Universal Subspaces for Local Unitary Groups of Fermionic Systems

Let (formula presented.) 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 |v₁⟩,..,|v_M〉 of V. Then the Slater determinants (formula presented.) form an o.n. basis of (formula presented.). Let (formula presented.) be the subspace spanned by all (formula presented.) contains no pair {2k−1,2k}, k an integer. We say that the (formula presented.) are single occupancy states (with respect to the basis |v₁⟩,..,|v_M⟩). We prove that for N = 3 the subspace S is universal, i.e., each G-orbit in V meets 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 (formula presented.) spanned by M(M−1)(M−5)/6 states (formula presented.). Moreover, the number M(M−1)(M−5)/6 is minimal.