How to calculate chern number from berry curvature?
Answers
There are several real-space formulas and a formula based on scattering theory. Let me discuss some of the real-space formulas.
The first is called, in physics anyway, the Bott index. This will work if the following hold: you are on a regular lattice; the Hamiltonian involves hopping terms only to nearest and next-nearest neighbors; the ratio of the norm of the Hamiltoninan to the gap size is about 10 or smaller.
To compute the Bott index, take a lattice model of about 15-by-15 and impose periodic boundary conditions. Now take the (diagonal) position observables, scale them so the width and length is 2π, and form exponentials U0=exp(iX) and V0=exp(iY). Do a full eigensolve of the Hamiltoninan and find the projector P corresponding to all energy below a certain gap. Now form invertible matrices U=PU0P+(I−P) and V=PU0P+(I−P). Compute all eigenvalues of VUV†U†, take the log of all of these, now sum them up. The real part of this complex number will be the Chern number (or minus the Chern number).
Be sure to repeat this with a larger system size, to get some evidence you are avoiding the effects of small system size. As to why, and when, the Bott index equals the Chern number, you can look here: "On the equivalence of the Bott index and the Chern number on a torus, and the quantization of the Hall conductivity with a real space Kubo formula" by Toniolo, Arxiv:1708.05912.,