A varying J is not neccessary

In the spin-fermion model, which is explicitly two-band, proposed by the blog owner for cuprate superconductors, three terms are considered relevant: the kinetic energy of doped carriers, the on-site AFM spin coupling between the carrier and the local Cu spin, and the AFM coupling (with energy J) between nearest local Cu spins. In this entry, I note that, the J might not be the same for every pair of Cu spins, in the presence of carriers. The reason is clear: the J actually stems from the virtual hopping of Cu electrons and such virtual hopping occurs through the shared O site. When carriers are there on the O site, the virtual hopping shall be affected and hence the J be affected. Nonetheless, it is still realistic to maintain a uniform J for these considerations: (1) the J, as non-uniform as it may be, is always positive, thus no special thing such as frustration could take place; (2) the non-uniformity must be very small, of the order of kintic energy /16, because each carrier in Zhang-Rice orbital becomes diluted by a factor of 1/4; (3) the overall effect may be just to enlarge J a bit (on average). So, this model with uniform J is still very physically reasonable.