  In this paper, we discuss the effects of the symmetries and related
topological orders to the thermalization of many-body localized system. We
consider the one-dimensional fermion chain system with open (or periodic)
boundary condition, whose boundaries are characterized by Sachdev-Ye-Kitaev
(SYK) intercation. Just like in the SYK model and the tendor models, there are
many-body quantum chaos in out-of-time-ordered correlation when the system is
being thermalized by the interactions (usually nonuniform and being randomly
distributed), and satisfies the eigenstate thermalization hypothesis (ETH).
While the continuous or discrete symmetries usually protect the related
topological orders against the thermalization, which may lead to the
localization of quantum states and generate large degeneracy. We discuss these
effects in terms of the fermionic or spin languages. In many-body localized
phase, a large number of degenerate ETH-violated states can be found in both
the frustration-free AKLT model or the integrable biquadratic (Wishart) SYK
model. The quantum scars appear when such degenerate states are embeded into an
ETH-satisfying spectrum of the Hilbert space enlarged by a much larger number
of bosonic flavors (e.g., SU(M) multiplets degeneracy). Usually, the emergence
of quantum chaos require large limits of boson flavor M and the number of
coupled (undegenerate) states (N; which related to Z N symmetry). Further, the
boson (or excitations) flavor number M can often to related to the fermion
number N through the duality transformations, in which case the Z M symmetry is
possible to generted by SU(M) model and leads to asymptotic degeneracy in
large-M limit.
