Suppose there are three particles: A, B, and C. Particles B and C are subjected to XXX Heisenberg interaction, while the spin of particle A interacts with both B and C. then we have:
$ H_{BC} = J_x \sigma_B^x \sigma_C^x + J_y \sigma_B^y \sigma_C^y + J_z \sigma_B^z \sigma_C^z $
$ H_{A-BC} = \left( \sigma_{A}^{x} \otimes \left( \sigma_{B}^{x} + \sigma_{C}^{x} \right) + \sigma_{A}^{y} \otimes \left( \sigma_{B}^{y} + \sigma_{C}^{y} \right) + \sigma_{A}^{z} \otimes \left( \sigma_{B}^{z} + \sigma_{C}^{z} \right) \right) $
Then the total hamiltoian is as follows:
$H = H_{BC} + H_{A-BC} $
Did I write all of the above correctly, or did I make any mistakes?
