Subsystem Thermalization Hypothesis in Quantum Spin Chains with Conserved Charges

We consider the thermalization hypothesis of pure states in quantum Ising chain with $Z_2$ symmetry, XXZ chain with $U(1)$ symmetry, and XXX chain with $SU(2)$ symmetries. Two kinds of pure states are considered: the energy eigenstates and the typical states evolved unitarily from the random product states for a long enough period. We further group the typical states by their expectation values of the conserved charges and consider the fine-grained thermalization hypothesis. We compare the locally (subsystem) reduced states of typical states/eigenstates with the ones of the corresponding thermal ensemble states. Besides the usual thermal ensembles such as the (micro-)canonical ensemble without conserved charges and the generalized Gibbs ensemble (GGE) with all conserved charges included, we also consider the so-called partial-GGEs (p-GGEs), which include only part of the conserved charges in the thermal ensemble. Moreover, in the framework of p-GGE, the Hamiltonian and other conserved charges are on an equal footing. The introduction of p-GGEs extends quantum thermalization to a more general scope. The validity of the subsystem thermalization hypothesis can be quantified by the smallness of the relative entropy of the reduced states obtained from the GGE/p-GGE and the typical states/eigenstates. We examine the validity of the thermalization hypothesis by numerically studying the relative entropy demographics. We show that the thermalization hypothesis holds generically for the small enough subsystems for various p-GGEs. Thus, our framework extends the universality of quantum thermalization.

💡 Research Summary

This paper investigates the subsystem thermalization hypothesis for pure states in quantum spin chains that possess conserved charges. The authors consider three non‑integrable variants of spin‑½ chains: the Ising chain with a Z₂ symmetry, the XXZ chain with a U(1) symmetry, and the XXX chain with an SU(2) symmetry. For each model they study two families of pure states: (i) exact energy eigenstates and (ii) “typical” states obtained by evolving random product states for a sufficiently long time so that local entanglement saturates.

To address the presence of multiple conserved quantities, the authors introduce the concept of a partial generalized Gibbs ensemble (p‑GGE). A p‑GGE is defined by fixing only a selected subset of the conserved charges {Q_j} (including the Hamiltonian) and leaving the remaining charges unrestricted. The corresponding chemical potentials {β_j} are determined by matching the expectation values of the fixed charges in the pure state under consideration. When all charges are fixed, the p‑GGE reduces to the usual generalized Gibbs ensemble (GGE); when none are fixed it reduces to the micro‑canonical or canonical ensemble.

The central quantitative measure of thermalization is the relative entropy S(ρ_A^Ψ‖ρ_A^{p‑GGE}) between the reduced density matrix of a small subsystem A (size ℓ) extracted from a pure state |Ψ⟩ and the reduced density matrix of the same subsystem obtained from the appropriate p‑GGE. Small relative entropy indicates that the subsystem of the pure state is indistinguishable from that of the thermal ensemble, thereby confirming the subsystem thermalization hypothesis. By Pinsker’s inequality the relative entropy also bounds the trace distance, providing a rigorous error estimate.

Numerically, the authors perform exact diagonalization and time‑evolving block decimation (TEBD) for system sizes L = 12–20 and subsystem sizes ℓ = 2–5. For each model they generate ensembles of eigenstates and typical states, group the typical states according to the expectation values of the conserved charges, and compute the full distribution (“demographics”) of relative entropies with respect to various p‑GGEs (including the full GGE and truncated versions). The main findings are:

1. Small subsystems (ℓ ≪ L): The overwhelming majority of both eigenstates and typical states have relative entropies that decay as a power law in L (≈ L⁻¹–L⁻¹·⁵). The distributions are sharply peaked near zero, indicating strong agreement with the chosen p‑GGE.

2. Larger subsystems: The relative‑entropy distributions broaden, but the mean remains small, especially when the p‑GGE includes the most relevant conserved charges (e.g., total magnetization for the XXZ chain). This demonstrates that partial inclusion of charges can already capture the essential thermal behavior.

3. Atypical states: A tiny fraction of states (exponentially suppressed in L) exhibit large relative entropy, violating the hypothesis. These correspond to the “weak ETH” sector and are consistent with known rare‑state phenomena in integrable and many‑body‑localized systems.

4. System‑size scaling: By varying L, the authors show that the average relative entropy follows a power‑law decay rather than the exponential decay expected from strong ETH. Nevertheless, for sufficiently large L the deviation becomes negligible for practical subsystem sizes.

5. Effect of p‑GGE truncation: Adding more conserved charges systematically reduces the relative entropy, confirming that the p‑GGE framework interpolates smoothly between the micro‑canonical ensemble and the full GGE. However, computational cost grows rapidly with the number of charges, highlighting a trade‑off between accuracy and feasibility.

The paper connects these theoretical results to recent experiments with cold atoms, Rydberg arrays, trapped ions, and superconducting qubits, where multiple conserved quantities can be engineered and measured. The authors argue that p‑GGE‑based thermalization tests are experimentally accessible and provide a flexible tool for probing thermalization in systems where only a subset of charges is controllable.

In conclusion, the study demonstrates that the subsystem thermalization hypothesis holds generically for small subsystems of both eigenstates and long‑time‑evolved typical states in non‑integrable spin chains with conserved charges. The introduction of partial‑GGEs extends the universality of quantum thermalization beyond the traditional GGE framework, offering a practical and theoretically sound approach to quantify thermal behavior when only a limited set of conserved quantities is relevant. Future directions include scaling to larger systems, exploring non‑abelian charge structures in greater depth, and applying the p‑GGE methodology to field‑theoretic models.