Topological Orders from Reflection Positive Frustration-free Hamiltonians

We establish a framework with reflection positivity as the first principle for establishing the boundary theory of topologically ordered quantum spin systems. For any reflection positive frustration-free Hamiltonian, We proved that the local topological quantum order (LTQO) condition of ground states on a disk holds if and only if the ground state on the sphere is non-degenerate. Furthermore, we show that the Osterwalder-Schrader reconstruction produces the boundary local net of operator algebras from the local ground states.