# Topolgical insulators order parameter

For topological insulators Is there any way to define order parameter for topological phase transitions?

• One should clarify that: Topological insulator is NOT topologically ordered, but topological insulator is ONLY symmetric-protected (U(1) charge and time reversal). Such a transition is not a topological phase transition as the nature of (fractional) quantum hall states. – wonderich Jun 9 '14 at 2:56
• You can not define a local order parameter. However, you can finds some topological invariants (http://arxiv.org/abs/1002.3895), which distinguish the phases. – user27964 Jun 9 '14 at 7:37
• Yes, the order parameter is the (global) topological invariant ($\nu_{0}=0,1$) introduced by Fu, Kane, and Mele: arxiv.org/abs/cond-mat/0607699 for time-reversal symmetric free fermion systems. In (say) Bi$_{1-x}$Sb$_{x}$ alloys, the topological invariant can be defined as a function of $x$. As you increase $x$ from zero, for some $x=x_{c}$, a phase transition will occur and $\nu_{0}$ will change from 0 to 1. – NanoPhys Jun 10 '14 at 16:31