Generalized Streda formula in the adiabatic heuristic principle

The adiabatic heuristic principle for the quantum Hall (QH) states are numerically demonstrated under the algebraic constraints of the Braid group on a torus. The many-body gap is smooth and finite when the Hamiltonian is adiabatically deformed by the flux attachement from the fractional QH state to the integer one. The many-body Chern number of the ground state multiplet works well as an adiabatic invariant for the gap while their topological degeneracy changes wildly. Assuming the invariance of the many-body Chern number, we have analytically proved a generalized Streda formula, which explains the wild behavior of the topological degeneracy in terms of the Chern number. [1]
[1] K. Kudo and Y. Hatsugai, Phys. Rev. B 102, 125108 (2020).