  Thanks to the nearsightedness principle, the low-energy electronic structure
of solids can be represented by localized states such as the Wannier functions.
Wannier functions are actively being applied to a wide range of phenomena in
condensed matter systems. However, the Wannier-functionbased representation is
limited to a small number of bands and thus cannot describe the change of
wavefunctions due to various kinds of perturbations, which require sums over an
infinite number of bands. Here, we introduce the concept of the Wannier
function perturbation, which provides a localized representation of
wavefunction perturbations. Wannier function perturbation theory allows
efficient calculation of numerous quantities involving wavefunction
perturbation, among which we provide three applications. First, we calculate
the temperature-dependent indirect optical absorption spectra of silicon near
the absorption edge nonadiabatically, i.e., differentiating phonon-absorption
and phonon-emission processes, and without arbitrary temperature-dependent
shifts in energy. Second, we establish a theory to calculate the shift spin
conductivity without any band-truncation error. Unlike the shift charge
conductivity, an exact calculation of the shift spin conductivity is not
possible within the conventional Wannier function methods because it cannot be
obtained from geometric quantities for low-energy bands. We apply the theory to
monolayer WTe$_2$. Third, we calculate the spin Hall conductivity of the same
material again without any band-truncation error. Wannier function perturbation
theory is a versatile method that can be readily applied to calculate a wide
range of quantities related to various kinds of perturbations.
