Figure 2: Composite bands and Wannier functions.

(a) We consider lattices in three and two dimensions (2D in the figure). Periodic boundary conditions (PBC) are used. The lattice contains N_c unit cells and each unit cell contains N_orb sites/orbitals. The vectors a_i (i=1, 2, 3) are the fundamental vectors of the Bravais lattice¹⁴ while the vectors b_α (α=1,…,N_orb) are the positions of the centres of the orbitals (Wannier functions) within a unit cell. A single orbital is specified by a triplet (or pair) of integers i=(i_x, i_y, i_z) and by the sublattice index α and is centred at the position vector r_{i α}=i_xa₁+i_ya₂+i_za₃+b_α. (b) The band structure is obtained by solving the Schrödinger equation with periodic potential V(r)=V(r+a_i). It consists of the band dispersions ɛ_{n k}, with n the band index and ħ k the lattice quasimomentum, and the periodic Bloch functions g_{n k}(r)=g_{n k}(r+a_i) (Bloch functions for brevity) obtained from the Bloch plane waves . We consider a composite band, that is, a subset of contiguous bands well separated in energy from other bands. The Chern numbers C_n for individual bands calculated from the Bloch functions may be nonzero (such as the flat band n=2 in the figure), but their sum equals zero . The Chern number refers to spin-resolved bands since the spin along a quantization axis (conventionally the z axis) is conserved. (c) The Wannier functions, defined as the Fourier transform of the Bloch functions, allow us to derive a tight-binding Hamiltonian that reproduces exactly a single band or a composite band of the original continuum Hamiltonian (see Supplementary Note 2). Since individual bands may be topologically nontrivial with nonzero Chern numbers, their Wannier functions w_n(r) are not exponentially localized²⁵, and the Peierls substitution in the effective Hamiltonian is therefore not justified^15,16. (d) By constructing Wannier functions as linear superpositions of Bloch waves of all bands in the composite band, exponentially localized Wannier functions w_α(r) can be created. The mixing of the different bands is provided by the unitary matrix U_α,n(k). This justifies the Peierls substitution for a composite band .