Universal Short Range Correlations in Bosonic Helium Clusters

TL;DR

This paper investigates universal short-range correlations in bosonic helium clusters using the generalized contact formalism, supported by Monte Carlo simulations and experimental data, revealing consistent two-body contact values.

Contribution

It introduces a formalism for universal n-body correlations in helium clusters and numerically evaluates contact values up to five particles.

Findings

Universal n-body correlations are demonstrated numerically.

Two-body contact values agree with experimental measurements.

Monte Carlo simulations support the theoretical formalism.

Abstract

Short-range correlations in bosonic Helium clusters, composed of $^4$He atoms, are studied utilizing the generalized contact formalism. The emergence of universal $n$-body short range correlations is formulated and demonstrated numerically via Monte Carlo simulations. The values of the $n$-particle contacts are evaluated for $n\le5$. In the thermodynamic limit, the two-body contact is extracted from available experimental measurements of the static structure factor of liquid $^4$He at high momenta, and found in a good agreement with the value extracted from our calculations.

Tables

Table 1. Table 1: The ground-state energies (in mK) of small 4 He clusters, with the LM2M2 pair-potential. The dimer energy is 1.30348 1.30348 1.30348 mK HiyKam12 .

$N$	Ref LazCar06	Ref HiyKam12	Ref BluGre00	Ref GuaKorNav06	This work
3	126.39	126.40	125.5(6)	124(2)	125.9(2)
4	557.7	558.98	557(1)	558(3)	557.4(4)
5			1296(1)	1310(5)	1300(2)
6			2309(3)	2308(5)	2315(2)
7			3565(4)	3552(6)	3571(2)
8			5020(4)	5030(8)	5041(2)
9			6677(6)	6679(9)	6697(2)
10			8495(7)	8532(10)	8519(3)

Table 2. Table 2: The asymptotic values of the reduced n 𝑛 n -body contacts C ~ n ( N ) = C n ( N ) / N superscript subscript ~ 𝐶 𝑛 𝑁 superscript subscript 𝐶 𝑛 𝑁 𝑁 \tilde{C}_{n}^{(N)}={C}_{n}^{(N)}/N of 4 He droplets.

$n$	2	3	4	5
${\tilde{C}}_n^{\infty }$	$230\pm 25$	$500\pm 60$	$1800\pm 300$	$5900\pm 1000$

Table 3. Table S1: The optimal parameters for the trial wave function (Eq. 18).

$N$	$p_0$	$p_1$	$p_2$	$p_5$
2	1.	0.0309	0.3002	0.9022
3	0.5	0.2136	0.1784	0.8884
4	0.3333	0.2547	0.1774	0.8898
5	0.25	0.2654	0.2975	0.877
6	0.2	0.2279	0.1076	0.8825
7	0.1667	0.2257	0.1166	0.8844
8	0.1429	0.2063	0.1559	0.8822
9	0.125	0.1972	0.1455	0.8829
10	0.1111	0.1744	0.1202	0.8839
15	0.0714	0.1379	0.1896	0.8823
20	0.0714	0.1046	0.2401	0.881
25	0.0417	0.093	0.2526	0.8795
30	0.0345	0.0796	0.2733	0.8789
35	0.0294	0.07	0.2727	0.8801
40	0.1029	0.0405	0.3052	0.884
45	0.0926	0.0364	0.3276	0.8808
50	0.1068	0.0262	0.3326	0.8798

Table 4. Table S2: The numerical values of the n 𝑛 n -body contacts in the N 𝑁 N -body system C n ( N ) superscript subscript 𝐶 𝑛 𝑁 {C}_{n}^{(N)} for 4 He clusters in the range N ∈ ( 2 , 50 ) 𝑁 2 50 N\in(2,50) , calculated using the VMC-DMC mixed estimate.

$N$	$C_2^{\left(N\right)}$	$C_3^{\left(N\right)}$	$C_4^{\left(N\right)}$	$C_5^{\left(N\right)}$
2	$1.00e+00\pm 0e+00$
3	$1.78e+01\pm 3e-02$	$1.00e+00\pm 0e+00$
4	$4.88e+01\pm 1e-01$	$6.55e+00\pm 4e-03$	$1.00e+00\pm 0e+00$
5	$9.11e+01\pm 3e-01$	$1.95e+01\pm 4e-02$	$6.19e+00\pm 6e-03$	$1.00e+00\pm 0e+00$
6	$1.43e+02\pm 6e-01$	$4.08e+01\pm 9e-02$	$1.93e+01\pm 8e-03$	$6.18e+00\pm 1e-02$
7	$2.01e+02\pm 8e-01$	$7.15e+01\pm 2e-01$	$4.45e+01\pm 1e-02$	$2.11e+01\pm 4e-02$
8	$2.67e+02\pm 1e+00$	$1.11e+02\pm 3e-01$	$8.31e+01\pm 8e-02$	$5.06e+01\pm 2e-01$
9	$3.38e+02\pm 2e+00$	$1.59e+02\pm 6e-01$	$1.39e+02\pm 9e-02$	$1.03e+02\pm 5e-01$
10	$4.15e+02\pm 2e+00$	$2.13e+02\pm 7e-01$	$2.07e+02\pm 5e-01$	$1.74e+02\pm 1e+00$
15	$8.54e+02\pm 6e+00$	$6.01e+02\pm 3e+00$	$8.39e+02\pm 2e+00$	$1.08e+03\pm 1e+01$
20	$1.36e+03\pm 1e+01$	$1.14e+03\pm 8e+00$	$1.92e+03\pm 3e+00$	$3.07e+03\pm 3e+01$
25	$1.93e+03\pm 2e+01$	$1.78e+03\pm 1e+01$	$3.37e+03\pm 6e+00$	$6.13e+03\pm 6e+01$
30	$2.52e+03\pm 2e+01$	$2.51e+03\pm 2e+01$	$5.14e+03\pm 9e+00$	$1.02e+04\pm 1e+02$
35	$3.15e+03\pm 3e+01$	$3.34e+03\pm 3e+01$	$7.30e+03\pm 9e+00$	$1.56e+04\pm 2e+02$
40	$3.83e+03\pm 4e+01$	$4.26e+03\pm 4e+01$	$9.83e+03\pm 6e+00$	$2.23e+04\pm 2e+02$
45	$4.48e+03\pm 5e+01$	$5.14e+03\pm 5e+01$	$1.23e+04\pm 1e+01$	$2.87e+04\pm 3e+02$
50	$5.16e+03\pm 6e+01$	$6.09e+03\pm 6e+01$	$1.50e+04\pm 1e+01$	$3.62e+04\pm 4e+02$

Equations

$$\underset{k\to \infty }{\lim }n(k)=C/k^4.$$

$$\Psi ({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N)\underset{r_{ij}\to 0}{\to }{\varphi }_2({\mathbf{r}}_{ij})A_2^{(N)}({\mathbf{R}}_{ij},\{{\mathbf{r}}_k{\}}_{k\ne i,j}).$$

$$C_2^{(N)}=\frac{N(N-1)}{2}⟨A_2^{(N)}|A_2^{(N)}⟩=(\genfrac{}{}{0ex}{}{N}{2})⟨A_2^{(N)}|A_2^{(N)}⟩.$$

$${\rho }_2^{(N)}(r)=⟨\Psi |{\hat{\rho }}_2^{(N)}(r)|\Psi ⟩\underset{r\to 0}{\to }{C}_2^{(N)}{\rho }_2(r)$$

$$\Psi \underset{r_{ijk}\to 0}{\to }{\varphi }_3({\mathbf{x}}_{ijk},{\mathbf{y}}_{ijk})A_3^{(N)}({\mathbf{R}}_{ijk},\{{\mathbf{r}}_l{\}}_{l\ne i,j,k})$$

$$C_3^{(N)}=\left(\genfrac{}{}{0ex}{}{N}{3}\right)⟨A_3^{(N)}|A_3^{(N)}⟩.$$

$${\hat{\rho }}_3^{(N)}(r)=\frac{1}{r^5}\sum _{i<j<k}\delta (r_{ijk}-r)$$

$${\rho }_3^{(N)}(r)=⟨\psi |{\hat{\rho }}_3^{(N)}(r)|\psi ⟩\underset{r\to 0}{\to }{C}_3^{(N)}{\rho }_3(r)$$

$${\rho }_n^{(N)}(r)\underset{r\to 0}{\to }{C}_n^{(N)}{\rho }_n(r),$$

$$n^{(N)}(\mathbf{k})\underset{k\to \infty }{\to }2C_2^{(N)}|{\tilde{\varphi }}_2(\mathbf{k}){|}^2$$

$$S(Q)\underset{Q\to \infty }{\to }1+\frac{2C_2^{(N)}}{N}\frac{4\pi }{Q}\int drr\sin (Qr){\rho }_2(r),$$

$$⟨V_2^{(N)}⟩=C_2^{(N)}⟨V_2^{(2)}⟩,$$

$${\rho }_2^{(N)}({\mathbf{r}}_{12})=\frac{1}{2}\int d{\mathbf{R}}_{12}\chi ({\mathbf{r}}_1,{\mathbf{r}}_2).$$

$$\chi (r)\underset{r\to 0}{\to }2\rho \frac{C_2^{(N)}}{N}{\rho }_2(r).$$

$${\tilde{C}}_n^{(N)}={\tilde{C}}_n^{\infty }+{\alpha }_n{N}^{-1/3}+{\beta }_n{N}^{-2/3}+\dots$$

$$E_{var}=\frac{⟨{\Psi }_T|H|{\Psi }_T⟩}{⟨{\Psi }_T|{\Psi }_T⟩}\ge E_0$$

$$\frac{\partial \Psi ({\mathbf{r}}_1\dots {\mathbf{r}}_N,\tau )}{\partial \tau }=\left(T+V-E_R\right)\Psi ({\mathbf{r}}_1\dots {\mathbf{r}}_N,\tau ).$$

$$f(r)=\exp \left[-(p_5/r{)}^5-(p_2/r{)}^2-p_{1}r\right]/r^{p_0}.$$

$$⟨\hat{O}⟩=2⟨\hat{O}{⟩}_{\text{DMC}}-⟨\hat{O}{⟩}_{\text{VMC}}$$

Full text

Interacting multiparticle systems where the interaction range is much smaller than any other characteristic length scale, such this associated with the density or the average momentum, can be studied using the zero range approximation. In this limit, the interaction details are neglected and the system acquires universal features depending only on its density $\rho$ and the scattering length $a_{s}$. When $a_{s}$ is small, the particles interact weakly and the system is amenable to perturbative treatment. When it is large, the particles are strongly correlated and one needs to resort to numerical methods to study the system properties.

About a decade ago, while studying two-component Fermi systems with large $a_{s}$, Tan has succeeded to show that many of its properties are governed by a single parameter, the so-called contact $C$, which measures the probability of two particles being in close proximity Tan08 . Following Tan’s work, different relations between various properties of such system and the contact, known as the Tan relations, were derived and verified experimentally with ultracold gases SteGaeDra10 ; SagDraPau12 ; WerTarCas09 ; KuhHuLiu10 . One example is the one-body momentum distribution $n(k)$ tail, determined to be

[TABLE]

The Pauli principle prevents identical fermions from approaching each other in a relative $s$-wave state. Consequently, three-body correlations are typically negligible in an ultracold two-component atomic Fermi gas. In contrast, such three-body coalescence is expected to play a decisive role in bosonic gases or for nucleons, where the spin-$\frac{1}{2}$ neutrons and protons form a four-component Fermi system. Indeed for bosonic systems, the tail of the momentum distribution is predicted to include a subleading $k^{-5}$ term, emerging from such three-body correlations BraKanPla11 . We note that other singular interactions, like Coulomb, also exhibit universal short-distance correlations Kim73 ; HofBarZwe13 .

To derive the Tan relations one may start with the observation that when two particles approach each other, the $N$-body wavefunction is factorized into a product of a universal 2-body function $\phi_{2}$ and a state-dependent function $A_{N}^{(2)}$ describing the residual system,

[TABLE]

Here $\bm{r}_{ij}=\bm{r}_{i}-\bm{r}_{j}$ is the interparticle distance and $\bm{R}_{ij}=(\bm{r}_{i}+\bm{r}_{j})/2$ is the pair’s center of mass coordinate. In the zero-range approximation the universal pair wave-function is given by $\phi_{2}(\bm{r}_{ij})=1/r_{ij}-1/a_{s}+O(r_{ij})$.

Recently, the contact formalism was generalized to systems where the zero-range approximation is not justified WeiBazBar15a ; WeiBazBar15b ; WeiCruBar18 ; Mil18 ; WerCas12ref . This is the situation, for example, in the atomic nucleus, where the interparticle distance is about $2.4\,\mathrm{fm}$, while the nuclear interaction range, estimated from the pion mass, is about $\hbar/m_{\pi}c\approx 1.4\,\mathrm{fm}$. This is also the situation in 4He atomic clusters, where the average interparticle distance within clusters with more than three atoms is about $5\,\mathrm{\AA}$, while the van der Waals length, characterizing the potential’s range, is about $5.4\,\mathrm{\AA}$.

In such cases, one would not expect to see a strong universality, i.e. relations which do not depend on the interaction details and are determined only by scattering parameters such as $a_{s}$. Still, given an interaction model strong at small distances, the wavefunction factorization (Eq. 2) remains valid since at close distance a correlated particle pair is barely influenced by the surrounding particles and therefore its wavefunction $\phi_{2}(\bm{r})$ should be the same regardless of the system size or state. We will call this situation weak universality.

It is instructive therefore to study the adaptation of Tan’s relations to weak universality. For instance, relations between the one and two-body momentum distributions as well as the two-body density were studied in nuclei WeiBazBar15b ; WeiCruBar18 . In the following we will investigate such relations for bosonic 4He clusters.

4He clusters have attracted a lot of attention. For a long time, the 4He trimer seemed to be the most promising candidate for experimental validation of the Efimov effect Efimov70 , as liquid Helium was for Bose-Einstein condensation. Recently 4He dimer and trimer densities were measured experimentally Zeller16 ; Voi14 . The results compare very well with theoretical calculations using 4He pair potential models. The dimer and trimer densities at short range play a crucial role in the contact formalism we study here. The atomic clusters exhibit a universal short-range $2,3$-body behavior stemming from the dimer and trimer wavefunctions, respectively. Moreover, this phenomenon also continues with the coalescence of more atoms inside these clusters, showing the emergence of $4,5,\ldots$-body universality.

In order to study the properties of 4He clusters we solve the $N$-body Schrödinger equation with the LM2M2 pair potential AziSla91 .

As we argued above, in the limit of vanishing interparticle distance $r\to 0$ we expect the wavefunction $\Psi$ to factorize as in Eq. (2) into a universal $2$-body function and a residual state dependent function. If true, this factorization holds also for $N=2$. Consequently, we can identify $\phi_{2}$ with the dimer wavefunction.

The resulting two-body contact is defined as the norm of the residual non-universal part of the wavefunction multiplied by the number of pairs,

[TABLE]

Using this definition, the pair density function at short distances attains an extremely simple form,

[TABLE]

where $\hat{\rho}^{(N)}_{2}(r)=\frac{1}{r^{2}}\sum_{i<j}\delta(r_{ij}-r)$, $\rho_{2}(r)\equiv\rho_{2}^{(2)}(r)=\int d\Omega_{2}|\phi_{2}(\bm{r})|^{2}$, and $\Omega_{2}$ is the solid angle.

In a bosonic system, coalescence of more particles should provide further factorizations of the wavefunction WerCas12 . When particles $i,j$ and $k$ come close together, the wavefunction is factorized as

[TABLE]

where the triplet wavefunction depends on the Jacobi coordinates $\bm{x}_{ijk}=\sqrt{1/2}(\bm{r}_{i}-\bm{r}_{j})$ and $\bm{y}_{ijk}=\sqrt{2/3}(\bm{r}_{k}-(\bm{r}_{i}+\bm{r}_{j})/2)$, and the factorization holds for small hyperradius $r_{ijk}^{2}=x_{ijk}^{2}+y_{ijk}^{2}$. Here $\bm{R}_{ijk}$ is the three body center of mass coordinate. In analogy with Eq. (3), the three-body contact in the $N$-body system is defined to be the number of triplets times the norm of the particular part of the wavefunction in three-body coalescence,

[TABLE]

The triplet density operator is defined as,

[TABLE]

and its expectation value in the $N$-body system is

[TABLE]

where $\rho_{3}(r)\equiv\rho_{3}^{(3)}(r)=\int d\Omega_{3}\;|\phi_{3}(\bm{x},\bm{y})|^{2}$, and $\Omega_{3}$ denotes the hyperangles associated with $\bm{x}$, and $\bm{y}$.

Similar factorization is assumed in the $n$-body coalescence, leading to the definition of the $n$-body contact, and to the $n$-body density function,

[TABLE]

where here $r=\sqrt{\sum_{i<j}^{n}(\bm{r}_{i}-\bm{r}_{j})^{2}/n}$ is the $n$-body hyperradius. This is one of the main results of this paper and in the following we shall show that this is indeed the case for $n\leq 5$ in atomic 4He droplets with $N$ atoms. In the mean time we note that with the above definition the contact for $n=N$ equals unity since $\rho_{n}(r)\equiv\rho^{(n)}_{n}(r)$.

Using this factorization, the zero-range result for the high momentum limit of the 1-body momentum distribution (Eq. 1), is now modified to get WerCas12a

[TABLE]

where $\tilde{\phi}_{2}(\bm{k})$ is the Fourier transform (FT) of $\phi_{2}(\bm{r})$. The high momentum limit of the static structure factor, which is proportional to the contact in the zero-range limit KuhHuLiu10 , gets now the form

[TABLE]

where $Q$ is the momentum transfer. It is also possible to relate the contact to the potential energy which, for a cluster of bosons interacting via 2-body forces can be written using the 2-body density $\langle V_{2}^{(N)}\rangle=\int d\bm{r}\rho_{2}^{(N)}(r)v(r)$. For a short range interaction we can replace $\rho_{2}^{(N)}$ by its asymptotic form, Eq. (4), relating the $N$-body potential energy to the 2-body contact and potential energy WerCas12a ,

[TABLE]

which generalizes the zero-range result of Ref. ZhaLeg09 .

The $N$ dependence - To understand the dependence of the $n$-body contact on the total particle number $N$ in the cluster, it is useful to relate the pair density $\rho_{2}^{(N)}$ to the $2$-body density $\chi(\bm{r},\bm{r}^{\prime})=\sum_{i\neq j}\langle\Psi|\delta(\bm{r}-\bm{r}_{i})\delta(\bm{r}^{\prime}-\bm{r}_{j})|\Psi\rangle$, namely

[TABLE]

In the limit $N\to\infty$ the system becomes homogeneous, $\chi(\bm{r}_{1},\bm{r}_{2})\to\chi(\bm{r}_{12})$ and therefore $\rho_{2}^{(N)}(\bm{r}_{12})=V\chi(\bm{r}_{12})/2=N\chi(\bm{r}_{12})/2\rho$ where $V$ is the volume of the system and $\rho=N/V$ is the density. Taking now the limit $r_{12}\to 0$ one can get the relation WerCas12a

[TABLE]

We know that in the thermodynamic limit $\chi$ and $\rho$ are finite. It follows that $C_{2}^{(N)}\propto N$ as $N\to\infty$. The same argument can be repeated for $n=3,4,5,\ldots$ leading to the general conclusion that for any $n$-body coalescence $C_{n}^{(N)}\propto N$ as $N\to\infty$. Equipped with this observation it seems natural to define a reduced contact $\tilde{C}_{n}^{(N)}\equiv C_{n}^{(N)}/N$. As the atomic He clusters behave very much like a cluster of rigid balls, we expect that the leading corrections to the above argument will depend on the ratio between surface particles $\propto N^{2/3}$ and volume particles $\propto N$. Consequently in the limit $N\to\infty$ the contacts are expected to have the following $N$ dependence

[TABLE]

The computational method - Throughout the years, a variety of numerical methods have been developed to solve the few-body Schrödinger equation. However, the increasing dimensionality and the hard-core nature of the 4He-4He pair potential make this problem hard to handle for most numerical methods. Here we use the Variational Monte Carlo (VMC) and Diffusion Monte Carlo (DMC) methods. Since these methods are well-known we will only describe them very briefly, for a comprehensive review see e.g. KalWhi08 .

Given a trial wave-function $\Psi_{T}$, the variational energy

[TABLE]

is an upper bound to the true ground-state energy $E_{0}$. In the VMC method the integrals in Eq. (16) are evaluated using the Monte Carlo numerical integration technique, typically the Metropolis algorithm MetRosRos53 . Using the variational principle (16), parameters characterizing $\Psi_{T}$ can be optimized, minimizing the trial energy or its variance.

DMC is an alternative approach to solve the Schrödinger equation through propagation of the solution in imaginary time $\tau=-it$,

[TABLE]

where $E_{R}$ is a reference energy. Eq. (17) is treated as a diffusion-reaction process for so-called walkers, distributed according to $\Psi$. As time propagates, $\Psi$ will be dominated by the eigenstate with the lowest energy which has a non-zero overlap with the initial state. All other eigenstates will decay exponentially faster. The ground state energy is the reference energy which conserves the walkers number.

Improved results are obtained by introducing a trial wavefunction to guide the diffusion process, therefore a typical DMC calculation starts with an optimized VMC wave-function. We adopt the trial wavefunction form of Ref. RicLynDol91 , $\Psi_{T}=\prod_{i<j}f(r_{ij})$ where

[TABLE]

Here $p_{5},p_{2},p_{1},$ and $p_{0}$ are variational parameters, which can be found in Ref. SM .

*Ground state energies - * To benchmark our Monte Carlo code we have calculated the ground-state energies of small 4He clusters with the LM2M2 pair-potential. Calculations were done with $4000$ walkers, using $10000$ blocks of $500$ iterations each. The first $100$ blocks were used for equilibration.

The 4He trimer ground state energy using this potential has been calculated using several few-body techniques. Most results agree with $B_{3}=126.0(5)$ mK NieFedJen98 ; BluGre00 ; RouYak00 ; MotSanSof01 ; LazCar06 ; GuaKorNav06 ; SalYarLev06 ; HiyKam12 , while different values also exist EsrLinGre99 ; GonRubMir99 .

Few calculations have been done for larger clusters. The tetramer energy was calculated in Refs. BluGre00 ; LazCar06 ; GuaKorNav06 ; HiyKam12 using the LM2M2 potential. In Ref. GatKieViv11 a soft-core potential was used while in Refs. PlaHamMei04 ; BazEliKol16 an effective field theory approach was followed. In both cases the interaction parameters were fitted to the LM2M2 potential. Larger clusters were investigated using the DMC method  BluGre00 ; GuaKorNav06 . In Table 1 we compare these calculations with our results, showing good agreement with the published binding energies.

The $n$-body density function – To calculate the $n$-body densities we have used a combination of VMC and DMC estimates,

[TABLE]

where $\langle\hat{O}\rangle_{\textrm{DMC}}=\langle\Psi_{T}|\hat{O}|\Psi\rangle/\langle\Psi_{T}|\Psi\rangle$ is the mixed DMC estimate, and $\langle\hat{O}\rangle_{\textrm{VMC}}=\langle\Psi_{T}|\hat{O}|\Psi_{T}\rangle/\langle\Psi_{T}|\Psi_{T}\rangle$ is the VMC estimate. This result is accurate to second order in the wavefunction $O(\delta\Psi^{2})$, $\delta\Psi=\Psi_{T}-\Psi$ CeperlyKalos1979 . Moreover, we checked our results with pure-estimator based on the descendant weighting method Kal70 , and found no significant change.

For the smaller clusters, the resulting $n$-body densities exhibit a typical bell shape, starting from zero at $r=0$, reaching a maximum value, and finally falling exponentially at large $r$. According to Eq. (4), we expect that at short distances the pair density function $\rho_{2}^{(N)}$ will coincide with the dimer density $\rho_{2}$ up to a scaling factor, the 2-body contact $C_{2}^{(N)}$, which we can extract fitting these two functions extract_contacts . This situation is expected to repeat itself for the 3-body density function, Eq. (8), and in general for any $n$-body density, Eq. (9).

Having extracted the contacts extract_contacts we are in position to demonstrate the validity of Eq. (9). To this end, we plot in Fig. 1 the normalized $n$-body densities $\rho_{n}^{(N)}/C_{n}^{(N)}$ as a function of the $n$-body radius $r/r_{m}$. $r_{m}=2.6965\rm{\AA}$ being the minimum 2-body potential locus. The plot contains results for 4He clusters with $N=n$ and $N=10,15,20,\ldots 50$ particles. Inspecting the plot we see that, indeed, for each $n$ there is a range $r_{n}$ such that for $r\leq r_{n}$ all the normalized densities collapse into a single curve. For the pair density this range is approximately $1.3r_{m}$ and it grows linearly with $n$, i.e. $r_{n}\approx n\;0.65r_{m}$.

The numerical values of the extracted contacts are presented in the supplementary material SM . Here we analyze the $N$ dependence of the $n$-body contacts. From Eq. (15) we expect $\tilde{C}_{n}^{(N)}=C_{n}^{(N)}/N$ to be finite in the thermodynamic limit. Our MC code was designed to study small He clusters with $N\leq 50$ particles, and is therefore ill-equipped to study this $N\to\infty$ limit. Instead, to estimate $\tilde{C}_{n}^{\infty}$ we fit our calculated contacts to Eq. (15). Doing so, we have found that, for $N\geq 10$, $3$ terms are enough to describe $C_{2}^{(N)},C_{3}^{(N)}$, and $4$ terms for $C_{4}^{(N)},C_{5}^{(N)}$. The asymptotic values of the reduced contacts are given in Table 2. The calculated contacts are plotted together with the asymptotic expansion in Fig. 2, where we observe that the calculated values are well reproduced by the asymptotic expansion.

Having calculated the 2-body contacts, the $Q\to\infty$ limit of the structure factor can be evaluated for any helium droplet and compared with experiment.

For liquid helium, the structure factor was measured using x-ray scattering RobHal82 ; WirHal87 , and neutron scattering techniques SveSeaWoo80 . Following the analysis of Donnelly and Barenghi DonBar98 we adopt the latter data set and compare it with the contact theory, Fig. 3. In the range $Q\geq 2\rm{\AA}^{-1}$, dominated by the short-range pair function $\phi_{2}$, we see a nice agreement between the two. The data fits contact values in the range $\tilde{C}_{2}^{\infty}\in(200,250)$ as predicted by our calculations, Tab. 2.

The dynamic structure factor $S(Q,E)$ of liquid 4He was recently measured by Prisk et al. Prisk2017 , using the neutron Compton scattering technique. In the impulse approximation, $S(Q,E)$ and consequently the neutron Compton profile can be calculated from the 1-body momentum distribution $n(\bm{k})$. Utilizing the contact relation (10), we analyzed these results. Overall we got reasonable agreement between the data and the theory for contact values $\tilde{C}_{2}^{\infty}=180\pm 40$, a value consistent with both the MC calculation and the static structure factor data.

Conclusion. Summing up, utilizing the generalized contact formalism, we have studied short-range correlations in bosonic Helium clusters composed of 4He atoms. Specifically, we have studied $n$-body coalescences, and the emergence of universal $n$-body short-range correlations. Employing the LM2M2 pair potential, VMC and DMC calculations were used to demonstrate and verify the universal nature of these correlations. For systems with up to $N=50$ particles, the values of the $n$-body contacts were evaluated numerically for $n\leq 5$. The thermodynamic limit was studied, extrapolating our numerical results. Comparing our prediction with the experimental two-body contact, extracted from available measurements of the structure factor of liquid 4He at high momenta, we have found a good agreement. It would be interesting to compare our predictions with detailed Monte Carlo simulations of Helium liquid.

The implications of the current formalism on the momentum distribution and the dynamic structure factors call for further experimental studies in the high momentum sector.

The variational parameters

In Table S1 we present the trial wave function parameters (Eq. (18)) optimized with VMC calculation for small 4He clusters $N\leq 50$.

The contacts

In Table S2 we present the contacts for small 4He clusters $N\leq 50$ calculated using a mixed VMC-DMC estimate. The table includes contacts ${C}_{n}^{(N)}$ for $n=2-5$ coalescing particles. As explained in the main body of this manuscript, the contacts were extracted from the calculated $n$-particle densities $\rho_{n}^{(N)}(r)$.