Exploring the Spin-1 XY Model: Understanding the Hamiltonian and Its Parameters

[chaksome]

I am reading a paper about quantum many-body scar based on the spin-1 XY model. I noticed that he write down the Hamiltonian as follows
$$
H=J \sum_{\langle i j\rangle}\left(S_{i}^{x} S_{j}^{x}+S_{i}^{y} S_{j}^{y}\right)+h \sum_{i} S_{i}^{z}+D \sum_{i}\left(S_{i}^{z}\right)^{2}
$$
which is a little bit different from what I've learned as
$$
H=J \sum_{\langle i j\rangle}\left(\left(1+\gamma_i\right)S_{i}^{x} S_{j}^{x}+\left(1-\gamma_i\right)S_{i}^{y} S_{j}^{y}\right)+h \sum_{i} S_{i}^{z}
$$
I think the $\gamma$ is a parameter characterizing the degree of anisotropy in the XY plane, so $\gamma = 0$ when we assume that the energy gap of the system is always closed. Besides, $h$ is a parameter characterizing the degree of the external field.
How about D, what does it represent? Why should we consider the term of the square of $S^z_i$(identity matrix)? Please help me out~Thanks a lot!

 

[azntoon]

The D term is known as a single-site anisotropy, and it is used to introduce an anisotropy in the system. In other words, it introduces a difference between the energy levels of the spin in the system. This term is important to study the effect of anisotropy on quantum many-body scar states.In addition, the D term allows you to change the shape of the energy spectrum for the system. This can be used to study the effect of different types of anisotropies on the system. For example, it can be used to study the effect of anisotropies on the phase transition of the system. Finally, the D term is also important to study the effect of single-site anisotropies on the ground state properties of the system. This can be used to study the effect of anisotropies on the entanglement properties of the system, such as entanglement entropy and entanglement spectrum.