Subsystem Thermalization Hypothesis in Quantum Spin Chains with Conserved Charges

TL;DR

This paper investigates the thermalization behavior of pure states in various quantum spin chains with conserved charges, introducing partial-GGEs to extend the scope of quantum thermalization and demonstrating its general validity.

Contribution

The study introduces partial-GGEs that include only some conserved charges, broadening the understanding of quantum thermalization in systems with multiple symmetries.

Findings

Thermalization holds for small subsystems across different p-GGEs.

Subsystem reduced states closely match those of the corresponding thermal ensembles.

The framework extends the universality of quantum thermalization to more general settings.

Abstract

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.