Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles

We consider nonequilibrium time evolution in quantum spin chains after a global quench. We show that global symmetries can invalidate the standard picture of local relaxation to a (generalised) Gibbs ensemble and provide a solution to the problem. We introduce in particular a family of statistical ensembles, which we dub "semilocal (generalised) Gibbs ensembles". The issue arises when the Hamiltonian possesses conservation laws that are not (pseudo)local but act as such in the symmetry-restricted space where time evolution occurs. Because of them, the stationary state emerging at infinite time can exhibit exceptional features. We focus on a specific example with a spin-flip symmetry, which is the commonest global symmetry encountered in spin-$1/2$ chains. Among the exceptional properties, we find that, at late times, the excess of entropy of a spin block triggered by a local perturbation in the initial state grows logarithmically with the subsystem's length. We establish a connection with symmetry-protected topological order in equilibrium at zero temperature and study the melting of the order induced either by a (symmetry-breaking) rotation of the initial state or by an increase of the temperature