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 ${\mathbb{R}}^d$ in thermal equilibrium. We prove positivity of the entanglement entropies with Rényi index $\gamma \le 1$ 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. $\text{49}$, 30LT04 (2016) [Corrigendum: $\text{50}$, 129501 (2017)], but being supported there only by the basic proof ideas.

arXiv commentchanges relative to v2: minor editing

References in corpus (8)

• Quantum entanglement
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Jensen's Operator Inequality
• Scaling of Rényi entanglement entropies of the free Fermi-gas ground state: a rigorous proof
• Wiener-Hopf operators in higher dimensions: the Widom conjecture for piece-wise smooth domains
• Functions of self-adjoint operators in ideals of compact operators
• Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature
• Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy

Cited by in corpus (2)

• Enhanced area law in the Widom-Sobolev formula for the free Dirac operator in arbitrary dimension
• An enhanced term in the Szegő-type asymptotics for the free massless Dirac operator