Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition. Consider for instance the case of 50 fermions in a ferromagnetic chain, for the sake of simplicity. A dimension of 12, let's say, for the χ χ c would be a reasonable choice, keeping the discarded eigenvalues at 0.0001 % of the total, as shown by numerical studies, meaning roughly 2 14 coefficients, as compared to the originally 2 50 ones.