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 ≤ M. We refer to the unitary group G of V as the local unitary (LU) group. We fix an orthonormal (o.n.) basis |v 1⟩,...,|v M 〉 of V. Then the Slater determinants $${e_{i_1,\cdots,i_N}:= |{v_{i_1}\wedge v_{i_2} \wedge\cdots\wedge v_{i_N}}\rangle}$$ with i 1 < ... < i N form an o.n. basis of $${\mathcal{V}}$$ . Let $${\mathcal{S}\subseteq\mathcal{V}}$$ be the subspace spanned by all $${e_{i_1,\cdots,i_N}}$$ such that the set {i 1,...,i N } contains no pair {2k−1,2k}, k an integer. We say that the $${|{\psi}\rangle \in\mathcal{S}}$$ are single occupancy states (with respect to the basis |v 1⟩,...,|v M ⟩). We prove that for N = 3 the subspace $${\mathcal{S}}$$ is universal, i.e., each G-orbit in $${\mathcal{V}}$$ meets $${\mathcal{S}}$$ , 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 $${\mathcal{W}\subseteq\mathcal{S}}$$ spanned by M(M−1)(M−5)/6 states $${e_{i_1,\ldots,i_N}}$$ . Moreover, the number M(M−1)(M−5)/6 is minimal.

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