Citation: Haah, Jeongwan. "A degeneracy bound for homogeneous topological order." SciPost Physics 10.1 (2021): 011.
Web: https://arxiv.org/abs/2009.13551
Tags: Error-correcting-codes, Higher-dimensional
In this paper, Haah proves that any topological system in which ground states on locally indistinguishable on disjoint unions of balls will have an upper bound on the growth of its groundspace dimension. He introduces the term "homogenous topological order" to refer to system in which for every operator acting on the ground space (that is, which commutes with the ground space projector) there is another operator which acts on the ground space the same way as the original operator but is localized on the compliment of finitely many pre-decided balls. He then shows that the logarithm of the group space dimension for homogenous topological systems grows at most like L^(d-2) where d is dimension.
The point is this paper is that fractons are homogenous topological order. In general, any frustration free, commuting, Pauli, Hamiltonian such that any Pauli operator which commutes with the Hamiltonian can be expresses as a produced of Hamiltonian terms will neccecarily be homogenous topological order. Fractons have all of these properties - at least, our known models of them do.
The main theorem is nice. Extensive degeneracy is a hallmark of fracton systems, so it's good to get a better grasp on the nature of this degeneracy. Seeing as the degeneracy for two-dimensional systems is O(1), this result can summarized in the d=2 special case a "there are no fractons in two dimensions".
A nice additional aspect of this paper is that the proof goes through for system with approximate homogenous topological order, as is shown in the last section.