Symmetry restoration in a fast scrambling system

TL;DR

This work probes how symmetry is restored during thermalization in a fast-scrambling quantum system by tracking entanglement entropy and entanglement asymmetry in the complex SYK model with a global U(1) charge. Using exact diagonalization, the authors show rapid volume-law entanglement growth compatible with subsystem ETH, followed by a finite-size plateau of entanglement asymmetry that encodes residual cross-sector coherence and incomplete symmetry restoration. They uncover a quantum Mpemba effect in which states prepared farther from symmetry relax faster, a phenomenon strengthened by disorder. A Pinsker-type bound links the asymmetry decay to differences in subsystem purities, identifying dephasing between charge sectors as the mechanism behind symmetry restoration. Collectively, the results establish entanglement asymmetry as a sharp, symmetry-resolved diagnostic of thermalization and finite-size effects in fast-scrambling quantum systems, with implications for holography and quantum simulations.

Abstract

Entanglement asymmetry -- used here as a direct probe of symmetry restoration -- provides a sharp diagnostic of post-quench dynamics. We test this idea in the complex Sachdev--Ye--Kitaev model with a conserved U(1) charge. Using exact diagonalization, we track the joint evolution of entanglement entropy and entanglement asymmetry after quenches from charge-asymmetric product states. We find rapid volume-law entanglement growth consistent with the subsystem eigenstate thermalization hypothesis, accompanied by a concurrent decay of entanglement asymmetry to a late-time plateau set by finite-size effects: small subsystems display near-complete restoration, while residual cross-sector weight yields a finite plateau. Notably, we uncover a quantum Mpemba effect: states prepared further from symmetry relax faster and approach lower residual asymmetry; disorder in the couplings renders this behavior more robust and monotonic across parameters. We further derive a Pinsker-type lower bound that ties the decay of asymmetry to differences in subsystem purity, identifying dephasing between U(1) charge sectors as the operative mechanism. These results establish entanglement asymmetry as a sensitive probe of symmetry restoration and thermalization, clarifying finite-size limits in fast-scrambling, closed quantum systems.

Figures (6)

• Figure 1: Schematic illustration of bipartition of the complex SYK model into subsystem $A$ and its complement $B$.
• Figure 2: Block diagonal structure of the ground state under the single realization of cSYK model, blue region are non-zero matrix elements. (left) Reduced density matrix $\rho_A$ of subsystem $B$ composed by $m=6$ sites. (right) Density matrix $\rho$ of the total system. Here, the number of total sites is $N=12$.
• Figure 3: The 50 times realizations average entanglement entropy (solid line) and its non-averaged single realization (dotted) as a function of time with various subsystem lengths $m$, where $\theta=\pi/2,~N=20$.
• Figure 4: The 50 times realizations average entanglement asymmetry (solid line) and its non-averaged single realization (dotted) as a function of time $\Delta S_A(t)$ with various subsystem lengths $m$, where $\theta=\pi/2,~N=20$.
• Figure 5: The Pinsker-type bound (dotted) of entanglement asymmetry, i.e. purity difference ${\rm Tr} (\rho_A^2-\rho_{A,Q}^2)$ and entanglement asymmetry (solid line) in a single realization at $m=8,~\theta=\pi/2,~N=20$.