Mean-field ferromagnetic Heisenberg model

One could write: $$ H=-frac{1}{2}sum_{i,j}J_{ij}left[langle S_irangle +(S_i - langle S_irangle)right]left[langle S_jrangle +(S_j - langle S_jrangle)right] approx -frac{1}{2}sum_{i,j}J_{ij}left[langle S_irangle langle S_jrangle +(S_i - langle S_irangle)langle S_jrangle +langle S_irangle(S_j - langle S_jrangle)right] = -frac{1}{2}sum_{i,j}J_{ij}left[S_ilangle S_jrangle +langle S_irangle S_j - langle S_irangle langle S_jrangleright] = -frac{1}{2}sum_{i,j}J_{ij}left[S_ilangle S_jrangle +langle S_irangle S_jright] + const = -frac{1}{2}sum_{i,j}left[J_{ij}langle S_jrangle +J_{ji}langle S_jrangle right]S_i + const = -sum_{i,j}B_i^{eff}S_i + const, $$ where $$ B_i^{eff} = frac{1}{2}sum_{j}left[J_{ij}langle S_jrangle +J_{ji}langle S_jrangle right]. $$ Note that strictly speaking the effective field is inhomogeneous. It can be assumed homogeneous only under specific assumptions about $J_{ij}$, e.g., that it depends only on the distance between the sites, but not on their absolute position or their arrangement on the lattice.

Fourier transform on a discrete lattice becomes a discrete transform, which depends on the specific lattice vectors. E.g., in 1D one could write : $$ b_k^{eff} = sum_j e^{ijk}B_i^{eff}. $$ (note that $i$ here is the imaginary unit) However, mean field expansion is known to give incorrect results in 1D, and the problem in question is probably done in multiple dimensions. Therefore one cannot say more without knowing more about the underlying lattice. Such expansions are extensively used in solid state physics, transforming from the crystal lattice to the reciprocal lattice.