Correlation Length Analysis about mphysproject

[pmorande27]

There is also an alternative way of getting the correlation length through $G(x)$ we can use the wall-to-point correlation, that is:
$$G_{w}(x_1) = \sum_{x_2}G(x_1,x_2)$$
We then fit G to the cosh model to obtain an estimate of the correlation length $\xi_w$. However, this is worse than the already implemented wall to wall correlation model as this one provides more stats per configuration. Also $G_w$ seems to be only approximately symmetric around the medium point in the lattice while $G_{ww}$ is exactly symmetric. More statistics should solve this issue.
It is good to have $\xi_w$ as the ratios between $\xi_w$ and $\xi_G$ are provided in Two dimensional SU(N) x SU(N) chiral models on the lattice by Rossi and Vicari

[pmorande27]

The errors for both $\xi_G$ and $G_w$ are more involved. The simplest analysis would involve propagation of errors from the errors on $G(x)$ which are directly accessible from the HMC simulation. This seems a reasonable choice but the variables might be correlated.
This has been implemented for $G_w(x_1)$

[pmorande27]

Increasing the number of configurations seems to improve/solve the problem on the second moment correlation length, I am currently testing it for SU(3) in $18 \times 18$ square lattice for $\beta = 1.08$ and comparing the value to that of Rossi and Vicari. The value after $10^4$ Thermalisation runs and $10^4$ Measurement Runs is $((0.9246192197158345+0.05578724056426441j) \pm(0.05934682171701872-0.00290907492629368j))$, after $10^4$ Thermalisation runs and $10^5$ Measurement Runs the result is $((1.0324631065326133+0.016871897949929604j)\pm (0.017183422734034358-0.00021680543643150615j))$ The value is closer much closer to the 1.003 given in the paper and the imaginary part is reduced too. I am currently waiting for a run with $10^4$ Thermalisation runs and $10^6$ Measurement Runs.

[pmorande27]

I have finalised the run with $10^4$ Thermalisation runs and $10^6$ Measurement Runs. The value obtained for the second moment correlation Length is $0.9853956686584021-0.01685379195961662j), \pm (0.0057039134139613605+7.709634606372116e-05j)$ the value is closer to the $1.003$ given by the paper (total difference is now $0.017$ when before it was $0.029$. The imaginary part is also reduced.

[pmorande27]

I also run the simulations for larger lattices and I found that the imaginary part of the second moment correlation length tends to stay in about $0.01$ units even when the real part increases. This indicates that the imaginary part could be due to numerical errors and statistical fluctuations.

[pmorande27]

There is also an alternative way of getting the correlation length through $G(x)$ we can use the diagonal wall-to-point correlation, that is:
$$G_{d}(x_1) = \sum_{x_2}G(x_1-x_2,x_2)$$
We then fit G to the cosh model to obtain an estimate of the correlation length $\xi_d$. This can be used to test rotational invariance of the model as the ratio of $\xi_w$ and $\xi_d$ should be close to 1.
It is good to have $\xi_w$ as the ratios between $\xi_w$ and $\xi_d$ are provided in Two dimensional SU(N) x SU(N) chiral models on the lattice by Rossi and Vicari

[pmorande27]

I have effectively fixed the Imaginary part issue in the calculation of $\xi_G$, to improve statistics I exploited the translational invariance of $G(x)$ such that $G(x) = \langle\frac{1}{N} ReTr U(x)U^\dagger (0)\rangle = G(x +y -d) =\langle \frac{1}{N} ReTr U(x)U^\dagger (0) \rangle$, therefore I used:
$G(x) = \frac{1}{N^2}\sum_y\langle \frac{1}{N} ReTr U(x+y)U^\dagger (y) \rangle = \frac{1}{N^2}\langle \sum_y\frac{1}{N} ReTr U(x+y)U^\dagger (y) \rangle $ The second step uses linearity. This increases the statistics by a factor of $N^2$ and now the imaginary part looks like noise $\approx 10^{-16}$, it also improves the results for $G_w$ and $G_d$