Universal Subspaces for Local Unitary Groups of Fermionic Systems

Let V = boolean AND V-N 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(i1), ... , i(N) := |v(i1). boolean AND v(i2) boolean AND ... boolean AND v(iN) > with i(1) < ... < i(N) form an o.n. basis of V. Let S subset of V be the subspace spanned by all e(i1), ... , i(N) such that the set {i(1), ... , i(N)} contains no pair {2k - 1, 2k}, k an integer. We say that the |psi > is an element of S are single occupancy states (with respect to the basis |v(1)>, ... , |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 W subset of S spanned by M(M - 1)(M - 5)/6 states e(i1), ..., i(N). Moreover, the number M(M - 1)(M - 5)/6 is minimal.