Quantum Mpemba effect in chaotic systems with conservation laws

1. Problem Statement

The Quantum Mpemba Effect (QME) is a counterintuitive phenomenon where a quantum system initialized further from its stationary (equilibrium) state relaxes faster than a system initialized closer to it. While observed in integrable and open quantum systems, the mechanism behind the QME in closed chaotic many-body systems remains poorly understood.

Existing explanations often rely on tuning overlaps with slow-decaying modes (in Lindbladian settings) or symmetry restoration in integrable models. However, a general physical explanation for the QME in chaotic systems with conservation laws, where relaxation is governed by hydrodynamics rather than simple exponential decay, was lacking. The authors aim to fill this gap by identifying a robust mechanism for QME in chaotic spin chains.

2. Methodology

The authors investigate two distinct models of chaotic spin-1/2 chains with periodic boundary conditions in the thermodynamic limit ($N \to \infty$):

1. U(1) Symmetric Floquet Model: A periodically driven system with a conserved magnetization ($\hat{Q} = \sum \hat{Z}_i$). The evolution operator is a product of unitary gates involving nearest-neighbor and next-nearest-neighbor interactions.
2. Mixed Field Ising (MFI) Model: A Hamiltonian system with conserved energy ($\hat{Q} = \hat{H}_{MFI}$).

Key Constraints and Setup:

• Stationary State: Both systems are tuned such that the conserved quantity's expectation value in the initial state is zero ($\langle \hat{Q} \rangle_0 = 0$). Consequently, the system relaxes locally to the infinite temperature Gibbs state ($\hat{\Lambda} \propto \hat{1}$).
• Initial States: The authors analyze specific product states with translational symmetry, parametrized by variables (e.g., $a$ in the Floquet model, $\theta$ in the MFI model). These states are chosen to have the same global conserved quantity but different local structures.
• Observables (Witnesses): To quantify the distance to equilibrium, the authors track:
  • Trace Distance ($D_A$): $D_A(t) = \frac{1}{2} \text{Tr}|\hat{\rho}_A(t) - \hat{\Lambda}_A|$.
  • Entanglement Entropy ($S_A$): Specifically the "missing" entropy $\Delta S_A = l \log 2 - S_A$.
  • Entanglement Asymmetry ($\Delta S_{A,Q}$): A measure of symmetry restoration.
• Numerical Approach: Finite-size simulations (up to subsystem size $l=6$) using Trotterization for the MFI model and exact evolution for the Floquet model.
• Analytical Approach: A hydrodynamic description using Langevin equations and coarse-grained diffusion modes to explain the power-law relaxation.

3. Key Contributions

• Identification of Hydrodynamic Mechanism: The paper establishes that in chaotic systems with conservation laws, relaxation is algebraic (power-law) rather than exponential due to hydrodynamic long-time tails.
• Initial State Dependence of Exponents: The authors demonstrate that the effective dynamical critical exponent ($z_{\text{eff}}$) governing the power-law decay ($t^{-1/z_{\text{eff}}}$) depends critically on the specific initial state.
• Symmetry-Protected Relaxation: They show that for highly symmetric initial states (e.g., $a=0$ in the Floquet model), the leading hydrodynamic mode (diffusive magnetization fluctuations) is suppressed because the initial fluctuations match the stationary state fluctuations. This forces the system to relax via sub-leading, faster-decaying modes.
• Robust QME Realization: By tuning the initial state to suppress the slowest decaying mode, the system can relax faster than a generic state that is initially "closer" to equilibrium (in terms of trace distance) but contains the slow mode.

4. Key Results

A. Trace Distance and Power-Law Regimes

• Generic States ($a \neq 0$): The trace distance decays as $D_A(t) \sim t^{-1/2}$ (corresponding to $z_{\text{eff}} \approx 2$). This is the standard diffusive behavior expected for conserved quantities.
• Symmetric States ($a = 0$): The leading $t^{-1/2}$ mode vanishes. The decay accelerates to $D_A(t) \sim t^{-3/2}$.
• Mpemba Crossing: Because the symmetric state starts with a larger distance (or comparable distance) but decays with a steeper power law, it crosses the relaxation curve of the generic state. A state initially "further" (or similarly distant) relaxes faster, creating the QME.

B. Entanglement Entropy and Asymmetry

• Entanglement Entropy: Follows similar scaling but with exponents halved ($\Delta S_A \sim t^{-1}$ for $a \neq 0$ vs. $t^{-3}$ for $a=0$).
• Entanglement Asymmetry:
  • In the Floquet model, the asymmetry decays as $t^{-3}$ for generic states and $t^{-5}$ for the symmetric state ($a=0$).
  • In the MFI model, where magnetization is not conserved, the asymmetry behaves similarly to the entropy, showing crossings but without the specific symmetry protection seen in the Floquet case.

C. Hydrodynamic Explanation

The authors derive these results using a coarse-grained hydrodynamic theory:

• The reduced density matrix is expanded in Pauli strings.
• Operators containing only conserved quantities (e.g., $\hat{Z}_i \hat{Z}_j$) decay diffusively ($\sim t^{-1/2}$).
• For the symmetric state ($a=0$), the initial fluctuations of these operators match the stationary state ($\langle \hat{Q}^2 \rangle_0 = \text{Tr}(\hat{Q}^2 \hat{\Lambda})$). Thus, the diffusive transport of these fluctuations is unnecessary, and the decay is dominated by sub-leading terms involving gradients of the magnetization ($\sim t^{-3/2}$).

5. Significance

• Theoretical Insight: This work provides the first comprehensive physical explanation for the QME in closed chaotic systems. It moves beyond the "tuning overlaps" argument of open systems to a mechanism based on hydrodynamic mode suppression due to initial state symmetries.
• Universality: The mechanism relies on the interplay between conservation laws and hydrodynamics, suggesting it is a generic feature of chaotic systems with conserved quantities, not limited to specific integrable models.
• Experimental Relevance: The findings suggest that by engineering specific initial states (e.g., specific spin configurations) in synthetic quantum matter (like cold atoms or superconducting qubits), one can induce faster thermalization, potentially enabling rapid state preparation protocols.
• Open Questions: The paper notes that while the Floquet model is well-understood via hydrodynamics, the MFI model exhibits anomalous exponents (sub-diffusive and super-diffusive) that are not yet fully explained by standard hydrodynamic theories, pointing to new physics in chaotic thermalization.

In summary, the paper demonstrates that the Quantum Mpemba Effect in chaotic systems is a robust phenomenon arising when specific initial states suppress the slowest hydrodynamic relaxation modes, allowing them to overtake generic states that are initially closer to equilibrium.