Zero and non-zero Chern number? In a crystal lattice, a non-zero Chern number shows non-trivial topology and a zero Chern number indicates trivial topology. But I am not able to differentiate the physical characteristics possessed by a crystal lattice with zero and non-zero Chern number.
Can anyone please explain, physically, what additional feature does a crystal with non-zero Chern number have over crystal with zero Chern number?
Or in other words, If a material is given how can we say that it have a non trivial topology by looking into the motion of electrons, or alignment of electrons or any other physical features?
 A: Whenever one classifies the topology of a  system by means of a Chern number, which is the case for systems breaking time-reversal symmetry, a non-trivial Chern number indicates the given material realizes the quantum anomalous Hall effect. In other words, without underlying external magnetic fields, a material with a non-trivial Chern number, a so-called Chern insulator, displays chiral edge states and a quantized longitudinal conductivity (which indeed is proportional to the non-zero Chern number). These edge states propagate ballistically, as these are topologically protected and therefore very robust, but most importantly they disperse linearly, as high-energy physics massless particles would do. So one might be tempted to think the magic arises at the boundary of such systems, but instead that is only what is called the “bulk-boundary correspondence”: topological edge states arise at boundaries between systems with trivial and non-trivial Chern numbers, i.e. at the boundaries between trivial and Chern insulators. More specifically, the difference between the Chern numbers of two materials corresponds to the number of states propagating along the boundary between them.
