Classifying fermionic states via many-body correlation measures

TL;DR

This paper introduces a rigorous classification of fermionic many-body states based on $k$-fermion correlations, using a measure called twisted purity, and explores its implications for computational physics methods.

Contribution

It establishes a mathematical framework linking fermionic correlations to computationally relevant state classes and provides explicit ansatzes for these classes.

Findings

States in $G_k$ have vanishing or nearly vanishing twisted purity $\,\, ext{for certain physical states}$

Explicit polynomial-parameter ansatzes are constructed for states in $G_k$

Connections to coupled-cluster wavefunctions are discussed

Abstract

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 $k$-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure $\omega_k$, a function of $k$-fermion reduced density matrix that we call twisted purity. A condition $\omega_k=0$ for a given $k$ puts the state in a class $G_k$ of correlated states. Sets $G_k$ are nested in $k$, and Slater determinants correspond to $k = 1$. Classes $G_{k=O(1)}$ are shown to be physically relevant, as $\omega_k$ 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 $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in $G_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.