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I've been looking at the BCS model through a few papers (listed at the bottom). I'm having trouble with many things but I think I've boiled it down to a few:

-- In the first paper below (Ortiz), the BCS's eigenstates are described using su(2) operators $\tau^\pm_\mathbf{\ell}$ and $\tau^z_\mathbf{\ell}$ where $\ell$ goes from $1$ to $L$. In particular, looking at Equation (57), what is the action of $\tau^+_\mathbf{\ell}$ on state $|\nu\rangle$?

I've tried representing $\tau^+_\ell$ in the language of angular-momentum ladder operators, but that didn't work because occupation numbers are bounded by $0$ on the bottom whereas angular momentum $m_j$ are bounded by $-j$. Bosonic-style things didn't work well either.

-- Do I represent $|\nu\rangle$ as $|n_1,n_2,...n_L;\nu_1,\nu_2,...,\nu_L\rangle$ where $n_\ell$ represent the occupation of particle pairs and $\nu_\ell$ are the unpaired particles? Both papers' usage of $|\nu\rangle$ confuse me. All we are given is that \begin{equation} \tau^-_\ell|\nu\rangle=0 \quad\quad \tau^z_\ell|\nu\rangle=\left(\frac{1}{2}\nu_\ell-\frac{2\tau_\ell+1}{4}\right)|\nu\rangle \end{equation} and that \begin{equation} |n_1,n_2,...,n_L,\nu\rangle\sim(\tau^+_1)^{n_1}(\tau^+_2)^{n_2}...(\tau^+_L)^{n_L}|\nu\rangle. \end{equation}

-- My goal is to write some Mathematica code which performs the diagonalization method outlined in these papers. Basically I want to input $L$, $N$, the interaction strength for the BCS, and pick some $|\nu\rangle$ and then the code spits out an eigenstate. Once I understand the action of $\tau^\pm_\ell$ and $\tau^z_\ell$ and/or the support of the eigenstates it should click into place.

https://arxiv.org/abs/cond-mat/0407429

https://arxiv.org/abs/nucl-th/0405011

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  • $\begingroup$ Are the $\tau$ operators representing the physical spin of an electron on site $l$, or are they representing the occupation number of a state more generally, for example via a Jordan-Wigner transform? $\endgroup$ Sep 2 at 13:23
  • $\begingroup$ @BySymmetry they are more general, like the JW transform. $\endgroup$ Sep 2 at 14:12

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