Abstract
The generalized Gibbs ensemble (GGE), which involves multiple conserved quantities other than the Hamiltonian, has served as the statistical-mechanical description of the long-time behavior for several isolated integrable quantum systems. The GGE may involve a noncommutative set of conserved quantities in view of the maximum entropy principle, and show that the GGE thus generalized (noncommutative GGE, NCGGE) gives a more qualitatively accurate description of the long-time behaviors than that of the conventional GGE. Providing a clear understanding of why the (NC)GGE well describes the long-time behaviors, we construct, for noninteracting models, the exact NCGGE that describes the long-time behaviors without an error even at finite system size. It is noteworthy that the NCGGE involves nonlocal conserved quantities, which can be necessary for describing long-time behaviors of local observables. We also give some extensions of the NCGGE and demonstrate how accurately they describe the long-time behaviors of few-body observables.
Highlights
The foundation of quantum statistical mechanics has seen a resurgence of interest in recent years [1,2,3,4] partly because well-isolated and -controlled artificial quantum systems have emerged as the ideal platform to reconsider the long-standing problem [5,6,7,8,9,10]
For the other initial state |ψiAni, as L increases, max of the CGGE decreases as ∝ 1/L because there are no characteristic peaks, which cannot be captured by the CGGE. These results show that the one-body NCGGE improves the generalized Gibbs ensemble (GGE) prediction quantitatively as a whole, but some of the local correlations such
Introducing noncommutative sets of conserved quantities and the observable projection idea, we have systematically shown that the NCGGE describes the long-time behavior of isolated quantum systems better than the conventional CGGE
Summary
The foundation of quantum statistical mechanics has seen a resurgence of interest in recent years [1,2,3,4] partly because well-isolated and -controlled artificial quantum systems have emerged as the ideal platform to reconsider the long-standing problem [5,6,7,8,9,10]. Spinless fermions or hard-core bosons under quasiperiodic potential cannot be described by the GGE due to the localization of single-particle eigenstates [45,46,47,48,49] Another example is the entanglement prethermalization in an interacting integrable system [50], where nonlocal conserved quantities play significant roles. We systematically study how the additional noncommutative conserved quantities affect the GGE and show that the GGE generalized (noncommutative GGE, NCGGE) describes the stationary states in isolated integrable systems better than the conventional CGGE. Understanding of why the (NC)GGE well describes the stationary states In this spirit, for a noninteracting model, we systematically construct the NCGGE that describes the stationary states without an error at finite system size for few-body observables.
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