Spin 1 Clifford Algebra

Question

For starters I'm not sure if what I'm looking for actually falls within Clifford Algebra,the name comes really from trying to find analogies with spin 1/2.

The system that I study (Trilayer Graphene) have a Bloch Hamiltonian of the form: \begin{align} L\otimes \mathbb{1}_3+ M\otimes (S_x+ S_y) \end{align} where L and M are 2x2 matrices (Pauli-vectors really) and $S_i$ are the spin 1 operators ($\hbar=1$):

\begin{align} S_x=\frac{1}{\sqrt{2}}\begin{pmatrix} 0&1&0\\ 1&0&1\\ 0&1&0 \end{pmatrix}\quad,\quad S_y=\frac{1}{\sqrt{2}}\begin{pmatrix} 0&-i&0\\ i&0&-i\\ 0&i&0 \end{pmatrix}\quad,\quad S_z=\begin{pmatrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{pmatrix} \end{align}

For spin 1/2 we have the three Pauli matrices that satisfy:

\begin{align} \sigma_i^2&=\mathbb{1}_2\qquad,\qquad \left\lbrace \sigma_i,\sigma_j\right\rbrace =2\delta_{ij}\mathbb{1}_2 \end{align}

which in turn makes life easy when we want to calculate eignvalues of operators comprised of linear combinations of $\sigma_i$'s. Essentially I'm looking for similar relations/ tricks.

solving the characteristic polynomial head on proved to be quite difficult so I'm thinking of different approaches and the past couple of days I've tried to find some algebraic method to diagonalize such a "pauli vector" or maybe find relations to ease the calculations.

So far I haven't come up with anything very useful and didn't find much information online (mostly it's either a construction of the matrices or research papers that I didn't manage to follow very well).

Is someone here aware of such a relation? besides the anti commutator that I can't see how would I use it for my needs (if I'm wrong about it please let me know :) )

Another approach I thought about but haven't tried out yet was to use $S_{\pm}$ as they are linear combinations of $S_x,S_y$ I'm not sure if that would lead anywhere.

In any case I would appreciate any thoughts you might have on approaches that might work

Answer 1

You are comparing two representations of su(2) : the 2D one is $\sigma_i/2$, and the 3D one (the adjoint) is $S_i$.

They obey identical commutation relations, but they have different relations for the anticommutator, and the quadratic Casimir: the sum over the squares of all three generators, $\sum_i\sigma_i ^2/4= \tfrac{3}{4} \mathbb{I}_2$, versus $\sum_i S_i^2= 2 \mathbb{I}_3$, the usual general spin s expression, $\sum_i T_i^2= s(s+1) \mathbb{I}_{2s+1}$.

For the doublet representation, your first relation is a special case of the anticommutator following it.

However, for the triplet representation, the anticommutator is a messier object, not always diagonal: $$ \{ S_i,S_j \}= \delta_{ij} (4/3){\mathbb I}_3 +M_{ij}, $$ where $M_{ij}$ is hermitean, symmetric, traceless, and trace orthogonal to all three generators $S_i$; in addition, $M_{ii}=0$, no index summation implied.