Rényi Entropies of the Free Fermi Gas in Multi-Dimensional Space at High Temperature

Abstract

We study the local and (bipartite) entanglement Rényi entropies of the free Fermi gas in multi-dimensional Euclidean space in thermal equilibrium. We prove positivity of the entanglement entropies with Rényi index γ ≤ 1 for all temperatures T > 0. Furthermore, for general γ > 0 we establish the asymptotics of the entropies for large T and large scaling parameter α > 0 for two different regimes—for fixed chemical potential and also for fixed particle density ρ > 0. In particular, we thereby provide the last remaining building block for a complete proof of our low- and high-temperature results presented (for γ = 1) in J. Phys. A: Math. Theor. 49, 30LT04 (2016); [Corrigendum. 50, 129501 (2017)], but being supported there only by the basic proof ideas.

In memory of Harold Widom (1932–2021)

Appendix: The Fermi Symbol for Fixed ρ and Large T

Our main aim is to derive formula (61) for a Hamiltonian h as specified in Sect. 5.1. After a change of variables and replacing μ with μ _ρ(T) formula (11) for the mean particle density takes the form

Since $T^{-1} h(T^{\frac {1}{2m}}\boldsymbol \xi )\to h_\infty (\boldsymbol \xi )$ as T →∞, for each ξ ≠ 0, the condition (12) requires that $\exp \big (-\mu _\rho (T)/T\big )\to \infty$. Consequently,

$$\displaystyle \begin{aligned} \varrho\big(T, \mu_{\rho}(T)\big) = \varkappa T^{\frac{d}{2m}} \exp\big(\mu_\rho(T)/T\big) \big(1+o(1)\big)\,, \end{aligned} $$

where ϰ is defined in (62). Using (12), this leads to (61).

As explained in the Introduction, the high-temperature limit under the condition (12) corresponds to the Maxwell–Boltzmann gas limit. This fact can be conveniently restated in terms of the so-called fugacity as follows. By (43) the integrated density of states $\mathcal N(T)$ of (13), satisfies for large T the relation $\mathcal N(T)\asymp T^{\frac {d}{2m}}$. So (61) implies $z_\rho (T)\asymp \rho \, \mathcal N(T)^{-1}\to 0$.