Classifying fermionic states via many-body correlation measures

Understanding the structure of quantum correlations in a many-body system is key to its computational treatment. For fermionic systems, correlations can be defined as deviations from Slater determinant states. The link between fermionic correlations and efficient computational physics methods is actively studied but remains ambiguous. We make progress in establishing this connection mathematically. In particular, we find a rigorous classification of states relative to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x03C9;</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>, a function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-fermion reduced density matrix that we call twisted purity. A condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x03C9;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> for a given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> puts the state in a class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> of correlated states. Sets <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> are nested in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>, and Slater determinants correspond to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>. Classes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:math> are shown to be physically relevant, as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x03C9;</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> vanishes or nearly vanishes for truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, we give an explicit ansatz with a polynomial number of parameters that covers all states in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.