SU(2)_Hubbard_model

[ALF-Import-Bot]

In GitLab by @fassaad on May 21, 2025, 08:04 UTC:

Migrated from GitLab: https://git.physik.uni-wuerzburg.de/ALF/ALF/-/issues/311

[ALF-Import-Bot]

In GitLab by @fassaad on May 21, 2025, 08:05 UTC:

Aim is to implement the following for for the Hubbard interaction:
$H_U = -\frac{U}{2}\sum_{i} \left( \vec{n}i \cdot \vec{\hat{m}}i\right)^2$ with $|\vec{n}| = 1$ and $\vec{\hat{m}}i=\vec{\hat{c}}^{\dagger}i \vec{\sigma} \vec{\hat{c}}^{}i$
With this we have: $\left( \vec{n}\cdot \vec{\hat{m}}\right)^2 = \sum{\alpha} n{\alpha}^2 \hat{m}{\alpha}^2 + \sum{\alpha > \beta=1}^3
n{\alpha} n_{\beta} \left{ \hat{m}{\alpha}, \hat{m}{\beta}\right} = \sum_{\alpha} n_{\alpha}^2 \hat{m}{\alpha}^2 = -2 |\vec{n}|^{2} \left( \hat{n}{\uparrow} - 1/2 \right)\left( \hat{n}_{\downarrow} - 1/2 \right)$

The goal is to implement:
$H_U = -\frac{U}{2}\sum_{i} \sum_{\alpha} n_{\alpha,i}^2 \hat{m}_{\alpha,i}^2$

[ALF-Import-Bot]

In GitLab by @fassaad on May 21, 2025, 10:54 UTC:

It seems that we generate a sign problem as soon as we have the vector n which is not along the x,y or z-direction. We should understand this.
We should check that for $n = e_x,e_y$ or $e_z$ that we always get the same energy, and that it compares with the generic Hubbard model code.