# Does the projected spin state of the $d+id$ mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

Consider the following $d+id$ mean-field Hamiltonian for a spin-1/2 model on a triangular lattice $$H=\sum_{<ij>}(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$$, with $\chi_{ij}=\begin{pmatrix} 0 & \Delta_{ij}\\ \Delta_{ij}^* & 0 \end{pmatrix}$, fermionic spinons $\psi_i=\binom{f_{i\uparrow}}{f_{i\downarrow}^\dagger}$, and the mean-field parameters $\Delta_{ij}=\Delta_{ji}$ defined on links have the same magnitudes and their phases differ by $\frac{2\pi}{3}$ with each other referring to the three bond-direction.

My question is, does the projected spin state $\Psi=P\phi$ have the TR symmetry? Where $\phi$ is the mean-field ground state of $H$, and $P$ removes the unphysical states with empty or doubly occupied sites.

Notice that from the viewpoint of Wilson loop, you can check that the Wilson loops $W_l=tr(\chi_{12}\chi_{23}\chi_{31})=0$ on each triangle plaquette, thus all the Wilson loops are invariant under the TR transformation $W_l\rightarrow W_l^*=W_l$. Thus, the TR symmetry should be maintained.

On the other hand, from the viewpoint of $SU(2)$ gauge-transformation, if there exist $SU(2)$ matrices $G_i$ such that $\chi_{ij}\rightarrow\chi_{ij}^*=G_i\chi_{ij}G_j^\dagger$, then the projected spin state $\Psi$ is TR invariant. But so far, I can not find out those $SU(2)$ matrices $G_i$. So can anyone work out the explicit form of those $SU(2)$ matrices $G_i$? Or they do not exist at all?

By the way, I think it would be awkward to explicitly write the form of state $\Psi$ to check the TR symmetry.
• @Jim Garrison Yes, I agree with you. If the nearest neighbor hopping $t$ is turned on, then the triangle Wilson loop $W_l$ will take a nonzero imaginary value $\propto it\Delta^2$ and $W_l$ is changed to $-W_l$ under TR operation, thus TR symmetry would be broken. – Kai Li Jan 26 '14 at 14:59
• @Jim Garrison But I want to know that whether the $SU(2)$ matrices mentioned in my question exist? And from which viewpoint(Wilson loop or $SU(2)$ matrices) you infer that the projected spin state has TR symmetry? – Kai Li Jan 26 '14 at 15:03
I just found that the solution of $SU(2)$ matrices is really simple.
When there is no hopping term, the projected spin state of the above $d+id$ mean-field Hamiltonian indeed has TR symmetry. Because there exist global $SU(2)$ matrices $G_i$ which implement the TR transformation, say $G_i=i\tau ^x$.