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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).

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My answer is in two parts.

First part. $SU(2)$ has representations of any dimension $2j+1$ with integer or half-integer j. Direct product of two $j=1/2$ representations is reducible to a direct sum $j=0$ (singlet) and $j=1$ (triplet). All remain representations of $SU(2)$ which defines the spin in the first place.

Now, if you have energy degeneracy in a 4 dimensional space, then it remains invariant under a much wider class of transformations -- it is a $SU(4)$ "singlet". Since in the quoted case the two $SU(2)$'s are different (real spin and valley pseudo spin) then extra symmetries $\subset SU(4)$ and $\not \subset SU(2) \otimes SU(2)$ are possible, which makes the buzzword of "approximate SU(4)" not necessarily empty.

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  • $\begingroup$ Ok, unless I misinterpret your words, what I now believe is correct. $SU(4)$ is the symmetry of two spin 1/2 particles. $\endgroup$ – ChickenGod Nov 17 '13 at 4:18

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