A bulk-edge correspondence between the second Chern number and the spectral flow

Abstract

A one-parameter family of semi-quantum Hamiltonians admitting the time-reversal symmetry is defined on the product S2(r1)×S2(r2) of two-spheres of respective radii r1 and r2, and a corresponding one-parameter family of quantum Hamiltonians is defined in terms of the angular momentum operators associated with respective spheres. The second Chern number of the eigen-vector bundle associated with a positive eigenvalue of the semi-quantum Hamiltonian is a piecewise constant function of the parameter with jumps at singular parameter values leading to the emergence of total degeneracy points for the eigenvalues. Correspondingly, the quantum Hamiltonian has edge-state eigenvalues which, as functions of the parameter, pass the zero energy level at the singular parameter values and the bulk-state eigenvalues prove to correspond to eigenvalues of the semi-quantum Hamiltonian. It is shown that the second-Chern number c2(T) at a non-singular parameter value T is in one-to-one correspondence with the spectral flow as a function of the end point of the interval (−∞,T). The bulk-edge correspondence is also shown to hold between the linearized semi-quantum and quantum systems, where the semi-quantum Hamiltonian is linearized at each of the degeneracy points of the eigenvalues and quantized to define linear quantum Hamiltonians. On this level, the spectral flow of each linear quantum Hamiltonian with a parameter μ proves to coincide with the index of the linear quantum Hamiltonian with μ=0.