# Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD)

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Consider a 2D gapped CSL with a nonzero Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$?

For example, see this article, in the last paragraph on page 7, the authors give the 4-fold GSD from the Chern number $m=\pm2$ for a CSL. I can not understand the explanation, can anybody present an intuitive illustration or a simple mathematical proof ? I will be very appreciated, thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy
Free arXiv version: arxiv.org/abs/1110.0116

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user Qmechanic

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Based on the paper, the answer is $|m|^2$. They suggest in their p.8, Eq.36, the effective theory is a Chern-Simons theory $$\frac{1}{4\pi}\int K_{IJ} a_I \wedge d a_J$$ with the $K_{IJ}$ bilinear K matrix as
$$K_{IJ}={\begin{pmatrix}m & 0\\ 0 & -m\end{pmatrix}}$$.

The up $m$ labels one sector and the lower $m$ labels the other sector. The degeneracy(GSD) is computed by a generalizing level-$k$ U(1) Chern-Simons theory(GSD=$k$) to a bilinear K matrix U(1)$^n$ Chern-Simons theory. GSD=$|\det(K)|=|m|^2$. This GSD result for GSD=$|\det(K)|$ is a well-known fact.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user Idear
answered Jan 22, 2014 by (1,455 points)
@ Idear I believe your answer is correct even though I don't know about the Chern-Simons theory.... So do we have an alternative algebraic or Hamiltonian approach to understand the GSD?

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy
Yes, a good question. the 1-form $a$ field represents so-called Wilson-line operator or simply anyons. In the lattice model, this Wilson-line operator or anyons can be regarded as the string or cycle on the lattice(such as string-net or toric-code, etc), See for example, page 4 of this paper: Boundary Degeneracy of Topological Order - Sec: Mutual Chern-Simons, Zk gauge theory, Toric code and String-net model and this paper: Quantum codes on a lattice with boundary

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user Idear

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