I have a question if you have an Hamiltonian given by [itex] H = \sum_{i,i+1} \sigma_i \cdot \sigma_{i+1} [/itex] where i can even or odd bonds so in a 1D lattice so if you have 4 sites(1 2 3 4 1) then (12) and (34) are even bonds and (23) and (41) are odd bonds. and I was checking if [itex] [H_{x even(12)} , H_{x even(34)}] [/itex] will they commute also do even and odd bonds commute i.e. [itex] [H_{x even} , H_{x odd}] [/itex]
Same as I defined above its a Heisenberg spin systems with [tex] H_{xeven} [/itex] and [tex] H_{xodd} [/itex] are both Heisenberg spin systems with spins defined for even and odd bonds. Here when I say bond I mean the distance between two atomic points in lattice. and alternative bonds are defined as even and odd. Also my ultimate goal is to calculate [tex] [ (\sigma_{1}^x \cdot \sigma_{2}^x + \sigma_{1}^y \cdot \sigma_{2}^y + \sigma_{1}^z \cdot \sigma_{2}^z) , (\sigma_{3}^x \cdot \sigma_{4}^x + \sigma_{3}^y \cdot \sigma_{4}^y + \sigma_{3}^z \cdot \sigma_{4}^z)] [/itex] so will it commute.