Asymptotic growth of the local ground-state entropy of the ideal Fermi gas in a constant magnetic field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $\mathbb R^2$ perpendicular to an external constant magnetic field of strength $B>0$. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $\mu\ge B$ (in suitable physical units). For this (pure) state we define its local entropy $S(\Lambda)$ associated with a bounded (sub)region $\Lambda\subset \mathbb R^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $\Lambda$ of finite area $|\Lambda|$. In this setting we prove that the leading asymptotic growth of $S(L\Lambda)$, as the dimensionless scaling parameter $L>0$ tends to infinity, has the form $L\sqrt{B}|\partial\Lambda|$ up to a precisely given (positive multiplicative) coefficient which is independent of $\Lambda$ and dependent on $B$ and $\mu$ only through the integer part of $(\mu/B-1)/2$. Here we have assumed the boundary curve $\partial\Lambda$ of $\Lambda$ to be sufficiently smooth which, in particular, ensures that its arc length $|\partial\Lambda|$ is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln(L)$ is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $\text L^2(\mathbb R^2)$ to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum R\'enyi entropies.


Introduction
Quantum correlations in many-particle ground states occur in a genuine and simple form for fermions without interactions between them. In this case all correlations are exclusively due to the Pauli-Fermi-Dirac statistics and are not affected by classical correlations. This certainly explains why the authors of many recent publications, devoted to the "trendy topic" of entanglement entropy, have considered ground states of free fermions in discrete or continuous position space. For these pure states its (bipartite spatial) entanglement entropy boils down to its local entropy associated with a bounded region Λ in the position space. An informal definition of the local entropy is given in the above abstract. For a formal definition, in the present context, see (4.2) (with α " 1) below. This local ground-state entropy may serve as a useful, but rough, single-number quantification of the correlations of all the particles in the region Λ with all those outside.
The local ground-state entropy is a complicated function(al) of Λ and difficult to study by analytic methods. Even without interactions, one can in general only hope for estimates and/or asymptotic results when the volume of the bounded region becomes large. As discovered by Gioev and Klich [8], one fascinating aspect of these type of asymptotic results is the connection to the quasi-classical evaluation of traces of (truncated) Wiener-Hopf operators (or Toeplitz matrices in the discrete, onedimensional, case), that is, to a conjecture of Harold Widom (respectively of Fisher and Hartwig). The "Widom conjecture" was finally proved by one of us in [29] and opened the gate to prove a conjecture by Gioev and Klich [8] about the precise asymptotic growth of the local ground-state entropy of free fermions in multi-dimensional Euclidean space, see [15].
Of course, it is physically relevant and mathematically interesting to determine such a precise asymptotics also for ground states of fermions subject to an external field or even with interactions between them. From a rigorous point of view, the latter seems currently to be out of reach. Concerning external scalar fields there are publications devoted to free fermions in a (random) potential [6,[19][20][21][22] or in a one-dimensional periodic potential [24]. As an aside, we mention that in the case of free fermions the large-scale behavior of the local entropy is not only known for the ground state, but also for the thermal equilibrium state at any temperature [16,17].
In the present paper, we (return to zero temperature and) consider the ground state of nonrelativistic, spinless, and electrically charged fermions in the Euclidean plane R 2 without interactions between them, but subject to an external magnetic field which is perpendicular to the plane and of constant strength B ą 0. This ground state became of interest in condensed-matter physics at first in the early 1930s for simplified explanations of the Landau diamagnetism and the De Haas-Van Alphen effect observed in metals, see [10,23]. The interest got revived and enhanced after the discovery of the (integrally) quantized Hall effect in certain quasi-two-dimensional semiconductor materials by Klaus von Klitzing in the year 1980, see [10,30]. To our knowledge, analytical contributions to the asymptotic growth of the local entropy of this ground state were made by Klich [13], by Rodríguez and Sierra [26,27], and recently by Charles and Estienne [4]. All these authors consider the case of the lowest Landau level only. In addition, [13,26,27] treat regions of simple geometric shape only. The important work of Rodríguez and Sierra [26] contains non-rigorous arguments, but their formula for the asymptotic coefficient M 0 ph 1 q (see (4.1) and (4.4) for the definition), confirmed in [4], has been a guide for us to arrive at the more general asymptotic coefficients presented here. After all, the simplicity of the coefficient M ℓ pf q of the sub-leading boundary-curve term in (2.13), see (2.9) and (2.10), is striking, given that it results from the 2m-fold integration in (3.20) for arbitrary exponent m ě 1.
By adapting Roccaforte's approach for translation invariant integral kernels [25] to those of the Landau-projection operators we extend results in [4,26] to rather general regions, to an arbitrary single Landau-level eigenspace, and even to the orthogonal sum of the first n`1 eigenspaces for arbitrary n ě 0. By the last extension we can allow for an arbitrary value of the chemical potential µ ě B (in suitable physical units) and, hence, for an arbitrary areal density of the particle number. Our proof consists of two basic steps. In the first step, we present the precise asymptotics of the trace of smooth functions of localized (or spatially truncated) Landau projections. By a suitable 2. Setting the stage and basic asymptotic results for smooth functions We denote the scalar products in the Euclidean plane R 2 and in the Hilbert space L 2 pDq of complex-valued, square-integrable functions on a Borel set D Ď R 2 by the same bracket x¨|¨y and use the same notation }¨} for the induced norms. Our convention is that the scalar product is anti-linear in the first and linear in the second argument.
Since in the ideal Fermi gas the indistinguishable (point) particles do not interact with each other, it is sufficient to consider the common Schrödinger operator for the kinetic energy of a single particle in the plane subject to a perpendicular constant magnetic field of strength B ą 0. This operator is known as the Landau Hamiltonian. It acts self-adjointly on a dense domain of definition in the single-particle Hilbert space L 2 pR 2 q and is given by (2.1) Here, we choose the symmetric gauge apxq " pa 1 pxq, a 2 pxqq :" px 2 ,´x 1 qB{2 for the vector potential a : R 2 Ñ R 2 generating the constant magnetic field (vector) perpendicular to the plane with Cartesian coordinates x " px 1 , x 2 q. Other gauges yield operators being unitarily equivalent to H. Moreover, here and in the following we are using physical units such that the particle mass and the particle charge equals 1{2 and 1, respectively. Similarly, we put the speed of light, Planck's constant (divided by 2π), and Boltzmann's constant all equal to 1.
Throughout the paper we use the symplectic 2ˆ2 matrix J :" p´1q j j!ˆℓ`k ℓ´j˙t j , k P t´ℓ,´ℓ`1, . . . u , t ě 0 of degree ℓ P N 0 , and the abbreviation L ℓ :" L p0q ℓ . For each degree ℓ we define an infinite-dimensional projection (operator) P ℓ on L 2 pR 2 q by the Hermitian integral kernel It is obviously C 8 -smooth and a Carleman kernel in the sense that it is square integrable with respect to y P R 2 for all x P R 2 , and vice versa. Now the spectral decomposition of the Landau Hamiltonian H may be written as As usual, this formula is meant in the sense of strong operator convergence on L 2 pR 2 q. It goes back to Fock [7] and Landau [14]. The projections P ℓ , now recognizable as spectral projections, depend on the chosen gauge through the last (complex-valued phase) factor in (2.2), but the set tB, 3B, 5B, . . .u of harmonic-oscillator like eigenvalues, in other words Landau levels, does not. The degree ℓ is now called Landau-level index. We will also need the projection P ďn :" ř 0ďℓďn P ℓ on the orthogonal sum of the first n`1 Landau-level eigenspaces P ℓ L 2 pR 2 q and mention the functional relation L ďn :" ř 0ďℓďn L ℓ " L p1q n . For later purposes we single out the (translation invariant) Gaussian part of the kernel by defining Now we are prepared to turn to the ground state of the ideal Fermi gas with the Landau Hamiltonian, see (2.1) and (2.3), as its single-particle Hamiltonian and with the chemical potential µ ě B as a real parameter. According to the grand-canonical formalism of quantum statistical mechanics [3,10] this (infinite-area) ground state is quasi-Gaussian (in other words, quasi-free) and, as such, characterized by its reduced single-particle density operator on L 2 pR 2 q given by the Fermi projection Here, Θ is Heaviside's unit-step function (defined by Θptq :" 1 if t ě 0 and zero otherwise), 1 denotes the identity operator on L 2 pR 2 q, and ν :" tpµ{B´1q{2u is the integer part of pµ{B´1q{2 ě 0. Now we consider a Borel set Λ Ď R 2 and the multiplication operator 1 Λ on L 2 pR 2 q corresponding to its indicator function 1 Λ on R 2 . Moreover, we introduce the local(ized) Landau projections P ℓ pΛq :" 1 Λ P ℓ 1 Λ , P ďn pΛq :" 1 Λ P ďn 1 Λ .
We ignore the uninteresting case µ ă B, because then the Fermi projection is the zero operator corresponding to a vanishing number of particles. In contrast, as a function of µ ě B the (mean) local areal density of the particle number, ρpxq, in the ground state characterized by the density operator (2.5) is non-zero and equal to the diagonal of its integral kernel, that is, Integration over the plane R 2 gives the (mean) total number of particles, pν`1qB|Λ|{2π, in its subset Λ with (Lebesgue) area |Λ|. So ν`1 corresponds to an integer value of the filling factor in the physics literature. In view of (2.5) it suffices in the following to consider the projection P ďn pΛq for arbitrary n P N 0 . Moreover, from now on we typically assume that Λ Ă R 2 is the union of finitely many bounded domains (open connected sets), such that their closures are pairwise disjoint. We call such a Λ a bounded region. If the boundary curve BΛ of Λ is C γ , γ P N Y t8u, then we say that Λ is a bounded C γ -region. Before we state our basic asymptotic results we recall the definition of the Hermite polynomials, H ℓ , of degree ℓ P N 0 . They satisfy the orthogonality relation ż An explicit formula is The Hermite functions ψ ℓ , defined by constitute an orthonormal basis of the Hilbert space L 2 pRq and are the (energy) eigenfunctions of the one-dimensional harmonic oscillator, that is, ψ 2 ℓ ptq`t 2 ψ ℓ ptq " p2ℓ`1qψ ℓ ptq , t P R . For ξ P R and a complex-valued function f on the closed unit interval r0, 1s as in Lemma 3 below we define λ ℓ pξq :" rf pλ ℓ pξqq´f p1qλ ℓ pξqs , ℓ P N 0 . (2.10) Obviously, each function λ ℓ takes values in r0, 1s and is (strictly) decreasing. We also need to introduce for each n P N 0 the one-parameter family of operators K n,ξ :" n ÿ ℓ"0 |ψ ℓ,ξ yxψ ℓ,ξ | " 1 rξ,8q n ÿ ℓ"0 |ψ ℓ yxψ ℓ | 1 rξ,8q , ξ P R . (2.11) The operator K n,ξ maps L 2 pRq self-adjointly on its pn`1q-dimensional subspace spanned by the first pn`1q truncated Hermite functions ψ ℓ,ξ :" ψ ℓ 1 rξ,8q . This operator is not a projection, but it satisfies 0 ď K n,ξ ď 1 rξ,8q ď 1. Its integral kernel is given by the sum ř 0ďℓďn ψ ℓ,ξ ptqψ ℓ,ξ pt 1 q, which can be evaluated explicitly, see (3.36) below.
Along with M ℓ pf q we also define for n P N 0 M ďn pf q :" ż R dξ 2π rtr f pK n,ξ q´f p1q tr K n,ξ s , M ď0 pf q " M 0 pf q . (2.12) Here the trace refers to operators on L 2 pRq. Now we are in a position to present our two basic asymptotic results.
THEOREM 1 (For the ℓth Landau level, ℓ P N 0 ). Let Λ Ă R 2 be a bounded C 3 -region in the sense defined below (2.6). Moreover, let f : r0, 1s Ñ C be a complex-valued continuous function on the closed unit interval with f p0q " 0, differentiable from the right at t " 0 and differentiable from the left at t " 1. Finally, let L ą 0 be a (dimensionless) scaling parameter. Then we have as L Ñ 8. The asymptotic coefficient is finite, that is, |M ℓ pf q| ă 8.
Here the trace refers to operators on L 2 pR 2 q and, as usual, opLq stands for some function of L with lim LÑ8 |opLq|{L " 0.
THEOREM 2 (For the first pn`1q Landau levels, n P N 0 ). Under the same assumptions as in Theorem 1 we have 14) as L Ñ 8. The asymptotic coefficient is finite, that is, |M ďn pf q| ă 8.
The finiteness of the coefficients M ℓ pf q and M ďn pf q are consequences of the following Lemma 3 and Lemma 4, because the smooth function f assumed in Theorem 1 and Theorem 2 satisfies the bound (2.15). The proofs of (2.13) and (2.14) are postponed until the proof of Lemma 4. In the next lemma and in the following, by C, c with or without indices, we denote various finite and positive constants, whose precise values are of no importance.
Concerning the other coefficient M ďn pf q we have the following LEMMA 4. Under the same assumption as in Lemma 3 we have }f pK n,ξ q´f p1qK n,ξ } 1 ď C δ e´δ qξ 2 , for every n P N 0 with an arbitrary 0 ă δ ă 1, and hence |M ďn pf q| ă 8.
Before proving this lemma we compile, for the reader's convenience, some basic properties of the Schatten-von Neumann classes, S p , 0 ă p ă 8, of compact operators, see [2,28]. By s n pTq with n P N, we denote the singular values of a compact operator T on an abstract (separable) Hilbert space, enumerated in decreasing order. Then, the operator T is said to belong to S p if it has the finite Schatten-von Neumann (quasi-)norm }T} p :" If p ě 1, then }¨} p is a norm. If 0 ă p ă 1, then it is a quasi-norm which satisfies the p-triangle inequality The class S 1 is the standard trace class. For T P S 1 its trace, tr T, is well-defined and satisfies |tr T| ď }T} 1 . If T ě 0, then tr T " }T} 1 . We also note that the usual (uniform) operator norm }¨} may be viewed as }¨} p in the limit p Ñ 8. Finally, we mention that }¨} p satisfies a Hölder-type inequality in the sense that Proof of Lemma 4. Let us now consider the operator K n,ξ " ř n ℓ"0 Q ℓ,ξ defined in (2.11), where we have put Q ℓ,ξ :" |ψ ℓ,ξ yxψ ℓ,ξ | Withgptq :" tp1´tq we then havẽ Since each operator Q ℓ,ξ is one-dimensional, we easily find that Consequently, for ξ ě 0, we have For the case ξ ă 0 we observe that xψ ℓ |ψ ℓ 1 y " 0, ℓ " ℓ 1 , so that xψ ℓ,ξ |ψ ℓ 1 ,ξ y "´xψ ℓ´ψℓ,ξ |ψ ℓ 1´ψ ℓ 1 ,ξ y, and hence Collecting the above bounds we conclude that Using now (2.15) we have }f pK n,ξ q´f p1qK n,ξ } 1 ď C}gpK n,ξ q} q q . Since the operator K n,ξ has finite dimension n`1, the right-hand side of the last inequality is bounded from above by C}gpK n,ξ q} q . Therefore (2.18) leads to the claimed result.
Proofs of Theorem 1 and Theorem 2. By linearity, Lemma 5 and Lemma 6 in the next section imply Theorem 1 and 2, respectively, for an arbitrary polynomial f (with f p0q " 0). So here we only need to show how to extend the claimed results (2.13) and (2.14) from polynomials to the smooth function f assumed in Theorem 1 and Theorem 2. This is by now standard and several versions of this extension are available, e.g. [24,25,29,31]. Here we follow the recent one in [24]. As a by-product we get the a-priori finiteness of the left-hand side of (2.13), see (2.19) and (2.20). The a-priori finiteness of the left-hand side of (2.14) follows similarly.
Using (2.13) forp andg, we arrive at the bound Since M ℓ ppq " M ℓ pf q`M ℓ pgpp´bqq ď M ℓ pf q`εM ℓ pgq, the right-hand side of the above inequality does not exceed M ℓ pf q`2εM ℓ pgq. Similarly, Since ε is arbitrary, formula (2.13) follows. In order to prove Theorem 2 we use the same argument as above for the operator P ďn pLΛq and the asymptotic coefficient M ďn pf q.

Underlying asymptotic results for polynomials
We have seen that the asymptotic results of Theorem 1 and Theorem 2 rely on corresponding results for polynomials f . By linearity, it suffices to consider natural powers of the corresponding projections. We begin with LEMMA 5. Let Λ Ă R 2 be a bounded C 3 -region and m P N. Then we have for any ℓ P N 0 Here, as usual, Op1q " OpL 0 q stands for some function of L with lim sup LÑ8 |Op1q| ă 8.
If m " 2 then the left-hand side of (3.4) is zero and its right-hand side is meant to be 0. By combining (3.2), (3.3), (2.2), and (3.4) the trace of P ℓ pΛq m can now be written as with the function f m defined by Now we insert the scaling parameter L ą 0 and apply Roccaforte's asymptotic expansion of Appendix A up to the first order ż |LΛ|´ˇˇLΛz`LΛ X py 1`L Λq X py 1`y2`L Λq X¨¨¨X py m`L Λq˘ˇ" as L Ñ 8. Here, A is the canonical arc-length measure on BΛ, n x is the inward unit normal vector at the point x P BΛ, and W is a function such that W psq ď Cs, s ě 0. We scale y by B´1 {2 and Λ by B 1{2 . Then we can set from now on B " 1 in the function f m and replace LΛ by ? L 2 BΛ. The parameter that tends to infinity in our asymptotic analysis is thus effectively ? L 2 B. For a given point on the boundary curve, x P BΛ, we decompose each vector y i P R 2 into a component parallel and a component perpendicular to the tangent (line) T x pBΛq -R at x P BΛ according to with the real numbers t i :" xy i |n x y and z i :"´xy i |Jn x y so that }y i } 2 " z 2 to be used on the right-hand side of (3.6). Here, z :" pz 1 , . . . , z m´1 q and t :" pt 1 , . . . , t m´1 q. Moreover, S is the pm´1qˆpm´1q matrix with entries By setting t 0 :" 0 the maximum in (3.7) can now be written as follows max 0, xy 1 |n x y, xy 1`y2 |n x y, . . . , Let us now introduce new variables pT 1 , . . . , T m´1 q by the sums We also define T m :" 0, t m :" t 1`¨¨¨`tm´1 , and z m :" z 1`¨¨¨`zm´1 . The change (3.8) from the (global) variables y i to the x-dependent (local) variables pz i , t i q corresponds to a translation and a rotation of the coordinate system. This implies that dy i " dt i dz i which is shorthand for the underlying invariance of the multi-dimensional Lebesgue measure. Once the integration with respect to all the variables z i and t i is done, the result will turn out to be independent of x P BΛ and the remaining integration with respect to x along the boundary curve BΛ simply yields the factor L ? B|BΛ|.
(3.16) [When it comes to integration we do not switch from the t-variables to the T -variables; moreover, we note that det S " 0, resp. " 1 if m is even, resp. odd.] The integral I m ptq can be viewed as the m-fold convolution product g T1,t1˚¨¨¨˚gTm,tm evaluated at 0. This suggests to introduce the (inverse) Fourier transform If ℓ " 0, then this integral can be calculated explicitly. But even then it turns out to be more convenient not to perform this integration at this point. Therefore, the pm´1q-fold integral (3.15) can be rewritten as an integral over the real line according to 19) and the term of the sub-leading order L in (3.14) becomes equal to (using the notation ω :" pω 1 , . . . , ω m q) Following Roccaforte [25] we now introduce mpě 2q subsets S, S 1 , . . . , S pm´1q of R m´1 by for 1 ď q ď m´1. The sets S q are pairwise disjoint and make up all of S in the sense that S " Ť m´1 q"1 S q , up to (hyperplane) sets of pm´1q-dimensional Lebesgue measure zero. In fact, t P S q implies that M ptq " ř q r"1 t r ą 0. And the conditions for 2 ď s ď q and for 1 ď p ď m´1´q ensure that the sets S q are indeed disjoint. Following again Roccaforte [25] we introduce variables τ :" pτ 1 , . . . , τ m´1 q adapted to the just introduced sets. We define for 1 ď s ď q and τ q`p :" q`p ÿ r"q`1 t r (3.24) for 1 ď p ď m´1´q.
On the set S q we have τ 1 " A pqq t 1 . Here τ 1 denotes the column tuple as the transpose of the (row) tuple τ and similarly for t 1 . And the pm´1qˆpm´1q matrix A pqq is defined in terms of its entries (3.25) Then we have det A pqq " 1 and on S q the comforting identity M ptq " τ 1 1`pτ 1 q¨¨¨1`pτ q q1´pτ q`1 q¨¨¨1´pτ m´1 q , t P S q , using the abbreviations 1˘for the indicator functions on the real line R for its two half-lines R˘. Now we consider the joint integration with respect to the m variables ξ and τ and apply the following changes of variables. Firstly, we change τ q`1 , . . . , τ m´1 to´τ q`1 , . . . ,´τ m´1 . Clearly, the τ integral is now over R m´1 . Secondly, we replace ξ by ξ´pτ 1`τm´1 q{2, and thirdly we replace ξ bý ξ. The negative of the argument in the product of exponentials in (3.20) then changes according to Here and in the following we are using the notation ù to present the results of changes of variables efficiently, without the explicit introduction of the underlying mappings. We prove (3.26) in Appendix B.1. The main advantage of the quadratic form (3.26) over that in the t-variables is that there are no mixed terms between the τ 's and the exponential can be factorized. The resulting term does not depend on q. This turns out to remain true with the Laguerre polynomials included as we will see next.
We perform the same changes of variables in the arguments of the Laguerre polynomials. For instance, if q " 1, theǹ Next we change τ 1 to τ 1´ξ so that the last expression equals For general q ě 2, see Appendix B.2. In the end, the product of the Laguerre polynomials equals ź The following remarkable identity will be proved in Appendix B.3, 1 ? 2π After performing the m-fold integration with respect to ω we obtain 2 m{2 p2 ℓ ℓ!q´1H 2 ℓ pξqp2 ℓ ℓ!q´1H 2 ℓ pτ 1 q¨¨¨p2 ℓ ℓ!q´1H 2 ℓ pτ m´1 q . To summarize, the boundary-curve term of the order L equals´L ?
B|BΛ|{p2πq times Finally, we turn to the leading area term of the order L 2 in (3.14), dt gpt 1 q¨¨¨gpt m´1 qgpt m q I m ptq .
Here, we use (3.19) for I m ptq and switch to the variables τ 1 , . . . , τ m´1 from (3.23) and (3.24) for q " 1, in all of R m´1 . We perform the same shifts in ξ and in the τ 's. Then the area term turns into by the normalization of the Hermite functions. Alternatively, the leading term can be obtained by replacing the x-integral in (3.5) by |Λ|. The remaining y-integration yields B{p2πq. This finishes the proof of Lemma 5.
The next lemma provides the basis for the proof of Theorem 2.
LEMMA 6. Under the same assumptions as in Lemma 5 we have for any n P N 0
Proof. By the same arguments as in the beginning of the proof of Lemma 5, the projection P ďn pΛq m is a trace-class operator and its trace can be calculated by integrating the diagonal of the m-fold iterated integral kernel of P ďn pΛq. Again, the case m " 1 is then obvious and we only need to consider the case m ě 2. We recall that P ďn px, yq " ř 0ďℓďn p ℓ px, yq. So in the proof of Lemma 5 we simply have to replace L ℓ with L ďn " ř 0ďℓďn L ℓ " L p1q n . For instance, expression (3.27) is replaced with the expression ź 1ďjďq´1 L ďn`p ω j´2 iτ j qpω j´2 iτ j`1 q{2˘ź q`2ďjďm L ďn`p ω j´2 iτ j´1 qpω j´2 iτ j q{2L ďn`p ω q´2 iξqpω q´2 iτ q q{2˘L ďn`p ω q`1´2 iξqpω q`1´2 iτ q`1 q{2L ďn`p ω q´2 iτ 1 qpω q´2 iτ m´1 q{2˘.

From smooth functions to the entropy functions
In this section we build on Theorem 1 and Theorem 2 with a suitable function f to derive the precise leading asymptotic growth of the local ground-state entropy with arbitrary Rényi index α ą 0. While the case α ą 1 is rather straightforward, non-smoothness in the case α ď 1 requires considerable attention. In the first subsection we define the local ground-state entropies and present our main result and related results. The second subsection prepares the ground for getting from smooth functions to the non-smooth functions needed in the case α ď 1. Proofs of our results are then given in the third subsection.
The proof is given in Subsection 4.3 after certain preparations in Subsection 4.2. The next theorem gives the precise asymptotic growth. It is our main result.
THEOREM 8 (Asymptotics of the local Rényi ground-state entropies). Let Λ Ă R 2 be a bounded C 3 -region and let the chemical potential satisfy µ ě B. Then the local α-Rényi ground-state entropy (4.2) obeys as L Ñ 8. The asymptotic coefficient M ďν ph α q is given by (2.12) with n " ν. It is finite and positive.
The proof is given in Subsection 4.3. It builds on Lemma 7, Theorem 2, and Subsection 4.2.
(i) The coefficient M ďν ph α q in (4.3) is in general not easy to calculate. The simplest case occurs when ν " 0. Then we have with λ 0 pξq " π´1 {2 ş 8 ξ dt expp´t 2 q being 1{2 of the complementary error function. The coefficient (4.4) was found in [26] for α " 1 and special regions Λ. The first proof of (4.3) for α " 1, ν " 0 (equivalently, ℓ " 0 in (4.7)), L 2 B P N, and general bounded C 8 -regions is due to Charles and Estienne in [4]. A numerical computation gives M 0 ph 1 q " 0.203 . . . , also in agreement with [26]. (ii) In the zero-field case B " 0, the leading term of the local Rényi ground-state entropy depends on its index α simply through the pre-factor p1`αq{α, see [15]. A numerical computation shows that in the case B " 0 the dependence on α is not so simple.
The remark immediately below Theorem 8 applies analogously to this theorem when (2.14) is replaced with (2.13). The proof of Theorem 10 builds on Theorem 1 and repeats the proof of Theorem 8 in Subsection 4.3 with P ďν replaced by P ℓ .
The next subsection contains estimates being crucial for the proof of the above results.

Estimates for singular values.
We begin by introducing certain operators T r,R on L 2 pR 2 q.
To this end, we denote by Dpx, Rq Ă R 2 the open disk of radius R ą 0, centered at the point x P R 2 and abbreviate 1 R :" 1 Dp0,Rq and similarly with R replaced by r ą 0. Then we define the operators T r,R :" T pℓq r,R :" 1 r P ℓ`1´1R˘, T r,0 :" 1 r P ℓ , ℓ P N 0 . (4.8) Here we assume that the magnetic-field strength has been "scaled out", so that B " 1 in formula (2.2). We interpret T r,R as an operator from L 2 pR 2 q into L 2 pDp0, rqq.
In order to estimate the singular values of T r,R we recall the short compilation below Lemma 4 and a classical result due to Birman and Solomyak, see [1,Theorem 4.7]. We quote the required fact in a form adapted to our purposes. PROPOSITION 11. Let Z : L 2 pR 2 q Ñ L 2 pDp0, rqq be an integral operator defined by a complexvalued kernel Zpx, yq obeying for some γ P N 0 . Then the singular values s n pZq of Z satisfy the bound s n pZq ď Cn´1`γ 2 N γ pZq , n P N , with a positive constant C dependent on r but independent of the kernel Z.
LEMMA 12. The operator T r,0 belongs to the Schatten-von Neumann class S p for all p ą 0. Moreover, if R ą r, then }T r,R } p ď C exp`´pR´rq 2 {8˘, (4.9) with some constant C dependent on r but independent of R.
Proof. In order to apply Proposition 11, we estimate for }x} ă r and s, t P N 0 with 0 ď s, t ď γ:ˇˇˇB with a constant C γ,ℓ depending on r. Here we have used the fact that L ℓ is a polynomial of degree ℓ. For R ą r and y R Dp0, Rq we conclude thaťˇˇˇB Thus the integral kernel T r,R px, yq :" 1 r pxqp ℓ px, yqp1´1 R pyqq of T r,R satisfies, for any γ P N, the bounds N γ pT r,R q ď C γ,ℓ , for any R ě 0 , where the constant C γ,ℓ is independent of R. For an arbitrary p ą 0 we now take γ ą 2p´1´1. Then by Proposition 11, T r,R P S p for all R ě 0 and the bound (4.9) holds for R ą r.
Lemma 12 is an important ingredient to bound quasi-norms of the operator 1 LΛ P ℓ p1´1 LΛ q. Although our main result Theorem 8 is proved for a bounded C 3 -region Λ, the next theorem even holds for a bounded Lipschitz region. Here, the boundary curve BΛ of Λ is Lipschitz continuous. THEOREM 13. Let Λ Ă R 2 be a bounded Lipschitz region and ℓ P N 0 . Moreover, let p P p0, 1s and L 0 ą 0 finite. Then there exists a constant C, depending only on Λ and L 0 , such that for any L ě L 0 , Proof. We begin with two useful observations. Firstly, we note that }1 LΛ P ℓ p1´1 LΛ q} p ď }T Lr,0 } p by (2.17) with some r ą 0, where the operator T r,0 is defined in (4.8). Thus, by Lemma 12, for every fixed L the left-hand side of the claim (4.10) is finite. Consequently, it suffices to prove (4.10) for L ě L 0 with an arbitrary choice of the finite L 0 . Secondly, denoting Λ 1 :" L 0 Λ, we can rewrite (4.10) as follows: }1 LΛ 1 P ℓ p1´1 LΛ 1 q} p p ď CL , L ě 1 , where C depends on Λ and the arbitrary L 0 . Now we can proceed with the proof. We cover Λ by finitely many disks Dpx k , r k q such that either (1) x k P BΛ and inside each disk Dpx k , 8r k q, with an appropriate choice of coordinates, the domain Λ is given locally by the epigraph of a Lipschitz function (see below), or (2) Dpx k , r k q Ă Λ and dist`Dpx k , r k q, Λ A˘ą 0, where Λ A :" R 2 zΛ denotes the complement of Λ. It is clear that we may assume that all radii r k are equal to each other. Moreover, by replacing Λ with Λ 1 " L 0 Λ with L 0 " r´1 k , we may assume that r k " 1. This equality holds throughout the proof.
Case (1): We fix one disk D :" Dpx k , 1q Ă R 2 , x k P BΛ, and denote r D :" Dpx k , 8q. Let Φ : R Ñ R be a Lipschitz function such that Λ X r D " tx " px 1 , x 2 q P R 2 : By M ě 0 we denote the Lipschitz constant for Φ, i.e.
It is clear that pLΛ X LDq Ă pLΛ X L r Dq " tx " px 1 , x 2 q : x 2 ą Φ L px 1 qu X L r D , Φ L ptq :" LΦptL´1q , t P R , and that the Lipschitz constant for Φ L also equals M . Without loss of generality we may assume that D " Dp0, 1q and Φp0q " 0. Now we construct a covering of LΛ X LD by open disks. Let D jk be a disk of radius 1, centered at the point z j,k :" pj{2, k{2q P LD, pj, kq P Z 2 . Clearly, such disks form an open covering for LD. To extract a convenient covering for LΛ X LD, we define two index sets: I 1 :" tpj, kq P Z 2 : k{2 ě Φ L pj{2q`2xM y, z jk P LDu , xM y :" a 1`M 2 , I 2 :" tpj, kq P Z 2 : |k{2´Φ L pj{2q| ă 2xM y, z jk P LDu .
Since Φ is Lipschitz, the number of indices in I 2 obeys |I 2 | ď CL with a constant independent of L.
The disks D jk , pj, kq P I 1 Y I 2 form a covering of the intersection LΛ X LD. Observe that for every point x " px 1 , L px 1 q| , so that R jk :" distpz jk , LΛ A q ě xM y´1`k{2´Φ L pj{2q˘ě 2 , pj, kq P I 1 . (4.11) Let pϕ jk q jk Ă C 8 0 pR 2 q be a partition of unity subordinate to the constructed covering. In the following, we use a superposed hat for the (bounded) multiplication operator p ϕ on L 2 pR 2 q uniquely corresponding to ϕ P C 8 0 pR 2 q. We estimate individually the quasi-norms } p ϕ jk 1 LΛ P ℓ p1´1 LΛ q} p , for p P p0, 1s. Let us consider firstly the set I 2 . For all pj, kq P I 2 we have The first inequality follows from (2.17). The second inequality holds since ϕ jk 1 LΛ ď 1 D jk and hence p ϕ jk 1 LΛ P ℓ 1 LΛ p ϕ jk ď 1 D jk P ℓ 1 D jk . Using the standard unitary equivalence of the Hamiltonian (2.1) under "magnetic" translations, we conclude that the right-hand side coincides with }1 D P ℓ } p , where D " Dp0, 1q, as before. By Lemma 12, this norm is bounded for all p ą 0, and hence } p ϕ jk 1 LΛ P ℓ p1´1 LΛ q} p ď C uniformly in j and k. Applying the p-triangle inequality (2.16) for quasi-norms, we get ÿ Suppose now that pj, kq P I 1 . Thus By the translation argument, the right-hand side coincides with }T 1,R jk } p , where the operator T 1,R is defined in (4.8). Consequently, by (4.9), Using the p-triangle inequality again, we obtain that ÿ pj,kqPI1 Employing (4.11), for any fixed j, the summation over k yields the estimate ÿ pj,kqPI1, j fixed exp`´ppR jk´1 q 2 {8˘ď ÿ kPZ e´c k 2 " C .
Since |j| ď 2L, the right-hand side of (4.12) does not exceed CL. Putting these estimates together we obtain Case (2): We fix one disk D " Dpx k , r k q such that distpD, Λ A q ě c, so that distpLD, LΛ A q ě cL. We cover LD by unit disks D j , j " 1, . . . , N with N ď CL 2 and distpD j , LΛ A q ě cL. As in the proof of Case (1), we introduce a smooth partition of unity pϕ j q j Ă C 8 0 pR 2 q subordinate to this covering, and estimate: Consequently, by the p-triangle inequality (2.16) (4.14) To complete the proof we add the estimates of the form (4.13) and (4.14) for all disks covering Λ, using the p-triangle inequality again.

On an improvement to sub-leading terms
Without going into details, we show here how one can improve the asymptotic expansions in Lemmata 5 and 6 to tr f pP ℓ pLΛqq " where f ptq " t n , n " 1, 2, . . . . This follows again by proving the corresponding statements for all natural powers of P ℓ pLΛq. Here, we use the expansion of Roccaforte in Proposition 14 up to the second order ε 2 . For any vector y "´zJn x`t n x P R 2 , written in the form (3.8), we have }y} 2´2 xy|n x y 2 " z 2´t2 . Then the Op ř j |y j | 2 q-term in (3.7) takes the form m´1 ÿ If we exchange the z and t variables, the integrand is seen to be almost anti-symmetric except for the sign in the exponent. This can be remedied by changing, for instance, t to´t. Hence, the integral in (5.3) vanishes by symmetry.
Then we need to show that B " 1. We distinguish between certain ranges of indices.
For j " q`1 we get in the end the expression pω´2iξqpω´2iτ q`1 q.
B.3. Proof of the identity (3.28). We start out with the representation of the Laguerre polynomial as a contour integral in the complex plane C, namely Here, the contour Γ is, say, a circle of radius ă 1, centered at the origin 0, and with counter-clockwise orientation. Then, for any given pair ξ, τ P R, we have 1 ? 2π H ℓ pξqH ℓ pτ q ℓ!´t 2¯ℓ , |t| ă 1 (B.14) now completes the proof of (3.28).