Reduced density matrix after a quantum quench

We consider the reduced density matrix (RDM) ${\ensuremath{\rho}}_{A}(t)$ for a finite subsystem $A$ after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of ${\ensuremath{\rho}}_{A}(t)$ is described by the RDM ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how ${\ensuremath{\rho}}_{A}(t)$ approaches ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law ${t}^{\ensuremath{-}3/2}$ in time. As the RDM completely determines all local observables within $A$, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.

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