# How to calculate spin texture in $k$-space?

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)

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

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

• What about the expectation value of the operator? Or what do you mean with 'calculate' $S_z$? Aug 1, 2021 at 20:46
• @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? Aug 1, 2021 at 22:27
• Then you should really state this. In general, also for possible next questions you may ask, be as precise as possible. Aug 2, 2021 at 8:26

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.