Pseudo-Hermitian Levin–Wen models from non-semisimple TQFTs

We construct large classes of exactly solvable pseudo-Hermitian 2D spin Hamiltonians. The ground states these systems depend only on the spatial topology system. identify state system a surface with value assigned to by non-semisimple TQFT generalizing Turaev–Viro model. A non-trivial example arises from subcategory representations quantum sl(2) where parameter is specialized root unity.