I think this is a FAQ when we are studying the rotation operations of lattice spin systems, but I can't find much references.
Background
Considering a Hamiltonian defined on a triangular lattice: \begin{aligned} H & =H_{ \pm}+H_z+H_{ \pm \pm}+H_{ \pm z} \\ H_{ \pm} & =J_{ \pm} \mathrm{H}_{ \pm}=J_{ \pm} \sum_{\langle i j\rangle}\left(S_i^{+} S_j^{-}+S_i^{-} S_j^{+}\right) \\ H_z & =J_z \mathrm{H}_z=J_z \sum_{\langle i j\rangle} S_i^z S_j^z \\ H_{ \pm \pm} & =J_{ \pm \pm} \mathrm{H}_{ \pm \pm}=J_{ \pm \pm} \sum_{\langle i j\rangle}\left(\gamma_{i j} S_i^{+} S_j^{+}+\gamma_{i j}^* S_i^{-} S_j^{-}\right) \\ H_{ \pm z} & =J_{ \pm z} \mathrm{H}_{ \pm z} =\mathrm{i} J_{ \pm z} \sum_{\langle i j\rangle}\left[\left(\gamma_{i j}^* S_i^z S_j^{+}-\gamma_{i j} S_i^z S_j^{-}\right)+(i \leftrightarrow j)\right]. \end{aligned}
Where $\gamma_{ij}=1, \mathrm{e}^{\mathrm{i} {2\pi}/3}, \mathrm{e}^{-\mathrm{i} {2\pi}/3} $ for bond ${i,j}$ along the real space basis $\vec{a}_1 = (1,0,0), \vec{a}_2 = (-1/2,\sqrt{3}/2,0), \vec{a}_3 = (-1/2,-\sqrt{3}/2,0)$ This is exactly the model in the paper https://scipost.org/SciPostPhys.4.1.003/pdf. The first two terms together are $XXZ$ model and the last two introduce anisotropic interactions.
The model Hamiltonian is invariant under a series of symmetry generators of the system. When we perform these operations, we should take care of both lattice space and spin space. The threefold $\mathcal{C}_3$ rotation is one of the symmetry generators, it transform the coordinates $x_1, x_2$ of a lattice point $\vec{r} = x_1 \vec{a}_1 + x_2 \vec{a}_2$ as
$$ (x_1, x_2)\rightarrow (-x_2, x_1 - x_2), $$
while it transforms the components of the spin operator as
$$ (S^x, S^y, S^z) \rightarrow \left(-\frac{1}{2}S^x -\frac{\sqrt{3}}{2}S^y , \frac{\sqrt{3}}{2}S^x- \frac{1}{2} S^y, S^z\right). $$
The transformation of spin component can be verified by $$S^x \rightarrow \mathrm{e}^{\mathrm{i} \frac{2\pi}{3} \cdot S^z} S^x \mathrm{e}^{-\mathrm{i} \frac{2\pi}{3} \cdot S^z} = -\frac{1}{2}S^x -\frac{\sqrt{3}}{2}S^y,$$ and it transforms the Ising basis $|\uparrow \uparrow\downarrow\uparrow\cdots\downarrow>$ as: $$ \mathrm{e}^{-\mathrm{i} \frac{2\pi}{3} \cdot S^z} |\uparrow \uparrow\downarrow\uparrow\cdots\downarrow> = \mathrm{e}^{-\mathrm{i} \frac{\pi}{3} \sigma^z} |\uparrow \uparrow\downarrow\uparrow\cdots\downarrow> = \mathrm{e}^{-\mathrm{i} \frac{\pi}{3} s_n} |\uparrow \uparrow\downarrow\uparrow\cdots\downarrow>, $$ where $s_n$ is the number of up spins minus number of down spins. It's the rotation operation in spin space.
My question is:
How does the real space rotation transform the Ising basis?
When we considering a finite size system with periodic boundary conditions on both $\vec{a}_1$ and $\vec{a}_2$ directions, the real space rotation seems to permute the lattice sites. So it that the spin rotation a permutation of uparrows and downarrows inside the Ising basis?
