I consider nonequilibrium time evolution in quantum spin chains after a global quench. I show that global symmetries can invalidate the standard picture of local relaxation to maximum-entropy statistical ensembles and provide a solution to the problem. 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. I focus on a specific example with a spin-flip symmetry and establish a connection with symmetry-protected topological order in equilibrium at zero temperature. In the second part of the talk I discuss some exceptional features of the infinite-time limit. In particular, the excess of entropy of a spin block triggered by a local perturbation in the initial state grows logarithmically with the subsystem’s length.
Finally, I discuss the melting of the order induced either by a (symmetry-breaking) rotation of the initial state or by an increase of the temperature.
References:  M. Fagotti, Phys. Rev. Lett. 128, 110602 (2022) [arXiv:2110.11322]  M. Fagotti, V. Marić, and L. Zadnik, arXiv:2205.02221.