Particle-Hole Pair Localization on the Fermi Surface and its Impact on the Correlation Energy

P article-Hole P air Lo calization on the F ermi Surface and its Impact on the Correlation Energy Niels Benedikter Universit` a de gli Studi di Milano, Via Cesar e Saldini 50, 20133 Milano, Italy External Scientific Memb er of Basque Center for Applie d Mathematics, A lame da de Mazarr e do 14, 48009 Bilb ao, Bizkaia, Sp ain ORCID: 0000-0002-1071-6091, e–mail: niels.benedikter@unimi.it Marc h 26, 2026 Abstract In recen t y ears it has b een sho wn ho w appro ximate b osonization can be used to justify the random phase appro ximation for correlation energy of in teracting fermions in a mean- field scaling limit. A t the core is the in terpretation of particle-hole excitations close to the F ermi surface at b osons. The main t wo approac hes ho wev er differ in emphasizing collectiv e degrees of freedom (particle-hole pairs delo calized o ver patches on the F ermi surface) or particle-hole pairs exactly lo calized in momen tum space. Both metho ds lead to equal precision for the correlation energy with regular interaction p oten tials. This p oses the question ho w big the influence of delocalizing particle-hole pairs really is. In the presen t note we show that a description with few, completely collective b osonic degrees of freedom only yields an upp er b ound of ab out 92% of the optimal v alue. Nev ertheless it is remark able that suc h a simple approach comes that close to the optimal bound. Contribution to the Pr o c e e dings of the Intensive Perio d Quantum Mathematics @ Polimi 2025 or ganize d by D. F ermi, M. Mosc olari, and A. Olgiati 1 In tro duction and Main Result W e consider a system of N spinless fermions in three dimensions, with Hamiltonian given in the mean-field scaling with an effective semiclassical parameter as introduced b y [NS81]: H N := − ℏ 2 N X i =1 ∆ x i + 1 N N X i 0 ) E N = E HF ( ω ) + ℏ κ X k ∈ Z 3 | k |  1 π Z ∞ 0 log  1 + 2 π κ ˆ V ( k )  1 − λ arctan 1 λ  d λ − π 2 κ ˆ V ( k )  + O ( N − 1 / 3 − α ) . T o se c ond or der in the inter action p otential, this is E N = E HF ( ω ) − ℏ π 2 (1 − log(2)) X k ∈ Z 3 | k | ˆ V ( k ) 2 + O  ˆ V ( k ) 3  . 2 P article-Hole T ransformation and Hartree-F o c k Theory W e consider L 2 a  ( T 3 ) N  as embedded in the fermionic F o c k space constructed ov er L 2  T 3  . Using the particle-hole transformation R ω on F o c k space (see [BD23, BPS14a] for a detailed discussion), where ω is the rank- N pro jection on to the N low est plane wa ves, we find R ∗ ω H N R ω = E HF ( ω ) + dΓ( uhu − v hv ) + Z T 3 × T 3 d x d y  a ∗ x a ∗ y ( uhv )( x, y ) + h.c.  + Q N . In this formula h is the Hartree-F o c k Hamiltonian (which dep ends on ω , see (6.27) for the explicit form). If ω pro jects on to a stationary p oin t of the Hartree-F o c k functional, then uhv = 0, so the ( a ∗ a ∗ + aa )-term v anishes. Consequen tly w e are lo oking for a trial state ξ ∈ F (con taining equal n umbers of particles and holes, equiv alent to R ω ξ b eing an N -particle state) such that ⟨ R ω ξ , H N R ω ξ ⟩ = E HF ( ω ) + ⟨ ξ ,  dΓ( uhu − v hv ) + Q N  ξ ⟩ < E HF ( ω ) . (2.4) The quartic terms are Q N = 1 2 N Z T 3 × T 3 d x d y V ( x − y )  E 1 + 2 a ∗ ( u x ) a ∗ ( v x ) a ( v y ) a ( u y ) + h a ∗ ( u x ) a ∗ ( u y ) a ∗ ( v y ) a ∗ ( v x ) + E 2 + h.c. i  4 where E 1 = a ∗ ( u x ) a ∗ ( u y ) a ( u y ) a ( u x ) − 2 a ∗ ( u x ) a ∗ ( v y ) a ( v y ) a ( u x ) + a ∗ ( v y ) a ∗ ( v x ) a ( v x ) a ( v y ) , E 2 = − 2 a ∗ ( u x ) a ∗ ( u y ) a ∗ ( v x ) a ( u y ) + 2 a ∗ ( u x ) a ∗ ( v y ) a ∗ ( v x ) a ( v y ) . The term E 1 can b e estimated using the n umber op erator, while E 2 do es not contribute to the exp ectation v alue due to a parit y argument (it creates particles in ± 1-steps, but the trial state that w e use contains only m ultiples of 2 particles or holes). Lemma 2.1. F or al l ξ ∈ F we have |⟨ ξ , 1 2 N Z T 3 × T 3 d x d y V ( x − y ) E 1 ξ ⟩| ≤ 2 N X k ∈ Z 3 | ˆ V ( k ) | ⟨ ξ , N 2 ξ ⟩ . Pr o of. W e use the F ourier decomp osition V ( x − y ) = P k ∈ Z 3 ˆ V ( k ) e ik ( x − y ) and Lemma 3.2 to get a b ound in terms of the op erator norm of an y b ounded op erator A , namely |⟨ ξ , dΓ( A ) ξ ⟩| ≤ ∥ A ∥⟨ ξ , N ξ ⟩ . Lemma 2.2. L et T b e an op er ator that c ommutes with i N , then ⟨ T Ω , E 2 T Ω ⟩ = 0 . Pr o of. W e can insert an i N in the righ t argument, since i N Ω = i 0 Ω = Ω: ⟨ T Ω , E 2 T Ω ⟩ = ⟨ T Ω , E 2 T i N Ω ⟩ = ⟨ T Ω , E 2 i N T Ω ⟩ = ⟨ T Ω , i N − 2 E 2 T Ω ⟩ = −⟨ T ( − i ) N Ω , E 2 T Ω ⟩ = −⟨ T Ω , E 2 T Ω ⟩ . W e contin ue with Q N = Q (0) N + 1 2 N Z T 3 × T 3 d x d y V ( x − y )  E 1 + E 2  (2.5) where Q (0) N = 1 2 N X k ∈ Z d ˆ V ( k ) Z T 3 × T 3 d x d y  2 a ∗ ( u x ) e ikx a ∗ ( v x ) a ( v y ) e − iky a ( u y ) + h a ∗ ( u x ) e ikx a ∗ ( v x ) a ∗ ( u y ) e − iky a ∗ ( v y ) + h.c. i  . W e introduce global b osonic particle-hole pair excitations by ˜ b ∗ k := Z T 3 d x a ∗ ( u x ) e ikx a ∗ ( v x ) = X p ∈ B c F h ∈ B F δ p − h,k a ∗ p a ∗ h . (2.6) Here B F is the F ermi ball and B c F = Z 3 \ B F . Due to uv = 0, we hav e b ∗ 0 = 0. Then Q (0) N = 1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k )  2 ˜ b ∗ k ˜ b k + ˜ b ∗ k ˜ b ∗ − k + ˜ b − k ˜ b k  . W e normalize the new op erators to strengthen the similarit y to b osonic creation and anni- hilation op erators, introducing b ∗ k := 1 n k ˜ b ∗ k , n 2 k := ∥ ˜ b ∗ k Ω ∥ 2 . (2.7) 5 k Figure 1: The tw o balls represent the pro jections e ikx ω e − ikx and ω , resp ectiv ely . The nor- malization constan t n 2 k is giv en by the num b er of lattice p oints in the lense marked in gray . Lemma 2.3 (Normalization Constan t) . We have n k = q tr e ikx (1 − ω ) e − ikx ω = ∥ ue ikx v ∥ HS . This c an b e c ompute d explicitly, yielding n k = n − k =  3 4 √ π  1 / 3 p | k | N ℏ + O (1) . (2.8) Pr o of. Using the CAR it is straightforw ard to find n 2 k = ∥ ˜ b ∗ k Ω ∥ 2 = ⟨ Ω , Z d xa ( v x ) e − ikx a ( u x ) Z d y a ∗ ( u y ) e iky a ∗ ( v y )Ω ⟩ = Z d x d y e − ikx ⟨ u x , u y ⟩ e iky ⟨ v x , v y ⟩ = tr e − ikx (1 − ω ) e ikx ω where w e used ⟨ u x , u y ⟩ = (1 − ω )( x, y ) and ⟨ v x , v y ⟩ = ω ( y , x ). T o show that n k is ev en as a function of k , we write n 2 k = tr e − ikx (1 − ω ) e ikx ω = tr ω − tr e − ikx ω e ikx ω n 2 − k = tr e ikx (1 − ω ) e − ikx ω = tr ω − tr e ikx ω e − ikx ω . By cyclicit y of the trace, the last expressions of the lines are the same. F or the computation of n 2 k , consider Figure 1: the trace of a pro jection is just its rank, whic h corresp onds to the num b er of p oin ts of Z 3 in the dark area in the figure. The v olume of the ov erlap of t wo balls, b oth with radius R , with centers displaced by a distance d is V lense = π 12 (4 R + d )(2 R − d ) 2 . The F ermi ball contains N mo des; thus 4 3 π R 3 = N , yielding radius k F =  3 4 π N  1 / 3 + O ( N 0 ). The v olume of the shaded region in Figure 1 is n 2 k = V ball − V lense = 4 3 π k 3 F − π 12 (4 k F + | k | )(2 k F − | k | ) 2 = π k 2 F | k | − π 12 | k | 3 = | k |  3 4 √ π  2 / 3 N ℏ − π 12 | k | 3 . By Gauss’ classical argument for counting lattice p oin ts, for ev ery k ∈ Z 3 w e think of ha ving a cub e of volume one attached. Including once all cub es completely con tained in the dark region, and once of all cub es that at least intersect with the dark region, w e obtain b oth an upp er and a low er b ound on the num b er of p oints which at leading order agree with the v olume of the dark region. 6 W e conclude that Q (0) N = 1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k  2 b ∗ k b k + b ∗ k b ∗ − k + b − k b k  . (2.9) W e do not hav e a simple formula for dΓ( uhu − v hv ) in terms of b ∗ k and b k , but we will see b elo w that w e can calculate its exp ectation v alue in the trial state anyw a y . 3 Almost-Bosonic Collectiv e Op erators The operators b k and b ∗ k in goo d appro ximation b eha ve like b osonic creation and annihilation op erators satisfying the CCR. Lemma 3.1 (Almost-CCR) . L et k , l ∈ Z d . The b -op er ators annihilate the vacuum Ω ∈ F , b k Ω = 0 . (3.10) F urthermor e they satisfy the appr oximate c anonic al c ommutator r elations [ b k , b ∗ l ] = δ k,l + E ( k , l ) , [ b ∗ k , b ∗ l ] = [ b k , b l ] = 0 , (3.11) wher e the err or term E ( k, l ) is an op er ator that c an b e estimate d using the fermionic p article numb er op er ator N by ∥E ( k , l ) ξ ∥ ≤ 1 n k n l ∥N ξ ∥ ∀ ξ ∈ F . (3.12) Note also that E ( k , l ) ∗ = E ( l , k ) . Pr o of. Straigh t-forward computations using the fermionic CAR, c. f., [BNP + 20]. W e need the following estimates; a pro of can b e found, e. g., in [BPS14a].) Lemma 3.2. F or every b ounde d op er ator O , we have ∥ d Γ( O ) ψ ∥ ≤ ∥ O ∥ ∥N ψ ∥ for every ψ ∈ F . If O is a Hilb ert-Schmidt op er ator, we also have the b ounds     Z dxdx ′ O ( x ; x ′ ) a x a x ′ ψ     ≤ ∥ O ∥ HS ∥N 1 / 2 ψ ∥ ,     Z dxdx ′ O ( x ; x ′ ) a ∗ x a ∗ x ′ ψ     ≤ 2 ∥ O ∥ HS ∥ ( N + 1) 1 / 2 ψ ∥ . (3.13) These b ounds directly imply estimates for the almost-b osonic operators b k and b ∗ k . Lemma 3.3. F or every k ∈ Z 3 we have, for al l ψ ∈ F , the estimates ∥ b k ψ ∥ ≤ ∥N 1 / 2 ψ ∥ , ∥ b ∗ k ψ ∥ ≤ 2 ∥ ( N + 1) 1 / 2 ψ ∥ . (3.14) Pr o of. These b ounds follow directly b y using Lemma 3.2 with the definition of the pair op erators (2.6) and recalling the Hilb ert-Sc hmidt and trace norms from Lemma 2.3. 7 4 Almost-Bosonic Quasifree T rial State The quadratic expression for the interaction given in (2.9) suggests that the we should think of the correlation energy as originating from a quadratic almost-b osonic Hamiltonian. Thus w e take our trial state as a quasifree state of the form exp( b ∗ b ∗ − bb )Ω. W e are going to calculate the expectation v alue of the Hamiltonian in that state and then optimize the c hoice of the Bogoliub o v transformation. More precisely , we define the following unitary op erator T ( λ ) := exp( λB ) , B := 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) b ∗ k b ∗ − k − h.c. (4.15) whic h mimics a bosonic Bogoliub o v transformation, but replacing bosonic creation op erators b y our quasi-b osonic pair creation op erators. F or simplicity we write T := T (1). Note that T (0) = I . Our trial state T Ω automatically has equal num b ers of particles and holes b ecause (4.15) contains only b ∗ - and b -op erators, which, b y (2.6), alwa ys create or annihilate a particle and a hole together. Lemma 4.1 (Almost-Bosonic Bogoliub o v T ransformation) . L et Ξ( k ) = Ξ( − k ) . F or l  = 0 , the c onjugation of b l and b ∗ l with T is given by T ∗ b l T = cosh(Ξ)( l ) b l + sinh(Ξ)( l ) b ∗ − l + E k (Ξ) , T ∗ b ∗ l T = cosh(Ξ)( l ) b ∗ l + sinh(Ξ)( l ) b − l + E ∗ k (Ξ) , (4.16) wher e cosh(Ξ)( l ) = 1 − 1 2! Ξ( l )Ξ( − l ) + · · · , sinh(Ξ)( l ) = Ξ( l ) − 1 3! Ξ( l )Ξ( − l )Ξ( l ) + · · · The op er ators b 0 and b ∗ 0 ar e invariant. The op er ators E k (Ξ) c an b e estimate d by ∥E l (Ξ) ψ ∥ ≤ sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t ) ψ ∥ e | Ξ( l ) | X k ∈ Z 3 \{ 0 } 2 | Ξ( k ) | n k n l . (4.17) Since Ξ( k ) = Ξ( − k ), we hav e cosh(Ξ)( l ) = cosh( | Ξ( l ) | ) and sinh(Ξ)( l ) = sinh( | Ξ( l ) | ) Ξ( l ) | Ξ( l ) | . (4.18) Pr o of. W e hav e T ∗ b l T − b l = Z 1 0 d λ d d λ  e − λB b l e λB  = Z 1 0 d λ T ( λ ) ∗ [ b l , B ] T ( λ ) = Z 1 0 d λ T ( λ ) ∗ 1 2 X k ∈ Z 3 \{ 0 } Ξ( k )  [ b l , b ∗ k ] b ∗ − k + b ∗ k [ b l , b ∗ − k ]  T ( λ ) = Z 1 0 d λ T ( λ ) ∗   Ξ( l ) b ∗ − l + 1 2 X k ∈ Z 3 \{ 0 } Ξ( k )  E ( k , l ) b ∗ − k + b ∗ k E ( − k , l )    T ( λ ) implying T ∗ b l T = b l + Ξ( l ) Z 1 0 d λ T ( λ ) ∗ b ∗ − l T ( λ ) + 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) Z 1 0 d λ T ( λ ) ∗  E ( k , l ) b ∗ − k + b ∗ k E ( − k , l )  T ( λ ) (4.19) 8 and in the same wa y T ∗ b ∗ l T − b ∗ l = Z 1 0 d λ T ( λ ) ∗ 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) ([ b ∗ l , b − k ] b k + b − k [ b ∗ l , b k ]) T ( λ ) = Z 1 0 d λ T ( λ ) ∗   − Ξ( l ) b − l − 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) ( E ( − k, l ) b k + b − k E ( k , l ))   T ( λ ) implying T ∗ b ∗ l T = b ∗ l − Ξ( l ) Z 1 0 d λ T ( λ ) ∗ b − l T ( λ ) − 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) Z 1 0 d λ T ( λ ) ∗ ( E ( − k , l ) b k + b − k E ( k , l )) T ( λ ) . (4.20) Plugging (4.19) and (4.20) into each other iteratively (we iterate only in the leading term, the error terms containing E we do not iterate), w e arrive at T ∗ b l T = cosh(Ξ)( l ) b l + sinh(Ξ)( l ) b ∗ − l + E sinh (Ξ)( l ) + E cosh (Ξ)( l ) =: cosh(Ξ)( l ) b l + sinh(Ξ)( l ) b ∗ − l + E l (Ξ) , where the error terms are E cosh (Ξ)( l ) = − X k ∈ Z d \{ 0 } Ξ( k ) 2 Ξ( l ) Z 1 0 d λ Z λ 0 d λ ′ T ( λ ′ ) ∗ A cosh ( k , l ) T ( λ ′ ) + Ξ( l )Ξ( − l )Ξ( l ) Z 1 0 d λ Z λ 0 d λ ′ Z λ ′ 0 d λ ′′ Z λ ′′ 0 d λ ′′′ T ( λ ′′′ ) ∗ A cosh ( k , l ) T ( λ ′′′ ) + · · · ! (a series similar to the one of the cos but with one factor replaced b y T ∗ A cosh T ) and E sinh (Ξ)( l ) = − X k ∈ Z 3 \{ 0 } Ξ( k ) 2 Z 1 0 d λT ( λ ) ∗ A sinh ( k , l ) T ( λ ) + Ξ( l )Ξ( − l ) Z 1 0 d λ Z λ 0 d λ ′ Z λ ′ 0 d λ ′′ T ( λ ′′ ) ∗ A sinh ( k , l ) T ( λ ′′ ) + · · · ! (a series similar to the one of the cosh but with one factor replaced b y T ∗ A sinh T ) with A cosh ( k , l ) = E ( − k , − l ) b k + b − k E ( k , − l ) , A sinh ( k , l ) = E ( k , l ) b ∗ − k + b ∗ k E ( − k , l ) . (The head term of the expansion v anishes as the expansion order tends to infinity .) W e now pro ceed to estimate E l (Ξ). First of all, for all ψ ∈ F we hav e ∥E l (Ξ) ψ ∥ ≤ ∥E cosh (Ξ)( l ) ψ ∥ + ∥E sinh (Ξ)( l ) ψ ∥ . Using the triangle inequality ∥E cosh (Ξ)( l ) ψ ∥ ≤ 1 2 X k ∈ Z d \{ 0 } | Ξ( k ) | | Ξ( l ) | Z 1 0 d λ Z λ 0 d λ ′ ∥ A cosh ( k , l ) T ( λ ′ ) ψ ∥ + | Ξ( l ) | 3 Z 1 0 d λ Z λ 0 d λ ′ Z λ ′ 0 d λ ′′ Z λ ′′ 0 d λ ′′′ ∥ A cosh ( k , l ) T ( λ ′′′ ) ψ ∥ + · · · ! 9 W e now giv e an estimate independent of λ for the norms on the righ t hand side of the previous equation, using (3.14) and the fact that N comm utes with E ( k , − l ) since E ( k, − l ) is a sum of t wo op erators dΓ( uAu ) and dΓ( v Av ) (dΓ-op erators conserve the n umber of particles): ∥ A cosh ( k , l ) T ( λ ) ψ ∥ = ∥ ( E ( − k, − l ) b k + b − k E ( k , − l )) T ( λ ) ψ ∥ ≤ 1 n k n l ∥N b k T ( λ ) ψ ∥ + ∥N 1 / 2 E ( k , − l ) T ( λ ) ψ ∥ ≤ 1 n k n l ∥ ( N + 2) b k T ( λ ) ψ ∥ + ∥E ( k , − l ) N 1 / 2 T ( λ ) ψ ∥ = 1 n k n l ∥ b k N T ( λ ) ψ ∥ + 1 n k n l ∥N N 1 / 2 T ( λ ) ψ ∥ ≤ 1 n k n l ∥N 1 / 2 N T ( λ ) ψ ∥ + 1 n k n l ∥N 1 / 2 N T ( λ ) ψ ∥ ≤ 2 n k n l ⟨ T ( λ ) ψ , N 3 T ( λ ) ψ ⟩ 1 / 2 . Th us ∥E cosh (Ξ)( l ) ψ ∥ ≤ X k ∈ Z d \{ 0 } | Ξ( k ) | sup t ∈ [0 , 1] 1 n k n l ⟨ T ( λ ) ψ , N 3 T ( λ ) ψ ⟩ 1 / 2 × | Ξ( l ) | Z 1 0 d λ Z λ 0 d λ ′ + | Ξ( l ) | 3 Z 1 0 d λ Z λ 0 d λ ′ Z λ ′ 0 d λ ′′ Z λ ′′ 0 d λ ′′′ + · · · ! ≤ X k ∈ Z d \{ 0 } | Ξ( k ) | 1 n k n l sup t ∈ [0 , 1] ⟨ T ( t ) ψ , N 3 T ( t ) ψ ⟩ 1 / 2 | Ξ( l ) | 1 2! + | Ξ( l ) | 3 1 4! + · · · ! . The series in the big parenthesis is summable. In the same wa y as the estimates ab o ve w e find ∥ A sinh ( k , l ) T ( λ ) ψ ∥ ≤ 4 n k n l ⟨ T ( λ ) ψ , ( N + 2) 3 T ( λ ) ψ ⟩ and from that ∥E sinh (Ξ)( l ) ψ ∥ ≤ 2 X k ∈ Z 3 \{ 0 } | Ξ( k ) | 1 n k n l sup t ∈ [0 , 1] ⟨ T ( t ) ψ , ( N + 2) 3 T ( t ) ψ ⟩ 1 / 2  1 + 1 3! | Ξ( l ) | 2 + . . .  Com bining, we find (4.17). 5 Calculating the In teraction Energy The exp ectation v alue of the interaction energy Q (0) N is easy to calculate now. Prop osition 5.1 (Interaction Energy) . We have ⟨ T Ω , Q (0) N T Ω ⟩ = 1 N Re X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k sinh( | Ξ( k ) | ) 2 + Ξ( k ) | Ξ( k ) | sinh( | Ξ( k ) | ) cosh( | Ξ( k ) | ) ! + ε 1 10 wher e the err or ε 1 ∈ C c an b e estimate d by | ε 1 | ≤ 1 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k " 4 sup t ∈ [0 , 1] ⟨ T ( t )Ω , ( N + 2) 3 T ( t )Ω ⟩ e 2 | Ξ( k ) |   X m ∈ Z d \{ 0 } | Ξ( m ) | n m n k   2 +  4 sinh( | Ξ( k ) | ) + 2 cosh( | Ξ( k ) | )  sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( m ) | n m n k # . Pr o of. W e apply the transformation rule (4.16) and then the annihilation of the v acuum (3.10) to obtain ⟨ T Ω , Q (0) N T Ω ⟩ = 1 2 N X k ∈ Z d \{ 0 } ˆ V ( k ) n 2 k ⟨ Ω , " 2  cosh(Ξ)( k ) b ∗ k + sinh(Ξ)( k ) b − k + E ∗ k (Ξ)  ×  cosh(Ξ)( k ) b k + sinh(Ξ)( k ) b ∗ − k + E k (Ξ)  +  cosh(Ξ)( k ) b ∗ k + sinh(Ξ)( k ) b − k + E ∗ k (Ξ)  ×  cosh(Ξ)( − k ) b ∗ − k + sinh(Ξ)( − k ) b k + E ∗ − k (Ξ)  + h.c. !# Ω ⟩ = 1 2 N X k ∈ Z d \{ 0 } ˆ V ( k ) n 2 k ⟨ Ω ,  sinh(Ξ)( k ) b − k + E ∗ k (Ξ)   sinh(Ξ)( k ) b ∗ − k + E k (Ξ)  +  sinh(Ξ)( k ) b − k + E ∗ k (Ξ)   cosh(Ξ)( − k ) b ∗ − k + E ∗ − k (Ξ)  + h.c. ! Ω ⟩ + c.c. Recalling that ∥ b ∗ − k Ω ∥ 2 = 1, w e extract the term stated in the Lemma (use (4.18) to simplify). The remain terms are errors, given by 1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k ∥E k (Ξ)Ω ∥ 2 +  sinh(Ξ)( k ) ⟨ b ∗ − k Ω , E k (Ξ)Ω ⟩ + c.c.  + ⟨E ∗ k (Ξ)Ω , E ∗ − k (Ξ)Ω ⟩ + sinh(Ξ)( k ) ⟨ b ∗ − k Ω , E ∗ − k (Ξ)Ω ⟩ + cosh(Ξ)( − k ) ⟨E k (Ξ)Ω , b ∗ − k Ω ⟩ ! + c.c. =: ε 1 . (5.21) Estimating the Error T erms. W e now sho w that ε 1 is smaller than the leading term (of order ℏ = N − 1 / 3 ) of the correlation energy . F rom Lemma 4.1 w e hav e       1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k ∥E k (Ξ)Ω ∥ 2       ≤ 2 N X k ∈ Z d \{ 0 } ˆ V ( k ) n 2 k sup t ∈ [0 , 1] ⟨ T ( t )Ω , ( N + 2) 3 T ( t )Ω ⟩ e 2 | Ξ( k ) |   X m ∈ Z d \{ 0 } | Ξ( m ) | n m n k   2 . (5.22) 11 In the same wa y       1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k ⟨E ∗ k (Ξ)Ω , E ∗ − k (Ξ)Ω ⟩       ≤ 2 N X k ∈ Z d \{ 0 } ˆ V ( k ) n 2 k sup t ∈ [0 , 1] ⟨ T ( t )Ω , ( N + 2) 3 T ( t )Ω ⟩ e 2 | Ξ( k ) |   X m ∈ Z d \{ 0 } | Ξ( m ) | n m n k   2 . (5.23) F urthermore, using (4.17) and (3.14) (with ∥ ( N + 1) 1 / 2 Ω ∥ = 1), w e hav e       1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k  sinh(Ξ)( k ) ⟨ b ∗ − k Ω , E k (Ξ)Ω ⟩ + c.c.        ≤ 1 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k | sinh(Ξ)( k ) |∥ b ∗ − k Ω ∥∥E k (Ξ)Ω ∥ ≤ 4 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k sinh( | Ξ( k ) | ) sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( m ) | n m n k . (5.24) The terms on the last line of the definition of ε 1 , (5.21), can b e con trolled in the same wa y ,       1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k sinh(Ξ)( k ) ⟨ b ∗ − k Ω , E ∗ − k (Ξ)Ω ⟩       ≤ 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k sinh( | Ξ( k ) | ) sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( m ) | n m n k (5.25) and       1 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k cosh(Ξ)( − k ) ⟨E k (Ξ)Ω , b ∗ − k Ω ⟩       ≤ 2 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k cosh( | Ξ( k ) | ) sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( k ) | n m n k . (5.26) The +c.c. add an ov erall factor 2. 6 Calculating the Kinetic Energy Calculating the exp ectation v alue of the kinetic energy requires a little more w ork since we do not hav e an appro ximation as a quadratic form of almost-b osonic op erators. W e start by estimating the corrections to the kinetic energy from the Hartree-F o c k op erator, which turn out to b e small. The Hartree-F o c k op erator is h ≃ − ℏ 2 ∆ + D + X, D = Z T 3 V ( x )d x, ∥ X ∥ ≤ C N , (6.27) where we use that for plane w av es the direct term D is a constant cancelling out from the differences, and the exchange term X is small in op erator norm. 12 Lemma 6.1. L et X ( x, y ) := N − 1 V ( x − y ) ω ( x, y ) b e the inte gr al kernel of the exchange op er ator, with ω ( x, y ) = X h ∈ B F e ih · ( x − y ) =: g ( x − y ) . Then its op er ator norm is b ounde d by ∥ X ∥ ≤ 1 N (2 π ) − 3 / 2 ∥ ˆ V ∥ L 1 ( T 3 ) . Pr o of. The in tegral kernel is translation in v arian t, X ( x, y ) = N − 1 V ( x − y ) g ( x − y ), so X is a con volution op erator, which in F ourier space is a m ultiplication op erator. By unitarit y ∥ X ∥ = ∥F X F − 1 ∥ = ∥ ˆ X ∥ L ∞ ( Z d ) . Due to the conv olution theorem we ha ve ˆ X ( k ) = 1 N [ ( V g )( k ) = 1 N (2 π ) − d/ 2 ˆ V ∗ ˆ g ( k ) and th us ∥ ˆ X ∥ L ∞ ( Z d ) = 1 N (2 π ) − d/ 2 sup k ∈ Z d   X l ∈ Z d ˆ g ( k − l ) ˆ V ( l )   ≤ 1 N (2 π ) − d/ 2 sup k ∈ Z d | ˆ g ( k ) | X l ∈ Z d | ˆ V ( l ) | . F urthermore, g is given as the F ourier transform of the F ermi ball, g = ˇ χ, χ ( k ) = X h ∈ B F δ h,k , so inv erting the F ourier transform and using that the Kronec ker deltas are b ounded b y one (recall that w e are on a torus) sup k ∈ Z 2 | ˆ g ( k ) | = 1. W e now conclude that direct and exchange term do not con tribute. Lemma 6.2 (Direct and Exc hange T erm) . L et H 0 := dΓ  u ( − ℏ 2 ∆) u − v ( − ℏ 2 ∆) v  . L et ξ b e the almost-b osonic quasifr e e trial state. Then we have ⟨ ξ , dΓ( uhu − v hv ) ξ ⟩ = ⟨ ξ , H 0 ξ ⟩ + E X , wher e the err or term satisfies |E X | ≤ 2(2 π ) − 3 / 2 ∥ ˆ V ∥ L 1 ( T 3 ) N ⟨ ξ , N ξ ⟩ . Pr o of. Since the direct term D is just a num b er, w e hav e dΓ( uD u − vD v ) = D ( N P − N H ) , where N P = P p ∈ B c F a ∗ p a p is the n umber of particle and N H = P h ∈ B F a ∗ h a h the num b er of holes. The exp onen tial defining the trial state creates and annihilates only equal num b er of particles and holes, therefore ⟨ ξ , ( N P − N H ) ξ ⟩ = 0. T o estimate the exc hange term we use the inequalit y |⟨ ξ , dΓ( A ) ξ ⟩| ≤ ∥ A ∥⟨ ξ , N ξ ⟩ v alid for an y b ounded op erator A as w ell as the previous lemma, so that w e obtain |⟨ ξ , dΓ( uX u − v X v ) ξ ⟩| ≤ 2 ( ∥ uX u ∥ + ∥ v X v ∥ ) ⟨ ξ , N ξ ⟩ ≤ 2 ∥ X ∥⟨ ξ , N ξ ⟩ ≤ 2(2 π ) − 3 / 2 ∥ ˆ V ∥ L 1 ( T 3 ) N ⟨ ξ , N ξ ⟩ , where w e used that the op erator norms are ∥ u ∥ = 1 = ∥ v ∥ . 13 F rom (2.7) and (2.6), we obtain ˜ b ∗ k = 1 n k X p ∈ B c F h ∈ B F δ p − h,k a ∗ p a ∗ h . W e expand into plane wa v es, recalling that v x ( y ) = v ( y , x ) = X h ∈ B F | f h ⟩⟨ f h | ( y , x ) = (2 π ) − d X h ∈ B F e ihy e ihx and u x ( y ) = u ( y , x ) = X p ∈ B c F | f p ⟩⟨ f p | ( y , x ) = (2 π ) − d X p ∈ B c F e ipy e − ipx . W riting the creation op erators in momen tum space as a ∗ p := a ∗ ( f p ) w e find b ∗ k = 1 n k X p ∈ B c F h ∈ B F δ p,h + k a ∗ p a ∗ h . Lemma 6.3 (Kinetic Energy Comm utator) . We have [ H 0 , b ∗ k ] = 1 n k X p ∈ B c F h ∈ B F δ p,h + k ℏ 2  p 2 − h 2  a ∗ p a ∗ h = ℏ 2 k · c ∗ k , with the ve ctor-value d op er ator c ∗ k given by c ∗ k := 1 n k X p ∈ B c F h ∈ B F δ p,h + k ( p + h ) a ∗ p a ∗ h . Likewise [ H 0 , b k ] = − ℏ 2 k · c k Pr o of. T rivial calculation using the CAR. Prop osition 6.4 (Exp ectation of Kinetic Energy) . We have ⟨ T Ω , H 0 T Ω ⟩ = X k ∈ Z 3 \{ 0 } ℏ 2 k · f ( k )sinh( | Ξ( k ) | ) 2 + 2 Re ε 2 , wher e f ( k ) := 1 n 2 k X p ∈ B c F h ∈ B F δ p,h + k ( p + h ) and ther e exists a c onstant C such that the err or term is c ontr ol le d by | ε 2 | ≤ 2 ℏ 2 X k ∈ Z 3 \{ 0 } " 2 | Ξ( k ) || k · f ( k ) | sinh( | Ξ( k ) | ) sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } 2 | Ξ( m ) | n m n k + X l ∈ Z 3 \{ 0 } | Ξ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l sup µ ∈ [0 , 1] ∥N 3 / 2 T ( µ )Ω ∥ ×  sinh( | Ξ( k ) | ) + sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( m ) | n m n k  # . 14 Pr o of. Noticing that H 0 Ω = 0, b y Duhamel we obtain ⟨ T Ω , H 0 T Ω ⟩ = Z 1 0 d λ ⟨ T ( λ )Ω , [ H 0 , B ] T ( λ )Ω ⟩ = Z 1 0 d λ 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) ⟨ T ( λ )Ω ,  [ H 0 , b ∗ k ] b ∗ − k + b ∗ k [ H 0 , b ∗ − k ]  T ( λ )Ω ⟩ − Z 1 0 d λ 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) ⟨ T ( λ )Ω , ([ H 0 , b − k ] b k + b − k [ H 0 , b k ]) T ( λ )Ω ⟩ = Z 1 0 d λ 1 2 X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 ⟨ T ( λ )Ω ,  k · c ∗ k b ∗ − k + b ∗ k ( − k ) · c ∗ − k  T ( λ )Ω ⟩ + c.c. = Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ T ( λ )Ω , c ∗ k b ∗ − k T ( λ )Ω ⟩ + c.c. (6.28) where in the last step w e used that Ξ( k ) = Ξ( − k ) and that c ∗ k and b ∗ − k comm ute since they consist of pairs of creation op erators. W e now use the almost-Bogoliub o v tranformation (4.16) (attention to include the factor λ to the Ξ) to transform the b ∗ − k op erator, arriving at Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ T ( λ )Ω , c ∗ k b ∗ − k T ( λ )Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , T ( λ ) ∗ c ∗ k T ( λ )  cosh( λ Ξ)( − k ) b ∗ − k + sinh( λ Ξ)( − k ) b k + E ∗ − k ( λ Ξ)  Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , T ( λ ) ∗ c ∗ k T ( λ )  cosh( λ Ξ)( − k ) b ∗ − k + E ∗ − k ( λ Ξ)  Ω ⟩ (6.29) The key observ ation is that b y the Duhamel formula we obtain a commutator of c ∗ k with B , and this commutator can b e expressed purely in terms of b ∗ –op erators, for whic h we kno w the almost-Bogoliub o v transformation rule. In detail, T ∗ ( λ ) c ∗ k T ( λ ) = c ∗ k + Z λ 0 d µ T ∗ ( µ )[ c ∗ k , B ] T ( µ ) = c ∗ k + Z λ 0 d µ T ∗ ( µ )   − 1 2 X l ∈ Z 3 \{ 0 } Ξ( l ) ([ c ∗ k , b − l ] b l + b − l [ c ∗ k , b l ])   T ( µ ) . (6.30) Similar to the commutator [ b ∗ k , b l ] = − δ k,l + O ( N − 2 / 3 ), also the commutator [ c ∗ k , b l ] can b e though t of as a leading Kroneck er delta (times a constan t) and tw o other terms to b e treated as errors; more precisely from Lemma 6.5 we hav e [ c ∗ k , b l ] = − δ k,l f ( k ) + E c ( k , l ) . Let us plug this commutator into (6.30) to get (we use again Ξ( k ) = Ξ( − k )) T ∗ ( λ ) c ∗ k T ( λ ) = c ∗ k + Ξ( k ) f ( k ) Z λ 0 d µT ∗ ( µ ) b − k T ( µ ) − 1 2 X l ∈ Z 3 \{ 0 } Ξ( l ) Z λ 0 d µT ∗ ( µ )  E c ( k , − l ) b l + b − l E c ( k , l )  T ( µ ) =: c ∗ k + Ξ( k ) f ( k ) Z λ 0 d µT ∗ ( µ ) b − k T ( µ ) + E kin (Ξ)( k ) . (6.31) 15 The terms E c and E kin are d -tup els of op erators. W e plug (6.31) into (6.29), use that c k acting on the v acuum v anishes, and then the almost-Bogoliub ov transformation rule to find Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ T ( λ )Ω , c ∗ k b ∗ − k T ( λ )Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · Z λ 0 d µ Ξ( k ) f ( k ) ⟨ Ω , T ∗ ( µ ) b − k T ( µ )cosh( λ Ξ)( − k ) b ∗ − k Ω ⟩ + Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , E kin (Ξ)( k )  cosh( λ Ξ)( − k ) b ∗ − k + E ∗ − k ( λ Ξ)  Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) × Z λ 0 d µ ⟨ Ω ,  cosh( µ Ξ)( − k ) b − k + sinh( µ Ξ)( − k ) b ∗ k + E − k ( µ Ξ)  cosh( λ Ξ)( − k ) b ∗ − k Ω ⟩ + Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , E kin (Ξ)( k )  cosh( λ Ξ)( − k ) b ∗ − k + E ∗ − k ( λ Ξ)  Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) Z λ 0 d µ ⟨ Ω , cosh( µ Ξ)( − k ) b − k cosh( λ Ξ)( − k ) b ∗ − k Ω ⟩ + ε 2 , where w e introduced the notation for the error term ε 2 := Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , E kin (Ξ)( k )  cosh( λ Ξ)( − k ) b ∗ − k + E ∗ − k ( λ Ξ)  Ω ⟩ + Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) Z λ 0 d µ ⟨ Ω , E − k ( µ Ξ)cosh( λ Ξ)( − k ) b ∗ − k Ω ⟩ . T o ev aluate the scalar pro ducts in the main term, recall that ⟨ Ω , b − k b ∗ − k Ω ⟩ = 1, so Z 1 0 d λ X k ∈ Z 3 \{ 0 } ℏ 2 k · Ξ( k ) ⟨ T ( λ )Ω , c ∗ k b ∗ − k T ( λ )Ω ⟩ = Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) Z λ 0 d µ cosh( µ Ξ)( − k )cosh( λ Ξ)( − k ) + ε 2 = Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) sinh( λ | Ξ( k ) | ) | Ξ( k ) | cosh( λ | Ξ( k ) | ) + ε 2 = X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) sinh( | Ξ( k ) | ) 2 2 | Ξ( k ) | 2 + ε 2 = 1 2 X k ∈ Z 3 \{ 0 } ℏ 2 k · f ( k )sinh( | Ξ( k ) | ) 2 + ε 2 . The in tegrals ov er µ and λ were computed explicitly . Plugging the last expression into (6.28) (remem b er that a factor of 2 arises from the +c.c.), we obtain the statemen t of the lemma. 16 Estimating the Error T erms. W e use the (3.14) and (4.17) to estimate the second line of ε 2 (notice that the in tegral ov er µ can b e b ounded by 1 and the in tegral ov er λ then turns cosh( λ | Ξ( k ) | ) into sinh( | Ξ( k ) | ) / | Ξ( k ) | )       Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 k · f ( k ) Z λ 0 d µ ⟨ Ω , E − k ( µ Ξ)cosh( λ Ξ)( − k ) b ∗ − k Ω ⟩       ≤ Z 1 0 d λ X k ∈ Z d \{ 0 } | Ξ( k ) | 2 ℏ 2 | k · f ( k ) | cosh( λ | Ξ( k ) | ) Z λ 0 d µ ∥E ∗ − k ( µ Ξ)Ω ∥∥ b ∗ − k Ω ∥ ≤ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 | k · f ( k ) | sinh( | Ξ( k ) | ) sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } 2 | Ξ( m ) | n m n k . Recall also from (7.37) that | k · f ( k ) | = O ( N 1 / 3 ). W e now consider the first line of ε 2 . First of all, we estimate ∥ k · E kin (Ξ)( k )Ω ∥ . Using Lemma 6.5 and (3.14), as w ell as the fact that E c ( k , l ) commutes with N , w e find ∥ k · E kin (Ξ)( k )Ω ∥ = 1 2 X l ∈ Z 3 \{ 0 } | Ξ( l ) | Z λ 0 d µ ∥  k · E c ( k , − l ) b l + b − l k · E c ( k , l )  T ( µ )Ω ∥ ≤ X l ∈ Z 3 \{ 0 } | Ξ( l ) | 2 Z λ 0 d µ   6 π  1 / 3 N 1 / 3 | k | n k n l ∥N b l T ( µ )Ω ∥ + ∥N 1 / 2 k · E c ( k , l ) T ( µ )Ω ∥  ≤ X l ∈ Z 3 \{ 0 } | Ξ( l ) | 2 Z λ 0 d µ   6 π  1 / 3 N 1 / 3 | k | n k n l ∥ b l N T ( µ )Ω ∥ + ∥ k · E c ( k , l ) N 1 / 2 T ( µ )Ω ∥  ≤ X l ∈ Z 3 \{ 0 } | Ξ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l sup µ ∈ [0 , 1] ∥N 3 / 2 T ( µ )Ω ∥ . 17 Using this estimate together with (3.14) and (4.17) we find       Z 1 0 d λ X k ∈ Z 3 \{ 0 } Ξ( k ) ℏ 2 k · ⟨ Ω , E kin (Ξ)( k )  cosh( λ Ξ)( − k ) b ∗ − k + E ∗ − k ( λ Ξ)  Ω ⟩       ≤ Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | ℏ 2 ∥ k · E kin (Ξ)( k )Ω ∥  cosh( λ | Ξ( k ) | ) ∥ b ∗ − k Ω ∥ + ∥E ∗ − k ( λ Ξ)Ω ∥  ≤ Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 ∥ k · E kin (Ξ)( k )Ω ∥ ×  cosh( λ | Ξ( k ) | ) + sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e λ | Ξ( k ) | X m ∈ Z 3 \{ 0 } λ | Ξ( m ) | n m n k  ≤ Z 1 0 d λ X k ∈ Z 3 \{ 0 } | Ξ( k ) | 2 ℏ 2 X l ∈ Z 3 \{ 0 } | Ξ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l sup µ ∈ [0 , 1] ∥N 3 / 2 T ( µ )Ω ∥ ×  cosh( λ | Ξ( k ) | ) + sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e λ | Ξ( k ) | X m ∈ Z 3 \{ 0 } λ | Ξ( m ) | n m n k  ≤ X k ∈ Z 3 \{ 0 } 2 ℏ 2 X l ∈ Z 3 \{ 0 } | Ξ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l sup µ ∈ [0 , 1] ∥N 3 / 2 T ( µ )Ω ∥ ×  sinh( | Ξ( k ) | ) + sup t ∈ [0 , 1] ∥ ( N + 2) 3 / 2 T ( t )Ω ∥ e | Ξ( k ) | X m ∈ Z 3 \{ 0 } | Ξ( m ) | n m n k  . T o get from the second last to the last line, we hav e estimated the λ in the m -sum by 1 and then in tegrated the cosh and the exp onen tial explicitly . Lemma 6.5. With c ∗ k as define d in L emma 6.3, we have [ c ∗ k , b l ] = − δ k,l f ( k ) + E c ( k , l ) with the function f : Z d → R 3 given by f ( k ) := 1 n 2 k X p ∈ B c F h ∈ B F δ p,h + k ( p + h ) . (6.32) F or any ve ctor m ∈ R 3 , the err or op er ator E c ( k , l ) is b ounde d by ∥ m · E c ( k , l ) ψ ∥ ≤ | m |  6 π  1 / 3 N 1 / 3 n k n l ∥N ψ ∥ ∀ ψ ∈ F . The op er ator E c ( k , l ) c ommutes with the fermionic numb er op er ator N . 18 Pr o of. Using the CAR we find [ c ∗ k , b l ] = 1 n k n l X p ∈ B c F h ∈ B F δ p,h + k ( p + h ) X ˜ p ∈ B c F ˜ h ∈ B F δ ˜ p, ˜ h + l [ a ∗ p a ∗ h , a ˜ h a ˜ p ] = 1 n k n l X p, ˜ p ∈ B c F h, ˜ h ∈ B F δ p,h + k ( p + h ) δ ˜ p, ˜ h + l  − δ h, ˜ h δ p, ˜ p + δ h, ˜ h a ∗ p a ˜ p + δ p, ˜ p a ∗ h a ˜ h  = − 1 n k n l δ k,l X p ∈ B c F h ∈ B F δ p,h + k ( p + h ) + 1 n k n l X p, ˜ p ∈ B c F h ∈ B F δ p,h + k ( p + h ) δ ˜ p,h + l a ∗ p a ˜ p + 1 n k n l X p ∈ B c F h, ˜ h ∈ B F δ p,h + k ( p + h ) δ p, ˜ h + l a ∗ h a ˜ h =: − δ k,l f ( k ) + E c ( k , l ) , in the last line defining the error operator E c ( k , l ). The first term of m · E c ( k , l ) can be written 1 n k n l X p, ˜ p ∈ B c F h ∈ B F δ p,h + k m · ( p + h ) δ ˜ p,h + l a ∗ p a ˜ p = 1 n k n l dΓ( A ) , where A p, ˜ p = X h ∈ B F δ p,h + k δ ˜ p,h + l m · ( p + h ) = m · (2 p − k ) χ ( p − k ∈ B F ) δ ˜ p,p − k + l . W e then hav e the usual estimate using the op erator norm of A , ∥ dΓ( A ) ψ ∥ ≤ ∥ A ∥∥N ψ ∥ . The op erator norm of A can b e controlled as follo ws: ∥ A ∥ = sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1 ∥ Ax ∥ = sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1   X p ∈ B c F   X ˜ p ∈ B c F A p, ˜ p x ˜ p   2   1 / 2 = sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1   X p ∈ B c F  m · (2 p − k ) χ ( p − k ∈ B F ) χ ( p − k + l ∈ B c F ) x p − k + l  2   1 / 2 = sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1   X p ∈ B c F  m · (2 p − k )  2 χ ( p − k ∈ B F ) χ ( p − k + l ∈ B c F ) x 2 p − k + l   1 / 2 ≤ sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1   X p ∈ B c F | m | 2 (2 | p | + | k | ) 2 χ ( p − k ∈ B F ) χ ( p − k + l ∈ B c F ) x 2 p − k + l   1 / 2 . (6.33) Due to the constraint p − k ∈ B F , we hav e | p | ≤  3 4 π  1 / 3 N 1 / 3 + | k | , where | k | ≤ diam supp ˆ V indep enden t of N and ℏ . Since w e are interested in large N , we can more simply write 19 | p | ≤  6 π  1 / 3 N 1 / 3 . W e obtain ∥ A ∥ ≤ | m |  6 π  1 / 3 N 1 / 3 sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1   X p ∈ B c F x 2 p − k + l   1 / 2 ≤ | m |  6 π  1 / 3 N 1 / 3 sup x ∈ ℓ 2 ( Z 3 ) ∥ x ∥ =1 ∥ x ∥ = | m |  6 π  1 / 3 N 1 / 3 . (6.34) W e conclude that ∥ dΓ( A ) ψ ∥ ≤ | m |  6 π  1 / 3 N 1 / 3 ∥N ψ ∥ . The second term of m · E c ( k , l ) can b e estimated in the same wa y . It is obvious that the error term comm utes with N b ecause it can b e written as a dΓ–op erator. 7 Optimizing the T rial State W e optimize the choice of the almost-Bogoliub o v transform with resp ect to Ξ. W e are later going to calculate n 2 k and k · f ( k ) explicitly , and then we will see that b oth α k and β k , and therefore the whole energy correction, are of order ℏ . This also sets the scale for our error b ounds: we hav e to show that all errors are at least as small as o ( ℏ ). Prop osition 7.1 (Minimal almost-Bogoliub o v Energy) . L et ˆ V ( k ) > 0 . The lowest ener gy among almost-Bo goliub ov trial states is found by taking for al l k the function Ξ 0 to b e Ξ 0 ( k ) = − 1 2 artanh  β k α k  = − 1 4 log 1 + β k α k 1 − β k α k ! . The minimal value of the functional define d in (7.36) b elow is inf Ξ E b osonize d (Ξ) = E b osonize d (Ξ 0 ) = X k ∈ Z 3 \{ 0 } 1 2  q α 2 k − β 2 k − α k  < 0 (7.35) with the c o efficients α k = ℏ 2 k · f ( k ) + 1 N ˆ V ( k ) n 2 k > 0 , β k = 1 N ˆ V ( k ) n 2 k > 0 . Note that k · f ( k ) > 0 , so α k > β k . Pr o of. F rom Lemma 5.1 and Lemma 6.4 we ha ve ⟨ T Ω ,  H 0 + Q (0) N  T Ω ⟩ = 1 N Re X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k sinh( | Ξ( k ) | ) 2 + Ξ( k ) | Ξ( k ) | sinh( | Ξ( k ) | ) cosh( | Ξ( k ) | ) ! + X k ∈ Z 3 \{ 0 } ℏ 2 k · f ( k )sinh( | Ξ( k ) | ) 2 + ε 1 + 2 Re ε 2 =: E bosonized (Ξ) + ε 1 + 2 Re ε 2 . (7.36) 20 W e minimize the functional E bosonized (Ξ). Due to the real part on the first line, the complex part of Ξ( k ) do es not matter, so w. l. o. g. Ξ( k ) ∈ R . W e can optimize for ev ery k separately . The co efficien ts 1 N ˆ V ( k ) n 2 k and ℏ 2 2 k · f ( k ) are b oth non-negative, and also sinh( | Ξ( k ) | ) 2 and sinh( | Ξ( k ) | ) cosh( | Ξ( k ) | ) are also non-negativ e. So the only wa y of obtaining a result smaller than zero is obtained for Ξ( k ) / | Ξ( k ) | = − 1, that is, for Ξ( k ) a negativ e function. W e hav e E bosonized (Ξ) = X k ∈ Z 3 \{ 0 } g k ( | Ξ( k ) | ) , g k ( x ) := α k sinh( x ) 2 − β k sinh( x ) cosh( x ) with the co efficien ts α k and β k as given ab o ve. Therefore w e hav e to minimize g k ( x ) with resp ect to x ≥ 0, with k as a parameter. W e determine the critical p oints: 0 ! = g ′ k ( x k ) = α k 2 sinh( x k ) cosh( x k ) − β k  cosh( x k ) 2 + sinh( x k ) 2  = α k sinh(2 x k ) − β k cosh(2 x k ) . This has the p ositiv e (since 0 < β k α k < 1) solution 2 x k = artanh  β k α k  . W e plug this into g k ( x ) = α k 1 2 (cosh(2 x ) − 1) − β k 1 2 sinh(2 x ) and use the tw o iden tities cosh(artanh( A )) = 1 / √ 1 − A 2 and sinh(artanh( A )) = A/ √ 1 − A 2 to obtain g k ( x k ) = 1 2 α k 1 p 1 − ( β k /α k ) 2 − α k − β k β k /α k p 1 − ( β k /α k ) 2 ! = 1 2  q α 2 k − β 2 k − α k  . This confirms (7.35). Lemma 7.2 (Constant for the Kinetic Energy) . The function f define d ab ove satisfies k · f ( k ) = | k | N 1 / 3  4 3 √ π  2 / 3 + O ( N 0 ) . (7.37) Mor e over β k = ℏ  3 4 √ π  2 / 3 ˆ V ( k ) | k | + O ( N − 1 / 3 ) , α k = ℏ | k |  4 3 √ π  2 / 3 + β k . Pr o of. W e hav e n 2 k k · f ( k ) = X p ∈ B c F h ∈ B F δ p − h,k k · (2 h + k ) ≃ 2 X h ∈ B F χ ( | h + k | > R ) k · h , where R =  3 4 π  1 / 3 N 1 / 3 is the leading order of the F ermi m omen tum. Now we use an integral appro ximation (by rescaling in the sum h = N 1 / 3 ˜ h , where ˜ h now corresp onds to Riemann cub es of side length N − 1 / 3 , the indicated errors are one p o wer of N − 1 / 3 smaller than the 21 main terms): 2 X h ∈ B F χ ( | h + k | > R ) k · h ≃ 2 N Z | ˜ h |≤ ( 3 4 π ) 1 / 2 d 3 ˜ h χ | ˜ h + k N 1 / 3 | >  3 4 π  1 / 3 ! k ·  N 1 / 3 ˜ h  (7.38) = 2 N 4 / 3 Z ( 3 4 π ) 1 / 3 0 r 2 d r Z π 2 +small 0 sin( θ )d θ Z 2 π 0 d φ χ | ˜ h + k N 1 / 3 | >  3 4 π  1 / 3 ! r cos( θ ) | k | (7.39) ≃ 2(2 π ) | k | N 4 / 3 Z π 2 0 sin( θ )d θ cos( θ ) Z ( 3 4 π ) 1 / 3 ( 3 4 π ) 1 / 3 −| k | cos( θ ) N − 1 / 3 r 3 d r (7.40) = 2(2 π ) | k | N 4 / 3 Z π 2 0 d θ sin( θ ) cos( θ ) 1 4    3 4 π  4 / 3 −  3 4 π  1 / 3 − | k | cos( θ ) N 1 / 3 ! 4   = 2(2 π ) | k | N 4 / 3  1 4 π | k | N − 1 / 3 + O ( N − 2 / 3 )  = | k | 2 N + O ( N 2 / 3 ) . (7.41) T o get from (7.38) to (7.39), we parametrized the vector ˜ h in spherical co ordinates b y its length r , the angle θ whic h is measured b et w een ˜ h and k , and the remaining rotation once around b y the angle φ . T o get from (7.39) to (7.40), w e wrote the condition of the c haracteristic function as r 2 + 2 r | k | N 1 / 3 cos( θ ) + | k | 2 N 2 / 3 >  3 4 π  2 / 3 , whic h is satisfied for (expanding to first order as N − 1 / 3 → 0) r ≥ s  3 4 π  2 / 3 − | k | 2 N 2 / 3 (sin θ ) 2 − | k | N 1 / 3 cos( θ ) ≃  3 4 π  1 / 3 − | k | N 1 / 3 cos( θ ) . Dividing (7.41) by n 2 k = | k | N ℏ  3 4 √ π  2 / 3 w e obtain the claimed formula for k · f ( k ). 8 Estimating the Error T erms W e hav e estimated all error terms through the exp ectation v alue sup t ∈ [0 , 1] ⟨ T ( t )Ω , ( N + 2) 3 T ( t )Ω ⟩ of the fermionic n umber op erator N . W e now estimate this exp ectation v alue; this follows the same strategy used to control the expectation v alues of the num b er op erator in [BPS14a, BPS14b, BJP + 16, PRSS17]. Prop osition 8.1 (Bound on the F ermionic Particle Number) . F or al l n ∈ N and for al l ψ ∈ F we have sup t ∈ [0 , 1] ⟨ T ( t ) ψ , ( N + 1) n T ( t ) ψ ⟩ ≤ e C n (Ξ) ⟨ ψ , ( N + 1) n ψ ⟩ , 22 wher e the c onstant is C n (Ξ) := 8 n (5 n ) X k ∈ Z 3 \{ 0 } | Ξ( k ) | . In p articular, for fixe d n and in the vacuum ψ = Ω , the b ound is of or der 1 . Pr o of. Using the CAR for the fermionic op erators w e obtain [ N , b ∗ k ] = 2 b ∗ k and b ∗ k b ∗ − k ( N + 4) = N b ∗ k b ∗ − k . (8.42) W e calculate the deriv ative of the exp ectation v alue:     d d t ⟨ T ( t ) ψ , ( N + 1) n T ( t ) ψ ⟩     =       ⟨ T ( t ) ψ , n − 1 X j =0 ( N + 1) j [ N , B ]( N + 1) n − j − 1 T ( t ) ψ ⟩       =       2 X k ∈ Z 3 \{ 0 } Ξ( k ) n − 1 X j =0 ⟨ T ( t ) ψ , ( N + 1) j b ∗ k b ∗ − k ( N + 1) n − j − 1 T ( t ) ψ ⟩ + c.c.       ; w e no w insert I = ( N + 5) n − 1 2 − j ( N + 5) j − n − 1 2 and commute in order to distribute the p ow ers of the n umber op erator equally onto b oth argumen ts of the scalar pro duct: =       2 Re X k ∈ Z 3 \{ 0 } Ξ( k ) n − 1 X j =0 ⟨ T ( t ) ψ , ( N + 1) j ( N + 5) n − 1 2 − j ( N + 5) j − n − 1 2 b ∗ k b ∗ − k ( N + 1) n − j − 1 T ( t ) ψ ⟩       =       2 Re X k ∈ Z 3 \{ 0 } Ξ( k ) n − 1 X j =0 ⟨ T ( t ) ψ , ( N + 1) j ( N + 5) n − 1 2 − j b ∗ k b ∗ − k ( N + 1) j − n − 1 2 ( N + 1) n − j − 1 T ( t ) ψ ⟩       ≤ 4 X k ∈ Z 3 \{ 0 } | Ξ( k ) | n − 1 X j =0 ∥ b k ( N + 5) n − 1 2 − j ( N + 1) j T ( t ) ψ ∥∥ b ∗ − k ( N + 1) j − n − 1 2 ( N + 1) n − j − 1 T ( t ) ψ ∥ ≤ 8 X k ∈ Z 3 \{ 0 } | Ξ( k ) | n − 1 X j =0 ∥N 1 / 2 ( N + 5) n − 1 2 − j ( N + 1) j T ( t ) ψ ∥ × ∥ ( N + 1) 1 / 2 ( N + 1) j − n − 1 2 ( N + 1) n − j − 1 T ( t ) ψ ∥ ≤ 8 X k ∈ Z 3 \{ 0 } | Ξ( k ) | n − 1 X j =0 ∥ ( N + 5) n/ 2 T ( t ) ψ ∥∥ ( N + 1) n/ 2 T ( t ) ψ ∥ ≤ 8 X k ∈ Z 3 \{ 0 } | Ξ( k ) | n ⟨ T ( t ) ψ , ( N + 5) n T ( t ) ψ ⟩ ≤ C n (Ξ) ⟨ T ( t ) ψ , ( N + 1) n T ( t ) ψ ⟩ . The claim no w follows through Gr¨ on wall’s Lemma. Lemma 8.2 (Collected Error Estimates) . L et c := 4  9 π 16  2 / 3 . Assume that the fol lowing 23 quantities ar e finite: A 1 := X k ∈ Z 3 \{ 0 }    log  1 + c ˆ V ( k )     A 2 := X k ∈ Z 3 \{ 0 } | ˆ V ( k ) | q 1 + c ˆ V ( k ) A 3 := X k ∈ Z 3 \{ 0 } | ˆ V ( k ) | q 1 + c ˆ V ( k ) p | k | A 4 := X k ∈ Z 3 \{ 0 }    log  1 + c ˆ V ( k )     q 1 + c ˆ V ( k ) p | k | A 5 := X k ∈ Z 3 \{ 0 }  1 + c ˆ V ( k )  1 / 4 p | k | . Then ther e exists a c onstant C > 0 , dep ending only on A 1 , A 2 , A 3 , A 4 , and A 5 , such that the c ol le ction of al l err or terms is | ε 1 + 2 Re ε 2 | ≤ C N . Pr o of. W e consider the Bogoliub o v transformation found in Prop osition 7.1, defined by Ξ 0 ( k ) = − 1 2 artanh  β k α k  = − 1 4 log 1 + β k α k 1 − β k α k ! , with β k , α k , and | k · f ( k ) | as computed in Lemma 7.2. According to (7.36) the sum of all error terms is ε 1 + 2Re ε 2 . Using Prop osition 8.1 we get sup t ∈ [0 , 1] ⟨ T ( t )Ω , ( N + 2) 3 T ( t )Ω ⟩ ≤ 8 e C 3 (Ξ 0 ) . En tering with this b ound into the estimates from Prop osition 5.1 and Prop osition 6.4 we get | ε 1 | ≤     1 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k " 32 e C 3 (Ξ 0 ) e 2 | Ξ 0 ( k ) |   X m ∈ Z d \{ 0 } | Ξ 0 ( m ) | n m n k   2 (8.43) +  4 sinh( | Ξ 0 ( k ) | ) + 2 cosh( | Ξ 0 ( k ) | )  √ 8 e C 3 (Ξ 0 ) / 2 e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | n m n k #     (8.44) and | ε 2 | ≤     2 ℏ 2 √ 8 e C 3 (Ξ 0 ) / 2 X k ∈ Z 3 \{ 0 } " 2 | Ξ 0 ( k ) || k · f ( k ) | sinh( | Ξ 0 ( k ) | ) e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } 2 | Ξ 0 ( m ) | n m n k (8.45) + X l ∈ Z 3 \{ 0 } | Ξ 0 ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l  sinh( | Ξ 0 ( k ) | ) + e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | n m n k  #     . (8.46) First observe that, with c = 4  9 π 16  2 / 3 , w e ha ve Ξ 0 ( k ) = − 1 4 log  1 + c ˆ V ( k )  , b y whic h w e find C 3 (Ξ 0 ) = 750 P k ∈ Z 3 \{ 0 }    log  1 + c ˆ V ( k )     = 750 A 1 . Next w e ha ve e 2 | Ξ 0 ( k ) | = q 1 + c ˆ V ( k ). 24 W e estimate the first line of ε 1 , as giv en in (8.43), b y     1 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k 32 e C 3 (Ξ 0 ) e 2 | Ξ 0 ( k ) |   X m ∈ Z d \{ 0 } | Ξ 0 ( m ) | n m n k   2     ≤ 1 N 5 / 3  4 3 √ π  2 / 3 32 e 750 A 1 X k ∈ Z 3 \{ 0 } | ˆ V ( k ) | q 1 + c ˆ V ( k )   X m ∈ Z d \{ 0 } | Ξ 0 ( m ) | p | m |   2 ≤ 1 N 5 / 3  4 3 √ π  2 / 3 2 e 750 A 1 A 2 A 2 1 , where w e used that, b ecause of | m | ≥ 1, P m ∈ Z d \{ 0 } | Ξ 0 ( m ) | √ | m | ≤ 1 4 A 1 . F or the second line of ε 1 , as given in (8.44), recall that sinh( | Ξ 0 ( k ) | ) ≤ e | Ξ 0 ( k ) | and cosh( | Ξ 0 ( k ) | ) ≤ e | Ξ 0 ( k ) | . Using these t wo estimates and then pro ceeding as ab ov e, we find     1 N X k ∈ Z 3 \{ 0 } ˆ V ( k ) n 2 k  4 sinh( | Ξ 0 ( k ) | ) + 2 cosh( | Ξ 0 ( k ) | )  √ 8 e C 3 (Ξ 0 ) / 2 e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | n m n k     ≤ 1 N 6 √ 8 e 375 A 1 X k ∈ Z 3 \{ 0 } | ˆ V ( k ) | p | k | q 1 + c ˆ V ( k ) 1 4 A 1 = 1 N 3 √ 2 e 375 A 1 A 3 A 1 . F or the first line of ε 2 , as giv en in (8.45), w e hav e     2 ℏ 2 √ 8 e C 3 (Ξ 0 ) / 2 X k ∈ Z 3 \{ 0 } 2 | Ξ 0 ( k ) || k · f ( k ) | sinh( | Ξ 0 ( k ) | ) e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } 2 | Ξ 0 ( m ) | n m n k     ≤ 2 ℏ 2 4 √ 8  4 3 √ π  2 / 3 e 375 A 1 X k ∈ Z 3 \{ 0 } | Ξ 0 ( k ) || k · f ( k ) | e 2 | Ξ 0 ( k ) | 1 N 2 / 3 p | k | X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | p | m | ≤ 1 N 4 / 3 √ 2 e 375 A 1  4 3 √ π  2 / 3 X k ∈ Z 3 \{ 0 }    log  1 + c ˆ V ( k )     | k · f ( k ) | p | k | q 1 + c ˆ V ( k ) A 1 ≤ 1 N √ 2 e 375 A 1  4 3 √ π  4 / 3 A 4 A 1 . F or the estimates for the second line of ε 2 , as giv en in (8.46), w e hav e     2 ℏ 2 √ 8 e C 3 (Ξ 0 ) / 2 X k ∈ Z 3 \{ 0 } X l ∈ Z 3 \{ 0 } | Ξ 0 ( l ) |  6 π  1 / 3 N 1 / 3 | k | n k n l  sinh( | Ξ 0 ( k ) | ) + e | Ξ 0 ( k ) | X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | n m n k      ≤ 1 N  6 π  1 / 3 2 √ 8 e 375 A 1  4 3 √ π  2 / 3 X l ∈ Z 3 \{ 0 } | Ξ 0 ( l ) | p | l | X k ∈ Z 3 \{ 0 } p | k | e | Ξ 0 ( k ) | ×   1 + X m ∈ Z 3 \{ 0 } | Ξ 0 ( m ) | p | m | 1 N 2 / 3 p | k |  4 3 √ π  2 / 3   ≤ 1 N  6 π  1 / 3 √ 2 e 375 A 1  4 3 √ π  2 / 3 A 1 A 5 1 +  4 3 √ π  2 / 3 A 1 4 ! where from the second to the third line w e used N − 2 / 3 | k | − 1 / 2 ≤ 1. Com bining these four estimates w e find the claimed result. 25 Ac kno wledgmen ts It is a pleasure to thank Benjamin Sc hlein and Marcello Porta for discussions. The author w as supp orted b y the Europ ean Union through the ERC StG FermiMa th , grant agreemen t nr. 101040991. Views and opinions expressed are ho wev er those of the author only and do not necessarily reflect those of the Europ ean Union or the Europ ean Researc h Council Executive Agency . Neither the Europ ean Union nor the granting authorit y can b e held resp onsible for them. Moreov er the author was partially supp orted b y Grupp o Nazionale p er la Fisica Matematica in Italy . References [BD23] Niels Benedikter and Da vide Desio. Tw o Commen ts on the Deriv ation of the Time-Dep enden t Hartree–F o c k Equation. In Mic hele Correggi and Marco F alconi, editors, Quantum Mathematics I , volume 57, pages 319–333. Springer Nature Singap ore, Singap ore, 2023. [Ben21] Niels Benedikter. Bosonic collective excitations in F ermi gases. R eviews in Math- ematic al Physics , 33(1):2060009, 2021. [BJP + 16] Niels Benedikter, V o jk an Jak ˇ si´ c, Marcello Porta, Chiara Saffirio, and Benjamin Sc hlein. Mean-field ev olution of fermionic mixed states. Communic ations on Pur e and Applie d Mathematics , 69(12):2250–2303, 2016. [BNP + 20] Niels Benedikter, Phan Th` anh Nam, Marcello Porta, Benjamin Schlein, and Rob ert Seiringer. Optimal Upper Bound for the Correlation Energy of a F ermi Gas in the Mean-Field Regime. Communic ations in Mathematic al Physics , 374(3):2097–2150, 2020. [BNP + 21] Niels Benedikter, Phan Th` anh Nam, Marcello Porta, Benjamin Schlein, and Rob ert Seiringer. Correlation energy of a weakly in teracting F ermi gas. In- ventiones mathematic ae , 225(3):885–979, 2021. [BPS14a] Niels Benedikter, Marcello Porta, and Benjamin Schlein. Mean–Field Ev olution of F ermionic Systems. Communic ations in Mathematic al Physics , 331(3):1087– 1131, 2014. [BPS14b] Niels Benedikter, Marcello Porta, and Benjamin Schlein. Mean-field dynam- ics of fermions with relativistic disp ersion. Journal of Mathematic al Physics , 55(2):021901, 2014. [BPSS23] Niels Benedikter, Marcello Porta, Benjamin Sc hlein, and Robert Seiringer. Corre- lation Energy of a W eakly In teracting F ermi Gas with Large In teraction P otential. A r chive for R ational Me chanics and Analysis , 247(4):65, July 2023. [CHN22] Martin Ra vn Christiansen, Christian Hainzl, and Phan Th` anh Nam. On the effectiv e quasi-bosonic Hamiltonian of the electron gas: Collectiv e excitations and plasmon mo des. L etters in Mathematic al Physics , 112(6):114, No vem b er 2022. [CHN23a] Martin Ra vn Christiansen, Christian Hainzl, and Phan Th` anh Nam. The Gell- Mann–Bruec kner F ormula for the Correlation Energy of the Electron Gas: A Rigorous Upp er Bound in the Mean-Field Regime. Communic ations in Mathe- matic al Physics , 401(2):1469–1529, July 2023. 26 [CHN23b] Martin Ravn Christiansen, Christian Hainzl, and Phan Th` anh Nam. The Random Phase Appro ximation for Interacting F ermi Gases in the Mean-Field Regime. F orum of Mathematics, Pi , 11:e32, January 2023. [CHN24] Martin Ravn Christiansen, Christian Hainzl, and Phan Th` anh Nam. The Corre- lation Energy of the Electron Gas in the Mean-Field Regime, May 2024. [GMB57] Murra y Gell-Mann and Keith A. Bruec kner. Correlation energy of an electron gas at high density . Phys. R ev. , 106:364–368, Apr 1957. [NS81] Heide Narnhofer and Geoffrey L. Sewell. Vlaso v hydrodynamics of a quantum mec hanical mo del. Communic ations in Mathematic al Physics , 79(1):9–24, 1981. [PRSS17] Marcello Porta, Simone Rademac her, Chiara Saffirio, and Benjamin Sc hlein. Mean Field Evolution of F ermions with Coulomb Interaction. Journal of Sta- tistic al Physics , 166(6):1345–1364, 2017. [Sa w57] Katuro Sa wada. Correlation Energy of an Electron Gas at High Density. Phys. R ev. , 106:372–383, Apr 1957. [SBFB57] Katuro Saw ada, Keith A. Brueckner, Nobuyuki F ukuda, and Rob ert Brout. Cor- relation Energy of an Electron Gas at High Density: Plasma Oscillations. Phys. R ev. , 108:507–514, 1957. 27