Alternative Spin Representation

[PaulWAyers]

Right now we use a spin representation where we map spins onto electron pairs. This is convenient since it maps spin-models onto seniority-zero Hamiltonians/wavefunctions. It is not convenient in the sense that the $\hat{S}_x$ and $\hat{S}_y$ operators break particle-number symmetry, so a general XYZ Heisenberg model is inaccessible.

There is an alternative approach. We can directly model the spin on each orbital as $S_k^+ = a_{k\alpha}^\dagger a_{k\beta}$ converting a down-spin electron in spatial orbital $k$ to an up-spin electron in spatial orbital $k$ and annhilating the wavefunction if the orbital is empty or has a up-spin electron already in it. Similarly, $S_k^- = a_{k\beta}^\dagger a_{k\alpha}$ changes an up-spin electron in orbital $k$ to a down-spin electron, and otherwise gives zero. $S^Z_k=\tfrac{1}{2}\left(a_{k\alpha}^\dagger a_{k\alpha}-a_{k\beta}^\dagger a_{k\beta} \right)$ gives $\pm \tfrac{1}{2}$ depending on whether the $k$-th orbital has a spin-up or spin-down electron.

This is a legitimate representation of the spin algebra because

$$ [S_k^+ , S_l^-] = 2\delta_{kl} S_k^Z $$

$$ [S_k^Z, S_l^{\pm}] = \pm \delta_{kl} S_k^{\pm} $$

The interesting state then is the maximum seniority state where every spatial orbital is singly-occupied. However, it seems that now a fully general XYZ model is allowable, because terms like $S_k^+ \pm S_l^-$ are now allowed. The key symmetry to maintain now is the maximum-seniority-sector, noting that we need to keep track of cases with different numbers of $N_\alpha$ and $N_{\beta}$ electrons (so when it comes time to solve the Schrödinger equation, it may be easiest to support this in the generalized framework).