I have been reading up on topological order and quantum phases which are continually being discovered in condensed matter systems. (Here's a great article...https://www.quantamagazine.org/physicists-aim-to-classify-all-possible-phases-of-matter-20180103/)
While I am not well versed in the topology behind it, I have a qualitative question. In a standard band insulator, the Hamiltonian of the system incorporates the kinetic and potential energy, $H=K+V$. There is the implicit assumption that the particles are gas like, i.e. non-interacting, as this actually maps a many body problem onto a single body problem, with quasiparticles of effective mass. As I recall, DFT calculations often sweep the more detailed "Exchange Correlation" effects under the rug as an approximated function, such that $H=K+V+XC$.
My question(s) is essentially this. If quantum phases are "collective excitations" of the system,
I assume this means that the magnitude of the $XC$ term in the Hamiltonian, which encapsulates exchange/correlation, is much larger relative to the kinetic $K$ and potential $V$ terms? Is there a more precise expression/condition for to express this?
In the article above, they state "The long-range entanglement patterns that arise are topological, or impervious to local changes, like the number of holes in a manifold". In this statement, I am a bit unclear about the term "local changes". In a physical system, are these local changes tied to the $K$ and $V$ terms in the Hamiltonian? I.e. could they be point or line defects in a crystalline semiconductor, or compositional fluctuations, or edges/surfaces/interfaces?
Am I correct to say that perturbations in $K$ or $V$, which would alter properties in the more standard band theory, become less influential as we transition away from that to a topological system, where $XC$ becomes the dominant term?
Thanks for clarification, if anyone has good references for this as well, greatly appreciated!