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Universal Subspaces for Local Unitary Groups of Fermionic Systems

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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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Correspondence to Lin Chen.

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Communicated by A. Winter

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Chen, L., Chen, J., Đoković, D.Ž. et al. Universal Subspaces for Local Unitary Groups of Fermionic Systems. Commun. Math. Phys. 333, 541–563 (2015). https://doi.org/10.1007/s00220-014-2187-6

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