# $Z_2$ topological index in spin liquid

What is $Z_2$ topological index in spin liquid system? How to understand its physical picture in condensed matter?

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Are you taking about the orientation dependence of the ground state? –  Ryan Thorngren Sep 30 '12 at 21:55
As written in the answer below, there are no such index for spin liquids (as there is for time-reversal invariant topological insulators). Are you thinking about the so-called $\mathbb Z_2$ spin liquids? The $\mathbb Z_2$ does not refer to a topological index, but rather to the fact that the low-energy dynamics are described by a $\mathbb Z_2$ gauge theory. There are also so-called $SU(2)$ and $U(1)$ spin liquids. See arxiv.org/abs/cond-mat/0107071 for a partial classification. –  Heidar Oct 1 '12 at 13:53
Yes, I was thinking about the the $Z_2$ spin liquids. Thanks for your answer and reference. –  Timothy Oct 3 '12 at 15:24

Different spin liquids are extremely rich, and they cannot be described by $Z_2$ topological index. So there is no $Z_2$ topological index for generic spin liquids.