I was wondering if there is a smart way in Mathematica to expand powers of dot products of spin-operators -for instance (Si * S j)^3 where S is the spin vector on site I and the same for the site j- into combinations of Splus, Sminus, S_z (where Splus and Sminus are the raising/lowering operators).
For the exponent 1, it is very easy by hand (by using the definition of Splus and Sminus):
Si * S j = Sz * Sz + 0.5(SplusSminus + Sminus*Splus)
I would like to obtain similar expansion for higher exponents, such as (Si * S j)^3 and (Si * Sj)^4.
Thank you very much.