# Definition for Chiral Spin Liquid

What is the definition of chiral spin liquid? Especially what does chiral mean here? I encounter a lot of terminologies with chiral. It seems they mean differently in different contexts. If you could generally answer what chiral means, it would be great.

A 2D chiral spin liquid (CSL) is a spin liquid state that breaks time-reversal and parity symmetries. For a CSL system with open boundaries, there may be chiral edge states propagate in only one direction, which is analogous to the case of integer quantum Hall effect.

Generally speaking, I think chiral, in a sense, means a particular direction, which is usually associated with broken time-reversal symmetry.

X.G. Wen first introduced the idea of chiral spin state (not necessarily spin liquid) in his paper. As Kai has mentioned above, the state breaks time reversal symmetry.

It's partially characterized, for example on a triangular lattice, by

$$E_{123} = $$

If the system has T, then $$E_{123}=0$$ since spin operator $$S$$ is odd under T. Thus, a nonzero $$E_{123}$$ suggests a CSL.

A recent technical and detailed classification based on PSG on CSL is here.