Abstract

We study disordered $SU(2)$-symmetric Heisenberg spin chains at high energy density.
We have found that in a broad range of disorder strengths and system sizes, Heisenberg chains exhibit a new kind of non-ergodic behavior. In this regime, the highly excited eigenstates exhibit scaling of entanglement entropy that is intermediate between the area-law characteristic of MBL states, and the volume-law found in thermalizing systems. This behavior stems from the tree tensor-network structure of the eigenstates obtained within strong disorder RG. Further, we found that for weak disorder, the behavior crosses over from non-ergodic to ergodic as the system size is increased. For stronger disorder, all system sizes accessible numerically exhibited non-ergodic behavior. To address the eventual fate of the non-ergodic phase in this case, we investigate resonances that endanger the stability of tree states. Our results strongly suggest eventual delocalization and ergodicity, albeit at very large system sizes; delocalization occurs via unconventional, multi-spin processes, which is yet another unique feature of disordered systems with non-Abelian symmetries.