I will discuss infinite-temperature properties of an infinite sequence of random quantum spin chains using a real-space renormalization group approach, and demonstrate that they exhibit non-ergodic behavior at strong disorder. The analysis is conveniently implemented in terms of SU(2)k anyon chains, including the random transverse-field Ising model as a special case. Highly excited eigenstates of these systems exhibit properties usually associated with quantum critical ground states. The excited-state fixed points are generically distinct from their ground state counterparts, and represent non-equilibrium critical phases of matter. I will briefly discuss related work on studying the interplay of symmetry-protected order and localization in one dimensional chains with "particle-hole symmetry".
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