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We discuss systems which have some, but not all of the hallmarks of topological phases. These systems topological character is not fully captured by a local order parameter, but they are also not fully described at low energies by topological quantum field theories. For such systems, we formulate the concepts of quasi-topological phases (to be contrasted with true topological phases), and symmetry-protected quasi-topological phases. We describe examples of systems in each class and discuss the implications for topological protection of information and operations. We explain why topological phases and quasi-topological phases have greater stability than is sometimes appreciated. In the examples that we discuss, we focus on Ising-type (a.k.a. Majorana) systems particularly relevant to recent theoretical advances and experimental efforts.
We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the proce
We describe topologically ordered and fracton ordered states on novel geometries which do not have an underlying manifold structure. Using tree graphs such as the $k$-coordinated Bethe lattice ${cal B}(k)$ and a hypertree called the $(k,n)$-hyper-Bet
We introduce topological phases of matter defined by skyrmions in the ground state spin -- or pseudospin -- expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by
Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While th
Topology in quantum matter is typically associated with gapped phases. For example, in symmetry protected topological (SPT) phases, the bulk energy gap localizes edge modes near the boundary. In this work we identify a new mechanism that leads to top