# Qualitative understanding of Hamiltonian Terms for Quantum Phases

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,

1. 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?

2. 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?

3. 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!

• 1. Usually multiple questions are not answered in this website. 2. What is XC ? 3. A non-interacting many-body problem is essentially a first quantised problem, and the band structure is a classical result of that regime. 4. Check out the other answers on this website to get partial answer to yours, then refine your question. 5. Once XC will be clarified, we could understand whether you want to discuss non-interacting topological systems or interacting topological systems. 6. Please precise which topological phase you want to discuss : topological insulator, superconductors, quantum phase, ... – FraSchelle Jan 6 '18 at 7:40
• Now, regarding the only question I could answer, namely your question 2, local in that context essentially means impurity. Topological insulators and superconductors (if you are concerned by these) require the description in the position space in addition to the band structure (in the reciprocal space then), because periodicity is broken by some edge. Then, as long as you do not alter the (discrete) symmetry of the system (as e.g. putting time-reversal breaking impurities in a system with time-reversal symmetry) – FraSchelle Jan 6 '18 at 7:47
• you may add any smooth potential $V(x)$ to the problem without destroying the edge states. This is only about non-interacting symmetry protected topological system, which some people would not call topological system. – FraSchelle Jan 6 '18 at 7:48
• Clarifications: By XC, I mean all of the terms of the Hamiltonian incorporating exchange and correlation effects. So everything except the particle's kinetic energy and the potential energy. I am not interested in any particular topological phase per se, but the overall concept. In reading more, is it correct to say that topological phases are distinctions in the ground state of the system, in which most of the particles reside and interact? – daFireman Jan 8 '18 at 17:41
• (Con't): The wave function now seems to be a combined function of all the particles (Hartee Fock), is it correct to say that topological order is essentially distinguishing between phases in this macroscopic ground state, and that the distinction arises in how the particles relate/are arranged with respect to one another? – daFireman Jan 8 '18 at 17:45