Grand Canonical-like Thermalization of Quantum Many-body Scars

Quantum many-body scar (QMBS) in kinetically constrained quantum systems challenges the conventional eigenstate thermalization hypothesis (ETH). We develop an effective open-system description for constrained dynamics and introduce the definition of quasiparticle number in the system. Based on this, we formulate a revised ETH that accounts for both diagonal and off-diagonal structures of local observables. By introducing the cross coherence purity (CCP), we obtain a unified characterization of off-diagonal matrix elements and show that the relevant density of states (DOS) is determined by the distribution of eigenstates on the energy–quasiparticle-number plane. We numerically verify an inverse relation between the CCP and this generalized DOS. Applied to the quantum many-body scar model, the revised ETH accurately predicts long-time averages and temporal fluctuations of local observables and explains their dependence on initial states. Our framework shows that the anomalous fluctuations and quasi-periodic dynamics of scar states arise naturally from low-DOS regions. These results provide a unified understanding of thermalization and QMBS in kinetically constrained systems.

💡 Research Summary

The paper addresses the puzzling phenomenon of quantum many‑body scars (QMBS) that appear in kinetically constrained quantum systems, such as the PXP model, and which violate the conventional eigenstate thermalization hypothesis (ETH). The authors propose a unified framework that treats constrained dynamics as an effective open‑system problem, introducing a quasiparticle number operator ˆN that quantifies the rate of information exchange with an auxiliary environment. By coupling the unconstrained Hamiltonian to two Lindblad channels, they construct a non‑Hermitian Hamiltonian H_N whose dynamics within the constrained subspace exactly reproduces the original unitary evolution, while the jump terms encode the kinetic blockade. This mapping motivates a grand‑canonical‑like description: each eigenstate |E_i⟩ is characterized not only by its energy E_i but also by its quasiparticle number N_i = ⟨E_i|ˆN|E_i⟩. Consequently, the diagonal part of ETH is revised to ⟨E_i|Ô|E_i⟩ ≈ O(E_i, N_i), a smooth function of both variables. Numerical analysis on a spin‑1 chain (length 10) shows that local observables (e.g., s_z₁ s_z₂) cannot be fitted by a single‑parameter function O(E) but are accurately captured by a two‑dimensional surface O(E,N). Scar states sit in regions of unusually low quasiparticle density, i.e., where the generalized density of states Ω(E,N) is sparse.
To treat off‑diagonal matrix elements, the authors introduce the cross‑coherence purity (CCP), a quantity that measures the coherence between different eigenstates. They demonstrate an inverse relationship between CCP and the generalized DOS: |⟨E_i|Ô|E_j⟩| ≤ Ω(E,N)^{‑1/2} · CCP^{‑1}. This bound simultaneously controls the amplitude of off‑diagonal contributions and the magnitude of temporal fluctuations in long‑time dynamics. Simulations confirm that regions with low Ω(E,N) exhibit large CCP, leading to pronounced quasi‑periodic revivals and large deviations of long‑time averages from canonical predictions.
Finally, the paper connects these findings to the emergence of a spectrum‑generating algebra (SGA). In low‑DOS sectors the eigenstates become approximately equally spaced, allowing an algebraic structure to arise naturally without imposing any fine‑tuned symmetry. Thus, the anomalous dynamics of scar states—large fluctuations, persistent revivals, and failure of canonical thermalization—are explained as generic consequences of a two‑parameter grand‑canonical ensemble and the associated statistical properties of Ω(E,N) and CCP. The work provides a comprehensive, thermodynamically grounded reinterpretation of QMBS and suggests that similar mechanisms may operate in other constrained many‑body systems.