Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\cdot | \hat{V} \in \ell ^1 (\mathbb {Z}^3)$$\end{document}|·|V^∈ℓ1(Z3). Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.


Introduction
The interacting high-density Fermi gas models a variety of important physical systems, in particular the behavior of electrons in alkali metals.The simplest approximation for the computation of its physical properties is mean-field theory, i. e., Hartree-Fock theory.Hartree-Fock theory only includes the minimal amount of quantum correlations unavoidable due to the antisymmetry requirement on the wave function of fermionic many-body systems.
In the present paper we consider corrections to the Hartree-Fock energy due to non-trivial quantum correlations (i.e., entanglement in the ground state).
According to [BP53], the dominant effect of correlations on the ground state energy should be described by the random-phase approximation (RPA), which may also be formulated as a partial resummation of the perturbation series [GB57] or as a theory of particlehole pairs behaving as bosonic quasiparticles [SBFB57].The latter point of view was recently used by [BNPSS21a,BNPSS20] (extending the second-order result of [HPR20]) to rigorously prove the validity of the random-phase approximation for the ground state energy, assuming the interaction potential to be small and its Fourier transform to have compact support.In the present paper, that result is generalized to arbitrarily large interaction potentials without restriction on the support.Our proof is a refinement of the method of [BNPSS21a, BNPSS20], a crucial point of which is to delocalize particle-hole pairs over patches on the Fermi surface, thus circumventing the Pauli principle and justifying the approximate bosonization of particle-hole pairs.This approach leads to a bosonic quasifree effective theory, from which the ground state energy can be computed.
The further predictions of this bosonic effective theory have been discussed in [Ben20] and it has also been proven to be a good approximation for the time evolution of the Fermi gas [BNPSS21a], refining the time-dependent Hartree-Fock approximation derived in [BSS18,BPS14a,BPS14b,BJPSS16].An alternative approach to the ground state energy, avoiding delocalization and thus closer in spirit to [SBFB57] has been developed recently in [CHN21]: still, also there an averaging over different particle-hole pairs is needed to justify the bosonization.In another context, the low-density Fermi gas, bosonization ideas have been applied by [FGHP21,Gia22a,Gia22b].
Let us turn to the mathematical description of our result.We consider a system of N fermions on the torus T 3 := R 3 /(2πZ 3 ) interacting through a potential V .The system is described on the Hilbert space L 2 a (T 3N ), consisting of all ψ ∈ L 2 (T 3N ) that are antisymmetric under exchange of particles, ψ(x σ(1) , . . ., x σ(N ) ) = sgn(σ)ψ(x 1 , . . ., x N ) for all permutations σ ∈ S N .The Hamiltonian is the linear self-adjoint operator (1.1) The interaction potential V is assumed to have non-negative Fourier transform V ≥ 0.
(For the interaction potential we use the convention that the Fourier transform is V (x) = k∈Z 3 V (k)e ik•x , unlike for the Fourier transform of wave functions which we normalize to be unitary.)Because of the antisymmetry of the wave functions, the sum of the Laplacians is typically of order N 5/3 , as may be seen most easily from the the non-interacting case V = 0, where the ground state is a Slater determinant of N plane waves f k (x) = (2π) −3/2 e ik•x , the momenta k ∈ Z 3 being located in a ball of radius proprtional to N 1/3 .To make both kinetic and potential energy scale extensively (i.e., proportionally to the number of particles N ) we set := N −1/3 and λ := N −1 .This is interpreted as a mean-field limit coupled to a semiclassical limit with effective Planck constant = N −1/3 → 0 as N → ∞; this scaling limit has been introduced by [NS81,Spo81] to derive the Vlasov equation from many-body quantum mechanics.We are interested in the ground state energy A first approximation for E N is the Hartree-Fock energy, defined by restricting the variational problem to Slater determinants, i. e., u j where {u j } N j=1 is an orthonormal family in L 2 (T 3 ) .
As already mentioned, for the non-interacting case V = 0, the Hartree-Fock and the manybody ground state energy are attained by the Fermi ball with the plane waves f k (x) := (2π) −3/2 e ik•x , for x ∈ T 3 and k ∈ Z 3 .Here, the Fermi ball B F is a set of N different momenta p ∈ Z 3 with p |p| 2 as small as possible.To simplify our analysis we assume that the Fermi ball is completely filled and thus uniquely defined, i. e., that B F = {k ∈ Z 3 : |k| ≤ k F }.This can be achieved by considering a sequence k F → ∞ and fixing N := |B F | as a function of k F .We find the relation k F = κN 1/3 between the two parameters, with κ = κ 0 + O(N −1/3 ) and κ 0 := (3/4π) 1/3 .Under the assumption of a complete Fermi ball and non-negative V , it was proven in [BNPSS21a, Theorem A.1] that the Hartree-Fock energy E HF N is still attained by the Fermi ball (1.2), even when V = 0.It follows that In this paper we focus on the correlation energy, defined as the difference E N − E HF N , due to many-body interactions among particles.The following theorem, our main result, provides an explicit formula for the dominant order (order ) of the correlation energy.
Theorem 1.1 (Main result: RPA correlation energy).Suppose V ∈ L 1 (T 3 ) with V ≥ 0 and Then there exists α > 0 such that where the RPA energy formula is (1.5) Remarks.(i) Unlike the result of [BNPSS21a], where V ℓ ∞ was assumed to be small, here we do not assume smallness of the interaction potential.
(ii) A further generalization is given in Appendix A: there, the upper bound of (1.4) is shown to hold assuming only V ≥ 0 and k∈Z 3 |k| V (k) 2 < ∞.Thanks to only the second power of the potential appearing, this almost covers the Coulomb potential.While our paper was under review, a new upper bound for the correlation energy has been established in [CHN22] for square integrable potentials; this includes potentials with Coulomb singularity.In this case, an additional second order contribution to the exchange energy, which is part of the error in our setting, becomes relevant.
In the next section we will introduce the correlation Hamiltonian which describes corrections to Hartree-Fock theory.In Section 3 we give a heuristic introduction to the bosonization method by which the correlation Hamiltonian can be approximately diagonalized.The remaining sections are dedicated to the steps of the rigorous implementation of this strategy, culminating in the proof of Theorem 1.1 in Section 9.

Correlation Hamiltonian
As the first step to the proof of Theorem 1.1, we apply a particle-hole transformation to the Hamiltonian, by which we obtain the correlation Hamiltonian which describes only the corrections to mean-field (Hartree-Fock) theory.This is an exact computation not involving any approximation.
We use second quantization on the fermionic Fock space F = n≥0 L 2 (T 3 ) ⊗an .On F, we use the well-known creation and annihilation operators satisfying canonical anticommutation relations, namely for all momenta p, q ∈ Z 3 we have {a p , a * q } = δ p,q , {a p , a q } = {a * p , a * q } = 0 . (2.1) As a simple consequence of (2.1), we find the operator norms a * p op ≤ 1 and a p op ≤ 1 for all p ∈ Z 3 .We define the vacuum vector Ω = (1, 0, 0, . . . ) ∈ F and the number-of-fermions operator N = p∈Z 3 a * p a p .We extend the Hamiltonian (1.1) to the full Fock space F setting The restriction of H N to the N -particle sector L 2 a (T 3N ) ⊂ F coincides with (1.1).To analyse the correlation energy E N − E HF N , it is convenient to factor out the Fermi ball (1.2) and focus on its excitations.This is achieved through a particle-hole transformation where we defined the number-of-holes operator N h := h∈B F a * h a h and the number-ofparticles operator N p := p∈B c F a * p a p .This shows that the N -particle sector L 2 a (T 3N ) ⊂ F is the image under R F of the eigenspace of N h − N p associated with the eigenvalue 0 (and thus R F defines a unitary map from the eigenspace χ(N h − N p = 0)F to L 2 a (T 3N )).We introduce the correlation Hamiltonian H corr by conjugating H N with R F and subtracting the energy of the Fermi ball (which, as already noted in [BNPSS21a, Theorem A.1], in our scaling limit and with V ≥ 0 equals the Hartree-Fock ground state energy).With (2.3) and the canonical anticommutation relations (2.1), a lengthy but straightforward computation leads to the correlation Hamiltonian with the main terms (2.5) and the error terms (2.6) Here we defined the delocalized particle-hole pair creation and annihilation operators b * (k) := and the non-bosonizable operators To prove Theorem 1.1, we improve the bosonization method introduced in [BNPSS20] for the upper bound and show that

Strategy of the Proof: Approximate Bosonization
The key idea is to derive, from the fermionic correlation Hamiltonian (2.4), a quadratic, approximately 1 bosonic, Hamiltonian which can be approximately diagonalized by a Bogoliubov transformation to obtain the ground state energy.
1 With approximate bosonization we refer to the fact that we construct operators that only up to an error term satisfy canonical commutator relations; this is in contrast to certain one-dimensional fermionic systems [ML65] and spin systems [CG12, CGS15, Ben17, NS19] which can be expressed through operators that satisfy the canonical commutator relations exactly.
Figure 1: Decomposition of (a shell around) the Fermi surface into patches.The vectors ωα (marked with dots) are the patch centers.The decomposition of the southern half sphere is obtained through reflection by the origin.See [BNPSS20] for the details of the construction.
The starting point is the observation that the particle-hole pair operators behave approximately as bosonic creation and annihilation operators, i. e., they approximately satisfy canonical commutator relations: Thus Q B can be understood as an approximately bosonic quadratic Hamiltonian.The terms X, E 1 , and E 2 do not have a bosonic interpretation and are going to be estimated as smaller errors.It remains to bosonize the kinetic energy H 0 .Because this step requires us to linearize the dispersion relation, we need to localize of the pair operators to patches B α , i. e., to M small regions covering a shell around the Fermi sphere in momentum space (see Fig. 1 for an illustration of the patch decomposition we have in mind; eventually the number of patches M will be chosen to tend to infinity as N → ∞): with a normalization constant n α (k) so that the one-pair states b * α (k)Ω have norm one.There is a catch here: the sum over pairs in (3.1) is only non-empty if the relative momentum k is pointing outward from the Fermi ball, so for about half of the possible values of α the operators b * α (k) vanish.To be sure that many particle-hole pairs contribute to the sum defining b * α (k), we introduce a cutoff by defining the index set (with δ > 0 to be optimized at the end) and combine the retained b * α (k)-operators into These operators again behave approximately bosonic in the sense that This provides important intuition on how to make the approximate bosonization rigorous: because n α (k) 2 counts the number of particle-hole pairs of relative momentum k in patch B α , we need the size of the patches to be sufficiently big and we need to bound the number of excitations counted by N in states close to the ground state.By virtue of the localization to patches we can linearize the dispersion relation e(p) locally in every patch, and thus find (the computation here shown for the case if we introduce the quadratic approximately bosonic operator While the substitution of H 0 by D B has here been motivated only in commutators with almost bosonic operators, a key step of our analysis is to justify this step also on general states close to the ground state.This step is explained in (3.8) to (3.11).Our further goal is to approximately (to order , the dominant contribution of the correlation energy) diagonalize the bosonic quadratic Hamiltonian D B + Q B by an approximately bosonic Bogoliubov transformation T , allowing us to read off the correlation energy.Given a state ψ ∈ F such that (N p − N h )ψ = 0 (think of the ground state of H corr ), and setting ξ := T * ψ, we write Through a suitable choice of the Bogoliubov kernel K(k) (a matrix indexed by the patch labels), the approximate Bogoliubov transformation diagonalizes approximately the quadratic Hamiltonian D B +Q B .On states with few particles (ie.with few excitations of the Fermi sea), we find as suggested by exact bosonic Bogoliubov theory that T ξ, with the intended E RPA N as in (1.5), and for the description of the possible bosonic excitation one obtains an effective Hamiltonian of the form To make these heuristics rigorous, apart from controlling the bosonic approximation (arising from the neglect of the error term in (3.2)) in the bosonic Bogoliubov diagonalization, we need to estimate the second and the third terms in (3.4).There are two obstacles.One is to give a meaning to the heuristics H 0 ≃ D B , which, a priori, holds only as in (3.3), at the level of commutators with the approximately bosonic operators.The other is to control the non-bosonizable term E 1 and the term E 2 which couples almost bosonic c-operators to non-bosonizable d-operators.(The exchange term X instead can be controlled by more elementary estimates.) Both problems were solved in [BNPSS21a] under the assumption that the interaction potential V is small and compactly supported in Fourier space.In the present work we overcome these limitations and prove the validity of the random-phase approximation for a much larger class of interaction potentials.The main achievements of the present paper, compared to [BNPSS20,BNPSS21a], are the following: • The combination H 0 − D B is approximately invariant under conjugation with the approximately bosonic Bogoliubov transformation because its action can be expanded in commutators: In the proof of the upper bound for the correlation energy, the vector ξ coincides with the vacuum, and the right-hand side is zero.For the lower bound this is not true, and we are left with controlling the negative term −D B .In [BNPSS21a], this was achieved by exploiting the positivity of H exc B in (3.6).More precisely, we proved that which, for small potential, is enough to control the r.h.s. of (3.8).In the present paper, we need a more refined analysis.In order to compare H exc B with D B , we need to diagonalize the matrix K(k) α,β appearing on the r.h.s. of (3.7) (because D B is already expressed through a diagonal matrix).This can be achieved through a second approximately bosonic Bogoliubov transformation having the form for an antisymmetric matrix L(k) α,β .If c * and c were bosonic operators, we could write Z = exp k∈Γ nor dΓ(L(k)) = k∈Γ nor Γ(e L(k) ) (where dΓ and Γ are the operators of bosonic second quantization) and its action on (3.7) would be simply (3.11) • In [BNPSS21a], we controlled the non-bosonizable error terms as, informally stated, explaining the necessity of the interaction potential being small to control this term by a positive H 0 .In the present paper instead we control E 1 more precisely.In particular, we prove that on states ξ close to the ground state of the correlation Hamiltonian, the following improved bound holds true (see Lemma 4.8): This means that the contribution of the non-bosonizable term E 1 to the energy is subleading with respect to E RPA N , which is of order .Concerning E 2 , by the Cauchy-Schwarz inequality we get (see Corollary 4.9) The first term in the bound is controlled by the improved bound (3.12), while the second term is controlled by positivity of T ξ, H 0 T ξ in (3.4), for N large enough without any smallness assumption on V .
• Furthermore, to implement this strategy, we improve the a-priori bounds on the number and the energy of excitations: our Lemma 4.1 and Corollary 4.2 generalize estimates of [BNPSS21a] to interaction potentials with V ≥ 0 and | • | V ∈ ℓ 1 (Z 3 ).Moreover, Lemma 4.3 now holds uniformly in k.
The rigorous implementation is the subject of all remaining sections.

A-Priori Estimates on Excitations of the Fermi Ball
The following lemma shows that vectors with total energy close to the ground state energy contain also only a small amount of kinetic energy.
Then there exists a C > 0 such that we have Hence, for every ψ ∈ L 2 a (T 3N ) with ψ = 1 and ψ, H N ψ ≤ E HF N + C the excitation vector Remark.In the present paper we will apply Lemma 4.1 to the ground state ψ gs , which by the variational principle even satisfies ψ gs , H N ψ gs ≤ E HF N .
Proof of Lemma 4.1.From V ≥ 0 we get 0 ≤ Switching to Fock space F and conjugating with R F , we conclude that We compare the r.h.s. of (4.1) with the Hartree-Fock energy (1.3).We have By assumption on V , this implies With (1.3) and (4.1) we conclude that The a-priori bound from Lemma 4.1 for the kinetic energy H 0 has several consequences.First of all, it gives control on the number of excitations of the Slater determinant.Here, it is useful to introduce gapped number-of-fermions operators which are easier to control than N .For ε > 0, we define the gapped number operator measuring the number of excitations with momenta further than a distance N −ε from the Fermi sphere.(The definition (4.2) differs slightly from the definition used in [BNPSS21a] but that is merely a matter of convenience.)

Corollary 4.2 (A-priori bounds on particle number).
There exists a constant C > 0 such that, on χ(N p − N h = 0)F, we have Assume furthermore that The bound for N is proven in [BNPSS21a, Lemma 2.4]; (4.4) follows using Lemma 4.1.
Furthermore, the estimate for H 0 from Lemma 4.1 allows us to bound the particle-hole pair operators b(k) and b * (k) introduced in (2.7).

Lemma 4.3 (Kinetic bound on particle-hole pairs).
There exists a constant C > 0 such that, for all k ∈ Z 3 , and moreover The bounds (4.5) and (4.6) have been established in [BNPSS21a, Appendix B] (and previously in [HPR20, Lemma 4.7]) for fixed k (which was sufficient since there only k in the compact support of V was relevant).Here, we improve the proof given in [BNPSS21a] to obtain uniformity in k.We use the following number theoretic result.
Proposition 4.4 (Lattice points in convex bodies, [Hux03]).Let K ⊂ R 2 be a smooth convex body and let RK be its dilation by a factor R > 0, Then, for any γ > 131/208, there exists Remark.The constant C K,γ in the estimate (4.9) depends on the curvature of the boundary of K.In particular, C K,γ is finite as long as the curvature is strictly positive.For us it is sufficient that (4.9) holds for some γ < 1.A simple proof for 2/3 < γ < 1 is given in [Hor03, Theorem 7.7.16](the condition 0 ∈ K given there can always be achieved by a translation).
The second factor is bounded by the kinetic energy as claimed, Therefore it is enough to show We need to further distinguish the cases The case p 2 − (p − k) 2 ≥ 4N 1/3 .We apply the argument used in [FLLS13, Eq. (5.13)].If η ∈ (0, 3 2C 0 ) then for q ∈ B η (p) we have Possibly choosing η > 0 still smaller, the balls B η (p) are disjoint for different p, and we obtain we conclude that F and p − k ∈ B F together imply the lower bound (recall that all momenta are elements of Z 3 ) (This is only an upper bound because (p 2 , p 3 ) ∈ Z 2 for which the r.h.s. of (4.12) is not integer do not contribute to B m (k)).On the (p 2 , p 3 )-plane, we define new variables (q 2 , q 3 ) by In terms of these new variables, we can rewrite (4.13) as We can therefore apply Proposition 4.4 to estimate the number of points (p 2 , p 3 ) ∈ Z 2 contained between the two ellipses described by (4.15).(From the assumptions Inserting this bound in (4.11) and choosing γ < 1 we arrive at To show (4.6), we proceed analogously.The only difference is that now the sum in (4.11) can be restricted to m ≤ CN 1/3−δ (here, the case From Lemma 4.3, we immediately obtain a bound on the operators b(k) and b * (k).For details, see [BNPSS21a, Lemma 2.3].

Corollary 4.5 (Kinetic bound on pair operators).
There exists a C > 0 such that for all Using the last corollary, we obtain an a-priori bound for the bosonizable interaction Q B .
Proof.We observe that, for any k ∈ Z 3 , by Corollary 4.5, After summing over k, this implies the desired estimate for Q B .
Finally, the a-priori bound for H 0 (and the resulting estimates on N and N ε from Corollary 4.2) imply that the error terms in (2.6) are negligible.First of all, the exchange operator X can be bounded with the following lemma, taken from [BNPSS21a, Lemma 2.5].
Lemma 4.7 (Exchange term).Assume V 1 < C. Then there exists a C > 0 such that for all ξ ∈ χ(N p − N h = 0)F we have The next lemma provides control on the error term E 1 in (2.6).It is one of the key achievements of the present paper.
Lemma 4.8 (Non-bosonizable interaction).Assume V 1 < ∞.Fix 0 < ε < 1/3 and 131/208 < γ < 1.Then there exists C > 0 such that for all ξ ∈ χ(N h − N p = 0)F we have Remark.With a localization argument, we will be able to restrict our attention to states for which N ≤ CN 1/3 and N δ ≤ CN δ (for the expectation value as stated in Corollary 4.2, but also for higher moments).Applying (4.16) for such states, choosing γ < 1 and ε > 0 small enough, we conclude that E 1 ≪ N −1/3 and therefore that E 1 does not contribute to the correlation energy, to leading order.
Proof of Lemma 4.8.Recall the definition (2.8) of the operators d * (k) and d(k where we introduced the notation With the canonical anticommutation relations (2.1), we obtain ξ, a * q 1 a q 1 ξ . (4.17) The second term can be estimated by Let us focus on the first term on the r.h.s. of (4.17).The first observation is that contributions with at least one of the four momenta q 1 , q 1 − σ 1 k, q 2 , q 2 − σ 2 k at distances larger than N −1/3+ε from the Fermi sphere, for an 0 < ε < 1/3 to be chosen later, can be bounded using a combination of N and of the gapped number operator N 1/3−ε defined in (4.2).In fact, considering for example the case (and dropping, for an upper bound, all other restrictions on q 1 and q 2 ), we have where we used a * p N = (N − 1)a * p for all p ∈ Z 3 .Thus where we defined the momentum sets Note that for q 1 ∈ A p k we have Inverting the roles of q 1 and q 1 − k, we also obtain 2q k is localized close to the equator of the Fermi sphere, thinking of the direction of k as defining the north pole).
Using the Cauchy-Schwarz inequality and a q 1 op ≤ 1, a q 1 −σ 1 k op ≤ 1, we conclude that the last term on the r.h.s. of (4.18) can be bounded by where we defined Proceeding as in the proof of Lemma 4.3 following (4.11),we find, for 131/208 < γ < 1, Inserting in (4.20) and using V 1 < ∞, we obtain With (4.18) this concludes the proof of Lemma 4.8.
Lemma 4.8 proves that the error term E 1 is negligible (in the ground state and, more generally, on low-energy states with correlation energy of order ).Together with Corollary 4.5, it also allows us to neglect the term E 2 in (2.6).The following corollary improves [BNPSS21a, Lemma 9.1] in not requiring smallness of V , and is also simpler to prove.Corollary 4.9 (Coupling of bosonizable and non-bosonizable terms).Assume V 1 < ∞ and V ≥ 0. With the error terms E 1 , E 2 defined as in (2.6), we have With Lemma 4.8, we conclude that for 131/208 < γ < 1 and ε > 0 small enough (choosing α = ε/4 in (4.22)), there exists a constant C > 0 such that Remark.The choice α = ε/4 optimizes the sum of the first and the last term on the r.h.s. of (4.23), counting (following the argument in the remark after Lemma 4.8 . The second term on the r.h.s. of (4.23) is of lower order if γ is chosen small enough.

Patch Decomposition and Almost Bosonic Operators
The bounds in last section allow us to approximate the correlation Hamiltonian (2.4) by H 0 + Q B , with H 0 and Q B defined in (2.5).The term Q B , arising from the interaction, is quadratic in the particle-hole pair creation and annihilation operators b * (k), b(k).It turns out that, on states with few excitations of the Fermi ball, the operators b * (k) and b(k) satisfy approximately bosonic commutation relations.
In order to express also the kinetic energy H 0 in terms of almost bosonic creation and annihilation operators, we have to decompose a layer around the Fermi sphere ∂B F into M patches {B α } M α=1 , for the number of patches M ∈ N to be chosen as a function of N at the end of the paper.Such a decomposition has been constructed in [BNPSS20].One starts by decomposing a half sphere in M/2 patches.The sidelengths of the patches are comparable (they are both of order N 1/3 /M 1/2 ).The patches have thickness in the radial direction (later we will impose stronger conditions).Furthermore, the patches are disjoint and separated by corridors, larger than R. We denote by ω α the center of the patch B α .Finally, the patch decomposition of the first half sphere is mirrored by the map k → −k onto the other half sphere.The construction is so that the area of the radial projection p α of the patch B α on the unit sphere S 2 has area 4π/M , up to corrections of order N −1/3 M −1/2 , and diameter bounded by C/ √ M , for all α = 1, . . ., M ; see [BNPSS20, Section 3.2] for the details.
For fixed k ∈ Z 3 with |k| < R, we are going to exclude patches in a small strip around the equator (thinking of the direction of k as defining the north direction) of the Fermi sphere.More precisely, for 0 < δ < 1/6, we define I k := I + k ∪ I − k , with Proof.The proof follows the argument given in [BNPSS20, Section 6]; only the control of the error terms needs to be refined in two respects.First, in order for the vector k to point from inside the Fermi ball to outside the Fermi ball even at the boundaries of the patch, we need N 2δ R 2 ≪ M , as can be verified by elementary geometry.This condition is illustrated in Fig. 2.
Second, the error term arising from the loss of particle-hole pairs near the boundary of the patch (thus proportional to the number of pairs in the patch of thickness |k| ≤ R not more than a distance |k| ≤ R from the patch boundary on the Fermi sphere) implies 3) The error term becomes o(1) since by assumption It will be convenient to combine modes associated with k and −k.To this end, we set for every k ∈ Γ nor .Here, we introduce the notation or (k 3 = k 2 = 0 and k 1 > 0) (5.5) so that Γ nor ∩ (−Γ nor ) = ∅ and Γ nor ∪ (−Γ nor ) = B R (0)\{0}.Note that compared to [BNPSS21a], in the definition of Γ nor we replaced the restriction k ∈ supp V by |k| < R, with the parameter R to be optimized at the end.
Our analysis is based on the observation that the pair operators c * α (k) and c α (k) behave approximately as bosonic creation and annihilation operators, on states with few excitations.This is established by the following lemma, taken from [BNPSS20, Lemma 4.1] and [BNPSS21a, Lemma 5.2].
where the error operator E α (k, ℓ) is controlled by the bounds and Another important property of the operators c * α (k) and c α (k) is that they can be controlled in terms of the gapped number of particles operator N δ introduced in (4.2), with δ > 0 the parameter introduced in (5.1) to exclude a strip around the equator of the Fermi sphere in the definition of the sets I k .The point is that, since we are away from the equator, k has a component orthogonal to the Fermi sphere, which makes sure that the momentum of either the particle or of the hole annihilated by c α (k) is at least at distance N −δ from the Fermi sphere.More precisely, we have the following lemma, whose proof can be found in [BNPSS21a, Lemmas 5.3 and 5.4] (the first estimate in (5.10) and in (5.12) are not stated explicitly in [BNPSS21a, Lemmas 5.3 and 5.4] but can be proven like the second bounds).
Lemma 5.3 (Bounds on pair operators).Assume M ≫ R 2 N 2δ and R ≪ N 1/6−δ/2 .For all k ∈ Γ nor we have (5.9) Moreover, for any f ∈ ℓ 2 (I k ), For k ∈ Γ nor , α ∈ I k and g : Z 3 × Z 3 → R, we define the weighted pair operator Similarly to (5.9) and (5.10), we find Furthermore (5.11) and, for f ∈ ℓ 2 (I k ), (5.12) 6 Reduction to an Almost Bosonic Quadratic Hamiltonian Comparing (2.7) with (5.4), we find for all k ∈ Γ nor (these are only approximate decompositions since, on the r.h.s., pairs in corridors and close to the equator are missing).Inserting this decomposition in (2.5) we find the following approximation for Q B , quadratic in c-and c * -operators: The difference between Q B and Q R B is estimated in the following lemma, which we take from [BNPSS21a, Lemma 4.1].Compared to [BNPSS21a], here we only need to compare Q B with Q R B since we already controlled E 2 in Corollary 4.9; therefore the bound also does not use E 1 .
Lemma 6.1 (Removing corridors and removing patches near the equator).Assume that Then there exists C > 0 such that for all ψ ∈ F we have Proof.We consider the difference where U k consists of all momenta p ∈ B c F with p − k ∈ B F that do not belong to any patch.
containing pairs close to the equator.Proceeding as in the proof of [BNPSS21a, Lemma 4.1] and using (4.6), we obtain and (again under the assumption that |k| < R) Here we estimated |U k \Y k | ≤ CR|k|N 1/3 M 1/2 (for |k| < R, the set U k \Y k contains momenta p ∈ Z 3 localized in a shell of thickness |k| around the Fermi sphere, so that either the projection of p or the projection of p − k onto the Fermi sphere falls in corridors of size R between patches).For |k| > R, on the other hand, we use Corollary 4.5.We conclude that b(k)− Proceeding as in the last part of the proof of [BNPSS21a, Lemma 4.1], using Corollary 4.5 and the assumption k∈Z 3 V (k)|k| < ∞, we arrive at the intended bound.
To understand how the kinetic energy H 0 , defined in (2.5), can be expressed through the patch-wise particle-hole creation and annihilation operators, we compute the commutator which suggests that, in a sense to be made precise, Based on this heuristic observation, we expect that the correlation Hamiltonian (2.4) can be approximated by with the quadratic (in c-and c * -operators) expression (6.6)

Approximate Bogoliubov Transformations
If the c-and c * -operators were exactly bosonic, we could write with the quadratic Hamiltonian (in the following discussion we omit the fixed argument k) and setting we can decompose Using the polar decomposition S 1 = O|S 1 | with an orthogonal matrix O and the positive matrix However, it will be important to further transform the block-diagonal matrix as to make the resulting quadratic Hamiltonian comparable with the bosonic kinetic energy D B , defined in (6.3).To reach this goal we have to look more closely at E, decomposing it further into blocks associated to the index sets I + k and I − k (associated with patches in the north and south hemisphere, respectively).Note that It will play an important role in the proof of Lemma 7.2 that, as a consequence of (5.1) and Lemma 5.1, we have To block-diagonalize E (w. r. t. the decomposition (where I is the I × I identity matrix) and observe This implies that The upper-left entry is clearly larger than the operator d.It seems more difficult to compare the lower-right entry with d (thus, it seems difficult to compare U T EU with D).To solve this problem, we define the I ×I matrix X := (d+2b) 1/2 d 1/2 and consider its polar decomposition X = AP , with A orthogonal and P := (X * X) 1/2 .Then, from (7.8), we have Using the easily-checked invariance of the matrix with blocks P on the diagonal w. r. t. conjugation with U we conclude that where we defined Inserting in (7.4), we arrive at If the c-and c * -operators were exactly bosonic we could therefore bring the quadratic operator (7.1) into a diagonal form comparable to the bosonic kinetic energy D B by means of the two Bogoliubov transformations 2 where (re-inserting now the dependence on k in the notation) we introduced the matrix Recall that O(k) and O(k) are orthogonal matrices, i. e., all their eigenvalues are on the unit circle.The function log denotes an arbitrary branch of the complex logarithm with Im log 1 = 0.The matrix L(k) is by definition antisymmetric, so that Z is a unitary operator on Fock space.If the c-and c * -operators were exactly bosonic, we would find Recall that tr P = tr E. Since P = (X * X) This comparison is not surprising in view of the discussion of the spectrum of E(k) in [Ben20].
There the problem is reduced to a rank-one perturbation of the matrix D(k); the perturbed eigenvalues are all larger than the corresponding unperturbed eigenvalues.However, E(k) and D(k) cannot be simultaneously diagonalized, so we do not have an operator inequality between E(k) and D(k).This problem is overcome here noting that E(k) can be diagonalized by a Bogoliubov transformation which leaves H 0 − D B (though not D B alone) invariant.
Since the c-and c * -operators are not exactly bosonic, we can expect (7.13) to hold only approximatively, on states with few excitations of the Fermi ball.To prove that this is indeed the case, we need some estimates on the kernels K(k) and L(k).The following bound for K(k) has already been shown in [BNPSS21b, Lemma 2.5].

Lemma 7.1 (Bogoliubov kernel).
There exists a C > 0 such that for all k ∈ Γ nor we have M for all α, β ∈ I k .
In particular K(k) HS ≤ C V (k).
The following bounds for the antisymmetric matrix L(k) are new.
Lemma 7.2 (Kernel of one-particle transformation).Suppose that the parameters δ, M, R used to define the patch decomposition in Section 5 are such that M ≫ R 2 N 2δ .Then there exists a C > 0 such that for all k ∈ Γ nor we have Remark.Since L(k) is the logarithm of an orthogonal matrix, we always have L(k) op ≤ 2π.From Lemma 7.2, we also have Proof.All matrices depend on k but in this proof we do not indicate this dependence explicitly.We split the bound in two parts by Since O is orthogonal we have O op = 1 and we only need to estimate O − 1 HS and O − 1 HS .The same applies for the operator norm.
Bound for O − 1 HS .From the definition (7.9), we get with A the orthogonal matrix arising from the polar decomposition of X = (d + 2b) 1/2 d 1/2 , i. e., A = X(X * X) −1/2 .We have To bound the second term on the r.h.s. of the last equation, we use the representation to write by means of a resolvent identity To control the norms in this integral (and similar norms that will arise in the rest of the proof), we use (7.6) so that, for j = 1, 2 and −1/2 ≤ k ≤ j − 1, we have is the inclination angle of the center ω α of the patch B α , measured w. r. t. the vector k.We consider then the sum on the r.h.s. of (7.21) as a Riemann sum for a surface integral on the northern hemisphere of the unit sphere, parametrized by the angles θ ∈ (0, π/2) and ϕ ∈ (0, 2π).To estimate the error in going from the Riemann sum to the integral, we set and compute its derivative, finding Let p α denote the surface area on the unit sphere S 2 covered by the patch B α .With slight abuse of notation, let us also write p α for the set of inclination angles θ ∈ (0, π/2) corresponding to points in p α .For all θ, θ ∈ p α we have |θ − θ| ≤ CM −1/2 (this being the order of the diameter of the patch).According to the definition (5.1) of the index set, for α ∈ I + k we have cos θ α ≥ R −1 N −δ .Thus for all θ ∈ p α we have where we recall the assumption M ≫ R 2 N 2δ .Moreover, by the mean value theorem (if necessary enlarging the set of angles p α to its convex hull in all the following supremuma to make sure that θ 0 is contained) . Thus for all θ ∈ p α we have We conclude that In particular, with j = 1, k = −1/2, we find To bound the other norm in the integral in (7.20), we write which implies, applying (7.22) with j = 1 and k = 0, Inserting this bound in (7.20) and integrating the variable s separately over the intervals [0, 1] and [1, ∞), we conclude that As for the first term on the r.h.s. of (7.17), we proceed analogously, writing We write b = g|v v|.We can bound d −1/2 v ≤ C, as well as and, using (7.22) with j = 2 and k = 1/2, Combined with (7.17) and (7.20), this implies Bound for ||O − 1|| HS .Recall that O arises from the polar decomposition (7.2) of S 1 , i. e., .
Using the orthogonal matrix U defined in (7.7) and the fact that O − 1 and U T (O − 1)U have the same spectrum we obtain To estimate the first norm on the r.h.s. of (7.23) we decompose We start with the first summand on the r.h.s. of (7.24).With an integral representation similar to (7.18) and using X * X − d 2 = 2d 1/2 bd 1/2 , we write it as Here we used (recalling We conclude therefore that Let us now consider the second summand on the r.h.s. of (7.24).Since for a constant c > 0, independent of N and k.The last two bounds imply that c ≤ (X * X) −1/4 d(X * X) −1/4 ≤ 1 and therefore that with To estimate the Hilbert-Schmidt norm of J, we expand, similarly as we did in (7.19), Writing again b = g|v v| and using the bounds and also (7.22) with j = 2, k = 0 to bound we arrive at J HS ≤ C V (k).Inserting in (7.28) and combining the resulting bound with (7.27), we conclude that We turn to the second term on the r.h.s. of (7.23).Similarly as for the first term The term on the first line can be bounded analogously as we did with the first term on the r.h.s. of (7.24).With Here we used, analogously to (7.26), the bounds (XX * ) 1/2 (d + 2b) −1 op ≤ 1 and On the other hand, we can bound and using again b = g|v v|, we get v, and therefore (proceeding as in the proof of (7.22)) arrive at v, This implies that (7.34) From (7.31), we conclude that Finally, let us consider the term on the second line of the r.h.s. of (7.30).Since which also implies that (XX * ) −1/4 (d + 2b)(XX * ) −1/4 ≥ 1.We define therefore Then we have 1 To estimate the Hilbert-Schmidt norm of W we write With the resolvent identity, we obtain and thus Using (7.22) with j = 2, k = 0, (7.33), and (7.34) we arrive at Applying also (7.32), (d + 2b) −1/2 v ≤ C and Inserting this bound in (7.37) and combining it with (7.35), we can bound (7.30) by Together with (7.29) and with (7.23), we obtain Using the bounds on the kernels K(k) and L(k), our next goal is to show that the unitary transformations T and Z defined in (7.11) act on the c-and c * -operators as bosonic Bogoliubov transformations, up to errors that are small on states with few excitations.(This will allow us to show that conjugation of the r.h.s. of (6.4) by T and Z produces approximately the r.h.s. of (7.13).)To reach this goal, we need to show first that conjugation with T and Z does not change the number operator N and the gapped number operators N δ substantially.We generalize the definition (7.11) for λ ∈ R to so that T = T 1 and Z = Z 1 .
Lemma 7.3 (Stability of number operators).Assume V 1 < ∞ and M ≫ N 2δ R 2 .Then for every m ∈ N there exists C > 0 such that for all λ ∈ [−1, 1] we have Conjugation with Z λ leaves the total number of particles constant, Moreover, for every m ∈ N there exists C > 0 such that, for all λ ∈ [−1, 1], we have Proof.The proof of (7.40) can be found in [BNPSS21a, Lemma 7.2] where it is stated under the additional assumption that V has a compact support; however, using Lemma 7.1 it easily extends to V 1 < ∞.
The invariance of N w.r. t.Z λ follows since the exponent commutes with N (the c *operator creates two fermions while the c-operator annihilates two fermions).
We still have to show (7.41).We consider the case m = 0; the extension to m > 0 is straightforward.We compute Using the weighted pairs operators introduced in Lemma 5.3 we have for a weight function g with values in {0, 1, 2}.Thus and by Cauchy-Schwarz .
Observe that Since C g is a positive matrix, we can use (7.15) to estimate Applying Lemma 5.3 and using V 1 < ∞, we find By Grönwall's lemma, we conclude that for all λ ∈ [−1, 1] we have We can now show that the unitary operators T and Z approximately act on c-and c *operators as bosonic Bogoliubov transformations, up to errors that are negligible on states with few excitations.The action of T is described in the next lemma, whose proof can be found in [BNPSS21a, Lemma 7.1].
In the next lemma, we control the action of Z in an analogous fashion.
Then for every ℓ ∈ Γ nor , γ ∈ I ℓ , and λ ∈ [−1, 1] we have where there exists a C > 0 such that for all ψ ∈ F we have Proof.Recall that L is antisymmetric; hence Z * λ has the same form as Z λ , but with L replaced by −L.For λ ∈ [−1, 1] we compute with the error operator E γ (ℓ, k) introduced in (5.6).In integral form, we obtain Iterating n 0 times, we find (with where, in the last line, for n = 0, we have L(ℓ) 0 γ,β = δ γ,β .Thus, completing the first sum to reconstruct the exponential, we have with error term for an arbitrary n 0 ∈ N.This error term can be estimated by We estimate . With Lemma 7.2, we obtain L(ℓ) n HS ≤ C n , uniformly in N and ℓ.From Lemma 5.3 then Similarly, using the invariance of N w.r. t. conjugation with Z τ , we find Let us finally consider the last term on the r.h.s. of (7.46).We have the bound (5.7), the relation N c α (k) = c α (k)(N − 2), and Lemma 5.3, we find With V 1 < ∞ and Lemma 7.3, we conclude that Since the r.h.s. of both (7.47) and (7.48) vanishes as n 0 → ∞ (and since (7.49) does not depend on n 0 ), we arrive at (7.45).

Linearization of the Kinetic Energy
We will use Lemma 7.5 to show that (7.13) and (7.14) hold approximately true on states with few excitations.What is still missing to conclude the argument explained in Section 2 is the invariance of H 0 − D B w. r. t. the action of the approximate Bogoliubov transformations (7.11).The proof is based on the fact that the commutators of H 0 and D B with the c *operators are approximately the same, as described by the following lemma.
Lemma 8.1 (Kinetic commutators).Let RM 1/2 ≤ N 1/3 .For all k ∈ Γ nor and all α ∈ I k , we have where there exists a C > 0 such that for all f ∈ ℓ 2 (I k ) and all ψ ∈ F we have Proof.The bounds for E lin α are shown as in [BNPSS21a, Lemma 8.2], keeping track of the k-dependence.From (2.1) we get where, using the weighted pair operators as in Lemma 5.3, Since B α has diameter of order N 1/3 M −1/2 on the Fermi surface and since p can be at most at distance |k| from the Fermi surface, we can bound (using the assumption The first two estimates in (8.2) follow from (5.11) and (5.12).The last bound in (8.2) is shown exactly as in [BNPSS21a, Eq. (8.6)], using the bound |Γ nor | ≤ CR 3 to sum over l ∈ Γ nor there.
The invariance w. r. t.T is established in the next lemma.This lemma can be shown as [BNPSS21a, Lemma 8.1], replacing bounds for E lin α and E B α with those established in Lemma 8.1 (and using the assumption k V (k)|k| < ∞).We skip further details.
Then there exists a C > 0 such that for all ψ ∈ F we have In the next lemma, we use (8.2) to show the approximate invariance of H 0 − D B w. r. t. the action of the transformation Z defined in (7.11).
Then there exists a C > 0 such that for all ψ ∈ F we have Proof.Recalling the definition (7.39) of the operators Z λ , we compute With (8.1) we obtain With Lemmas 7.2 and 7.3 we obtain (since Integrating over λ ∈ [0, 1] we arrive at the desired bound.
9 Proof of Theorem 1.1 We use the following proposition for localization in particle number sectors of Fock space.It is taken from [LNSS15, Prop.6.1] (given there for bosonic Fock space, but inspection of the proof shows that the symmetry/antisymmetry of the wave function does not play any role).
Proposition 9.1 (Particle number localization).Let A be a non-negative operator on F with P i D(A) ⊂ D(A) and P i AP j = 0 if |i − j| > ℓ, where Then, there exists a C > 0 (one can take We turn to the proof of our main result.
Proof of Theorem 1.1.The main work is for the proof of the lower bound; the upper bound follows from the same operator estimates but using a specific trial state, for which the errors are easier to control.
Lower bound.Let ψ gs be a normalized ground state vector for the Hamilton operator H N in (1.1).Since the Hartree-Fock energy arises from a restriction of the many-body variational problem to a smaller set, we have Let ξ gs = R * ψ gs denote the excitation vector associated with ψ gs , defined through the unitary particle-hole transformation (2.3).From the definition (2.Next we localize w. r. t. the number of particles.We choose smooth functions f and g as in Proposition 9.1 and set f N := f (N /C 0 N 1/3 ), g N := g(N /C 0 N 1/3 ) for a constant C 0 > 0 large enough, to be fixed below.We set A = H corr + C , with C > 0 large enough.From Lemma 4.1 we get A ≥ 0. From the definition (2.4) of H corr , combined with the bounds in Corollary 4.6 for the operator Q B , in Lemma 4.7 for the exchange operator X and in Corollary 4.9 for the error term E 2 , we conclude that Since H 0 and E 1 both commute with N , it also follows that A diag ≤ C(H 0 + E 1 + ).From Proposition 9.1 (since, with the notation introduced in the proposition, P i AP j = 0 if |i − j| > 4), we find We apply this bound to the ground state ξ gs .From the a-priori bounds in (9.1), we obtain From (9.2), we have, fixing C 0 large enough, g N ξ gs 2 = ξ gs , g 2 (N /C 0 N 1/3 )ξ gs ≤ 1 C 0 N 1/3 ξ gs , N ξ gs ≤ 1 2 .
Hence f N ξ gs 2 ≥ 1/2 and, from (9.4), ξ gs , H corr ξ gs ≥ ξ, H corr ξ − CN −1 (9.5) where we defined ξ = f N ξ gs / f N ξ gs ∈ χ(N p − N h = 0)F (particle number localization leaves the space invariant, since N p and N h commute with N ).Like ξ gs , the localized vector ξ satisfies ξ, H corr ξ ≤ C and therefore by Lemma 4.1 we get ξ, H 0 ξ ≤ C .(9.6) The advantage of working with ξ is that it satisfies stronger bounds (compared with ξ gs ) on the number of particles.In fact, we find for every m ∈ N and ε > 0 (to prove the second estimate, we used [N , N ε ] = 0).From (9.5), to conclude the proof of the lower bound, it is therefore enough to show that ξ, H corr ξ ≥ E RPA N −CN −1/3−α , for sufficiently small α > 0 and for all ξ ∈ χ(N p −N h = 0)F satisfying (9.6) and (9.7).For such vectors, it follows from Lemma 4.7, Corollary 4.9 and Lemma 6.1 that, for any sufficiently small ε, δ > 0 and for N 2δ ≪ M ≪ N 2/3−2δ , ξ, H corr ξ ≥ ξ, (H 0 + Q R B )ξ − C N −1/3 + N −ε/4 + N −(1−γ)/3+5ε/4 + N −δ/2 + R 1/2 M 1/4 N −1/6+δ/2 + R −1/2 (9.8) with the quadratic expression Q R B defined in (6.1) (notice that the definition of Q R B depends on δ).Using the notation introduced in (6.3) and in (6.5), we can write We now focus on the second term on the r.h.s. of (9.9).Writing ξ = T Zη, we compute first the action of T .We proceed here as in the proof of [BNPSS21a, Lemma 10.Comparing with (7.4), we find K(k) = O(k)E(k)O(k) T .The first error term in the square brackets on the r.h.s. of ((9.13)) arises from [BNPSS21a, Eq. (10.10)], a bound which holds under the assumption V 1 < ∞; this follows from the observation that [BNPSS21a, Eq. (10.9)] can be improved to Proceeding similarly to bound the last two terms on the r.h.s. of (9.15), we obtain With Lemma A.2 and Lemma A.3, we can go back to the computation of the expectation value on the r.h.s. of (A.4).We control the expectation of the error term E 1 with the bound established in [BNPSS21b, Eq. (4.10)].With (A.5) and estimating we find, for a constant C R depending on the cutoff R > 0, The expectation value of E 2 in our trial state vanishes for parity reasons exactly as in [BNPSS20, Lemma 5.2].Applying Lemma A.3 and (A.11) and using the fact that X ≤ 0, from (A.4) we get In order to obtain an upper bound for the expectation of the operator H 0 + Q R B , we proceed as in the proof of Theorem 1.1, now with V (k) replaced everywhere by V (k)χ(|k| ≤ R).We conclude that Fixing M = N α , choosing α > 0 small enough and then R = R(N ) so that R(N ) → ∞ as N → ∞ at a sufficiently slow pace, we obtain (A.2).This concludes the proof of the generalized RPA upper bound, Theorem A.1.

(5. 1 )
Given k ∈ Z 3 , |k| < R and α ∈ I + k , we introduce the particle-hole pair creation operatorb * α (k) := 1 n α (k) p : p∈B c F ∩Bα p−k∈B F ∩Bα constant n α (k) 2 := p : p∈B c F ∩Bα p−k∈B F ∩Bα1 counting the number of particle-hole pairs of relative momentum k in B α .The normalization constant n α (k) should be large (the more summands contribute to (5.2), the less the b *operators are affected by the Pauli principle, and the more bosonic they behave).The following lemma is a variation of [BNPSS20, Prop.3.1] and [BNPSS21a, Lemma 5.1].Lemma 5.1 (Number of pairs per patch).Assume that N 2δ R 2 ≪ M ≪ N 2 3 −2δ R −4 .Then for all k ∈ Z 3 with |k| < R and α ∈ I k , we have

Figure 2 :
Figure 2: Illustration for the condition N 2δ R 2 ≪ M of Lemma 5.1.The angle between patch center and patch boundary is θ 1 ∼ 1/ √ M .The angle between the tangent at the center and at the boundary is θ 2 = θ 1 by elementary geometry.We know k • ωα ≥ N −δ by definition of I k .This means that the angle between k and the tangent at the center (being perpendicular to ω α ) is at least of order ∼ N −δ /R.To have k pointing from the inside to the outside of the Fermi ball even at the boundary we need N −δ /R ≫ 1/ √ M .Lemma 5.2 (Approximate bosonic CCR).Let k, ℓ ∈ Γ nor .Let α ∈ I k and β ∈ I ℓ .Then p∈B c F ∩(B F +k)∩Bα (e(p) + e(p − k))a * p a * p−k .With e(p) + e(p − k) = 2 p 2 − 2 (p − k) 2 ≃ 2 κ|k • ωα | (with ωα = ω α /|ω α | the normalized vector pointing to the center of the α-th patch), we obtain symplectic conjugation of the 2|I k | × 2|I k | matrix defining the quadratic Hamiltonian (7.1) is sufficient to obtain a block-diagonal matrix (with |I k | × |I k | blocks OEO T ) corresponding to a "diagonal" quadratic Hamiltonian in the sense of containing only terms of the form c * c and none of the form c * c * or cc.