Measures of complexity and entanglement in fermionic many-body systems

TL;DR

This paper explores measures of complexity and entanglement in many-fermion systems, proposing intrinsic and unique metrics like natural orbitals and entanglement spectrum, demonstrated through nuclear fission simulations.

Contribution

It introduces intrinsic complexity measures based on natural orbitals and entanglement spectrum, addressing ambiguities in traditional Slater determinant-based complexity.

Findings

Natural orbitals provide a unique characterization of wave function complexity.

Orbital entanglement entropy vanishes for Slater determinants.

The entanglement spectrum offers a comprehensive complexity measure.

Abstract

There is no unique and widely accepted definition of the complexity measure (CM) of a many-fermion wave function in the presence of interactions. The simplest many-fermion wave function is a Slater determinant. In shell-model or configuration interaction (CI) and other related methods, the state is represented as a superposition of a large number of Slater determinants, which in case of CI calculations reaches about 20 billion terms. Although in practice this number has been used as a CM for decades, it is ill defined: it is not unique, and it depends on the particular type and the number of single-particle wave functions used to construct the Slater determinants. The canonical wave functions/natural orbitals and their corresponding occupation probabilities are intrinsic properties of any many-body wave function, irrespective of the representation, and they provide a unique solution to characterize the CM. The non-negative orbital entanglement entropy, which vanishes for a Slater determinant, provides the simplest CM, while a more complete measure of complexity is the entanglement spectrum. We illustrate these aspects in the case of a complex non-equilibrium time-dependent process, induced nuclear fission described within a real-time Density Functional Theory framework extended to superfluid systems, which can describe simultaneously the long-range and the short range correlations between fermions.