# Finding the classical antiferromagnetic ground state for the Kagomé lattice

I am attempting to do the following exercise in Altland and Simons Condensed Matter Field Theory:

"Show that the classical antiferromagnetic ground state of the Kagomé lattice – a periodic array of corner-sharing Stars of David – has a continuous spin degeneracy generated by local spin rotations."

Here, the lattice is governed by the antiferromagnetic Heisenberg Hamiltonian $$H = J\sum_{\langle i j\rangle} \mathbf{S}_i\cdot\mathbf{S}_j$$, where $$J > 0$$.

I am not sure where to start and would appreciate some hints. Some questions I have are:

1. Does solving the exercise require knowing the actual classical ground state? If so, how do you go about finding the classical ground state? The only thing I can think of is taking derivatives, but that already seems quite complicated.
2. Is there a way to show the ground state satisfies certain symmetries? If the ground state arrangement is somewhat symmetric, I can see how you may be able to spin each individual Star of David on the lattice to demonstrate the degeneracy, but it's not clear to me how you would prove that the ground state satisfies the symmetry necessary to demonstrate the degeneracy in this manner.

The full exercise is:

Employing only symmetry considerations, identify a possible classical ground state of the triangular lattice Heisenberg antiferromagnet. (Hint: construct the classical ground state of a three-site plaquette and then develop the periodic continuation.) Show that the classical antiferromagnetic ground state of the Kagomé lattice–a periodic array of corner-sharing Stars of David–has a continuous spin degeneracy generated by local spin rotations. How might the degeneracy affect the transition to an ordered phase?

2. The best way to think about the symmetries is to have your trial ground state from the first part of the question in hand, and then asking yourself what arbitrary choices you made to arrive there. Starting from any ground state, the set of all other periodic ground states of the same form should be indexed by $$SO(3)$$, the rotations in spin space of every spin on the whole lattice at the same time.
A classical state is a choice of a vector on the unit 2-sphere for each site. Let's start with the three-site plaquette and call the vectors $$\vec{s}_i$$ for $$i=1,2,3$$. The classical Hamiltonian has a global $$SO(3)$$ symmetry from rotating all the spins together, so our first choice doesn't matter: we'll take $$\vec{s}_1 = \hat{z}$$. The second and third spins will have polar angles $$\theta_2, \theta_3$$, and we'll take $$\phi_2 = 0$$ and $$\phi_3 \equiv \phi$$ without loss of generality, also by symmetry. We can write the total energy as \begin{align} E_{cl,n=3} &= J S^2 \sum_{\langle ij \rangle} \vec{s}_i \cdot \vec{s}_j\\ &= J S^2 \left( \cos\theta_2 + \cos \theta_3 + \cos\theta_2 \cos \theta_3 + \sin\theta_2\sin\theta_3 \cos\phi \right) \end{align}
The derivative of this thing has to be zero with respect to $$\theta_2$$, $$\theta_3$$, and $$\phi$$. There aren't too many cases to work out if you start with $$\phi$$. The result is that $$E = - 3J S^2 /2$$, which is achieved for $$\theta_2 = \theta_3 = 2\pi/3$$ and $$\phi = \pi$$; that is, for the three spins in an equilateral triangle along a great circle, which seems like a pretty reasonable answer to guess (tbh I found this with Mathematica's Minimize rather than derivatives but it's all the same). You can extend this pattern periodically along the infinite kagomé lattice without hitting any problems. There were two arbitrary (continuous-valued, excluding discrete lattice reflections) choices that we made along the way: we picked $$\vec{s}_1$$ as $$\hat{z}$$, and we picked $$\phi_2 = 0$$. This means that the other possible classical ground states are enumerated precisely by acting on this one with elements of $$SO(3)$$.
From this perspective that result may seem a little banal, because of course you had to pick a direction for $$\vec{s}_1$$ and another for $$\vec{s}_2$$. Things get much more interesting in the quantum case, where you can get stuff like valence-bond states where neighboring spins entangle themselves into singlets that are invariant under local rotations. But that's neither here nor there.