Eigenstate phases with finite on-site non-Abelian symmetry

We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group, S3, and numerically study a particular two-parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from S3 to Z3 in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the S3 symmetry down to Z3. Our observed phases are consistent with the possibilities outlined by A. C. Potter and R. Vasseur [Phys. Rev. B 94, 224206 (2016)2469-995010.1103/PhysRevB.94.224206], namely thermal/ergodic and many-body-localized spin-glass (MBL-SG) phases. We also speculate on the nature of an intermediate region between the thermal and MBL + SSB regions where full S3 symmetry exists.