# Diagonalizing the BCS model as a Gaudin model

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

• 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? Sep 2 at 13:23
• @BySymmetry they are more general, like the JW transform. Sep 2 at 14:12