Understanding Hamiltonian with Even/Odd Bonds

[adityaphysics]

I have a question if you have an Hamiltonian given by
$H = \sum_{i,i+1} \sigma_i \cdot \sigma_{i+1}$
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

$[H_{x even(12)} , H_{x even(34)}]$
will they commute also do even and odd bonds commute i.e.
$[H_{x even} , H_{x odd}]$

 

[DrDu]

How do you define $H_{xeven}$ and $H_{xodd}$?

 

[adityaphysics]

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.