Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems

Abstract

We present exact, closed-form results for the non-stabilizerness of random pure states subject to a U(1) symmetry constraint. Using stabilizer entropy as our non-stabilizerness monotone, we derive the average and the variance for U(1)-constrained Haar random states. We show that the presence of a conserved charge leads to a substantial suppression of non-stabilizerness (magic) compared to the unconstrained case, and identify a qualitative difference between entanglement and magic response. In the thermodynamic limit, stabilizer entropy exhibits a different leading-order scaling close to a vanishing relative charge density, implying that magic is more robust to charge density fluctuations than entanglement entropy. We test our analytical predictions against midspectrum eigenstates of two chaotic many-body systems with conserved $U(1)$ charge: the complex-fermion Sachdev-Ye-Kitaev (cSYK) model and a Heisenberg XXZ chain with next-to-nearest-neighbour couplings and conserved magnetization. We find an excellent agreement for the non-local cSYK model and systematic deviations for the local XXZ chain, highlighting the role of interaction locality.

Figures (3)

• Figure 1: (a) Schematic of the random-state ensemble: states are sampled uniformly within a fixed $U(1)$ charge sector $q$, modeling eigenstates of $U(1)$-symmetric chaotic Hamiltonians. (b),(c) Analytical predictions (curves) versus numerical data for the order-2 stabilizer purity (b) and stabilizer entropy (c) of Haar and Haar$\times U(1)$ ensembles ($L=14$, $q=0$, $2.5\times10^5$ samples). Closed-form expressions for the $U(1)$ constrained ensemble mean $\mathbb{E}_{U_q}[\Xi_2]$ and the variance $\Delta_{U_q}^2$ of stabilizer purity are among our main results. The unconstrained ensemble results are previously known. (d) Average stabilizer entropy versus charge sector $q$ for Haar$\times U(1)$ (red) and Haar (blue).
• Figure 2: Disorder-averaged stabilizer entropy of cSYK eigenstates as a function of the charge sector $q$ for different system sizes $L$. Each data point is averaged over disorder realizations (for more detail see SM SI) and all eigenstates within the sector, regardless of energy density, since all of states exhibit chaotic behaviour in both entanglement and non-stabilizerness content Liu_Chen_Balents_2018Jasser_Odavic_Hamma_2025. Faded points show single-realization data. Solid lines denote our analytical prediction for constrained Haar-random states, while dashed lines correspond to the unconstrained Haar ensemble. The two differ by an $\mathcal{O}(1)$ offset. Numerical results agree with the theory across all sectors, except $q = -L +2$ where the Hamiltonian is trivial (see main text).
• Figure 3: Stabilizer entropy of XXZ-NNN$\times U(1)$ eigenstates as a function of the magnetization sector $q$ for different system sizes $L$. Each data point is averaged over all eigenstates (mid-spectum states) within the sector in the energy density window $-0.25 < E/L < 0.25$; missing data at large positive $q$ reflect the absence of states with those magnetization quantum numbers in the selected energy interval. Solid lines denote our analytical prediction for constrained Haar-random states, while dashed lines correspond to the unconstrained Haar ensemble.