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Semiclassical limit for almost fermionic anyons

In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a"perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic behavior in its momentum distribution. Our proof is based on coherent states, Husimi functions, the Diaconis-Freedman theorem and a quantitative version of a semi-classical Pauli pinciple.


Model and main results
1.1. Introduction. In two dimensional systems there are possibilities for quantum statistics other than the bosonic one and the fermionic one, so called intermediate or fractional statistics. Particles following such statistics, termed anyons (as in anything in between bosons and fermions), can arise as effective quasi-particles in correlated many-body quantum systems confined to two dimensions. They have been conjectured [2,43,55] to be relevant for the fractional quantum Hall effect (see [25,31,29] for reviews and [5,51] for recent experimental developments). They can also be simulated in certain cold atoms systems with synthetic gauge fields [18,11,63,65,66,13,67,68].
The statistics of anyonic particles is labeled by a single parameter α ∈ [0, 2[ where the cases α = 0, 1 correspond to ordinary bosons and fermions, respectively. Since the many-anyon problem is not exactly soluble (even in the absence of interactions other than the statistical ones) except for the latter special values, it makes sense to study limiting parameter regimes where reduced models are obtained as effective descriptions.
The "almost bosonic" limit α → 0 is studied in [24,42], in the joint limit of the particle number N → ∞, α ∼ N −1 , which turns out to be the relevant one for the non-trivial statistics to have an effect at leading order [14,13]. Here we study an "almost fermionic limit" α → 1 in dependence with the particle number. The relevant scaling in dependence of N leads to a problem resembling the usual mean-field/semi-classical limit for interacting fermions [37,21,6,4,19,23].
Our starting point is a gas of anyonic particles evolving in R 2 subjected to a magnetic field B e = curlA e and a trapping external potential To model anyonic behavior we introduce the statistical gauge vector potential felt by particle j due to the influence of all the other particles (1.1) The associated magnetic field is The statistics parameter is denoted by α, corresponding to a statistical phase e iπ(1+α) under a continuous simple interchange of two particles. In this so-called "magnetic gauge picture", 2D anyons are thus described as fermions, each of them carrying an Aharonov-Bohm magnetic flux of strength α. Instead of looking at the ground state of an non-interacting Hamiltonian acting on anyonic wave-functions we thus study the following Hamiltonian (−i ∇ j + A e (x j ) + αA(x j )) 2 + V (x j ) acting on L 2 asym (R 2N ). (1.3) Note that in the above we view the anyons statistics parameter α counting from the fermionic representation of anyons. The limit α → 0 for (1.3) is then formally equivalent to α → 1 in the bosonic representation alluded to above [24,42]. We are interested in a joint limit N → ∞, α → 0 in which the problem would simplify but still display signatures of anyonic behavior, i.e. a non-trivial dependence on α. Since we perturb around fermions, the Pauli principle implies that the kinetic energy grows faster than N . This behavior is encoded in the Lieb-Thirring inequality [41,40] which, for any N -particle normalized antisymmetric wave function Ψ N with support in a bounded domain Ω N , gives where d is the dimension of the physical space. Because of this rapid growth of the kinetic energy, the natural mean-field limit [6,7,21] for fermions involves a scaling ∼ N −1/d . Here with d = 2 we get a kinetic energy proportional to N 2 and thus, in expectation value −i∇ j ∼ N 1/2 . Investigating the N -dependence of each term of (1.3) we thus expect for any where the terms in parenthesis accounts for kinetic energy, including the influence of (1.1), and the second term is the energy in the external potential V . To obtain a non trivial limit when α → 0, a natural choice is to take Such an approach combines two limits (see also Remark 1.2 below), the semi-classical → 0 and the quasi-fermionic α → 0.
1.2. Extended anyons: energy convergence. The Hamiltonian (1.3) is actually too singular. Consequently we introduce a length R over which the magnetic flux attached to our particles is smeared. In our approach, R → 0 will make us recover the point-like anyons point of view. This "extended anyons" model is discussed in [48,62,10]. Anyons arising as quasi-particles in condensed-matter systems are not point-like objects. The radius R then corresponds to their size of the quasi-particle, i.e. a length-scale below which the quasi-particle description breaks down. We introduce the 2D Coulomb potential generated by a unit charge smeared over the disc of radius R w R (x) = (log | . | * χ R ) (x), with the convention w 0 = log | . | (1. 7) and χ R (x) a positive smooth function of unit mass R and R 2 χ = 1, χ(x) = 1/π 2 |x| ≤ 1 0 |x| ≥ 2. (1.8) We take χ to be smooth, positive and decreasing between 1 and 2. We will recover the magnetic field (1.2) (in a distributional sense) in the limit R → 0 by defining We henceforth work with the regularized Hamiltonian of N anyons of radius R It is essentially self-adjoint on L 2 asym (R 2N ) (see [52,Theorem X.17] and [3]). The bottom of its spectrum then exists for any fixed R > 0, we denote it Under our assumptions H R N will actually have compact resolvent, so the above is an eigenvalue. However, this operator does not have a unique limit as R → 0 and the Hamiltonian at R = 0 is not essentially self-adjoint, see for instance [46,15,16,1,9]. Nevertheless, in the joint limit R → 0 and N → ∞ we recover a unique well-defined (non-linear) model. It is convenient to expand (1.10) and treat summands separately (p A j ) 2 + V (x j ) "Kinetic and potential terms" Mixed two-body term" "Singular two-body term". (1.13) The fourth term of the above, being N times smaller than the others in the regime (1.5), will easily be estimated. Further heuristics will only involve the mixed two-body term and the three-body term. These being of the same order, we expect (as in other types of mean-field limits, see e.g. [57] and references therein) that, for large N , particles behave independently, and thus j =k where ρ is the density of particles. Therefore, it is convenient to define, given a one-body density ρ normalized in L 1 (R 2 ) Natural mean-field functionals associated with (1.13) are then where the argument is a trace-class operator Similar Hartree-like functionals were obtained in the almost-bosonic limit [24,42]. Here we simplify things one step further by considering a semi-classical analogue (1.17) with m(x, p) a positive measure on the phase-space R 4 of positions/momenta and Our convention is that m satisfies the mass constraint The latter is imposed at the classical level by the requirement that a classical state (x, p) cannot be occupied by more than one fermion. By the Bathtub principle [35,Theorem 1.14] we can perform the minimization in p explicitly (this parallels the considerations in [21]). We find minimizers of the form where ρ minimizes the Thomas-Fermi energy We define the minimum of the Thomas-Fermi functional Our first purpose is to obtain the above as the limit of the true many-body energy. We shall prove this under the following assumptions The external potentials entering (1.10) satisfy • |A e | 2 ∈ W 1,∞ (R 2 ) ∩ L 2 (R 2 ) and, for some s > 1 and c, C, where the many-anyon ground state energy E R (N ) and the Thomas-Fermi energy are defined in (1.12) and (1.22), respectively.
The study of the regime (1.5) resembles the usual Thomas-Fermi limit for a large fermionic system [37,21,61,34], but with important new aspects: • The effective interaction comprises a three-body term, and a two-body term (see Equation (1.13)) which mixes position and momentum variables. • The limit problem (1.17) comprises an effective self-consistent magnetic field A [ρ].
• One should deal with the limit R → 0, α → 0 and → 0 at the same time as N → ∞. Regarding the last point, we make the Remark 1.2 (Scaling of parameters). Extracting a multiplicative factor N from the energy, the parameter regime (1.5) we consider is equivalent to provided we replace the external potential V (x) by N V (x) and a similar replacement for the external vector potential A e . The scaling (1.23) better highlights the quasi-fermionic character of the limit we take. We however find it preferable technically to work with (1.5), where the semi-classical aspect is more apparent.
Regarding the scaling of R, it would be highly desirable (and challenging) to be able to deal with a dilute regime η > 1/2 where the radius of the magnetic flux is much smaller than the typical inter-particle distance. We could probably relax our constraint η < 1/4 to η < 1/2 if we assume a Lieb-Thirring inequality for extended anyons of the form for any fermionic wave-function Ψ N with one-particle density ρ Ψ N . In the above we would need a C α,R uniformly bounded from below when N → ∞, α → 0 and R ≫ N −1/2 . Such an inequality is made very plausible by the results of [30] on the homogeneous extended anyon gas and the known Lieb-Thirring inequalities for point-like anyons [44,45,46,47]. We plan to return to this matter in the future. ⋄ The limit ground state energy e TF does not reveal any anyonic behavior, i.e. it does not depend on α. This is however deceptive, for the behavior of the system's ground state does depend on α and shows the influence of the non-trivial statistics. This will become apparent when we state the second part of our result, the convergence of ground states. It is expressed in terms of Husimi functions, constructed using coherent states. We discuss this next.
For every (x, p) in the phase space R 2 × R 2 we define the squeezed coherent state It has the property of being localized on a scale √ x in space and p in momentum, these two scales being related by = x p . (1.26) The usual coherent state [21, Section 2.1] is the particular case √ x = p = √ . Any such state saturates Heisenberg's uncertainty principle, and is thus as classical as a quantum state can be. Indeed, for the two observables x 1 and p 1 = −i∂ x 1 evaluated in this state we get In our proofs we will take x ≪ p → 0, which is convenient because our Hamiltonian is singular in x but not in p.
We define the -Fourier tranform Its Fourier transform (see the calculation in Appendix C) makes apparent the localization in momentum space we just claimed.
To F x,p we associate the orthogonal projector P x,p P x,p := |F x,p F x,p |. (1.30) Then we have the well-known The proof is recalled in Appendix C, see also [35,Theorem 12.8].  [28,12,60] describes how many particles are distributed in the k semi-classical boxes of size √ x × p centered at its arguments (x 1 , p 1 ), . . . , (x k , p k ). A wave function Ψ N being given we define or, in terms of fermionic annihilation and creation operators (see (2.2) below) where a and a * satisfy the canonical anticommutation relations [59,26] a * (f )a(g) + a(g)a * (f ) = g, f a * (f )a * (g) + a * (g)a * (f ) = 0.
We will also use the k-particle density matrices of Ψ N , i.e. the operator γ (k) (1.33) We can also express the Husimi function in terms of the k-particle density matrices of above (1.34) These objects are furthermore related to the density marginals in space and in momentum, respectively defined as |F [Ψ N ](p 1 , . . . , p N )| 2 dp k+1 . . . dp N . (1.36) We can now express the convergence of states corresponding to Theorem (1.1) using the above objects. The minimizer of the Vlasov energy (1.17) is (1.37) Its marginals in space and momentum are respectively given by where λ TF is a Lagrange multiplier ensuring normalization, and We will prove the Consequently we also have the convergence of the k-particles marginals in position and momentum weakly as measures.
The convergence of Wigner functions follows from the Husimi functions convergence, see [21, Theorem 2.1]. The result above does not depend on the choice of function f used to construct the coherent states [21,Lemma 2.8]. Likewise, the result is the same with any choice of scaling of x , p → 0 as long as (1.26) is satisfied. In our proof of energy convergence it will however be convenient to make a particular choice, and in particular use squeezed coherent states, x ≪ p .
Recall that the semi-classical Thomas-Fermi energy does not depend on α, but the Vlasov minimizer does. Anyonic features are retained e.g. in the limiting object's momentum distribution (1.39).

1.5.
Outline and Sketch of the proof. Our general strategy is inspired by works on mean-field limits for interacting fermions [37,38,39,22,32,61], in particular by the method of [21]. Several improvements are required to handle the singularity of the anyonic Hamiltonian that emerges in the limit R → 0. In particular • we replace the use of the Hewitt-Savage theorem by that of its' quantitative version, the Diaconis-Freedman theorem. • we implement the Pauli principle quantitatively at the semi-classical level.
To obtain an energy upper bound (Section 2) we use a Slater determinant as trial state, leading by Wick's theorem to a Hartree-Fock-like energy. We then discard exchange terms to get to the simpler Hartree energy (1.14). Finally, a suitable choice of the trial state's one-body density matrix γ allows to take the semi-classical limit and establish an upper bound in terms of the Vlasov energy (1.17).
For the correponding lower bound we first express (Section 3) the energy semi-classically, in terms of the Husimi functions. The remainder terms are shown to be negligible when N → ∞. The use of squeezed coherent states with x ≪ p takes into account the singularities of the Hamiltonian, more severe in position than in momentum space.
Section 4 contains the main novelties compared to [21]. There we tackle the meanfield limit of the semi-classical functional expressed in terms of Husimi functions. We use the Diaconis-Freedman theorem [17] to express the latter as statistical superpositions of factorized measures, with a quantitatively controled error. We then estimate quantitatively the probability for the Diaconis-Freedman measure to violate the Pauli-principle (1.20). This leads to the sought-after energy lower bound and completes the proof of Theorem 1.1. The convergence of states Theorem 1.4 follows as a corollary.
Acknowledgments. Funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement COR-FRONMAT No 758620) is gratefully acknowledged.

Energy upper bound
In this section we derive an upper bound to the ground state energy (1.12). We first introduce a Hartree-Fock trial state and apply Wick's theorem to calculate its energy. We next discard lower contributions (exchange terms) to obtain a regularized (R > 0) Hartree energy in Lemma 2.4 and use Lemma 2.5 to send the regularization to 0. We conclude by using a particular sequence of semi-classical states, which will make the Hartree energy converge to the Thomas-Fermi one. All in all this will prove the 2.1. Hartree-Fock trial state. We recall the definition of creation and annihlation operators of a one-body state f ∈ L 2 (R 2 ) (for more details about second quantization see [59,26]) Sums are over the permutation group and sgn(σ) is the signature of a permutation. Let now {ψ j } j=1...N ∈ L 2 (R 2 ) be an orthonormal family. We use it to construct a Slater determinant Ψ SL N (also called Hartree-Fock state) Denoting a † j = a † (ψ j ) and a j = a(ψ j ) we have: Hence the associated 1-particle reduced density matrix (1.33) and density are with Tr 2→N the partial trace. Replacing the last formula in (1.34) gives us the 1-particle reduced Husimi measure We will use Wick's theorem, see [50,Corollary IV.6] or [59].

Theorem 2.2 (Corollary of Wick's theorem).
Let a ♯ 1 , . . . , a ♯ 2n be creation or annihilation operators. We have that where the sum is over all pairings, i.e. permutations of the 2n indices such that term by term as in (1.13). Until the end of this section · means · SL . We will use the following notation (recall (1.11)) Recalling the definition of the Hartree energy functional we have the following With Ψ SL as above and γ the associated one-particle density matrix (2.4) we have In the first line we have denoted by U 12 the exchange operator For the two-and three-body terms we express higher density matrices of Ψ SL using Wick's theorem. To this end, let (ψ j ) j ∈ L 2 (R 2 ) be an orthonormal basis, and a j , a † j the associated annihilation and creation operators.
Two-body terms. We have with the exchange operator defined as in (2.9). Using [59, Lemma 7.12] we have Similarly we obtain (2.12) Three-body term. Using [59, Lemma 7.12] again we write Applying Wick's theorem (2.2) we obtain for the 3-body density matrix of the Slater determinant, where the exchange operators U ij are natural extensions of (2.9) to the three-particles space. Gathering the above expressions yields We now want to discard the exchange terms and the singular two-body terms to reduce the above expression to the Hartree functional.

Lemma 2.4 (From Hartree-Fock to Hartree).
Recalling the notation (1.11), assume that Proof. We recall the bound for the density matrix (2.4) of a Slater determinant. It follows that Next we use Cauchy-Schwarz to obtain and hence, since both sides commute with U 12 , Next, using (2.17) again, and we obtain a similar bound on the last term of (2.8) noticing that We now want to take R → 0 in (2.16) and show that E af R can be replaced by E af . Lemma 2.5 (Convergence of the regularized energy).
The functional E af R (1.14) converges pointwise to E af (1.15) as R → 0. More precisely, for any fermionic one-particle density matrix γ N with and similarly for E af , which is the case R = 0.
Hence a use of the Cauchy-Schwarz inequality implies We now expand the two energies of (2.19) and begin by estimating the squared terms using the Hölder and Yound inequalities for a sequence γ N with uniformly bounded Hartree energy. The last step consists in using constructing such a sequence γ N , whose energy will behave semi-classicaly.

2.2.
Semi-classical upper bound for the Hartree energy. We use [21, Lemma 3.2], whose statement we reproduce for the convenience of the reader.
Lemma 2.6 (Semi-classical limit of the Hartree energy).
with support in the square C r = (−r/2, r/2) 2 such that ρ ≥ 0 and Cr ρ = 1. Let A ∈ L 4 (C r ) be a magnetic vector potential. If we define Cr is the magnetic Dirichlet Laplacian in the cube and = 1/ and weakly-* in L ∞ (R 2 ), strongly in L 1 (R 2 ) and L 2 (R 2 ). Furthermore, the same properties stay true if we replace γ N by γ N the projection onto the N lowest eigenvectors of The convergences we claim for N −1 ρ γ N are stronger than in the statement of [21, Lemma 3.2]. They easily follow from the proof therein.
We are now able to complete the proof of Proposition 2.1. We take and γ N the associated rank-N projector, as in the above lemma. Therefore we know by the above theorem that strongly in L ∞ (R 2 ), by the weak Young inequality [35,Chapter 4]. We thus have We now use (2.22) to take the limit of the trace and (2.23) to take the limit of the potential term (V ∈ L 1 loc (R 2 ) by assumption). Combining with the bounds from Lemmas 2.5 and 2.4 we finally obtain completing the proof of Proposition 2.1.

Energy lower bound: semi-classical limit
In this section we first derive some useful a priori bounds and properties of the Husimi functions. Next we express our energy in terms of the latter and obtain a classical energy approximation on the phase space, plus some errors terms that we show to be negligible. We will use this expression in Section 4 to construct our lower bound.
Proof. We expand the Hamiltonian and use the Cauchy-Schwarz inequality for operators to get We estimate the second term of the above using [24, Lemma 2.1]: Recalling that α = N −1 we get , we thus deduce (3.1). We obtain (3.2) by using the Lieb-Thirring inequality [40,41] Tr −∆γ The proof is similar to considerations from [21]. We recall it in Appendix C for completeness.
In addition when N tends to infinity and We refer to [21, Lemma 2.2] for the proof.
3.3. Semi-classical energy. We now define a semi-classical analogue of the original energy functional, given in terms of Husimi functions. We denote dx ij = dx i dx j and x ij = (x i , x j ): Ψ N (x 12 , p 12 )dx 12 dp 12 Ψ N (x 123 , p 123 )dx 123 dp 123 .
Our aim is to show that it correctly captures the leading order of the full quantum energy.

Proposition 3.4 (Semi-classical energy functional).
Pick some ε > 0 and set Pick a sequence Ψ N of fermionic wave-functions such that Under Assumption 1.1 we have We first derive an exact expression, whose extra unwanted terms will be estimated below.  with and Proof. We expand the whole energy as in (1.10) and express each term using Husimi's function (1.32). We begin with calculations involving the p variable. Using (1.36) and (3.5) we can write, for the purely kinetic term, Ψ N (p)dp · R 2 p|G p (p)| 2 dp .
We now use that G p (−p) = G p (p) which makes |G p (p)| 2 an even function (recall that G p is the Fourier tranform of F x (1.28)) to discard the last term of the above. On the other hand, using (1.36) we find For the magnetic cross term (3.16) we use (C.2) to obtain, for any u ∈ L 2 (R 2 ), Then since F x,0 and |u| 2 are real we obtain Combining the spectral decomposition of the one-particle reduced density matrix with the above we get Now for the mixed two-body term (3.12): For the terms which do not involve the variable p we only discuss the three-body term (3.14), the others being treated in the same way. We apply Lemma (3.2) We now show that all errors in (3.15) are smaller than the classical energy E C . This will conclude the proof of Proposition 3.4. The following lemma will deal with all convolutions involved in the estimates.

Lemma 3.6 (Convolution terms).
For any function W on L ∞ R 2 , consider the n-fold convoluted product of W with |F x | 2 .
We have the estimate Proof. For any x, y ∈ R 2 there exists some z ∈ R 2 such that using the radial symmetry of y → e −|y| 2 to discard the term of order √ x . It follows that for any n ≥ 2 Proof of Proposition (3.4). Let Ψ N be a sequence of fermionic wave functions such that . We will systematically use the bounds (3.2) and (3.1) to deal with our error terms. Recall that the main term in the energy (3.15) is of order N .
Estimate of (3.16). We have where ||W || S p = Tr [|W | p ] 1 p is the Schatten p-norm. We used (3.1) to bound the kinetic term and we now use Lemma 3.6 to deduce Estimate of (3.17). We have where we used Cauchy-Schwarz's inequality and the estimate (3.19). For the second term of (3.17) we use Assumption 1.1 and Lemma 3.6 to write by using Young's inequality r s−2 ≤ s − 2 s r s + 2 s the choice (3.9) and the a priori bound (3.3).
Estimate of (3.18). We have recalling that f is a fixed function and the choice (3.9).
Estimate of (3.14). We denote where we used [

Energy lower bound: mean-field limit
This section is dedicated to a lower bound on (1.6). We have reduced the problem to a classical one in the previous section. We now perform a mean-field approximation of the classical energy (3.8), showing that its infimum is (asymptotically) attained by a factorized Husimi function. The crucial point is to keep track of the semi-classical Pauli principle (1.20).
We first recall the Diaconis-Freedman theorem. We use it to rewrite many-particle probability measures as statistical superpositions of factorized ones. We then estimate quantitatively the probability for the measures in the superposition to violate the Pauli principle. This is the main novelty compared to the approach of [21]. We finally prove the convergence of the energy and states, Theorems 1.1 and 1.4.

The Diaconis-Freedman measure.
For Z N ∈ R 4N a set of points in the phase space, we define the associated empirical measure The following originates in [17].

Theorem 4.1 (Diaconis-Freedman).
Let µ N ∈ P sym (R 4N ) the set of symmetric probability measures over R 4N . Define the probability measure P µ N over P(R 4 ) and set ρ ⊗n dP (ρ). (4.5) The measure P obtained in the limit is that appearing in the Hewitt-Savage theorem [27]. See [56, 54, Theorems 2.1 and 2.2] for proofs and references. We will use the following consequence of the Diaconis-Freedman construction.
We now apply the above constructions to the probability measure to obtain an approximation of the marginals (cf Lemma 3.3) (4.12) We use the notation W 12 and W 123 respectively defined in (2.6) and (2.7) to denote the two and the three body operators. In the former case, this is with a slight abuse of notation: we replace p A by the classical vector p 1 + A e (x) in the phase space.

Lemma 4.3 (Energy in terms of the Diaconis-Freedman measure).
Let m N ∈ P sym (R 4N ) be as in (4.11). Let P DF N ∈ P(P(R 4 )) be the associated Diaconis-Freedman measure, as in Theorem 4.1. Evaluating each term of (3.11) gives f,Ψ N (x 12 , p 12 )dx 12 dp 12 = Proof. To apply the Diaconis-Freedman theorem (4.1) to the measures m (i) f,Ψ N , i = 1, 2, 3 they have to be normalized as in (4.12). We then use the measure m  (1) Applying the Cauchy-Schwarz inequality to the two particles term we obtain As for the three-particles term we have

4.2.
Quantitative semi-classical Pauli principle. Lemma (4.3) allows to write the energy as an integral of E R V over P(R 4 ) plus some negligible error terms. Assuming R = N −η for some η < 1/4, we get from Proposition 3.4 and the above lemma: µ(x, p)dp. Note that defined in (1.17). However, P DF N only charges empirical measures which do not satisfy the Pauli principle (3.6). To circumvent this issue we divide the phase-space in hyperrectangles labeled (Ω m ) m∈N . We take them to have side-length l x in the space coordinates and l p in the momentum coordinates, ensuring that The parameter β > 0 and the lengths l x , l p will be chosen in the sequel. If each Ω m contains less than (1 + ε)(2π) −2 |Ω m | points (for some small ε) our measure will approximately satisfy the Pauli principle. We thus have to estimate the probability for a box to have the right density of points. This is the purpose of the following lemmas. We denote for a given measure µ: (4.25) Let m N be as in (4.11) and P DF N the associated Diaconis-Freedman measure. Recall the tiling (4.23) and assume (4.24) with β < 1. For any Note that the condition β < 1 means l 2 x l 2 p ≥ N −1 . Since the average interparticle distance in phase-space is N −1/4 , our hyperrectangles typically contain a large number of particles.
We start the proof of the above theorem with the with S(n, k) the Stirling number of the second kind, the number of ways to partition a set of n objects into k non-empty subsets.
We refer to [8] for combinatorial definitions and estimates.
Proof of Lemma 4.5. The measure µ N constructed from a µ N ∈ P sym (R 4N ) in Theorem 4.1 satisfies (4.9). Hence µ (n) with Γ n,k the set of ordered samples of n indices amongst N (allowing repetitions) with exactly k distinct indices. Integrating on each side of (4.28) and using the symmetry of (recall Lemma 3.3) Now we calculate the cardinal |Γ n,k |. For a given k, choose k indices among N (without order) to, next, be distributed in n boxes. The first step gives us We now need a rough control of the sum appearing in (4.27) when the set Ω is not too small. Take |Ω m | = N −β with β < 1. Choose some for some exponent δ ′ > 0 depending only on δ.

Now we can complete the
Proof of Theorem 4.4. We call the set of all empirical measure violating the Pauli principle in Ω m by a finite amount. We have, using (4.3) that, for any n ≤ N , Next, using Lemma 4.5 we obtain Choosing n as in Lemma 4.6 and inserting the latter result we deduce and the result follows.

4.3.
Averaging the Diaconis-Freedman measure. We divide R 4 into hyperrectangles Ω m = Ω mx × Ω mp as in (4.23). We set |Ω m | = |Ω mx ||Ω mp | = l 2 x l 2 p = N −β (4.32) for some 0 < β < 1 to be fixed later. We know from Theorem 4.4 that the probability for one such hyperrectangle to violate the Pauli principle is exponentially small. This means that the contribution to (4.20) of empirical measures . . . , z j ) ∈ Γ (see (4.31)) will be negligible (by the union bound), provided the number of hyperrectangles is not too large. The next steps of our proof are thus • to reduce estimates to the contribution of a finite set of phase-space.
• in the latter set, to average empirical measures to replace µ in (4.20) by true Pauliprinciple abiding measures (in the sense of (1.20)).
As for the first step, let L > 0 be a number of the form for some n ∈ N and define the hypercube of side 2L centered at the origin. This way the number of Ω m -hyperrectangles contained within S L is 2 4 n 4 . We first discard the energetic contribution of S c L .
Lemma 4.7 (Reduction to the energy in the phase-space hypercube S L ). Let σ > 0 and µ a probability measure satisfying (4.34)

Its energy can be bounded from below as
The parameter σ will tend to 0 at the very end of the proof (after we take N → ∞, L → ∞, R → 0 while ensuring R 6 inf(L 4 , L 2s ) → ∞).
Proof. First, the positivity of the integrand gives Then we split A R [µ] into a contribution from S L and one from its complement, Next we use that ab ≤ a 2 2σ + σb 2 2 to deduce where we used Young's inequality combined with [24, Lemma 2.1]. The measure µ satisfies (4.34) so, using Assumption 1.1, which concludes the proof.
We now turn to the averaging procedure turning empirical measures based on configurations from Γ c (recall (4.31)) into bounded measures satisfying the Pauli principle (1.20): Note that Ave [µ] lives on S L by definition. We next quantify the energetic cost of the above procedure: Theorem 4.9 (Averaging the empirical measure). Assume β < 1 in (4.24) as well as l x ≪ R, l p ≪ 1 in the limit N → ∞. (4.38) Assume that (4.34) holds for µ = Emp Z N . Then, for any σ > 0, . Proof. In this proof we denote µ = Emp Z N for brevity. We first use Lemma 4.7 to write We expand E R V (4.21) and proceed term by term. Recall from (4.33) that S L = ∪ 2 4 n 4 m=1 Ω m . We first calculate We denote By the mean value theorem applied in each Ω m there exists a point and we have, using Assumption 1.1 to control the variations of A e , we may recognize Riemann sums and use Assumption 1.1 again to deduce using Young's inequality We next treat the two body operator W 12 defined in (2.6) (understanding now p as a momentum variable rather than an operator) in the same way. We set dx = dx 1 dx 2 , dp = dp 1 dp 2 .
By the mean-value theorem, for each ( (4.44) Expanding the functions around the points z i of the sum there exists a c i ∈ Ω m(i) such that with J ∇w R the Jacobian matrix of ∇w R . We denote where ∂ u is the radial derivative explicitly calculated in (B.8). We then use (B.1) and estimate where we used that A e and its derivatives are in L ∞ by assumption and use the convention z i = (x i , p i ) systematically. We denote with Ω mx (x) ⊂ R 2 and Ω mp (p) ⊂ R 2 the square of our grid (4.32) that x belongs to (respectively Ω mp (p) the square p belongs to). Hence (x 2 , p 2 )dx 1 dx 2 dp 1 dp 2 =: I + II.
Recognizing a Riemann sum similarly as in (4.43) yields, using the properties of W (see (B.8)), where we also used the weak Young and the Cauchy-Schwarz inequality. Similarly, since ∇w R ∈ L 2,w , We next turn to the three-body term (2.7) We denote (with W as in (4.45)) G(x 1 , x 2 , x 3 ) = sup {G(y 1 , y 2 , y 3 ), y 1 ∈ Ω mx (x 1 ), y 2 ∈ Ω mx (x 2 ), y 3 ∈ Ω mx (x 3 )} and observe that G gives a pointwise upper bound to gradients of W 123 in any variable. Hence, arguing as above where we used the weak Young inequality twice. Collecting the previous inequalities we conclude inserting our assumptions on the various parameters involved l 2 x = N −β+ε ≪ R 2 and l 2 p = N −ε . We have also used the fact that, since Finally, arguing as in Appendix A below (using (4.47) again), and we can absorb an error term in the main term to obtain .
A last use of the bathtub principle gives Thus the proof is concluded by absorbing one last error term in the main expression.

4.4.
Convergence of the energy. Going back to our lower bound (4.20), assuming R = N −η , 0 < η < 1/4, we have Γ ε is as in (4.31), and we have used the positivity of the integrand to go to the second line of (4.48). We next pick l x , l p to enforce for some β < 1. This is certainly compatible with our other requirements, since R ≫ N −1/4 by assumption. We can then use Theorem 4.9 to deduce .
(4.50) Next, denote, for some γ > 0 the set of measures whose mass lie mostly in S L and note that, as per Lemmas A.3 and A.4, with a suitable c > 0 we have that Hence, with this choice, with a self-explanatory notation for P DF m N . There remains to estimate the probability of the "bad events" in the last inequality. First, there is a δ > 0 such that We used the union bound, the bound on the probability of violating the Pauli principle in a single box from Theorem 4.4 and the relation (4.33).
On the other hand, using Markov's inequality, (3.1), Lemma 3.2 and (4.6) we find There remains to insert (4.54) and (4.55) in (4.53). We can now set L = N κ with κ > 0 a suitably large, fixed number and τ = N κ ′ with κ ′ a suitably small, fixed number. This makes all errors depending on N negligible because, crucially, (4.54) contains an exponentially small term in N . In particular, recalling (4.52), γ → 0 with such a choice of a parameters. Hence we can pass to the limit in (4.53), first letting N, R −1 , L → ∞ with the previously specified scalings, and finally let ε, σ → 0, obtaining the energy lower bound claimed in 1.
It follows from (4.56) and results of Appendix A that m N is tight. Applying the last part of Theorem 4.1 we thus obtain that the Diaconis-Freedman measure P DF m N weakly converges to a probability measure P ∈ P(P(R 4 )) (this is the Hewitt-Savage measure). It follows from Theorem 4.4 (or [21, Theorem 2.6]) that P is concentrated on probability measures satisfying µ ≤ (2π) −2 . Combining with Lemma 3.3 and (4.12) we have that, for any fixed weakly as measures. We now identify the limit measure P to conclude the proof of Theorem 1.4.
To this aim it is convenient to write the integral above as where l = max(l x , l p ) is the largest dimension of the box Ω m . We thus have where the o N (1) is uniform in ∇φ L ∞ by (4.60).
It follows that, if we take a sequence µ n → µ when N → ∞, (4.61) We turn to the convergence of Q N defined in (4.58). For any continous bounded function Φ ∈ C b (P(R 4 )) over probability measures, Combining with (4.56) and letting σ → 0 after N → ∞ we conclude Recall that P is concentrated on probability measures satisfying µ ≤ (2π) −2 and that e TF is the infimum of E V [µ] over those. Hence P must be concentrated on the unique minimizer of E V [µ], and this concludes the proof.
Appendix A. Properties of the Vlasov functional In this appendix we etablish some of the fundamental properties of the functional E V and some useful bounds on the vector potential A R [ρ] associated to a measure ρ. Proof. We recall that the marginal in p of µ n is It follows from applying the bathtub principle [35,Theorem 1.14] in the p variable that Hence we may assume that ρ n L 2 is uniformly bounded and that ρ n ⇀ ρ weakly in L 2 R 2 . We then deduce from the weak Young inequality [35,Chapter 4] that We recall that the weak L p (R d ) space is the set of functions with the associated norm where Leb is the Lebesgue measure. We expand the energy and use that ab ≤ a 2 2σ + σb 2 2 to get and hence Thus (µ n ) is tight and, up to extraction, converges strongly in L 1 R 4 . In order to obtain the convergence of A 2 [ρ n ]µ n we write and treat each resulting term separately. The first term yields using that ||µ n || L 1 ≤ C and the weak Young inequality. For the second term of (A.4) and for the last one so A[ρ n ] converges strongly in L 2 . We have || p A 2 µ n || L 1 ≤ C so p A µ n converges weakly in L 2 and by weak-strong convergence we deduce that the cross term converges. We conclude using Fatou's lemma for V and the kinetic term lim inf Lemma A.2 (Existence of minimizers). There exists a minimizer for the problem Proof. We consider a minimizing sequence (µ n ) n converging to a candidate minimizer µ ∞ . By Lemma (A.1) we have We also found during the previous proof that (µ n ) n must be tight, hence For any measure µ ≤ (2π) −2 such that R 4 µ ≤ C we have that Proof.
1/2 where we have used the triangle inequality. Moreover we have that by the weak Young inequality and because Using that the minimizer in p µ TF = (2π) −2 1 1 |p A + A[ρ]| ≤ 4πρ is explicit we have and the minimization problem is now formulated in terms of µ(x, p)dp.
This gives and concludes the proof.
Lemma A.4 (Dependence on the upper perturbed constraint).
So, evaluating the energy for any (i, j) ∈ {1, 2} 2 . We also have the estimates with χ R defined as in (1.8). We call u 1 and u 2 the two components of the vector u and u its norm. Using Newton's theorem [35,Theorem 9.7] we write where φ R (u) = ∂ u χ R (u) is bounded with compact support. We observe that regardless of whether u is smaller or greater than 2R we have and get (B.3). Moreover We compute the two derivatives of the second component of (B.10) (B.12) We do the same with the first component of ∇ ⊥ w R and get (B.1). If we differentiate once again we get which is also the case for the other component and derivative. We deduce We can also compute which gives (B.4). To get (B.5) we combine (B.9) and (B.12). For the third inequality (B.6) we expand a little bit our first expression so the first component of the gradient is The rest of the proof consists of the computation of the above term using basic inequalities.

Appendix C. Computations for squeezed coherent states
We give for completeness three proofs that we skipped in the main text.
Proof of Lemma 1.3. For any u ∈ L 2 (R 2 ), F x,p , u = It follows that 1 (2π ) 2 R 2 R 2 ψ, P x,p ψ dxdp = 1 (2π Fourier transform of F . We need to calculate the Fourier Transform of the Gaussian but we also have dG 1 (p) dp = − p p G 1 (p) which gives the result.