Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion

We prove an upper bound on the energy density of the dilute spin-12 Fermi gas capturing the leading correction to the kinetic energy 8 πaρ ↑ ρ ↓ with an error of size smaller than aρ 2 ( a 3 ρ ) 1 / 3 − ε for any ε > 0, where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260)


Introduction and main results
We consider an interacting Fermi gas of N particles interacting via a two-body interaction v which we assume to be non-negative, radial and of compact support.In units where = 1 and the particle mass is m = 1/2 the Hamiltonian is given by where ∆ j denotes the Laplacian on the j'th coordinate.For spin-1 2 fermions in some domain Λ = Λ L = [−L/2, L/2] 3 one realises the Hamiltonian on the space L 2 a (Λ N , C 2 ) = N L 2 (Λ, C 2 ).Since the Hamiltonian is spin-independent we can specify definite values for the number of particles with each spin, i.e.N σ particles of spin σ ∈ {↑, ↓} and N ↑ + N ↓ = N.In this setting the Hamiltonian is realized on the space H N ↑ ,N ↓ = L 2 a (Λ N ↑ )⊗L 2 a (Λ N ↓ ).The ground state energy on the space L 2 a (Λ N , C 2 ) is then given by minimizing in N σ (satisfying N ↑ +N ↓ = N) the ground state energies on the spaces H N ↑ ,N ↓ .
This system was previously studied in [FGHP21; Gia22a;LSS05] where it is shown that for a dilute system in the thermodynamic limit e(ρ ↑ , ρ ↓ ) = lim where ρ = ρ ↑ + ρ ↓ , a is the (s-wave) scattering length of the interaction v and ε(a 3 ρ) = o(1) in the limit a 3 ρ ≪ 1.The existence of the thermodynamic limit follows from [Rob71].Moreover the limit doesn't depend on the boundary conditions.The leading term 3 5 (6π 2 ) 2/3 (ρ ↓ ) is the kinetic energy of a free Fermi gas.The next term 8πaρ ↑ ρ ↓ is the leading correction coming from the interaction.This term may be understood as coming from the energy of a pair of opposite-spin fermions times the number of such pairs.The energy correction arising from interactions between fermions of the same spin is of order a 3 p ρ 8/3 , where a p denotes the p-wave scattering length (see [LS23]) and so much smaller.
The first proof of this result was given by Lieb, Seiringer and Solovej [LSS05].Their proof gives the explicit error bounds −C(a 3 ρ) 1/39 ≤ ε(a 3 ρ) ≤ C(a 3 ρ) 2/27 for some constant C > 0. These error bounds were later improved in [FGHP21] and very recently in [Gia22a], where in particular the "optimal" upper bound ε(a 3 ρ) ≤ C(a 3 ρ) 1/3 is shown.The works [FGHP21;Gia22a] however deal with more regular potentials than the quite general setting studied in [LSS05], where it is assumed that the interaction is non-negative, radial and compactly supported.In [FGHP21;Gia22a] the interaction is additionally assumed to be smooth.In particular interactions with a hard core are not treated in [FGHP21;Gia22a].
The upper bound of order aρ 1/3 is optimal in the sense that this is the order of the conjectured next term in the expansion.Namely the Huang-Yang term [HY57], see [Gia22a;Gia22b].
Our main theorem is the "almost optimal" upper bound ε(a 3 ρ) ≤ C δ (a 3 ρ) 1/3−δ for any δ > 0 for some δ-dependent constant C δ > 0 under the same assumptions as in [LSS05], i.e. weaker than that of [FGHP21;Gia22a].In particular we allow for v to have a hard core.A central ingredient in the proof is to prove a rigorous version of a fermionic cluster expansion adapted from [GGR71].This is analogous to what is done in [LS23] for spin-polarized fermions.

Precise statements of results
To give the statement of our main theorem, we first define the scattering length(s) of the interaction v. Definition 1.1 ([LY01, Appendix A; SY20, Section 4]).The s-and p-wave scattering lengths a and a p are defined by 4πa = inf The minimizing f 's are the s-and p-wave scattering functions.They are denoted f s0 and f p0 respectively.
We collect properties of the functions f s0 and f p0 in Lemma 2.4.With this we may then state our main theorem.
The essential steps of the proof are (1) Show the absolute convergence of a fermionic cluster expansion adapted from the formal calculations of [GGR71].For this we need the "Fermi polyhedron", a polyhedral approximation to the Fermi ball, described in [LS23, Section 2.2].The calculation of the fermionic cluster expansion is given in Section 3 and the absolute convergence in shown in Section 4.
(2) Bound the energy of a Jastrow-type trial state.For this step, the central part is computing the values of all diagrams of a certain type exactly and using these exact values up to some arbitrary high order.This is somewhat similar to the approach in [BCGOPS22] for the dilute Bose gas.This calculation is part of the proof of Lemma 5.1.
Remark 1.3 (Higher spin).With not much difficulty one can extend the result to higher spin and with a spin-dependent interaction v σσ ′ = v σ ′ σ .The result for S ≥ 2 spin values {1, . . ., S} is e (ρ 1 , . . ., ρ S ) ≤ 3 5 (6π 2 ) 2/3 S σ=1 where a σσ ′ is the s-wave scattering length of the spin σ -spin σ ′ interaction v σσ ′ and a = sup σ<σ ′ a σσ ′ .For conciseness of the proof we will only give it for S = 2, i.e. for spin-1 2 fermions.We will however give comments on how to adapt the individual (non-trivial) steps of the proof to the higher spin setting.These comments are given in Remarks 3.3, 4.2, 5.4 and 5.6.
The paper is structured as follows.In Section 2 we give some preliminary computations and recall some properties of the scattering functions and Fermi polyhedron from [LS23;LY01].Next, in Section 3 we give the calculation of a fermionic cluster expansion adapted from [GGR71].Subsequently, in Section 4 we find conditions for the absolute convergence of the cluster expansion formulas of Section 3. Finally, in Section 5 we use the formulas of Section 3 to bound the energy of a Jastrow-type trial state.

Preliminary computations
We first give a few preliminary computations.We will construct a trial state using a box method of glueing trial states in smaller boxes together in Section 5.4.In the smaller boxes we will need to use Dirichlet boundary conditions, however in Section 5.4 we will construct trial states with Dirichlet boundary conditions out of trial states with periodic boundary conditions.(See also [LS23].)Thus, we will use periodic boundary conditions in the box Λ First we establish some notation.

Notation
• We write x i and y j for the spatial coordinates of particle i of spin ↑ respectively particle j of spin ↓.
We write z i to mean either x i or y i if the spin is not important.
We write additionally z (i,↑) = x i and z (i,↓) = y i .
• For a set A we write Z A = (z a ) a∈A for the coordinates of the vertices with labels in A.
(Similarly for X A and Y A .) In particular we write Z [n,m] = (z n , . . ., z m ) for the coordinates of particles n, n + 1, . . ., m.
• We write C for a generic (positive) constant, whose value may change line by line.If we want to emphasize the dependence on some parameter A we will denote this by C A .
Define the rescaled and cut-off scattering functions f s and f p as where and the scattering function f s0 and f p0 are defined in Definition 1.1.(They are radial functions, see Lemma 2.4, so f s and f p are well-defined.)We prefer to write b instead of its value ρ −1/3 to keep apparent dependences on b.For b = ρ −1/3 then b > R 0 , the range of v, for sufficiently small a 3 ρ so f s , f p are continuous.(Note that the metric on the torus Λ is given by d(x, y) = |x − y|.We will abuse notation slightly and denote by | • | also the absolute value of some number, the norm on R 3 or the size of some set.) To simplify notation we write for µ, ν ∈ V ∞,∞ and similar for all quantities derived from f s and f p .
For the trial state ψ N ↑ ,N ↓ defined in Equation (2.5) below we denote for µ, ν ∈ V ∞,∞ Nσ are the one-particle density matrices of ψ N ↑ ,N ↓ .Next, we introduce the (non-normalized) Slater determinants D N ↑ and D N ↓ as where P σ F is the "Fermi polyhedron", a polyhedral approximation to the Fermi ball described in Section 2.3, see also [LS23, Section 2.2], and #P σ F = |P σ F | denotes the number of points in P σ F .Finally, for any (normalized) state ψ ∈ L 2 a (Λ N ↑ ) ⊗ L 2 a (Λ N ↓ ) we will normalize reduced densities as follows (for n + m ≥ 1). (2.4) and for the trial state ψ N ↑ ,N ↓ we write ρ We will fix the Fermi momenta k σ F such that the ratio k ↑ F /k ↓ F is rational, see Remark 2.2.This is a restriction on which densities ρ σ can arise from the trial state ψ N ↑ ,N ↓ , see Remark 2.3.We extend to all densities in Section 5.4.The dilute limit will be realized as

Computation of the energy
We consider the trial state where Note that for (real-valued) functions F, G we have By symmetries of the Fermi polyhedron, see Definition 2.1, we have that D N ↑ and D N ↓ are real-valued.Thus, using Equation (2.6) for F = µ<ν f µν and G = D N ↑ D N ↓ for each of the derivatives ∇ x i , ∇ y j we get arises as 2 = σ∈{↑,↓} 1.)The terms are grouped according to how many s-wave f 's appear.In terms of the reduced densities we thus get (2.7) We find formulas for the reduced densities in Section 3. Before doing so, we first recall some properties on the "Fermi polyhedron" P σ F and the scattering functions f s , f p .

Properties of the "Fermi polyhedron" and the scattering functions
In this section we recall a few properties of the "Fermi polyhedron" from [LS23, Section 2.2 and Lemma 4.9] and scattering functions from [LY01, Appendix A].
The "Fermi polyhedron" is defined in [LS23, Definition 2.7].We give here only a sketch of the definition and state a few properties needed for our purposes.For a full discussion with proofs we refer to [LS23, Section 2.2 and Appendix B].
The Fermi polyhedron P σ F is then defined as 2π is rational and large for σ ∈ {↑, ↓}.
Remark 2.2.We choose k σ F such that k ↑ F /k ↓ F is rational since we need L with 2π rational for both values of σ ∈ {↑, ↓}.
Remark 2.3.The free parameters are the Fermi momenta k σ F , the length of the box L and the number of corners of the polyhedra s σ .The particle numbers are then N σ = #P σ F and the particle densities are ) .Not all densities ρ σ0 arise this way.We need some argument to consider general densities ρ σ0 .This is discussed in Section 5.4.Essentially by continuity and density of the rationals in the reals we can extend results for the densities arising as ρ σ = N σ /L 3 to general densities ρ σ0 .

Gaudin-Gillespie-Ripka expansion
In this section we calculate reduced densities of the trial state ψ N ↑ ,N ↓ .The ideas behind this calculation are mostly contained in (the formal calculations of) [GGR71].The calculation we give here is a slight generalization thereof including the spin.Additionally, we give conditions for the final formulas (given in Theorem 3.2) to hold, i.e. we give conditions for their absolute convergence.The argument here is in spirit the same as that of [LS23, Section 3].Here it is slightly more involved as we have to take into account the different spins.In [LS23, Section 3] there is only one value of the spin.
In the calculations below one may replace the functions f s , f p and the one-particle density matrices γ (1) Nσ by more general functions.We discuss this in Remark 3.5 below.

Calculation of the normalization constant
We first compute the normalization constant C N ↑ ,N ↓ .Recall the definition of the trial state ψ N ↑ ,N ↓ in Equation (2.5).Write f 2 µν = 1 + g µν for all the f -factors and factor out the product µ<ν f 2 µν = µ<ν (1 + g µν ).We are then led to define the set G p,q as the set of all graphs on p black and q white vertices such that each vertex has degree at least 1, i.e. has an incident edge.We label the black vertices as V ↑ p = {(1, ↑), . . ., (p, ↑)} and the white vertices as V ↓ q = {(1, ↓), . . ., (q, ↓)}.For an edge e = (µ, ν) we write g e = g µν and define , where we introduced (with similar notation as in [GGR71; LS23]) ∆ p,q := ρ (p,q) .(Recall the definition of ρ (p,q) in Equation (2.4).)A simple calculation using the Wick rule then shows (recall the definition of γ µν in Equation (2.3)) Nσ the p-particle reduced density of 1 √ Nσ! D Nσ .If either p > N ↑ or q > N ↓ then ∆ p,q = 0, since the matrices [γ (i,↑),(j,↑) ] i,j∈N and [γ (i,↓),(j,↓) ] i,j∈N have ranks N ↑ and N ↓ respectively.Thus we may extend the p-and q-sums to ∞.Now, expanding the determinant ∆ p,q and the W p,q we group the permutations and the graph together in a diagram.We will for the calculation of the reduced densities need a slightly more general definition, which we now give.
Definition 3.1.The set G n,m p,q is the set of all graphs with p internal black vertices, n external black vertices, q internal white vertices and m external white vertices, such that there are no edges between external vertices, and such that all internal vertices has degree at least 1.That is, all internal vertices are incident to at least one edge and external vertices may have degree 0. As above we label the black vertices as V ↑ p+n = {(1, ↑), . . ., (p + n, ↑)} where the first n are the external vertices.The white vertices are labelled V ↓ q+m = {(1, ↓), . . ., (q + m, ↓)}, where the first m are the external vertices.In case n = m = 0 we recover G 0,0 p,q = G p,q .If we need the vertices to have certain labels we will write G B * ,W * B,W (or similar with only some of p, q, n, m replaced by sets) for the set of all graphs with internal black vertices B, external black vertices B * , internal white vertices W and external white vertices W * , where p,q is the set of all diagrams on p internal black vertices, n external black vertices, q internal white vertices and m external white vertices.Such a diagram is a tuple D = (π, τ, G) where π ∈ S p+n , τ ∈ S q+m (viewed as directed graphs on the black and white vertices respectively) and G ∈ G n,m p,q .We will refer to the edges in G as g-edges and the graph G as a g-graph.Moreover, we refer to the edges in both π and τ as γ-edges.
The value of the diagram If p = 0 and/or q = 0 there are no integrations in the x i and/or y j variables.
A diagram D = (π, τ, G) is said to be linked if the graph G with union all edges of π, τ and G is connected.The set of linked diagrams is denoted L n,m p,q .In case m = n = 0 we write D 0,0 p,q = D p,q , L 0,0 p,q = L p,q and Γ 0,0 D = Γ D (i.e.without a superscript).
In terms of diagrams we have We may decompose any diagram D = (π, τ, G) into its linked components.For this, note that its value Γ D factors over its linked components.Moreover, each linked component has at least 2 vertices, since they have degree at least one.Thus, Note that there are no γ-edges between vertices of different colours (i.e. with different spin).
Here the factor 1 k! comes from counting the possible ways to label the k linked components and the factors 1 p ℓ !q ℓ !come from counting the possible ways of labelling the vertices in the different linked components (and using the factor 1 p!q! already present).If we assume that the sum p,q:p+q≥2 1 p!q! D∈Lp,q Γ D is absolutely convergent, (more precisely we assume that p,q:p+q≥2 1 p!q! D∈Lp,q Γ D < ∞,) then we may interchange the p, q-sum with the p ℓ , q ℓ -sums.The absolute convergence is proven in Theorem 3.2 below.Thus, under the conditions of Theorem 3.2, we have (3.2)

Calculation of the reduced densities
For the calculation of the reduced densities we need to keep track of also the external vertices.First, we have the formula (for n + m ≥ 1) where we extended the p, q-sums to ∞ as in Section 3.1 above and used that the p = q = 0 term gives Note here that the p, q-sum does not require p + q ≥ 2, since the internal vertices may connect to external ones.As for the normalization constant in Section 3.1 we decompose each diagram D into its linked components.Here we have to keep track of which linked components contain which external vertices.To do this we introduce the set The set Π n,m κ parametrizes all possible ways for the diagram D ∈ D n,m p,q to have exactly κ many linked components containing at least 1 external vertex each.Note that for κ > n + m we have Π n,m κ = ∅, since we require that for all λ we have B * λ = ∅ or W * λ = ∅.Denoting then k the number of linked components with only internal vertices we get the following.
(3.5) (Note that the linked components with external vertices may have 0 or 1 internal vertices, i.e. the p * λ , q * λ -sums do not require p * λ + q * λ ≥ 2.) The factorial factors come from counting the different labellings: The factors 1 k! and 1 κ! from the labellings of the clusters and the factors from labelling the internal vertices of the different clusters exactly as in Section 3.1 above.
If we assume absolute convergence of all the Γ n ′ ,m ′ -sums with n ′ ≤ n and m ′ ≤ m (i.e. that p,q≥0 < ∞) then we may interchange the p, q-sum with the p * λ , q * λ -and p ℓ , q ℓ -sums.We then get p,q≥0 Thus by Equations (3.2) and (3.3) we conclude the formula under the assumption of absolute convergence.

Summary of results
With the calculation above we may then state the following theorem.
Theorem 3.2.For integers n 0 , m 0 ≥ 0 there exist constants for any n ≤ n 0 and m ≤ m 0 with n + m ≥ 1.In particular, then As particular cases we note that for n + m = 1 we have by translation invariance that We give the proof of Theorem 3.2 below.
Remark 3.3 (Higher spin).One may readily generalize the computation above to a general number of spins S. For this one introduces vertices of more colours and diagrams with such, i.e. the sets of graphs and diagrams G n 1 ,...,n S p 1 ,...,p S , D n 1 ,...,n S p 1 ,...,p S , L n 1 ,...,n S p 1 ,...,p S and the values Γ n 1 ,...,n S

D
. The condition of absolute convergence is completely analogous.
Remark 3.4.The condition for the absolute convergence is not uniform in the volume, hence the need for a box method as given in Section 5.4.The condition of absolute convergence is additionally the reason for introducing the Fermi polyhedron.This is discussed in [LS23, Remark 3.5].If one did not introduce the Fermi polyhedron and instead used the Fermi ball the factor s(log N) 3 in the assumption of Theorem 3.2 should be replaced by N 1/3 .Remark 3.5 (General f and γ).In the computation above we may replace the specific functions f s , f p by more general functions Moreover, for the absolute convergence we may additionally replace the one-particle densities γ (1) Nσ by general functions γ σ (z i − z j ).(One then defines γ µν as in Equation (2.3) above.)In the computation above we crucially used that [det γ µν ] µ,ν∈Vp,q = 0 for appropriately large p, q in order to extend the p, q-sums to ∞.If for the general γ σ this is not valid, this step of the computation above is not valid.The rest of the calculation starting from what one gets out of this step is however still valid.That is, the calculation in Section 3.1 is valid starting from Equation (3.1) and the calculations in Equations (3.5) and (3.6) in Section 3.2 are valid.
We give the proof of Lemma 3.6 in Section 4 below for the case S = 2.The proof for general S is a straightforward modification, but notationally more cumbersome.The case S = 1 is treated in [LS23, Section 3.1].Theorem 3.2 is an immediate corollary.
4 Absolute convergence of the Gaudin-Gillespie-Ripka expansion In this section we give the proof of Lemma 3.6 for the case S = 2.The proof is similar to that of [LS23,Theorem 3.4].We need to prove (for all n, m and uniformly in X n , Y m ) Equation (3.12) if Equations (3.10) and (3.11) are satisfied.To simplify notation we define where as above γσ (k) = L −3 ´Λ γ σ (x)e −ikx dx.Equation (3.11) then reads that γ ∞ I g I γ is sufficiently small.We give the proof in two steps.First we consider the case n = m = 0.

Absolute convergence of the Γ-sum
In this section we show that p,q≥0 p+q≥2 under the relevant conditions.Defining clusters as connected components of G we split the sum into clusters as in [LS23, Section 3.1].Denoting the sizes of the clusters by (n ℓ , m ℓ ), ℓ = 1, . . ., k (meaning that the cluster ℓ has n ℓ black vertices and m ℓ white vertices) we get ) where C p,q ⊂ G p,q denotes the subset of connected graphs.The factorial factors arise from counting the possible labellings exactly as in Section 3.
The last line of Equation (4.1) is what we will call the truncated correlation.We give a slightly more general definition for later use.ρ for any connected graphs G ℓ ∈ C B ℓ ,W ℓ .The definition does not depend on the choice of the graphs G ℓ .
If the underlying sets B 1 , . . ., B k , W 1 , . . ., W k are clear we will also use the notation .
The truncated correlations are studied in [GMR21, Appendix D].To better compare to the definition in [GMR21] we note the following.
In Equation (4.2) we may view (π, τ ) together as a permutation of all the vertices (both black and white).Moreover, if we instead sum over all permutations π ′ ∈ S ∪ ℓ B ℓ ∪∪ ℓ W ℓ we have that any π ′ not coming from two permutations π, τ on the black (respectively white) vertices contributes 0, since any γ-factor between vertices of different spins is 0. That is, ρ In [GMR21, Equation (D.53)] is shown the formula for the truncated correlation where A denotes the set of anchored trees, µ A is a probability measure and N (r) is an explicit matrix.The set A ((B 1 ,W 1 ),...,(B k ,W k )) of anchored trees is the set of all directed graphs on the vertices ∪ ℓ B ℓ ∪ ∪ ℓ W ℓ such that each vertex has at most one incoming and at most one outgoing edge, and such that upon identifying all vertices inside each cluster the resulting graph is a (directed) tree.The matrix N (r) satisfies the bound Proof (sketch) of Equation (4.4).Write γ σ (z It is then explained in the proof of [GMR21, Lemma D.6] how to adapt this argument to bound det N (r).
Combining Equations (4.3) and (4.4) we conclude the bound ρ With the truncated correlation we may write the last line as ρ (N ,M) t , where That is, .
To bound this we use the tree-graph bound [Uel18], see also [PU09, Proposition 6.1].By assumption Equation (3.10) is satisfied and thus [Uel18] G∈Cp,q e∈G g e ≤ C p+q where T p,q ⊂ G p,q denotes the subset of trees.(To see this note that C p,q (respectively T p,q ) is the set of connected graphs (respectively trees) on p + q vertices with the colours of the vertices just serving as a handy reminder of the edge-weights g e .)By moreover using the bound on the truncated correlation in Equation (4.5) we conclude that p,q≥0 p+q≥2 (4.7) To do the integrations we note that the graph T with edges the union of (g-)edges in T 1 , . . ., T k and (γ-)edges in A is a tree on all the ℓ n ℓ + ℓ m ℓ many vertices.One then integrates the coordinates one leaf at a time (meaning that the index of the corresponding coordinate is a leaf of the graph T ) and removes a vertex from the graph after integrating over its corresponding coordinate.
To be more precise suppose that ν 0 is a leaf of T .Then the variable z ν 0 appears exactly once in the integrand.Either in a factor g µν 0 (in which case the z ν 0 -integral gives ´|g| ≤ I g by the translation invariance) or in a factor γ µν 0 (in which case the z ν 0 -integral gives ´|γ| ≤ I γ by the translation invariance).The final integral gives L 3 by the translation invariance.There are k − 1 factors of γ and ℓ (n ℓ + m ℓ − 1) = p + q − k factors of g.Thus we get p,q≥0 p+q≥2 Moreover, T n,m = (n + m) n+m−2 ≤ C n+m (n + m)! by Cayley's formula.Finally, we may bound the binomial coefficients (n+m)!n!m! ≤ 2 n+m .Thus p,q≥0 p+q≥2 for γ ∞ I g I γ small enough.This shows that p,q:p+q≥2 1 p!q! D∈Lp,q Γ D is absolutely convergent under this assumption.Next, we bound the Γ n,m -sum for n + m ≥ 1.

Absolute convergence of the Γ n,m -sum
In this section we prove that (for n + m ≥ 1 and uniformly in X n , Y m ) p,q≥0 if Equation (3.10) is satisfied and γ ∞ I g I γ is sufficiently small.
We do the same splitting into clusters (connected components of G) as in Section 4.1 above.There is however a slight complication: One needs to keep track of in which clusters the external vertices lie.This is exactly parametrized by the set Π κ n,m (defined in Equation (3.4)).Denoting the sizes (number of internal vertices) of the clusters containing external vertices by (n * λ , m * λ ) and the sizes of clusters only containing internal vertices by (n ℓ , m ℓ ) and introducing C n,m p,q ⊂ G n,m p,q as the subset of connected graphs (and similarly (4.8)For k = 0 the n 1 , m 1 , . . ., n k , m k -sum should be interpreted as an empty product, i.e. as a factor 1. Similarly for p = 0 and/or q = 0 the empty product of integrals should be interpreted as a factor 1.
The last line in Equation (4.8) is the truncated correlation , where and ⊕ means concatenation of vectors, i.e.
where we abused notation slightly and wrote B * 1 + n * 1 for the union of the vertices B * 1 and the n * 1 internal black vertices of the graph G * 1 .(Similarly for the other terms.)We use as in Section 4.1 the tree-graph bound and the bound on the truncated correlation in Equation (4.5).For the clusters with external vertices we add 0-weights to the disallowed edges as in [LS23, Section 3.1.3],i.e. for G ∈ C n,m p,q define ge = 0 e = (i, j) with i, j external vertices g e otherwise.
Then we may readily apply the tree-graph bound [Uel18] with edge-weights ge : where T p,q ⊂ G p,q and T n,m p,q ⊂ C n,m p,q denotes the subsets of trees.Thus p,q≥0 (4.9) To do the integrations we bound some g-and γ-factors pointwise.Recall first, that there are κ clusters with external vertices.We split the anchored tree into pieces according to these clusters as follows.
We may view the anchored tree A as a tree on the set of clusters.If κ = 1 set A 1 = A. Otherwise iteratively pick a γ-edge on the path in A between any two clusters with external vertices and bound it by and remove it from A. This cuts the anchored tree A into pieces.Doing this κ − 1 many times we get κ anchored trees A 1 , . . ., A κ with each exactly one cluster with external vertices.That is, Next, in each cluster with external vertices, say with label λ 0 , we do a similar procedure of splitting the cluster into pieces according to the external vertices.
In the cluster λ 0 there are Otherwise iteratively pick a g-edge on the path in T * λ 0 between any two external vertices and bound it by using Equation (3.10) for q = 2. Remove the edge e from T * λ 0 .This cuts the tree T * λ 0 into pieces.Doing this | with each exactly one external vertex.That is, We do this procedure for all the κ many clusters with external vertices.Then the graph T with edges the union of all (g-or γ-)edges in T * λ,ν , T ℓ , A λ (for λ ∈ {1, . . ., κ}, ℓ ∈ {1, . . ., k} ) is a forest (disjoint union of trees) on the set of vertices V n+ λ n * λ + ℓ n ℓ ,m+ λ m * λ + ℓ m ℓ with each connected component (tree) having exactly one external vertex.Moreover, we have the bound Since each connected component of T is a tree we may do the integrations one leaf at a time exactly as for the Γ-sum in Section 4.1 above.To bound the value we count the number of γ-and g-factors that are left.
The number of γ-integrations is exactly the number of γ-factors.There are k + κ many clusters, so A has k + κ − 1 many edges.In constructing A 1 , . . ., A κ we cut κ − 1 many edges, thus there is k many γ-factors left and so there are k many γ-integrations in Equation (4.10).The remaining The integrals may be bounded by ´|γ| ≤ I γ and ´|g| ≤ I g as in Section 4.1.Moreover, since each connected component of T has one external vertex, which is not integrated over, there are no volume factors from the last integrations in any of the connected components of T .That is, We use this to bound the integrations in Equation (4.9).Additionally we need to bound the number of (anchored) tree.In [GMR21, Appendix D.5] it is shown that since we have k + κ many clusters and n + m + λ (n * λ + m * λ ) + ℓ (n ℓ + m ℓ ) many vertices in total.Moreover, #T n,m p,q ≤ #T p+n,q+m = (p + q + n + m) p+q+n+m−2 ≤ (p + q + n + m)!C p+q+n+m by Cayley's formula as in Section 4.1.These bounds together with Equation (4.9) then gives p,q≥0 Multinomial coefficients may be bounded as (p 1 +...+p k )!
(4.12)For some c n,m > 0 we have that if γ ∞ I g I γ < c n,m the sums are convergent and we get p,q≥0 This shows the desired.We conclude the proof of Lemma 3.6 for the case S = 2. Remark 4.2 (Higher spin).For the case of higher spin S ≥ 3, the computations are essentially the same.
For later use we define for all diagrams some values characterising their sizes.
p,q .Define the number k = k(D) as the number of clusters entirely within internal vertices (i.e. the same k as in the computations above) and κ = κ(D) as the number of clusters containing at least one external vertex (i.e. the same κ as in the computations above).Define then n * g = n * g (D) and n g = n g (D) as where n * λ , m * λ , n ℓ , m ℓ are the sizes of the different clusters exactly as in the computations above.(Then n g + n * g + 2k = p + q.)For a diagram D the number n g + n * g is the "number of added vertices" in the following sense.A diagram with n + m external vertices and k clusters entirely within internal vertices has at least n + m + 2k many vertices, since each cluster (with only internal vertices) has at least 2 vertices.Then n g + n * g is the number of vertices a diagram has more than this minimal number.Note that in the special case of consideration with the scattering functions f s , f p and the one-particle density matrices γ (1) by Equations (2.9) and (3.13), see also the proof of Theorem 3.2.Then, by following the arguments above (see in particular Equations (4.11) and (4.12)), we have (for p + q = 2k 0 + n g0 ) for any n, m with n + m ≥ 1.We think of s as s ∼ (a 3 ρ) −1/3+ε for some small ε > 0. Thus increasing n g0 by 1 we decrease the size of the diagram by (a 3 ρ) 1/3 , and increasing k 0 by 1 we decrease the size of the diagram by (a 3 ρ) ε .(Recall that b = ρ −1/3 .)

Energy of the trial state
In this section we use the formulas in Equation (3.8) to calculate the energy in Equation (2.7).
We will refer to a term in Equation (2.7) where ρ (n,m) Jas appears as a (n, m)-type term.

2-body terms
In this section we consider the terms in Equation (2.7) where a two-particle density (ρ (n,m) Jas with n + m = 2) appears.We consider first the term with m = n = 1.

(1, 1)-type terms
We consider the term ¨ρ(1,1) The formula in Equation (3.8) reads for ρ (1,1) Jas as follows. ρ (1,1) p,q≥0 p,q = ∅ for p = q = 0.The second summand is an error term.We bound it as follows.Lemma 5.1.There exists a constant c > 0 such that if sab 2 ρ(log N) 3 < c, then for any integer K there exists a constant C K > 0 such that p,q≥0 p+q≥1 We give the proof at the end of this section.Using Equation (5.2) and Lemma 5.1 we get for any integer K By Definition 1.1 we have + O(L 3 sa 3 ρ 3 log(b/a)(log N) 3 ).
(5.3) Finally, we give the Proof of Lemma 5.1.We split the diagrams into three groups using the numbers n * g , n g and k from Definition 4.3: (B2) only s-wave g-factors.
Remark 5.2.The diagrams of types (A) and (B1) are those for which the bound in Equation (4.13) is good enough to show that these diagrams give contributions to the energy density ≤ Ca 2 ρ 7/3 .Naively using the bound in Equation (4.13) for the diagrams of type (B2) we only get that these are bounded by ρ 2 (a 3 ρ) ε with b = ρ −1/3 and s chosen as described immediately after Equation (4.13).We will calculate the value of all the (infinitely many) diagrams of type (B2) below and use this exact calculation for all diagrams up to some arbitrary high order.This is an essential step in proving the "almost optimal" error bound in Theorem 1.2.It is similar to the approach in [BCGOPS22] for the dilute Bose gas.
The contribution of all diagrams of type (A) (with n g + n * g ≥ 1) is ≤ Cab 2 ρ 3 by Equation (4.13) if sab 2 (log N) 3 is sufficiently small (recall Theorem 3.2).For diagrams of type (B) note that we have k 0 ≥ 1, since any summand p, q has p + q ≥ 1.Moreover, for diagrams of type (B1), at least one factor ´|g s | ≤ Cab 2 should be replaced by ´|g p | ≤ Ca 3 log b/a (recall the bounds in Equation (3.13)).Thus, again by Equation (4.13), we may bound the size of all diagrams of type (B1) by Csa 3 ρ 3 log(b/a)(log N) 3 .More precisely we have p,q≥0 p+q≥1 if sab 2 ρ(log N) 3 is sufficiently small.It remains to consider the diagrams of type (B2), where n g + n * g = 0 and only s-wave g-factors appear.These diagrams have g-graph as in Figure 5.1b.Note that in particular p = q = k(D) for any such diagram.
(1, ↑) We now evaluate all these diagrams.We give an example calculation of the (unique) diagram of smallest size, and then do the computation in full generality.The diagram of smallest size is the diagram in Figure 5.1a.Its value is N ↑ (x 2 ; x 1 )γ (1) N ↓ (y 1 ; y 2 )γ (1) where ĝs (k) = L −3 ´gs (x)e −ikx dx denotes the Fourier transform and we used the translation invariance to evaluate the g s -integral.We have the bound (recall Equation (3.13)) 2 ) effectively kills one of the four k σ j -sums.The remaining k σ j -sums have at most N σ ≤ N many summands.We conclude the bound (uniformly in x 1 , y 1 ) Γ 1,1 D small ≤ Cab 2 ρ 3 .For the general diagram in Figure 5.1b we may use the same method.We then have x 1 e x j ˆdy j g s (x j − y j )e x 1 e .
Again, each factor L 3 ĝs we may bound by Cab 2 .Moreover, since the diagram is linked we have for each j that π −1 (j) = j and/or τ −1 (j) = j.(Otherwise the vertices {(j, ↑), (j, ↓)} would be disconnected from the rest.)Thus, each characteristic function is non-trivial, and hence effectively kills one of the k σ j -sums.Each surviving k σ j -sum has at most N σ ≤ N many summands.Thus (uniformly in x 1 , y 1 ) Γ 1,1 D ≤ ρ 2 (Cab 2 ρ) k for any diagram D of type (B2) with k clusters of internal vertices, i.e. with g-graph as in Figure 5.1b.For any integer K we have some finite K-dependent number of diagrams with k ≤ K. Concretely let ν k 0 < ∞ be the number of type (B2) diagrams with k = k 0 .Thus, using Equation (4.13) for diagrams with k > K, we get p,q≥0 p+q≥1 (5.4) for some constant C K > 0 if sab 2 ρ(log N) 3 is sufficiently small.
Remark 5.3 (Upper bound on number of diagrams -why we can't pick K = ∞).For an upper bound on the number of diagrams we first find an upper bound on the number of graphs.All the underlying graphs look like Figure 5.1b, but the labelling of the internal vertices may be different.We are free to choose which white (internal) vertex connects to (2, ↑) and so on.In total there are thus q! = k! many possible graphs.Next, to bound the number of diagrams with any given g-graph we may forget the constraint that the diagram has to be linked and consider all choices of π ∈ S k+1 and τ ∈ S k+1 instead of just those, for which the diagram is linked.For both π and τ there are then (k + 1)! many choices.Thus for each graph G there is at most (k + 1)! 2 many linked diagrams of type (B2) with g-graph G. Thus there are at most k!(k + 1)! 2 diagrams of type (B2) with k clusters of internal vertices.With this bound the sum is not convergent.This prevents us from taking K = ∞ in Equation (5.4) and using the exact calculations for all (infinitely many) diagrams of type (B2).
Remark 5.4 (Higher spin).For S ≥ 3 values of the spin the evaluation of the diagrams is the same, but the combinatorics of counting how many diagrams there are for each given size is more complicated.Still, there is only some finite K-dependent number of diagrams with k(D) ≤ K and thus (the appropriately modified version of) Equation (5.4) is valid if sab 2 ρ(log N) 3 < c S for some constant c S > 0.
Consider next ξ 0 .We do a Taylor expansion to second order around the diagonal.For the zero'th order we have ξ 0 (x 1 = x 2 ) + ξ ≥1 (x 1 = x 2 ) = 0 exactly as in [LS23, Equation (4.13)].The first order vanishes by the symmetry in x 1 and x 2 .Finally, we may bound the second derivatives ∂ i x 1 ∂ j x 1 ξ 0 by following the same procedure as in [LS23, Proof of Lemma 4.1, Equations (4.15) to (4.20)].This crucially uses the bounds in Equation (2.9).We give a brief sketch of this argument.
Write (recalling Equation (4.8) and using that the k = 0 term together with ρ 2 ↑ give the two-particle density ρ (2,0) by Wick's rule) The only dependence on x 1 is in the γ-factors.Then the second derivatives ∂ i x 1 ∂ j x 1 ξ 0 may be computed exactly as in [LS23, Equations (4.15) and (4.17)]:One or two of the γ-factors will gain the derivatives ∂ i x 1 and ∂ j x 1 .We then use the (appropriately modified) truncated correlation (Equation (4.2)) and (the appropriately modified version of) Equation (4.3).The derivative γ-factors may end up in the matrix N (r), in which case the bound in Equation (4.4) gains a power of ρ 1/3 for each derivative, see [LS23, Equation (4.16)], or they may end up in the anchored tree, in which case we need to bound their integrals.Then, using the computations of Section 4.2, we bound, analogously to [LS23, Equation (4.20)], where #∂ denotes the number of derivatives in ∂, i.e. #1 = 0, #∂ i x 1 = 1 and #∂ i x 1 ∂ j x 1 = 2. Using Equation (2.9) to bound the integrals we conclude that if N σ is sufficiently large and sab 2 (log N) 3 is sufficiently small.By Taylor's theorem we conclude the desired.

3-body terms
In this section we bound the 3-body terms of Equation (2.7).
Remark 5.6 (Higher spin).For higher spin we also have terms of type (1, 1, 1).These may be bounded exactly as the (2, 1)-type terms with two s-wave factors.

Box method
We extend to the thermodynamic limit using a box method exactly as in [ N,L denotes the Hamiltonian on a box of sides L with periodic boundary conditions.We are free to choose the the parameter d.We will choose it some large negative power of a 3 ρ.
the Fourier coefficients.Then by the Gram-Hadamard inequality [GMR21, Lemma D.1] where T ℓ ∼ A λ means that T ℓ and A λ share a vertex.(Equivalently they are part of the same connected component of T .)