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

+ 3 like - 0 dislike
1527 views

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

+ 1 like - 0 dislike

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.