Entanglement entropies and fermion signs of critical metals

TL;DR

This paper investigates the relationship between fermion sign structures and entanglement entropy in critical metals, revealing a crossover from volume law to Fermi-liquid behavior influenced by fractal nodal surfaces.

Contribution

It introduces a wavefunction ansatz with long-range backflow correlations to model critical metals and analyzes how fractal fermion sign structures affect entanglement scaling.

Findings

Fermion sign structures exhibit fractal behavior with a diverging length scale at criticality.

Entanglement entropy shows a crossover from volume law to Fermi-liquid scaling.

Volume scaling of entanglement is robust against variations in backflow parameters.

Abstract

The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron systems. Only recently, it has been suggested that fermion signs are fundamental for the universal behavior of critical metallic systems and crucially enhance their degree of quantum entanglement. In this work we explore potential connections between emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis is based on a wavefunction ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants. Such wavefunctions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces -- a representation of the fermion sign structure in many-particle configurations space -- show fractal behavior up to a length scale $\xi$ that diverges at a critical backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on $\xi$, the number of fermions and the exponent of the backflow. For the same wavefunctions we numerically calculate the second R\'enyi entanglement entropy $S_2$. Our results show a cross-over from volume scaling, $S_2\sim \ell^\theta$ ($\theta=2$ in $d=2$ dimensions), to the characteristic Fermi-liquid behavior $S_2\sim \ell\ln \ell$ on scales larger than $\xi$. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.