 Summary

I wanted to simulate the Ising model and it was okay until I wanted to get the fluctuations <M^2>.
In fact, first, I wanted to obtain the magnetization M :
$$ M = \sum_i \sigma ^ z_i $$
and it worked, I got indeed the magnetization M. However I don't succeed unfortunately to obtain the fluctuations <M^2>.
I wanted to simulate the Ising model and it was okay until I wanted to get the fluctuations <M^2>.
In fact, first, I wanted to obtain the magnetization M :
$$ M = \sum_i \sigma ^ z_i $$
and it worked, I got indeed the magnetization M by writing these lines :
sigmaxop = []
sites = []
for i in range (L):
# \ sum_i sigma ^ z (i)
sigmaxop.append ((sigmaz) .ToList ())
sites.append ()
with L being the length of the lattice (ie the number of spins).
However I would like now to obtain the average fluctuations :
$$ \langle M^2\rangle = \sum_ {i, j} \langle\sigma_i \sigma_j\rangle $$
To obtain this average fluctuation, I try to modify the part of the code above (that allowed me to obtain M) to get the fluctuations <M^2> but I didn't succeed : I don't obtain <M^2>.
A priori for every lattice
$$m_j=\sum_k \sigma_k$$
and the average of magnetization is
$$<M>=\sum_j m_j$$
So
$$<M^2>=\sum_j m^2_j$$
Unfortunately after trying again I am still stuck with my code. It doesn't work.
Could someone help me to modify/rewrite this part of code to obtain the fluctuations <M^2>, please?
In fact, first, I wanted to obtain the magnetization M :
$$ M = \sum_i \sigma ^ z_i $$
and it worked, I got indeed the magnetization M by writing these lines :
sigmaxop = []
sites = []
for i in range (L):
# \ sum_i sigma ^ z (i)
sigmaxop.append ((sigmaz) .ToList ())
sites.append ()
with L being the length of the lattice (ie the number of spins).
However I would like now to obtain the average fluctuations :
$$ \langle M^2\rangle = \sum_ {i, j} \langle\sigma_i \sigma_j\rangle $$
To obtain this average fluctuation, I try to modify the part of the code above (that allowed me to obtain M) to get the fluctuations <M^2> but I didn't succeed : I don't obtain <M^2>.
A priori for every lattice
$$m_j=\sum_k \sigma_k$$
and the average of magnetization is
$$<M>=\sum_j m_j$$
So
$$<M^2>=\sum_j m^2_j$$
Unfortunately after trying again I am still stuck with my code. It doesn't work.
Could someone help me to modify/rewrite this part of code to obtain the fluctuations <M^2>, please?