%0 Journal Article
%J Communications in Mathematical Physics
%D 2015
%T Universal Subspaces for Local Unitary Groups of Fermionic Systems
%A Lin Chen
%A Jianxin Chen
%A Dragomir Z. Djokovic
%A Bei Zeng
%X 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 $\ket{v_1},...,\ket{v_M}$ of $V$. Then the Slater determinants $e_{i_1,...,i_N}:= \ket{v_{i_1}\we v_{i_2}\we...\we v_{i_N}}$ with $i_1<...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 $\cW\subseteq\cS$ spanned by $M(M-1)(M-5)/6$ states $e_{i_1,...,i_N}$. Moreover the number $M(M-1)(M-5)/6$ is minimal.
%B Communications in Mathematical Physics
%V 333
%P 541 - 563
%8 2014/10/10
%G eng
%U http://arxiv.org/abs/1301.3421v2
%N 2
%! Commun. Math. Phys.
%R 10.1007/s00220-014-2187-6
%0 Journal Article
%J Journal of Mathematical Physics
%D 2014
%T Unextendible Product Basis for Fermionic Systems
%A Jianxin Chen
%A Lin Chen
%A Bei Zeng
%X We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace $\wedge^ N \bC^M$. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space $\wedge^ 2 \bC^M$ of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example of complex FUPB of $N=2,M=4$ with minimum cardinality $5$. In contrast, we show that a real FUPB does not exist for $N=2,M=4$ . Finally we provide a systematic construction for FUPBs of higher dimensions using FUPBs and UPBs of lower dimensions.
%B Journal of Mathematical Physics
%V 55
%P 082207
%8 2014/01/01
%G eng
%U http://arxiv.org/abs/1312.4218v1
%N 8
%! J. Math. Phys.
%R 10.1063/1.4893358