What are spin and valley symmetries in graphene? I have been assigned a presentation on a part of a paper (http://arxiv.org/abs/1303.6942). My task is to present on the spin and valley symmetries in graphene, and relate it back to the paper above.
However, after looking at many papers detailed spin and valley degeneracy lifting and symmetry breaking, I still can't get my head round what spin and valley symmetries in graphene actually are.
So if anyone could give me a description of what these are and maybe any other relevant properties that are worth presenting on that would be incredibly helpful.
I apologise for the vagueness of my question, if I could be more specific I probably wouldn't need to ask in the first place!
 A: At low energies, close to the Dirac points we can obtain the following effective Hamiltonians:
\begin{align*}
\begin{array}{ll}
\mathcal{H}_{\mathbf{K}} = \hbar v_{F}\left(\Pi_{x}\sigma^{x} + \Pi_{y}\sigma^{y}\right) & \text{at the $K$ point}
\\
\mathcal{H}_{\mathbf{K^{\prime}}} = -\hbar v_{F}\left(\Pi_{x}\sigma^{x} - \Pi_{y}\sigma^{y}\right) & \text{at the $K^{\prime}$ point}
\end{array}
\end{align*}
(where $\Pi_{i}$ is the gauge invariant momentum $\Pi_{i} = p_{i} + eA_{i}(\mathbf{r})$)
We can combine these to write:
\begin{align*}
H_{valley} &= \hbar v_{F} \left(\begin{array}{cc}\boldsymbol{\Pi}.\boldsymbol{\sigma} & 0 \\ 0 & -\boldsymbol{\Pi}.\boldsymbol{\sigma}\end{array}\right) 
\\
&= \hbar v_{F}\sigma^{3}\otimes\mathbf{q}.\boldsymbol{\sigma}
\end{align*}
Here thetensor product can be thought of as valley$\otimes$sublattice, so the valley pseudospin takes the place of the usual spin we see in the massless Dirac Hamiltonian.
The spin itself is added to the Hamiltonian then, say writing it as $H_{S}$, and they (spin and valley pseudospin) are completely independent from each other. We can write the total Hamiltonian as:
\begin{align*}
H_{tot} = H_{S} \otimes H_{valley}
\end{align*}
Each spin has an associated $SU(2)$ symmetry, and due to their independent natures these combine to give an overall $SU(4)$ symmetry (since we can have entangled operators). The $SU(4)$ symmetry is generated by $\sigma^{i} \otimes \mathbb{1}$, $\mathbb{1} \otimes \sigma^{i}$,$\sigma^{i} \otimes \sigma^{j}$ for $i,j \in \{1,2,3\}$ where the first component of the tensor product is associate with the spin, and the second is the valley pseudospin.
Hope this clears some things up.
A: The hexagonal Graphene lattice can be considered as a superposition of two identical sub-lattices set off by one one carbon-carbon bond length. As a result, it has two sets of wavevectors k,that are picked out by the lattice, inequivalent (since the two sublattices really are distinct) but otherwise identical (since it's semantics to say which sublattice is primary and which is secondary) and this is known as “valley degeneracy”.
