# Difference between S_x S_x +S_y S_y and S+S- +S-S+ for MPO

I am trying to create an MPO with 3 or more spin interactions of type Sx Sy Sz with x, y and z at different sites and then find correlation of type Sx Sy Sz.
As a benchmark, first I am trying to study Heisenberg model and correlation function.

1.I am getting a different value of correlation function if I use Sx Sx +Sz Sz instead of usual S+S- +S-S+ for MPO. Why am I getting different results for correlation? Is there anything which I can do such that using Sx, Sy in MPO will provide me same results as S+S-?
1. Also when I am trying to calculate correlation of terms like Sx Sx or Sy Sy, should I convert them also to S+S- form?
Using Sx Sy terms instead of S+ S- term in MPO will simplify my code if I have 3 or more spin interaction .
Please suggest something which can help me understand and simplify things.
Thank you,

Here is snapshot of my code:
(int j = 1; j <= N-4; j += 1)
{
ampo += -J3,"Sx",j,"Sy",j+1,"Sz",j+2;
ampo += -J3,"Sy",j,"Sz",j+1,"Sx",j+2;
ampo += -J3,"Sz",j,"Sx",j+1,"Sy",j+2;

    ampo +=   J3,"Sx",j,"Sz",j+1,"Sx",j+2;
ampo += J3,"Sy",j,"Sx",j+1,"Sz",j+2;
ampo += J3,"Sz",j,"Sy",j+1,"Sx",j+2;

ampo +=   -J3,"Sx",j+1,"Sy",j+2,"Sz",j+3;
ampo +=  -J3,"Sy",j+1,"Sz",j+2,"Sx",j+3;
ampo +=  -J3,"Sz",j+1,"Sx",j+2,"Sy",j+3;

ampo +=   J3,"Sx",j+1,"Sz",j+2,"Sx",j+3;
ampo += J3,"Sy",j+1,"Sx",j+2,"Sz",j+3;
ampo += J3,"Sz",j+1,"Sy",j+2,"Sx",j+3;


answered by (37.8k points)

Hi, thanks for the question.

So I believe the answer to your first question is that S+S- + S-S+ is proportional to SxSx + SySy, not SxSx+SzSz. So unless your wavefunction is perfectly spin-rotationally invariant you will in general get a different answer when measuring these operators which are different.

You don’t have to convert operators to S+ S- form if you aren’t conserving the total Sz quantum number. However if you are conserving total Sz, you must input your Hamiltonian in terms of operators which change total Sz by a well-defined amount, so mainly Sz, S+, and S- (and not Sx or Sy).

Best regards,
Miles

commented by (170 points)
Thanks a lot  for your reply, Miles. Yes you are right, Sx Sx +Sy Sy is proportional to S+S- + S-S+, I made a mistake while writing. My Hamiltonian is SU(2) symmetric, I got different result if I use ampo +=   J3,"Sx",j,"Sz",j+1,"Sx",j+2; format  than replacing Sx and Sy terms by combination of S+ +iS-/ S+ +iS-.
regards,
Ajit