Stability of the enhanced area law of the entanglement entropy

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 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.


Introduction and Result
Entanglement properties of the ground state of quasi-free Fermi gases have received considerable attention over the last two decades, see e.g. [BR04, KM04, Wol06, GK06, HLS11, LSS14, PS14, ARS15, EPS17, LSS17, ARNSS17, PS18b, MPS19,LSS20]. 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 non-trivial 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 H 0 := −∆ given by the Laplacian in d ∈ N space dimensions. Its entanglement entropy was suggested [Wol06,GK06,Gio06,HLS11] to obey a logarithmically enhanced area law, was expected [GK06,Gio06,HLS11] to be determined by Widom's conjecture [Wid82]. This was finally proved in [LSS14] based on celebrated works by Sobolev [Sob13,Sob15]. 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 Λ L . If a periodic potential is added to H 0 , and the Fermi energy falls into a spectral band, the logarithmically enhanced area law (1.1) is still valid, as was proven in [PS18b] for d = 1.
If H 0 is replaced by another Schrödinger operator H with a mobility gap in the spectrum and if the Fermi energy falls into the mobility gap, then the ln L-factor is expected to be absent in the leading asymptotic term of the entanglement entropy. Such a phenomenon is referred to as an area law, namely S E (H, Λ L ) ∼ L d−1 as L → ∞. It was first observed by Bekenstein [Bek73,Bek04] in a toy model for the Hawking entropy of black holes. An area law also holds if H models a particle in a constant magnetic field [LSS20]. Area laws are proven to occur for random Schrödinger operators and Fermi energies in the region of dynamical localisation [PS14,EPS17,PS18a]. The proofs rely on the exponential decay in space of the Fermi projection for E in the region of complete localisation. It should be pointed out that spectral localisation alone is not sufficient for the validity of an area law. This has been recently demonstrated [MPS19] for the random dimer model if the Fermi energy coincides with one of the critical energies where the localisation length diverges and dynamical delocalisation takes over.
Due to the complexity of the problem, there does not exist a mathematical approach which allows to determine the leading behaviour of the entanglement entropy for general Schrödinger operators H. All that is known is what happens for the examples discussed above. The experts in the field have conjectured for a decade that, given H with a "reasonable" potential, a possibly occurring enhancement to the area law for S E (H, Λ L ) should not be stronger than logarithmic. Even though no counterexamples are known so far, proving the conjecture turned out to be a very difficult task which has not been solved yet. As an aside, we mention that for interacting quantum systems, stronger enhancements to area laws than logarithmic are known in peculiar cases. In fact, spin chains (d = 1) can be designed in such a way as to realise any growth rate up to L [RRLS14,MS16].
In this paper we undertake a first step towards a proof of the conjecture. We establish an upper bound on the entanglement entropy corresponding to H = −∆ + V which grows like L d−1 ln L as L → ∞, provided the potential V is bounded and has compact support. Compactness of the support is the crucial restriction of our result. It could be relaxed to having a sufficiently fast decay at infinity, but we have chosen not to focus on this for reasons of simplicity. The main technical input in our analysis is a limiting absorption principle for H. Since H has absolutely continuous spectrum filling the non-negative real half-line, one expects S E (H, Λ L ) to obey an enhanced area law for Fermi energies E > 0. Therefore, a corresponding lower bound, which grows also like L d−1 ln L as L → ∞, is of interest, too. These findings are summarised in Theorem 1.1, which is our main result. The proof of the upper bound is much more involved than that of the lower bound. Both bounds require the representation of the Fermi projection as a Riesz projection with the integration contour cutting through the continuous spectrum. Such a representation may be of independent interest. We prove it in the Appendix in a more general setting for operators for which a limiting absorption principle holds.
Let H := −∆ + V be a densely defined Schrödinger operator in the Hilbert space L 2 (R d ) with bounded potential V ∈ L ∞ (R d ). According to [Kli06] there exists a trace formula for the entanglement entropy which we take as our definition (1.3) Here, Λ ⊂ R d is a Borel set, we write 1 A for the indicator function of a set A and, in abuse of notation, 1 <E := 1 ]−∞,E[ for the Fermi function with Fermi energy E ∈ R. We also introduced the entanglement-entropy function h : and use the convention 0 log 2 0 := 0 for the binary logarithm. We recall that The main result of this paper is summarised in (1.5) Remarks 1.2.
(i) The constant Σ l can be expressed in terms of the coefficient Σ 0 in the leading term of the unperturbed entanglement entropy S E (H 0 , Λ L ) for large L, cf.
(1.7) In particular, this constant is independent of V . The numerical prefactor in (1.7) can be improved by using the alternative approach described in Remark 2.5. In d = 1 dimension, however, we only obtain a constant Σ u which also depends on V , because there is an additional contribution from (2.68).
(iii) At negative energies there is at most discrete spectrum of H. Thus, if E < 0 the Fermi function can be smoothed out without changing the operator 1 <E (H). Therefore, the operator kernel of 1 <E (H) has fast polynomial decay, and S E (H, Λ L ) = O(L d−1 ) follows as in [PS14,EPS17]. In other words, the growth of the entanglement entropy is at most an area law. The same holds at E = 0 because eigenvalues cannot accumulate from below at 0 due to the boundedness of V and its compact support.

Proof of Theorem 1.1
We prove the upper bound of Theorem 1.1 in Section 2.2 and the lower bound in Section 2.3. Section 2.1 contains results needed for both bounds.
where |A| 2 := A * A for any bounded operator A, and the superscript c indicates the complement of a set. This observation leads us to consider von Neumann-Schatten norms of operator differences which is done in Lemma 2.3 and Lemma 2.4. Lemma 2.3 allows to deduce the lower bound in Theorem 1.1, whereas the upper bound requires more work due to the presence of the additional logarithm. Lemma 2.6 will tackle this issue.
In order to show the crucial Lemma 2.3, we need two preparatory results. The first one is about the decay in space of the free resolvent in Lemma 2.1.
Then, there exists a constant C 1 ≡ C 1 (d, V ) > 0 such that for any z ∈ C \ R and any Here, · p denotes the von Neumann-Schatten norm for p ∈ [1, ∞[.
Since the Hilbert-Schmidt norm of an operator can be computed in terms of the integral kernel, we get For every n ∈ Z d ∩ Λ c ℓ 0 , every x ∈ Γ n and every ξ ∈ suppV ⊆ Λ ℓ 0 , we infer that |x − ξ| R(d)/|z| 1/2 . Therefore the Green's-function estimate (2.4) yields This implies the lemma.
As a second preparatory result for one of our central bounds we require We fix an energy E > 0 and consider two compact subsets Γ, Γ ′ ⊂ R d . Then we have the representation (2.10) The right-hand side of (2.10) exists as a Bochner integral with respect to the operator norm, and the integration contour γ is a closed curve in the complex plane C which traces the boundary of the rectangle z ∈ C : | Im z| E, Re z ∈ [−1 + inf σ(H), E] once in the counter-clockwise direction.
PROOF. The lemma follows from the corresponding abstract result in Theorem A.1 in the appendix. Indeed, according to [JM17], both H and H 0 fulfil a limiting absorption principle at any E > 0, with X being the position operator, · · · := 1 + | · · · | 2 the Japanese bracket and Π c (H (0) ) the projection onto the continuous spectral subspace of because the potential V is bounded and compactly supported [RS78, Cor. on p. 230].
The statement of the next lemma is a crucial estimate that will be needed for both the upper and the lower bound in Theorem 1.1.
(2.12) PROOF. We fix E > 0. To estimate the difference between the perturbed and the unperturbed Fermi projections we express them in terms of a contour integral as stated in Lemma 2.2. We set where ℓ 0 is defined in Lemma 2.1 and img(γ) denotes the image of the curve γ in Lemma 2.2. We obtain for all m, 14) The Bochner integral exists even with respect to the Hilbert-Schmidt norm, as will follow from the estimates (2.18) and (2.24) below. We point out that (2.24) relies again on the limiting absorption principle (2.11).
In order to estimate the integral in (2.14) we apply the resolvent identity twice to the integrand. The integrand then reads This implies the Hilbert-Schmidt-norm estimate Lemma 2.1 already provides bounds for the first and third factor on the right-hand side of (2.16). To estimate the second factor we employ two different methods, depending on the location of z on the contour. Therefore we split the curve γ into two parts. We denote by γ 1 the right vertical part of γ with image img(γ 1 ) = z ∈ C : Re z = E, | Im z| min{E, 1} . The remaining part of the curve γ is denoted by γ 2 . Let us first consider the curve γ 2 . We observe dist z, σ(H (0) ) min{1, E} for all z ∈ img(γ 2 ).
(2.17) Therefore, the middle factor in the second line of (2.16) is bounded from above by Hence, according to Lemma 2.1 we estimate (2.16) by We now turn our attention to γ 1 , the part of the contour that intersects the continuous spectrum of H. Writing 1 = Π pp (H) + Π c (H) and recalling σ pp (H) ⊂ ] − ∞, 0], see the end of the proof of Lemma 2.2, we infer for every z ∈ img(γ 1 ) \ R. The second term on the right-hand side admits the uniform upper bound (2.21) Here, we used the compact support of V and introduced the abbreviation C LA ≡ C LA (d, E, V ) < ∞ for the supremum on the left-hand side of (2.21). It is finite because of the limiting absorption principle (2.11) for H.
In addition, we need a lower bound for the decay rate of the exponential in (2.6) along the curve γ 1 . We write img(γ 1 ) ∋ z = E + iη with |η| min{1, E}. Then, with α : [0, ∞[ → [0, 1], x → sin 1 2 arctan x . We note that sin y y(1 − y 2 /6) for all y 0, arctan x π/2 and arctan x x/2 for all x ∈ [0, 1]. Therefore, we infer the existence of a constant ζ 1 ≡ ζ 1 (E) > 0 such that Here, we introduced the constant (2.25) We are now able to estimate the contour integral in (2.14) with the help of the bounds (2.18) and (2.24) (2.27) In order to prove the lemma for any L ∈ N, we first consider the case of L L 0 . In this case we have (2.28) Following [Sim82, Thm. B.9.2 and its proof], we infer the existence of a constant C S ≡ C S (d, V, E) > 1 such that 1 <E H (0) 1 Γn 1 C S (2.29) holds uniformly in n ∈ Z d . By applying the binomial inequality (a + b) 2 2a 2 + 2b 2 for a, b ∈ R and the inequality A 2 2 A 1 for any trace-class operator A with A 1, we estimate the right-hand side of (2.28) by (2.30) In the other case of L > L 0 we partition Λ L and Λ c L into a disjoint union of smaller cubes (2.32) The first term on the right hand side of (2.32) is estimated by (2.28) and (2.30). To bound the double sum in (2.32), we use (2.26) together with the binomial inequality to obtain (|n| + |m|) 2 . (2.33) By definition of L 0 , we have |l| |u| − √ d |u|/2 for every l ∈ Ξ i ∪ Ξ a and every u ∈ Γ l ⊆ Λ c L 0 . We therefore estimate (2.33) in terms of the integrals (2.34) We estimate (2.34) using spherical coordinates and the inclusions Λ c L ⊆ B c L and Λ L \  . (2.38) The estimate (2.29) implies that the operator is trace class for all L ∈ N with norm A L 1 2(2L) d C S . Moreover, A L 2 2 C 2 2 for all L ∈ N by Lemma 2.3. This proves the claim with   . We point out that the right-hand side of (2.45) does not exceed the upper bound e −1/2 because of A − B A + B (2/3) e −1/2 . Together with the monotonicity of f , we conclude from (2.44) that tr{f (|A|)} 2 n∈N f a n (B) + a n (A − B) . (2.46) Next, we claim that for all x, y 0 with x + y < 1. The first estimate follows from the binomial inequality together with − log 2 [(x + y) 2 ] 0 for x + y < 1, the second estimate from (x + y) 2 x 2 , respectively (x + y) 2 y 2 , and the fact that − log 2 is monotonously decreasing. Combining (2.46) and (2.47), we arrive at tr{f (|A|)} 4 n∈N f a n (B) + f a n (A − B) . ( where f was defined in Lemma 2.6. In order to apply Lemma 2.6, we will decompose the compact operator 1 Λ c L 1 <E H (0) 1 Λ L into a part bounded by e −1/2 /3 in norm and a finite-rank operator. To this end, we introduce N (0) ≡ N (0) (d, V, E, L) := min n ∈ N : a n 1 Λ c L 1 <E (H (0) )1 Λ L e −1/2 /3 − 1, (2.50) the number of singular values of 1 Λ c L 1 <E H (0) 1 Λ L which are larger than e −1/2 /3. We define F (0) as the contribution from the first N (0) singular values in the singular-value decomposition of 1 Λ c L 1 <E H (0) 1 Λ L , whence rank(F (0) ) = N (0) and F (0) 1. The remainder fulfils Q (0) e −1/2 /3 by definition of N (0) . We note the upper bound (2.52) Using Lemma 2.3, we further estimate N in terms of unperturbed quantities (2.53) The identity (2.3) and the lower bound in (A.10) imply 1 Λ c L 1 <E (H 0 )1 Λ L 2 2 S E (H 0 , Λ L ) so that we obtain for later usage. We deduce from (2.45) and rank(F ) = N that for all n ∈ N a n+N (Q + F ) a n (Q) + a N +1 (F ) = a n (Q) e −1/2 /3. which implies n∈N f a n (δQ + δF ) C 3 e 2d ln L. (2.68) The entanglement entropy of a free Fermi gas exhibits an enhanced area law, (2.69) The second term on the right-hand side is uniformly bounded in L according to Lemma 2.3. For the first term, it was shown in [LSS14, Eq. (7)] that the leading behaviour of the asymptotic expansion in L is of order L d−1 ln L. Hence, Finally, Eqs.

Appendix A. Auxiliary results
The following representation (A.2) of the Fermi projection in terms of a Riesz projection with the integration contour cutting through the continuous spectrum may be of independent interest.
Here, Π c (K) denotes the projection onto the continuous spectral subspace of K. Let A 1 , A 2 be two bounded operators on H such that A 1 B −1 < ∞ and B −1 A 2 < ∞.
Finally, we assume that there are no eigenvalues of K near E, i.e. dist σ pp (K), E > 0. Then we have the representation The right-hand side of (A.2) exists as a Bochner integral with respect to the operator norm · · · , and the integration contour γ is a closed curve in in the complex plane C which, for s > 0, traces the boundary of the rectangle z ∈ C : | Im z| s, Re z ∈ [−1 + inf σ(K), E] once in the counter-clockwise direction.
PROOF. Let ε > 0 and let γ ε be the curve γ without the vertical line segment from E − iε to E + iε. Since (K − z) −1 is uniformly bounded for z in the image of γ ε , it suffices to verify that in order to show the existence of the right-hand side of (A.2) as a Bochner integral with respect to the operator norm. But uniformly in η ∈ [−ε, ε], and the estimate (A.3) holds. It remains to prove the equality in (A.2). Let ϕ, ψ ∈ H. Since the contour integral along γ exists in the Bochner sense with respect to the operator norm, we equate ϕ, where we introduced the complex spectral measure µ ϕ,ψ := ϕ, 1 • (K)ψ of K and used Fubini in the last step. On the other hand, we apply the residue theorem to conclude which is justified because E is not an eigenvalue of K. The right-hand side of (A.6) equals (A.7) The explicit computation, using symmetry, holds for every real λ = E. Therefore, dominated convergence implies that the second limit in (A.7) vanishes. Here, we used again that E is not an eigenvalue of K. Since ϕ and ψ are arbitrary, the theorem follows from (A.5) to (A.7).
Remark A.2. Theorem A.1 readily generalises from Fermi projections to spectral projections of more general intervals.