This is a simple question. Please forgive me, as I am a lowly experimentalist.
Suppose we have two free spin 1/2 particles, i.e. a 4-fold degenerate system. What is the set of symmetry operations on this system? Is it $SU(2) \times SU(2)$, $SU(4)$, or something else? Or am I misunderstanding all of this group jargon entirely?
My current understanding is that $SU(2)$ rotates a single spin 1/2 particle, and $SU(2) \times SU(2)$ rotates both particles (but not necessarily with the same axis and angles). Furthermore, when we do this addition of angular momentum magic, we are taking $SU(2) \times SU(2)$ and decomposing it into irreducible representations of $SO(3)$ because we want to rotate the spins together (with the same axis and angle). Am I wrong about any of this?
I ask this because people in the graphene field say that a "fourfold spin–valley degeneracy lead[s] to an approximate SU(4) isospin symmetry." This was confusing to me because I previously thought that two spin 1/2 degrees of freedom led to $SU(2) \times SU(2)$ symmetry. However, now I'm led to believe that $SU(4)$ describes the symmetries of a 4-fold degenerate system, and that $SU(2) \times SU(2) \subset SU(4)$ with some entangled states not represented by rotating two spins (i.e., if I prepare two spin up particles, I can't get every possible state by simply rotating them).