Nonstabilizerness in Stark many-body localization

TL;DR

The study probes how nonstabilizerness, or magic, evolves in a disorder-free Stark MBL system by analyzing a tilted transverse-field Ising chain. It combines analytical Schrieffer-Wolff arguments with numerical simulations to show slow, initial-state–dependent growth of the stabilizer Rényi entropy M2 deep in the strong tilt, signaling an ETH–SMBL crossover alongside entanglement dynamics. A key result is the emergence of a diagonal effective description that suppresses long-range processes, naturally explaining the slow magic growth and finite-size plateaus. The work also outlines a feasible trapped-ion experimental protocol to extract both M2 and half-chain entanglement from common randomized measurements, highlighting nonstabilizerness as a practical complexity probe for disorder-free ergodicity breaking and fragmentation.

Abstract

Quantum many-body disorder-free localization can suppress transport while still allowing the buildup of computationally costly non-Clifford resources. In a transverse-field Ising chain realizing disorder-free Stark many-body localization, we show that the stabilizer Rényi entropy remains nonzero and grows slowly to a finite plateau deep in the strong Stark-field regime, with strong initial-state selectivity. As the Stark field strength increases, long-time magic and entanglement consistently signal a crossover from ergodic to constrained localized dynamics. These results establish nonstabilizerness (``magic'') as a practical complexity probe for disorder-free ergodicity breaking and constrained localization, with direct relevance to benchmarking and designing near-term quantum simulators, and fill a gap in the understanding of nonstabilizerness in disorder-free many-body localization.

Figures (4)

• Figure 1: Time evolution of the SRE-2. SRE-2 $M_2$ as a function of dimensionless time $Jt$ with system size $L\!=\!10$ for four initial states: (a) $X$-polarized, (b) $Y$-polarized, (c) $Z$-polarized, and (d) Bloch sphere random product state. Colored curves correspond to different field strengths $F$ (Cf. legends). The black dash-dotted line indicates the Haar-random saturation value $M_2^{\mathrm{Haar}}$ for the corresponding system size $L\!=\!10$.
• Figure 2: Dynamics and scaling of the SRE-2. (a-d) Time evolution of the SRE-2 $M_2$ versus dimensionless time $t$ for (a) $X$-polarized, (b) $Y$-polarized, (c) $Z$-polarized, and (d) random Bloch product initial states. The colored curves correspond to system sizes $L=6, 8, 10, 12, 14$, with color intensity increasing from light ($L\!=\!6$) to dark ($L\!=\!14$). Black dashed curves in (a, b, d) indicate fits to the logarithmic growth model Eq. \ref{['eq:log_growth']}. (e, f, h) Scaling of the fitted saturation values $M_{2,\mathrm{fit}}^{\mathrm{Sat}}$ versus system size $L$ for the (e) $X$-polarized, (f) $Y$-polarized, and (h) random Bloch initial states. The black dashed line represents the theoretical Haar value $M_2^{\mathrm{Haar}}\!\approx\!L-2$. (g) Saturation magic $M_2^{\mathrm{sat}}$ versus $L$ for varying field strengths $F$. The data shows a crossover from volume-law like scaling at small $F$ to area-law behavior at large $F$.
• Figure 3: ETH-SMBL crossover in magic and entanglement. (a) $\Delta M_2$ versus field strength $F$ for system sizes $L\!=\! 8,10,12,14$. Inset: finite-size collapse of $\Delta M_2$ versus $(F-F_c)L^{1/\nu}$ with $F_c\!\approx\!0.19$ and $\nu\!\approx\!0.53$. (b) Half-chain entanglement entropy $S_{L/2}$ versus $F$ for the same sizes. Inset: finite-size collapse of $S_{L/2}$ versus $(F-F_c)L^{1/\nu}$ with $F_c\!\approx\!0.23$ and $\nu\!\approx\!0.46$. (c) Parametric relation between SRE $M_2$ and entanglement $S_{L/2}$ for a $Y$-polarized initial state at fixed $L\!=\!10$ and several SMBL $F$ values with Gaussian-smoothed time traces. The black dashed curve is a polynomial fit $M_2(S_{L/2})$ with Eq. \ref{['eq:log_growth']}, and the dash-dotted line marks the Haar saturation value. (d) Same as (c) for Bloch sphere random product initial states, plotting the rescaled SRE-2 $M_2/f(F)$, where $f(F)$ is $L$-independent rescaled function; the dashed curve is a fit $M_2(S_{L/2})$.
• Figure 4: Circuit for digitally simulating the tilted transverse-field Ising model using a second-order Strang step repeated $k$ times, followed by local single-qubit Clifford randomized measurements to extract $S_2(A,t)$ and $M_2(t)$ from the same bitstring data.