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arXiv:0706.1855 · Journal of Physics A Mathematical and Theoretical

ConnectingN-representability to Weyl's problem: the one-particle density matrix forN= 3 andR= 6

Mary Beth Ruskai

Published Oct 23, 2007• 43 citations
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Abstract

An analytic proof of the necessity of the Borland–Dennis conditions for 3-representability of a one-particle density matrix with rank 6 is given. This may shed some light on Klyachko's recent use of Schubert calculus to find general conditions for N -representability.
DOI
10.1088/1751-8113/40/45/f01

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The NNN-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The NNN-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric NNN-particle functions, is considered. New necessary conditions are given for both NNN-completeness and NNN-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.
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For a quantum-mechanical system of NNN identical fermions, the NNN-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric NNN-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, NNN, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be NNN-representable is that its natural spin-orbital occupation numbers be equal in pairs.
In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8). Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.