Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 2 / a / Solution

The continuous internal symmetry is a common rotation around $z$:Both the transverse exchange and the Dzyaloshinskii-Moriya interaction contain one raising and one lowering spin operator, so the phases cancel. Equivalently, $[H,{\sum }_j{S}_j^z]=0$ and the continuous group is U(1).

The discrete internal symmetry is time reversal, $\mathcal{T}{\mathbf{S}}_j{\mathcal{T}}^{-1}=-{\mathbf{S}}_j$, together with $\mathcal{T}i{\mathcal{T}}^{-1}=-i$. Each term in $H$ contains two spin operators, so it is invariant. The operation is antiunitary; treating the reversal of all three spin components as a unitary rotation would violate the spin commutation relations. It reverses the magnetization and interchanges the two easy-axis ordered orientations. A convenient equivalent representative, composed with a $z$ rotation, is $R_z(\pi )\mathcal{T}$, which reverses $S^z$ while preserving $S^x,S^y$ as operators.

The statement about one continuous and one discrete internal symmetry describes generic parameters. At $D=0$, additional unitary $\pi$ rotations about transverse axes are allowed; at $D=u=0$, the continuous internal symmetry enlarges to full spin rotation symmetry. Spatial translations of the uniform chain are separate symmetries of the sites.