Universal Subspaces for Local Unitary Groups of Fermionic Systems

Let $\mathcal{V}={\wedge }^{N}V$ be the $N$-fermion Hilbert space with $M$-dimensional single particle space $V$ and $2N\le M$. We refer to the unitary group $G$ of $V$ as the local unitary (LU) group. We fix an orthonormal (o.n.) basis $\mathrm{\mid }{v}_1⟩,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }{v}_M⟩$ of $V$. Then the Slater determinants $e_{i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_N}:=\mathrm{\mid }{v}_{i_1}\text{we}{v}_{i_2}\text{we}\mathrm{.}\mathrm{.}\mathrm{.}\text{we}{v}_{i_N}⟩$ with $i_1<\mathrm{.}\mathrm{.}\mathrm{.}<i_N$ form an o.n. basis of $\text{cV}$. Let $\text{cS}\subseteq \text{cV}$ be the subspace spanned by all $e_{i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_N}$ such that the set $\{i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_N\}$ contains no pair $\{2k-1,2k\}$, $k$ an integer. We say that the $\mathrm{\mid }\psi ⟩\in \text{cS}$ are single occupancy states (with respect to the basis $\mathrm{\mid }{v}_1⟩,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }{v}_M⟩$). We prove that for N=3 the subspace $\text{cS}$ is universal, i.e., each $G$-orbit in $\text{cV}$ meets $\text{cS}$, and that this is false for N>3. If $M$ is even, the well known BCS states are not LU-equivalent to any single occupancy state. Our main result is that for N=3 and $M$ even there is a universal subspace $\text{cW}\subseteq \text{cS}$ spanned by $M(M-1)(M-5)\mathrm{/}6$ states $e_{i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_N}$. Moreover the number $M(M-1)(M-5)\mathrm{/}6$ is minimal.