A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions

摘要: State sum constructions, such as Kuperberg's algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights, to different parts a closed triangulated manifold. Here we extend this construction including matter fields build both two and three space-time dimensions. The introduces new weights the vertices they correspond Potts spin configurations described an $\mathcal{A}$-module with inner product. Performing on manifold boundary obtain transfer matrices which are decomposed into product operators acting vertices, links plaquettes. vertex plaquette similar ones appearing quantum double models (QDM) Kitaev. link operator couples fields, it reduces usual interaction terms known $\mathbb{Z}_2$ theory fields. lead Hamiltonians that frustration-free exactly solvable. According choice initial input, group module, interesting have kind ground state degeneracy depends number equivalence classes module under action. Some confined flux excitations bulk become deconfined at surface. These edge modes protected energy gap provided operator. properties also appear "confined Walker-Wang" 3D having surface states. Apart from there sector immobile can be thought defects Ising model. We only consider bosonic paper.