R\'enyi entropies of the free Fermi gas in multi-dimensional space at high temperature

Abstract

We study the local and (bipartite) entanglement R\'enyi entropies of the free Fermi gas in multi-dimensional Euclidean space $\mathbb{R}^d$ in thermal equilibrium. We prove positivity of the entanglement entropies with R\'enyi index $\gamma\leq1$ for all temperatures $T>0$. Furthermore, for general $\gamma>0$ we establish the asymptotics of the entropies for large $T$ and large scaling parameter $\alpha>0$ for two different regimes $-$ for fixed chemical potential $\mu\in\mathbb{R}$ and also for fixed particle density $\rho>0$. In particular, we thereby provide the last remaining building block for a complete proof of our low- and high-temperature results presented (for $\gamma=1$) in J. Phys. A: Math. Theor. $\textbf{49}$, 30LT04 (2016) [Corrigendum: $\textbf{50}$, 129501 (2017)], but being supported there only by the basic proof ideas.