(a) We consider lattices in three and two dimensions (2D in the figure). Periodic boundary conditions (PBC) are used. The lattice contains Nc unit cells and each unit cell contains Norb sites/orbitals. The vectors ai (i=1, 2, 3) are the fundamental vectors of the Bravais lattice14 while the vectors bα (α=1,…,Norb) 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=(ix, iy, iz) and by the sublattice index α and is centred at the position vector ri α=ixa1+iya2+iza3+bα. (b) The band structure is obtained by solving the Schrödinger equation with periodic potential V(r)=V(r+ai). It consists of the band dispersions ɛn k, with n the band index and ħ k the lattice quasimomentum, and the periodic Bloch functions gn k(r)=gn k(r+ai) (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 Cn 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 wn(r) are not exponentially localized25, and the Peierls substitution in the effective Hamiltonian is therefore not justified15,16. (d) By constructing Wannier functions as linear superpositions of Bloch waves of all bands in the composite band, exponentially localized Wannier functions wα(r) can be created. The mixing of the different bands is provided by the unitary matrix Uα,n(k). This justifies the Peierls substitution for a composite band .