Fracton order: relation to and features beyond TQFT

In the recent study of quantum error correction codes, a class of new 3+1D models were discovered which host both features that are similar to the usual topological models and ones that are drastically different. On the one hand, similar to topological models, these so-called ‘fracton’ models have ground state degeneracy that is stable against any local perturbation. There are also fractional point excitations in the bulk, which cannot be created or destroyed on their own. On the other hand, the ground state degeneracy is not a topology dependent constant. Instead it grows with system size. At the same time, the fractional excitations have restricted motion, i.e. they cannot move freely in the full three dimensional space. It is obvious that the fracton models are beyond the framework of TQFT, but are also closely related to it in many important ways. In this talk, I will review some of the most interesting examples of fracton models and explain how we propose to interpret their nontrivial features as depending on not only the topology of the underlying manifold, but also its ‘foliation’ structure, where each foliation leaf carries a 2+1D TQFT of its own. While our understanding of the fracton models are very limited at this point, they are pointing to a way to generalize TQFT in higher dimensions.

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