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My understanding is that the Toric code is a model with topologically non-trivial ground state. The ground state is degenerate on a Torus and is robust to local perturbations. The model has anyonic excitations and a non-zero topological entanglement entropy.

Topologically non-trivial phases seem to be distinguished from the trivial ones by a topological invariant. For example the Su-Schrieffer–Heeger (SSH) model has two phases which are distinguished by a winding number.

My question is: Is there a topological invariant for the Toric code? If yes, then what is it? If not, then is it wrong to say that topologically non-trivial phases are distinguished from trivial ones by a topological invariant? Another (perhaps unrelated) question is the following: does the Toric code have zero modes if it is put on a finite lattice with edges?

(In this question, I am referring to to the Toric code in 2D)

Jeet Shah
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    You may want to see this answer by Xiao-Gang Wen discussing the distinction between "classical topology" and "quantum topology." The winding number in the SSH model is an example of a classical topological invariant. The toric code model exhibits topological order, and is described by "quantum topology." Roughly, you can think about the "topological invariant" being the topological ground state degeneracy as well as the fusion and braiding rules of the point-like topological excitations. – d_b Dec 02 '21 at 20:07
  • To understand the topological nature of toric code, you can try braiding $e$ and $m$ around each other. – Fatimah Rita Ahmadi Nov 25 '22 at 23:27

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