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)
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Appendix: The Fermi Symbol for Fixed ρ and Large T
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,
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 \(z_\rho (T)\asymp \rho \, \mathcal N(T)^{-1}\to 0\).
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Leschke, H., Sobolev, A.V., Spitzer, W. (2022). Rényi Entropies of the Free Fermi Gas in Multi-Dimensional Space at High Temperature. In: Basor, E., Böttcher, A., Ehrhardt, T., Tracy, C.A. (eds) Toeplitz Operators and Random Matrices. Operator Theory: Advances and Applications, vol 289. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-13851-5_21
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