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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)

enter image description here

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?

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  • $\begingroup$ What about the expectation value of the operator? Or what do you mean with 'calculate' $S_z$? $\endgroup$
    – Tobias Fünke
    Commented Aug 1, 2021 at 20:46
  • $\begingroup$ @Jakob Yes. I want to calculate the expectation value of $S_z$. I know for $n-th$ band, it will be $\langle S_z \rangle ^{(n)} = \langle n | S_z | n\rangle$. Here, I get $|n\rangle$ numerically which is $1\times 6$ column matrix. I have to define $S_z$ operator in matrix form. It should be $6\times 6$, I think. For a spin-1/2 particle, $S_z = \hbar/2 \sigma_z$, where $\sigma_z$ is Pauli $2\times 2$ matrix. So, my question is how to define $S_z$ operator for such a system where wavefunction is $1\times 6$ vector? $\endgroup$
    – Luqman Saleem
    Commented Aug 1, 2021 at 22:27
  • $\begingroup$ Then you should really state this. In general, also for possible next questions you may ask, be as precise as possible. $\endgroup$
    – Tobias Fünke
    Commented Aug 2, 2021 at 8:26

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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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