The Gell-Mann–Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρlogρ+c2ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{1}\rho \log \left( \rho \right) +c_{2}\rho $$\end{document} in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.


Introduction
Although interacting Fermi gases have been studied extensively from the beginning of quantum mechanics, their rigorous understanding remains one of the major issues of condensed matter physics.From first principles, a system of N fermions in R 3 can be described by a Schrödinger equation in R 3N , subject to the anti-symmetry condition between the variables due to Pauli's exclusion principle.However, this fundamental theory becomes very complex when N → ∞, leading to the need of various approximations.Justifying these approximations is an important task of mathematical physics.
One of the most basic approximations for fermions is the Hartree-Fock (HF) theory.In HF theory, the particles are assumed to be independent, namely the HF energy is computed by restricting the consideration to Slater determinants.In spite of its simplicity, the HF theory is used very successfully in computational physics and chemistry to compute the ground state energy of atoms and molecules.The accuracy of the HF energy (in comparison to the full quantum energy) for large Coulomb systems was investigated in the 1990s by Fefferman and Seco [12], Bach [1], and Graf and Solovej [15].
On the other hand, for the electron gas (e.g.jellium, a homogeneous electron gas moving in a background of uniform positive charge), the HF theory is essentially trivial in the high density limit since the HF energy only contains an exponentially small correction to the energy of the Fermi state, the ground state of the non-interacting gas [14].Therefore, computing the correlation energy 1 , namely the correction to the HF energy, is a crucial task to understand the effect of the interaction.It was already noticed by Wigner in 1934 [23] and confirmed by Heisenberg in 1947 [17] that it would be very challenging to accomplish this task within perturbation theory due to the long-range property of the Coulomb potential.Nevertheless, a remarkable attempt in this direction was done by Macke in 1950 [18] when he used a partial resummation of the divergent series to predict the leading order contribution c 1 ρ log (ρ) of the correlation energy (with density ρ → ∞).
A cornerstone in the correlation analysis of the electron gas is the random phase approximation (RPA) which was proposed by Bohm and Pines in the 1950s [7,8,9,19].As an important consequence of the Bohm-Pines RPA theory, the electron gas could be decoupled into collective plasmon excitations and quasi-electrons that interacted via a screened Coulomb interaction.The latter fact justified the independent particle approach commonly used for many-body fermion systems.The justification of the RPA was a major question in condensed matter and nuclear physics in the late 1950s and 1960s.An important justification was given by Gell-Mann and Brueckner in 1957 [13] when they formally derived the RPA from a resummation of Feynman diagrams where each term separately diverges but the sum is convergent.More precisely, by considering the diagrams corresponding to the interaction of pairs of fermions, one from inside and one from outside the Fermi state, Gell-Mann and Brueckner were able to produce the leading order contribution c 1 ρ log (ρ) + c 2 ρ of the correlation energy.
Soon after the achievement of Gell-Mann and Brueckner, Sawada [21] and Sawada-Brueckner-Fukuda-Brout [22] proposed an alternative approach to the RPA where the pairs of electrons are interpreted as bosons, leading to an effective Hamiltonian which is quadratic in terms of the bosonic creation and annihilation operators.Note that within the purely bosonic picture, quadratic Hamiltonians can be diagonalized by Bogolubov transformations [6], and hence their spectra can be computed explicitly.Therefore, the Hamiltonian approach in [21,22] is conceptually more transparent than the resummation method in [13].Unfortunately the analysis in [21,22] only gives the contribution c 1 ρ log (ρ) of the correlation energy because the exchange contribution of order ρ is missed in the purely bosonic picture.
Recently, the bosonization argument in [21,22] has been revisited and made rigorous in the mean-field regime with smooth interaction potentials [16,2,3,4,10,5,11].In principle, if the interaction is sufficiently weak, then the non-bosonizable terms of the interaction energy are negligible, and the quasi-bosonic Hamiltonian can be analyzed with great precision.In particular, the correlation energy has been successfully computed to the leading order [2,3,10,5].However, the boundedness of interaction potentials is crucial for all of these works, and extending the analysis to the electron gas remains a very interesting open question.
In the present paper, we will give the first rigorous upper bound to the correlation energy of the electron gas in the mean-field regime.Our bound is consistent with the Gell-Mann-Brueckner formula c 1 ρ log (ρ) + c 2 ρ for jellium in the high density limit [13].Although our trial state argument is inspired by the bosonization method in [21,22], we are able to capture correctly the exchange contribution by carefully distinguishing the purely bosonic picture and the quasibosonic one.On the mathematical side, we will use the general method in our recent work [10], but several new estimates are needed to deal with the singularity of the potential.The matching lower bound in the mean-field regime, as well as the corresponding result in the thermodynamic limit, remain open, and we hope to be able to come back to these issues in the future.
On the technical side, the key idea of [10] is that while the bosonic property of fermionic pairs holds only in an average sense, this weak bosonic property is sufficient to extract correctly the correlation energy by implementing a quasi-bosonic Bogolubov transformation.The main contribution of the present paper is to show that this approach is also sufficient to extract the exchange correction to the purely bosonic computation.On the other hand, another bosonization method has been proposed in [2], where the bosonic property of fermionic pairs is strengthened by using suitable patches on the Fermi sphere for the quasi-bosonic creation and annihilation operators, making the comparison with the purely bosonic computation significantly easier.In fact, as explained in [5], the approach in [2] can be extended to give the leading order of the correlation energy upper bound for potentials satisfying V 2 k |k| < ∞.Although this condition only barely fails for the Coulomb potential, there is a huge difference to the Coulomb case.While for V 2 k |k| < ∞ the bosonic correlation contribution is of order k F and the exchange correlation is of lower order o(k F ), for the Coulomb potential the exchange contribution raises to the order k F , whereas the bosonic correlation behaves as k F log(k F ), which makes the Coulomb case much more challenging (here k F is the radius of the Fermi ball).In particular, the method in [2,5] does not seem to capture the exchange contribution which is indeed important for the Coulomb potential.

Main Result
Let T 3 = [0, 2π] 3 with periodic boundary conditions.Let V : T 3 → R be defined by V (x) = 1 (2π) 3  For k F > 0, let N = |B F | be the number of integer points in the Fermi ball B F = B (0, k F ) ∩ Z 3 and consider the mean-field Hamiltonian on the fermionic space H N = N h with h = L2 T 3 2 .The leading order of the ground state energy of H N is given by the Fermi state It is straightforward to find (see e.g.[10,Eqs. (1.10) and (1.20)]) where for every k ∈ Z 3 * , we denoted the lune associated to k by Our main result concerns the corrections to the ground state energy.For every k ∈ Z 3 * , define We will prove the following: Theorem 1.1.As k F → ∞ it holds that inf σ (H N ) ≤ E FS + E corr,bos + E corr,ex + C where is the bosonic contribution and is the exchange contribution, for a constant C > 0 depending only on Some remarks on our result: 1. Consider the Coulomb potential, Vk = g |k| −2 for a constant g > 0. Following the analysis of [15], we find that inf σ where E FS contains the kinetic energy of order k 5 F , the direct interaction energy of order k 5 F and the exchange interaction energy of order k 3 F .Furthermore, it is straightforward to adapt the proof in [14] to see that the difference between E FS and the HF energy is exponentially small as k F → ∞.Therefore our result really concerns the correlation energy, which we bound from above by E corr,bos ∼ −k F log (k F ) and E corr,ex ∼ k F (1.9) plus the error term of order In fact, it is easy to verify (1.10) To see the leading order behavior E corr,ex ∼ k F in (1.9), one may use that λ k,p ∼ |k| max{|k|, k F } (in an average sense) and that we have and hence the asymptotic behavior E corr,bos ∼ −k F log(k F ) in (1.9) follows similarly.
Note that the correlation energy E corr,bos + E corr,ex in Theorem 1.1 is exactly the mean-field analogue of the Gell-Mann-Brueckner formula c 1 ρ log (ρ) + c 2 ρ for jellium in the thermodynamic limit [13].Indeed, substituting k −1 F Vk → 4πe 2 |k| −2 and (2π) 3 → the volume Ω, E corr,bos agrees with [22,Eq. (34)] which is equivalent with [13, Eq. ( 19)] (accounting also for spin).In the thermodynamic limit, the right-hand side of (1.12) always diverges, no matter if we have the mean-field scaling or not, but the full expression on the left-hand side converges in either case.Furthermore, we also obtain the exchange contribution E corr,ex , which is the analogue of [13, Eq. ( 9)], which is completely absent from the bosonic model of [22].With the same substitutions as above, the exchange contribution takes the form V 2 k |k| < ∞, and so is less singular than the Coulomb potential, then the bosonic contribution E corr,bos is of order k F , while the exchange contribution is o (k F ).In this case, the upper bound  [10,5] (see also [2] and [3] for previous results on the upper and lower bounds, respectively, when Vk is finitely supported).
In comparison, the Coulomb potential is much more challenging to analyze, since it leads to an additional logarithmic factor in the bosonic contribution, and lifts the exchange contribution to the order k F .On the mathematical side, working with the Coulomb potential thus requires a substantial refinement of the bosonization method compared to the existing works.

3.
Although the case of the greatest physical interest is the Coulomb potential, our result covers a far greater class of singular potentials: Under the condition kZ 3 * V 2 k < ∞, the error term , and so Theorem 1.1 is always a meaningful result.

Overview of the Proof
We will construct a trial state by applying a quasi-bosonic Bogolubov transformation to the Fermi state ψ FS .We will follow the general formulation of the bosonization method in [10].We quickly recall this here for the reader's convenience, after which we explain the new components of the proof and the structure of the rest of the paper.

Rewriting the Hamiltonian
We will use the second quantization formalism in which we associate to every plane wave state u p of equation (1.4) the creation and annihilation operators c * p = a * (u p ) and c p = a(u p ) on the fermionic Fock space.They obey the canonical anti-commutation relations (CAR) The Hamiltonian H N of equation (1.3) can then be written as Note that the Fermi state ψ FS obeys (B c F denoting the complement of B F with respect to Z 3 ) and so it follows by the CAR that the kinetic energy of the Fermi state is We define the localized kinetic operator H ′ kin by where we for the last identity used the "particle-hole symmetry" From the last identity of equation (1.19) it is clear that H ′ kin is non-negative.
We normal-order H int with respect to ψ FS : Using the CAR and the fact that p∈Z 3 c * p c p = N = N on H N , it factorizes as we can write Using the CAR again it is easy to compute that whence (using also that Vk = V−k ) Note that the first sum is finite as and we summarize the calculations above in the following: Proposition 1.2.It holds that where E FS = ψ FS , H N ψ FS and the cubic and quartic terms, C and Q, are defined by We will prove that the cubic and quartic terms are negligible, and so the main contribution to the correlation energy comes from the bosonizable terms (1.27) We will write these in terms of quasi-bosonic operators, which will lead us to define a quasibosonic Bogolubov transformation that serves to effectively diagonalize them.
The Quasi-Bosonic Quadratic Hamiltonian We define the excitation operators b The name is due to the fact that b * k,p acts by annihilating a state with momentum p − k ∈ B F and creating a state with momentum p ∈ B c F , i.e. it excites the state p − k to the state p.For the purpose of computations it is convenient to also introduce a basis-independent notation for the quasi-bosonic operators.Consider for k ∈ Z3 * the auxilliary space ℓ 2 (L k ), which we will consider only as a real vector space, with standard orthonormal basis (e p ) p∈L k .For any k ∈ Z 3 * and ϕ ∈ ℓ 2 (L k ) we define the generalized excitation operators b k (ϕ) and b * k (ϕ) by Note that the assignments ϕ → b k (ϕ), b * k (ϕ) are both linear (as we only consider ℓ 2 (L k ) as a real vector space).In this notation we simply have that b k (e p ) = b k,p .A short calculation using the CAR shows that these operators are quasi-bosonic in the following sense: where the exchange correction ε k,l (ϕ; ψ) is given by ϕ, e p e q , ψ δ p,q c q−l c * p−k + δ p−k,q−l c * q c p .
Note that in the purely bosonic picture the exchange correction is absent.In our quasi-bosonic case, these corrections are small but non-zero; it will be important to keep careful track of them as it is these that gives rise to the exchange contribution E corr,ex .
For any operators A, B on ℓ 2 (L k ), we define the associated quadratic operators and (1.31) Defining the operator P k on ℓ2 (L k ) by we can express the interaction part of the bosonizable terms as The localized kinetic operator H ′ kin cannot be written exactly in a quadratic quasi-bosonic form, but due to the commutation relation (see [10,Eq.(1.76)]) and the quasi-bosonicity of the b * k,p operators, it is sensible to consider it analogous to a quadratic operator of the form where the operators h k on ℓ 2 (L k ) are simply defined by h k e p = λ k,p e p .In all we thus consider the bosonizable terms as being analogous to a quasi-bosonic quadratic operator as (1.36) The Quasi-Bosonic Bogolubov Transformation If the quadratic Hamiltonian on the right-hand side of equation (1.36) was exactly bosonic, it could be diagonalized by a Bogolubov transformation.Motivated by this we define such a transformation in the quasi-bosonic setting, while keeping careful track of the additional terms arising from the exchange correction. Let * , be a collection of symmetric operators satisfying Then we define the associated quasi-bosonic Bogolubov kernel K on H N by It is obvious from the second equation that K is skew-symmetric; K thus generates a unitary transformation e K : H N → H N -the quasi-bosonic Bogolubov transformation.
We consider the case k∈Z 3 * K k We choose the operators (K k ) such that e K would diagonalize the right-hand side of equation (1.36) if it was exactly bosonic.As explained in [10, Section 3] the diagonalizing kernel is Keeping careful track of the quasi-bosonic corrections, the action of e K on the bosonizable terms are as follows: Theorem 1.4.Let H eff be as in (1.27).Assume k∈Z 3 * V 2 k < ∞.Then e K is well-defined and e K H eff e −K = E corr,bos + H ′ kin + 2 where for any symmetric operators A k , B k : This result is essentially the same as [10,Proposition 5.7], except that we now do not introduce a momentum cut-off and assume only that k∈Z 3

Outline of the Paper
Now we come to the main part of the paper.We will choose as our trial state Ψ = e −K ψ FS .As mentioned the cubic and quartic terms are negligible, so the energy of our trial state energy is by Theorem 1.4, to leading order, The main task will thus be to extract the exchange contribution E corr,ex from this last term.The outline of the paper is as follows: In Section 2 we show that e K is well-defined by proving that K is bounded under the condition We do this by employing a type of higher-order fermionic estimate, resulting in a bound of the form which will also be crucial in allowing us to control N E later.
In Section 3 we establish various bounds on the one-body operators K k , A k (t) and B k (t).This is conceptually similar to the one-body analysis in our previous paper [10], but we must refine several estimates in order to establish control using only the assumption that k∈Z 3 * V 2 k < ∞.In Section 4 comes the main new work: We engage in a detailed study of the exchange terms ) so that we can extract E corr,ex from the last term of equation (1.40), first in the form and then analyze this expression further to obtain the leading order of this, which is precisely E corr,ex as given in Theorem 1.1.
Finally in Section 5 we control the non-bosonizable cubic and quartic terms, and bound the number operator N E and its powers by a Gronwall argument.We end the paper by concluding Theorem 1.1.
Acknowledgements.We would like to thank the editor and the referees for helpful suggestions.MRC and PTN acknowledge the support from the Deutsche Forschungsgemeinschaft (DFG project Nr. 426365943).

The Bogolubov Kernel
We consider the kernel K defined by (1.38).We prove the following: * , be a collection of symmetric operators.Then provided l∈Z 3 * K l 2 HS < ∞, the expression defines a bounded operator K : H N → H N , and for any Ψ, Φ ∈ H N we have Moreover, it was shown in [10] (see also Theorem 3.1) that the kernels in (1.39) satisfy K k HS ≤ C Vk , and hence the boundedness of K follows from the assumption and focus on the boundedness of K. Since for any Ψ, Φ ∈ H N we may estimate by the Cauchy-Schwarz inequality 3) The operator appearing under the root can be written as Let us estimate the following general expression, with some coefficients A p,q,r , (2.5)

A Higher Order Fermionic Estimate
Note that the Cauchy-Schwarz inequality trivially implies that but this is non-optimal for fermionic states.The "standard fermionic estimate" states that which can be proved by appealing to the CAR as follows: One can imagine generalizing this to quadratic expressions of the form p,q A p,q c p c q , but this fails since the CAR only yields a commutation relation for such expressions, and not an anticommutation relation.However, for cubic expressions, such as p,q,r A p,q,r c * p c q c r , the CAR does yield an anticommutation relation, allowing the trick to be applied.The anticommutator is of course not constant, but rather a combination of quadratic, linear and constant expressions, but this still yields a reduction in "number operator order", which will be crucial for our estimation of e K N m E e −K later on.We will need the following basic anticommutator: Lemma 2.2.For any p, p ′ ∈ B c F and q, q ′ , r, r ′ ∈ B F it holds that We can now conclude the desired bound: Proof: As in the proof of the standard fermionic estimate (2.8), we have Hence, by the identity of Lemma 2.2, we bound the left-hand side by We estimate the different types of expressions appearing above.Firstly, by the standard fermionic estimate (2.8), and likewise for the other two terms on the first line of equation (2.9).For the terms on the second line we similarly estimate The terms on the third line of equation (2.9) all factorize in a manifestly non-positive fashion, and so can be dropped, while for the fourth line (2.12) Lastly, the terms on the fifth line are seen to simply be constant and easily bounded by , whence the proposition follows.We can now conclude the following bound for K, which in turn implies Proposition 2.1.
Proof: By (2.3) and (2.4), combined with the estimate of Proposition 2.3, we can bound The sum inside the first square root is exactly equal to l∈Z 3 * K l 2 HS .

Analysis of the One-Body Operators
In this section we analyze the operators K k , A k (t) and B k (t) which appear in Theorem 1.4, obtaining the following: Moreover, for all p, q ∈ L k and t ∈ [0, 1], for a constant C > 0 independent of all relevant quantities.
The analysis of this section is similar to that of [10,Section 7], but compared to that section, the estimates of this section are considerably more precise: We quantify the error of the upper bound on e p , (−K k ) e q , obtain elementwise estimates for A k (t) and B k (t) (rather than only estimates for the norm • ∞,2 as in [10]), and determine the leading term of the operator 1 0 B k (t)dt which will be needed to extract the exchange contribution in the next section.

Matrix Element Estimates for K-Quantities
To ease the notation we will abstract the problem slightly: Instead of ℓ 2 (L k ) we consider a general n-dimensional Hilbert space (V, •, • ), let h : V → V be a positive self-adjoint operator on V with eigenbasis (x i ) n i=1 and eigenvalues (λ i ) n i=1 , and let v ∈ V be any vector such that x i , v ≥ 0 for all 1 ≤ i ≤ n, and let P w (•) = w, • w be the projection onto w ∈ V .Theorem 3.1 will then be obtained at the end by insertion of the particular operators h k and P k .
We define K : V → V by As ≥ h we see that K ≤ 0. In [10, Section 7.2] we proved the following result.
Proposition 3.2.For all 1 ≤ i, j ≤ n it holds that Below it will be more convenient to consider the hyperbolic functions sinh (−2K) and cosh (−2K) rather than e −2K and e 2K .The previous proposition implies the following for these operators: Proof: These bounds follow from Proposition 3.2 and the identities Now we extend our elementwise estimates to more general operators.These estimates are similar to those of Proposition 7.10 of [10], but more precise.First we consider K itself: Proposition 3.4.For any 1 ≤ i, j ≤ n it holds that Proof: From the identity which follows by the Mercator series, we thus have that Noting that Proposition 3.2 in particular implies that x i , 1 − e 2K x j ≥ 0 for all 1 ≤ i, j ≤ n, whence also x i , 1 − e 2K m x j ≥ 0 for any m ∈ N, we may estimate which is the lower bound.This similarly implies that x i , (−K) m x j ≥ 0 for all 1 ≤ i, j ≤ n, m ∈ N, so the upper bound now also follows from Proposition 3.2 by noting that The proof of Proposition 3.4 is complete.
The fact that x i , (−K) m x j ≥ 0 for all 1 ≤ i, j ≤ n, m ∈ N, has the important consequence that for any such i and j, the functions t → x i , sinh (−tK) x j , x i , (sinh (−tK) + tK) x j , x i , (cosh (−tK) − 1) x j (3.6) are non-negative and convex for t ∈ [0, ∞), as follows by considering the Taylor expansions of the operators involved.This allows us to extend the bounds of Corollary 3.3 to arbitrary t ∈ [0, 1]: Proposition 3.5.For all 1 ≤ i, j ≤ n and t ∈ [0, 1] it holds that Proof: By the noted convexity we immediately conclude the upper bounds and by non-negativity of x i , (sinh (−tK) + tK) x j and Proposition 3.4, the lower bound Lastly we can apply the non-negativity of the hyperbolic operators to conclude the bound for e tK − 1 as

Matrix Element Estimates for A(t) and B(t)
We now consider operators A(t), B(t) : V → V defined by We decompose these as We begin by estimating the e tK P v e tK terms: Proposition 3.6.For all 1 ≤ i, j ≤ n and t ∈ [0, 1] it holds that Proof: Writing we see that Now, by Proposition 3.5 we can for any 1 whence the claim follows.
Note that for x i , e tK P v e tK x j this in particular implies the bound We now consider A h (t) and B h (t): Proposition 3.7.For all 1 ≤ i, j ≤ n and t ∈ [0, 1] it holds that Proof: The estimates of Proposition 3.5 imply that The latter estimate only relied on the inequality which is also true for C K (t), so the terms whence Combining equation (3.16) and Proposition 3.7 we conclude the following: Analysis of {K, B(t)} and We end by estimating {K, B(t)} and 1 0 B(t)dt, the latter of which will be needed for the analysis of the exchange contribution in the next section.Proposition 3.9.For all 1 ≤ i, j ≤ n and t ∈ [0, 1] it holds that Proof: Using the Propositions 3.4 and 3.8 we see that This estimate is also valid for | x i , B(t)Kx j | whence the claim follows.
Proposition 3.10.For all 1 ≤ i, j ≤ n it holds that we can estimate using the Propositions 3.6 and 3.7 that

Insertion of the Particular Operators h k and P k
Recall that the particular operators we must consider are h k , P k : ℓ 2 (L k ) → ℓ 2 (L k ) defined by p∈L k e p . (3.25) For these we have that In [10] the following estimates for sums of the form p∈L k λ β k,p were proved: In particular, it holds that Inserting these quantities into the statements of the Propositions 3.4, 3.8 and 3.9 yields Theorem 3.1, noting also that by Proposition 3.4

Analysis of the Exchange Terms
In this section we analyze the exchange terms, by which we mean the quantities of the expression which appears in Theorem 1.4.The name is apt as these enter our calculations due to the presence of the exchange correction ε k,l (p; q) of the quasi-bosonic commutation relations (see Lemma 1.3).To be precise, we will consider in this section the operators , and the effect of the integration will be handled in the next section.The main result of this section is the following estimates for them: Theorem 4.1.For any Ψ ∈ H N and t ∈ [0, 1] it holds that for a constant C > 0 depending only on k∈Z 3 * V 2 k .The constant terms in the final estimate of the theorem give the exchange contribution It is not generally negligible for singular potentials V , and the leading behavior is given by by Proposition 4.2.It holds that , where

Analysis of ε k Terms
Let us first consider terms of the form k∈Z 3 * ε k (A k ), where we recall that When summing over k ∈ Z 3 * , we can split the sum into two parts and interchange the summations as follows: Recalling that q on H N , we can then immediately conclude that By the estimates of the previous section we thus obtain the first estimate of Theorem 4.1: Proof: By Theorem 3.1 we have that for a constant C > 0 independent of all quantities, so

Analysis of E 1 k Terms
We consider terms of the form Recalling that ε k,l (e p ; e q ) = − δ p,q c q−l c * p−k + δ p−k,q−l c * q c p (4.9) we see that E 1 k (A k ) splits into two sums as The two sums on the right-hand side have the same "schematic form": They can be written as where the index set is either the lune S k = L k or the corresponding hole states S k = L k − k, and depending on this index set the variables p 1 , p 2 , p 3 , p 4 are given by To bound a sum of the form k∈Z 3 * E k 1 (A k ) it thus suffices to estimate the two schematic forms

Preliminary Estimates
We prepare for the estimation of these schematic forms by deriving some auxilliary bounds for the operators involved.Recall that for any Proof: By the triangle and Cauchy-Schwarz inequalities we immediately obtain and the bound for b * k (ϕ)Ψ now follows from (4.16) and the fact that It is straightforward to see that N k ≤ N E .Moreover, by rearranging the summations, on H N .We also note that for any Ψ ∈ H N and p ∈ Z 3 as follows by the inequality (considering p ∈ B c F for definiteness) To analyze the commutator term b −l (K −l e p 4 ) , c * p 3 we calculate a general identity: For any so for our particular commutator we obtain b −l (K −l e p 4 ) , c *

.23)
It will be crucial to our estimates that the prefactors obey the following: Proposition 4.5.For any k, l ∈ Z 3 * and p ∈ S k ∩ S l it holds that Proof: Recall that p 1 , p 2 , p 3 , p 4 are given by From this we see that for any p ∈ S k ∩ S l where the assumption that p ∈ S k ∩ S l enters to ensure that 1 , respectively.Importantly this also implies that, when combined with such an indicator function, we also have the identity

.26)
The claim now follows by applying these identities to the estimates which are given by Theorem 3.1.
Below we will only use the simpler bound but for the E 2 k terms the more general ones will be needed.

Estimation of k∈Z
Now the main estimate of this subsection: Proposition 4.6.For any collection of symmetric operators (A k ) and Ψ ∈ H N it holds that Proof: Using the triangle and Cauchy-Schwarz inequalities and Proposition 4.4 we estimate k cp 2 Ψ (N −l + 1) and the first bound now follows by recalling that K l 2 HS ≤ C Vl .For the second we have by the equations (4.23) and (4.28) that where we used A k e p 1 λ Proof: By Theorem 3.1 we have Inserting these estimates into Proposition 4.6 yields the claim.

Analysis of E 2 k Terms
Now we come to the terms We will analyze these similarly to the we find that E 2 k (B k ) splits into two sums as and again these share a common schematic form, namely where the momenta are now the summand decomposes into 8 schematic forms as Of these it should be noted that only the last one is proportional to a constant (i.e.does not contain any creation or annihilation operators).As the rest annihilate ψ FS , it follows that (when summed) the constant term yields precisely ψ FS , E 2 k (B k )ψ FS , whence bounding the other terms amounts to estimating the operator as in the statement of Theorem 4.1.

Estimation of the Top Terms
We begin by bounding the "top" terms By the quasi-bosonic commutation relations, the commutator term reduces to where we used that p 1 = p 4 and p 2 = p 3 when k = l.Now, the exchange correction of the second sum splits as The estimates for the top terms are as follows: Proposition 4.8.For any collection of symmetric operators (B k ) and Ψ ∈ H N it holds that for a constant C > 0 depending only on Proof: The first term we can estimate as in Proposition 4.6 by E Ψ .For the commutator term we have By the matrix element estimate for K k of Theorem 3.1 we have for any p ∈ L k that whence the claim follows as K l HS ≤ C Vl .

Estimation of the Single Commutator Terms
For the single commutator terms and we note that by equation (4.22), the commutator b l (K l e p 4 ) , c * p 3 is given by

.51)
The prefactors again obey an estimate as in Proposition 4.5: Proposition 4.9.For any k, l ∈ Z 3 * and p ∈ S k ∩ S l it holds that The proof is essentially the same as that of Proposition 4.5 (indeed, this proposition can be obtained directly from the former by appropriate substition, but some care must be used since the p i 's differ in their definition).
For the single commutator terms we again only need the simpler bound but the full one will be needed for the double commutator terms below.Now the estimate: Proposition 4.10.For any collection of symmetric operators (B k ) and Ψ ∈ H N it holds that for a constant C > 0 depending only on Proof: As in the second estimate of Proposition 4.6 we have so the second term can be bounded as

Estimation of the Double Commutator Terms
Finally we have the double commutator terms An identity for the iterated commutators is obtained by applying the identity of equation (4.22) to itself: For any k, l The estimates are the following: Proposition 4.11.For any collection of symmetric operators (B k ) and Ψ ∈ H N it holds that Proof: For these estimates we consider only the case S k = L k for the sake of clarity, i.e. we let (p 1 , p 2 , p 3 , p 4 ) = (p, −p + l, −p + k, p) ; (4.58) the case S k = L k − k can be handled by similar manipulations.Using the identity of equation ( 4.57) we start by estimating (by the bound of Proposition 4.9) where we used k∈Z 3 * 1 L l (p 3 + l)λ −1 l,p 3 +l ≤ q∈L l λ −1 l,q ≤ Ck F .From (4.57) we have , e p 3 +l e p 2 +k , B k e p 1 cp 3 (4.60)so the second term can be similarly estimated as Finally, from (4.51) and (4.54) we see that b l (K l e p 4 ) , c * The E 2 k bound of Theorem 4.1 now follows: Proposition 4.12.For any Ψ ∈ H N and t ∈ [0, 1] it holds that for any B k , and as our estimate for B k (t) in Theorem 3.1 is the same as that for A k (t), the bounds follow exactly as those of Proposition 4.7.Insertion into the Propositions 4.8, 4.10 and 4.11 yields the claim.

Analysis of the Exchange Contribution
Finally we determine the leading order of the exchange contribution.To begin we derive a general formula for a quantity of the form where, using equation (4.22) in the form the terms A and B are given by δ p+q,k+l e p , B k e q e q , K l e p and similarly δ −p−q,k+l e p+k , B k e q+k e q+l , K l e p+l .
Although non-obvious, there holds the identity A = B. To see this we rewrite both terms: First, for A, we note that the presence of the δ p+q,k+l makes the L l of the summation p, q ∈ L k ∩ L l redundant: For any p, q ∈ B c F there holds the equivalence δ p+q,k+l e p , B k e q e q , K l e p = p,q∈L k e p , B k e q e q , K p+q−k e p .(4.71) A similar observation applies to B: For any p, q ∈ B F we likewise have e p+k , B k e q+k e −p−k , K −p−q−k e −q−k = p,q∈L k e p , B k e q e q , K p+q−k e p where we lastly used that the kernels K k obey e −p , K −k e −q = e p , K k e q = e q , K k e p , k ∈ Z 3 * , p, q ∈ L k .(4.74) In all we thus have the identity δ p+q,k+l e p , B k e q e q , K l e p (4.75) = − p,q∈L k e p , B k e q e q , K p+q−k e p .
Our matrix element estimates of the last section now yield the following: , where Proof: Since all the one-body operators are real-valued we can drop the Re (•) and apply the above identity for B k (t)dt e q e q , (−K l ) e p .Now, note that E corr,ex can be written as since, much as in Proposition 4.5, the δ p+q,k+l implies the following identity for the denominators: In conclusion we thus see that We estimate A and B. By the matrix element estimates of Theorem 3.1 we have that (using our freedom to replace λ l,p + λ l,q by λ k,p + λ k,q ) where we applied the inequality q∈L k λ −1 k,q ≤ Ck F and also used that Proposition 3.11 implies that for a C > 0 independent of all quantities.By Cauchy-Schwarz we can further estimate for the bound of the statement.By similar estimation also and the claim follows likewise.

Estimation of the Non-Bosonizable Terms and Gronwall Estimates
In this section we perform the final work which will allow us to conclude Theorem 1.1.
The main content of this section lies in the estimation of the non-bosonizable terms, by which we mean the cubic and quartic terms The cubic terms C will not present a big obstacle to us: As was first noted in [2] (in their formulation), the expectation value of these in fact vanish identically with respect to the type of trial state we will consider.The bulk of the work will thus be to estimate the quartic terms.We prove the following bounds: With these all the general bounds are established.As all our error estimates are with respect to N E and powers thereof, it then only remains to control the effect which the transformation e K has on these.By a standard Gronwall-type argument this control will follow from the estimate of Proposition 2.4, and we then end the paper by concluding Theorem 1.1.

Analysis of the Cubic Terms
Expanding the Re (•), the cubic terms are The operators B k can be written simply as As a consequence, B k maps the eigenspace these annihilate and create one hole or excitation, respectively, whence It follows that C maps the eigenspace we thus see that C maps each subspace into the other.On the other hand, since our transformation kernel K is of the form we note that K maps each

Analysis of the Quartic Terms
Now we consider the quartic terms We begin by rewriting these: Recalling the decomposition D k = D 1,k + D 2,k above, we calculate and similarly Noting that D 1,k = 0 for |k| > 2k F , as then B F ∩(B F + k) = ∅, we thus obtain the decomposition where G is the one-body operator the long-range terms Q LR are given by and the short-range terms Q SR are Estimation of G and Q LR G and the long-range terms are easily controlled: First, interchanging the summations we can write G as .16)This implies the following: for a constant C > 0 independent of all quantities.
Proof: For any p ∈ Z 3 we estimate by Cauchy-Schwarz where we lastly used that k → min {|k| , k F } −1 is radially decreasing and that (B F + p) ∩ Z 3 * contains at most |B F | points.As it is well-known that k∈B(0,R)\{0} |k| −1 ≤ CR 2 as R → ∞ the bound follows.
Q LR can be handled in a similar manner: for a constant C > 0 independent of all quantities.
Proof: Consider the first term in the parenthesis of (5.13):For any k ∈ Z 3 * we can estimate As e.g.

Analysis of Q SR
Lastly we come to Recall that the transformation K can be written as (5.20) To determine e K Q SR e −K we will need the commutator [K, we deduce (with the help of Lemma A.1) that where we for the third inequality substituted p → q and k → −k in the second sum.By the identity of equation (4.22) the anti-commutator is given by b l (K l e q ), c * p = 2 c * p b l (K l e q ) + 1 L l (p) K l e q , e p cp−l (5.25) which is inserted into the previous equation for the claim.
We bound the G operator as follows: Proposition 5.6.For any Ψ ∈ H N it holds that Proof: Using Proposition 4.4 we estimate the sum of the first term of G as Now, the K k HS estimate of Theorem 3.1 and Cauchy-Schwarz lets us estimate for a constant C > 0 independent of all quantities, so in all the first term of G obeys Similarly, for the second term (using simply that cp−l Op = 1 at the beginning)

Gronwall Estimates
We now establish control over the operators e K N m E e −K for m = 1, 2, 3. Consider first the mapping t → e tK N E e −tK : Noting that for any Ψ ∈ H N This proves the bound for N E ; for N 2 E we will as in [10] apply the following lemma: The assumption that Summing over both k and −k, we obtain the desired identity.
To state the commutator of K with Q k 2 terms we note the identity where we introduced the convenient notation where we also used that by the assumptions on A k (t) and B k (t) (A.12) The proof of Proposition A.6 is complete.
From this we can also easily deduce the action of e K on H ′ kin : Proposition A.7.It holds that Proof: By the Propositions A.3 and A.5 we see that We are now equipped to conclude Theorem 1.4.By the two previous propositions, we see that

. 12 )
Note that in either case p 1 , p 3 only depend on p and k, while p 2 , p 4 depend only on p and l.Additionally, p 1 is always an element of L k and p 4 is always an element of L −l .Since b k,p = c * p−k c p = cp−k cp it is easily seen that [b, c] = 0, so in normal-ordering (with respect to ψ FS ) the summand of equation (4.11) we find b

Proposition 4 . 4 .
p∈L k ϕ, e p b k,p = p∈L k ϕ, e p c * p−k c p .(4.15) Denote N k = p∈L k b * k,p b k,p .We can bound both b k (ϕ) and b * k (ϕ) as follows: For any k

.37) Again p 1 , p 3 3 *+ 2 p 2 * * cp 3 −p 2
only depend on p and k while p 2 , p 4 only depend on p and l.We normal order the summand: Asb k (B k e p 1 ) c * p 2 cp 3 , b * l (K l e p 4 ) = c * p 2 b k (B k e p 1 ) {c p 3 , b * l (K l e p 4 )} + b k (B k e p 1 ) , c * p 2 {c p 3 , b * l (K l e p 4 )} = 2 c * p 2 b k (B k e p 1 ) b * l (K l e p 4 ) cp 3 + c * p 2 b k (B k e p 1 ) b l (K l e p 4 ) , c * p 3 * + 2 b k (B k e p 1 ) , c * p 2 b * l (K l e p 4 ) cp 3 + b k (B k e p 1 ) , c * p 2 b l (K l e p 4 ) , c * p 3 * (4.38) = 2 c * p 2 b * l (K l e p 4 ) b k (B k e p 1 ) cp 3 + 2 c * p 2 [b k (B k e p 1 ) , b * l (K l e p 4 )] cp 3 + c * p 2 b l (K l e p 4 ) , c * p 3 * b k (B k e p 1 ) + c * p 2 b k (B k e p 1 ) , b l (K l e p 4 ) , c * p b * l (K l e p 4 ) b k (B k e p 1 ) , c * p 2 cp 3 + 2 b l (K l e p 4 ) , b k (B k e p 1 ), c * b l (K l e p 4 ) , c * p 3 * b k (B k e p 1 ) , c * p 2 + b k (B k e p 1 ) , c * p 2 , b l (K l e p 4 ) , c * cp 3 , b * l (K l e p 4 ) b k (B k e p 1 ) = c * p 2 {c p 3 , b * l (K l e p 4 )} b k (B k e p 1 ) (4.39) = 2 c * p 2 b * l (K l e p 4 ) b k (B k e p 1 ) cp 3 + c * p 2 b l (K l e p 4 ) , c * p 3 * b k (B k e p 1 )

1 0e
p cp−l c−q+l c−q+k cp−k .Proof: By the fundamental theorem of calculus e K Q SR e −K = Q SR + tK [K, Q SR ] e −tK dt (5.23) and as noted [K, Q SR ] = 2 Re K, Q SR .Using equation (5.22) we compute that G := K, Q SR is given by

e tK 2
Re E 1 k (h k ) e −tK dt.(A.14) Applying Proposition A.6 now yields the claim.
The claim follows by p∈L l λ −1 l,p ≤ Ck F .
* E 1 k (A k (t)) ofTheorem 4.1 now follows by our matrix element estimates: Proposition 4.7.For any Ψ ∈ H N and t ∈ [0, 1] it holds that e q+k e q+l , K l e p 4 c * q+l cq+k which are both of the schematic form q∈S ′ k ∩S ′ l B k e p 1 , e q 1 e q 4 , K l e p 4 c * q 2 cq 3 .To estimate k,l∈Z 3 * p∈S k ∩S l c * p 2 ε k,l (B k e p 1 ; K l e p 4 ) cp 3 it thus suffices to consider k,l∈Z 3 * p∈S k ∩S l q∈S ′ k ∩S ′ lB k e p 1 , e q 1 e q 4 , K l e p 4 c * e p 2 +k e p 3 +l , K l e p 4 hence K preserves H even N and H odd N , and so too does the transformation e −K .As any eigenstate Ψ ∈ H N of N E is contained in either H even N or H odd N , and these are orthogonal, we conclude the following: Proposition 5.2.For any eigenstate Ψ of N E it holds that 1−t)K dt where for t ∈ [0, 1], tK k h k e tK k + e −tK k h k e −tK k , h 2 k (t) = 1 2 e tK k h k e tK k − e −tK k h k e −tK k .