 18, 2025v2 · Jan 6, 2026
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Abstract

Disorder-free quantum many-body localization can strongly suppress transport while still enabling the dynamical buildup of computationally costly non-Clifford resources. In a tilted transverse-field Ising chain realizing disorder-free Stark many-body localization, we use the stabilizer Rényi entropy to quantify quantum magic (nonstabilizerness) and find that it remains finite and grows anomalously slowly over extended time windows before saturating to a size-dependent plateau deep in the strong-tilt regime, with pronounced initial-state selectivity. Upon increasing the Stark gradient, the long-time magic and half-chain entanglement exhibit consistent finite-size crossing behavior, indicating a crossover from ergodic dynamics to constrained localization. These results establish stabilizer-based magic as a practical complexity diagnostic of disorder-free ergodicity breaking and constrained dynamics, and provide an experimentally accessible route to benchmarking and designing near-term quantum simulators.

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Disorder-free quantum many-body localization can strongly suppress transport while still enabling the dynamical buildup of computationally costly non-Clifford resources. In a tilted transverse-field Ising chain realizing disorder-free Stark many-body localization, we use the stabilizer Rényi entropy to quantify quantum magic (nonstabilizerness) and find that it remains finite and grows anomalously slowly over extended time windows before saturating to a size-dependent plateau deep in the strong-tilt regime, with pronounced initial-state selectivity. Upon increasing the Stark gradient, the long-time magic and half-chain entanglement exhibit consistent finite-size crossing behavior, indicating a crossover from ergodic dynamics to constrained localization. These results establish stabilizer-based magic as a practical complexity diagnostic of disorder-free ergodicity breaking and constrained dynamics, and provide an experimentally accessible route to benchmarking and designing near-term quantum simulators.
Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-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.
Quantum magic---the resource that separates universal quantum computation from efficiently simulable Clifford circuits---has emerged as a diagnostic of many-body quantum states, yet the stabilizer Rényi entropy (SRE) that quantifies it remains largely inaccessible in interacting fermion systems. Estimating the global SRE requires a sum over exponentially many Majorana strings, which in determinant quantum Monte Carlo can be sampled without a sign problem only for the interacting density matrix obtained after averaging the auxiliary fields: sampling strings and fields simultaneously incurs a sign problem even for sign-problem-free models, so that the known sign-free alternative is a nested Monte Carlo that does not scale. We propose instead the two-point SRE---a practical stabilizer-Rényi correlator built from rank-2 SREs of one- and two-site reduced density matrices, which are fixed exactly by single-particle Green functions and density correlations already measured in standard simulations---and show that it obeys universal finite-size scaling at fermionic quantum critical points. In the one-dimensional half-filled spinless ttt-VVV chain, the correlator distinguishes algebraic and exponential decay regimes and tracks the inverse-logarithmic finite-size drift characteristic of the Berezinskii--Kosterlitz--Thouless transition. On the honeycomb lattice, sign-problem-free quantum Monte Carlo yields Gross--Neveu--Ising scaling with finite anomalous dimension at zero temperature and two-dimensional Ising collapse at the thermal transition. Our results suggest stabilizer-Rényi microscopy as a spatially resolved, quantitatively universal probe of critical correlations in interacting fermionic matter, on par with conventional order-parameter correlators and accessible to quantum-simulator measurements.
We discuss properties of interacting spinless fermions in a tilted optical lattice as modeled by Heisenberg spin chain in magnetic (or electric) field changing linearly across the chain. The system shows genuine features of strong localization for a sufficiently large tilt with distinct features different from disorder induced localization. In particular, Stark many-body localization is characterized in time dynamics by a linear increase of the entanglement entropy followed by a saturation at a constant, time independent value -- both the number (classical) and the configuration (quantum) entropies saturate to a constant value. Similarly, the correlation with the initial state (the so called imbalance) stabilizes at a constant value (modulo large fluctuations coming from short time Bloch-type oscillations). The presence of an additional harmonic potential profoundly affects the dynamics of the model -- its behavior resembles standard MBL characteristics -- the logarithmic growth of the entanglement entropy in particular. In that case, we find strong revival phenomena associated with the manifold of equally spaced states, degenerate in the absence of the harmonic potential.
The stabilizer Rényi entropy is a quantitative measure of non-stabilizerness, or magic, and has typically been found to scale extensively with system size NNN (i.e., Θ(N)Θ(N)Θ(N)) for a variety of many-body quantum states. In this note, we study the stabilizer Rényi entropy of the ground state of the spin-12\frac{1}{2}21​ Dyck-Fredkin chain and its ttt-deformation, a local frustration-free model with unusual spectral-gap scaling. Exploiting the combinatorial structure, we carry out numerically exact finite-size calculations, which indicate asymptotic behavior depending on ttt: Θ(N)Θ(N)Θ(N) for t<1t<1t<1, Θ(log⁡N)Θ(\log N)Θ(logN) at t=1t=1t=1, and Θ(1)Θ(1)Θ(1) for t>1t>1t>1. The scaling at t=1t=1t=1 could be another manifestation of the unconventional criticality of the model, while the contrast with the behavior of the entanglement entropy suggests that non-stabilizerness might provide a new window into quantum many-body systems.
Quantum magic, or non-stabilizerness, is an important quantum resource that characterizes computational power beyond classically simulable Clifford operations and is therefore essential for achieving quantum advantage. While previous studies have explored non-stabilizerness dynamics in random circuits and under time-independent generators, here we extend the study of its universal dynamics to time-dependent driving across quantum phase transitions. In particular, we show that the stabilizer Rényi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate in slow processes. Moreover, we show that the logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. Our results are explicitly demonstrated by exact results in the transverse-field Ising model and by analytical approximations in long-range Kitaev models.
We investigate many-body localization of interacting spinless fermions in a one-dimensional disordered and tilted lattice. The fermions undergo energy-dependent transitions from ergodic to Stark many-body localization driven by the tilted potential, which are manifested by the appearance of mobility edges between delocalized states and Stark many-body localized states even when the disorder is weak. We can concretely diagnose these transitions rather than crossovers by finite-size scaling of energy-level statistics. Moreover, in the Stark many-body localization, the entanglement entropy obeys the area law scaling, in analogy to that in the conventional many-body localization.