Table 1:
Symmetry-breaking orders can be probed/defined through linear responses. But topological order cannot be probed/defined through linear responses. We need topological probes to define topological orders.
Order
Experiment
Crystal order
X-ray diffraction
Ferromagnetic order
Magnetization
Antiferromagnetic order
Neutron scattering
Superuid order
Zero viscosity and vorticity quantization
Topological order
Topological degeneracy
(Global dancing pattern)
non-Abelian geometric phase