Low-energy effective Hamiltonian and end states of an inverted HgTe nanowire

The band inversion transition in a cylindrical HgTe nanowire is inducible via varying the nanowire radius. Here we derive the low-energy effective Hamiltonian describing the band structure of the HgTe nanowire close to the fundamental band gap. Because both the $E_1$ and $H_1$ subbands have quadratic dependence on $k_z$ when the gap closes, we need to consider at least three subbands, i.e., the $E_1$, $H_1$, and $H_2$ subbands, in building the effective Hamiltonian. The resulting effective Hamiltonian is block diagonal and each block is a 3 × 3 matrix. End states are found in the inverted regime when we solve the effective Hamiltonian with open boundary condition.