Stability of the Enhanced Area Law of the Entanglement Entropy
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nnales Henri Poincar´ e
Stability of the Enhanced Area Law of the Entanglement Entropy Peter M¨ uller
and Ruth Schulte
Abstract. We consider a multi-dimensional continuum Schr¨ odinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schr¨ odinger operators.
1. Introduction and Result Entanglement properties of the ground state of quasi-free Fermi gases have received considerable attention over the last two decades, see, for example, [1, 2,8–10,15,16,18,20–22,24,25,27,37]. Here, entanglement is understood with respect to a spatial bipartition of the system into a subsystem of linear size proportional to L and the complement. Entanglement entropies are a common measure for entanglement. Often, the von Neumann entropy of the reduced ground state of the Fermi gas is considered. Its investigations give rise to nontrivial mathematical questions and to answers that are of physical relevance. This is true even for the simplest case of a quasi-free Fermi gas, namely the free Fermi gas with (single-particle) Hamiltonian H0 := −Δ given by the Laplacian in d ∈ N space dimensions. Its entanglement entropy was suggested [14–16,37] to obey a logarithmically enhanced area law, (1.1) SE (H0 , ΛL ) = Σ0 Ld−1 ln L + o Ld−1 ln L , Ruth Schulte was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868.
3640
P. M¨ uller and R. Schulte
Ann. Henri Poincar´e
as L → ∞. Here, E > 0 stands for the Fermi energy, which characterises the ground state, and ΛL := L · Λ is the scaled version of some “nice” bounded subset Λ ⊂ Rd , which is specified in Assumption 1.1(i). The leading-order coefficient E (d−1)/2 |∂Λ| , Σ0 ≡ Σ0 (Λ, E) := (1.2) 3 · 2d π (d−1)/2 Γ (d + 1)/2 where |∂Λ| denotes the surface area of the boundary ∂Λ of Λ, was expected [14– 16] to be determined by Widom’s conjecture [35]. This was finally proved in [20] based on celebrated works by Sobolev [32,33]. The occurrence of the logarithm ln L in the leading term of (1.1) is attributed to the delocalisation or transport properties of the Laplacian dynamics. It leads to long-range correlations in the ground state of the Fermi gas across the surface of the subsystem in ΛL . If a periodic potential is added to H0 , and the Fermi energy falls into a spectral band, the logarithmically enhanced area law (1.1) is still valid, as was proven in [27] for d = 1. odinger operator H with a mobility If H0 is replaced by another Schr¨ gap in the spectrum and if the Fermi energy falls into the mobility gap, then the ln L-factor is expected to be
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