Solving one-body ensemble N-representability problems with spin

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> of orbitals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>&amp;#x03C6;</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> according to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> <mml:mo>&amp;#x2264;</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>&amp;#x2264;</mml:mo> <mml:mn>2</mml:mn> </mml:math> . In this work, we first refine the underlying one-body <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold-italic">w</mml:mi> </mml:math> of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">&amp;#x03A3;</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">w</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&amp;#x2282;</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . These constraints are independent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> and the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> of orbitals, while their dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> </mml:math> is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.