  Topological insulators are substances which are bulk insulators but which
carry current via special "topologically protected" edge states. The
understanding of long range topological order in these systems is built around
the idea of a Berry connection, which is a gauge connection obtained from the
phase of the electron wave function transported over momentum space rather than
coordinate space. The phase of a closed Wilson loop of the Berry connection
around the Brillouin zone defines a topological order parameter which labels
discrete flux vacua. The conducting states are surface modes on the domain
walls between discrete vacua. Evidence from large-$N_c$ chiral dynamics,
holographic QCD, and Monte Carlo observations has pointed to a picture of the
QCD vacuum that is very similar to that of a topological insulator, with
discrete quasivacua labelled by $\theta$ angles that differ by mod $2\pi$. In
this picture, the domain walls are membranes of Chern-Simons charge, and the
quark condensate consists of surface modes on these membranes, which are
delocalized and thus support the long range propagation of Goldstone pions. The
Berry phase in QED2 describes charge polarization of fermion-antifermion pairs,
while in 4D QCD it describes the polarization of Chern-Simons membranes.
