Spin Texture in $k$-Space – How to Calculate

computational physicscondensed-matterlattice-modelsolid-state-physicstight-binding

I have a triangular lattice model. In $k$-space, it is written as:
$$
H = \sum_k\sum_\sigma \Psi_{k\sigma}^+ h_k\Psi_{k\sigma}
$$
where $\Psi_{k\sigma} = [a_{k\uparrow}, b_{k\uparrow}, c_{k\uparrow},a_{k\downarrow}, b_{k\downarrow}, c_{k\downarrow}]^T$; $a_{k\sigma},b_{k\sigma},c_{k\sigma}$ are sublattice in the unitcell, and $h_k$ is $6\times6$ matrix.

We can numerically diagonalize $h_k$ and calculate band-structure, I did it in MATLAB and got (showing here only the lowest band)

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Now, I want to calculate spin texture, for example, the magnitude of say z-polarized spin at each point in k-space. One example is given in Figure 8 of Ref: arXiv:2008.10815 as
enter image description here

All in all, I want to know how to calculate $S_z$ at each point in $k$-space when we have wavefunctions and eigenvalues?

Best Answer

I guess I figured it out. What I need is expectation value of $\hat S_i$ operators $i=\{x,y,z\}$. The operator $\hat S_i = \hat I_3 \bigotimes \hat\sigma_i$ where $\hat\sigma_i$ are Pauli matrices.

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