Many Body Scars as a Group Invariant Sector of Hilbert Space

We present a class of Hamiltonians H for which a sector of the Hilbert space invariant under a Lie group G, which is not a symmetry of H, possesses the essential properties of many-body scar states.
A particular class of examples concerns interacting spin-1/2 fermions on a lattice consisting of N sites (it includes deformations of the Fermi-Hubbard model as special cases), and we show that it contains two families of N+1 scar states. One of these families, is comprised of the well-known η-pairing states.
We find another family of scar states which is U(N) invariant.
Both families and most of the group-invariant scar states produced by our construction in general, give rise to the off-diagonal long range order which survives at high temperatures and is insensitive to the details of the dynamics. Such states could be used for reliable quantum information processing because the information is stored non-locally, and thus cannot be easily erased by local perturbations. In contrast, other scar states we find are product states which could be easily prepared experimentally.
The dimension of scar subspace depends on G and can be made exponentially large.