We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of , where is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb–Schultz–Mattis (LSM) theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron–Dana–Zak relation between the Hall conductance and the filling factor.
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Our notation differs here from the tradition of reserving this symbol for the smaller algebra of observables that are strictly supported in Z.
As opposed to the “symmetry protected” topological order, which is not considered here.
The adjective “topological” cannot be omitted here. In the case of spontaneous symmetry breaking with a local order parameter, the index can be rational; see Sec. VI.
It is often stressed that this is a “large” gauge transformation, referring to the fact that it is not connected to the identity within the gauge group. However, on the lattice, it is not straightforward to make this distinction precise.