We report a diffusion Monte Carlo study of A4He2 and A4He3He trimers’ structural properties, were A is one of the alkali atoms 6,7Li, 23Na, 39K, 85Rb or 133Cs. Some of them are in a pure halo state, characterized by large spatial extent and universality, while some are close to the halo limit. The theoretical analysis of these trimers enables insight on how structural properties of weakly bound systems change when approaching the halo edge. For that purpose, two-variable distributions of inter-particle separations and angles were calculated. Extreme spatial extensions of some trimers with 3He confirm their halo nature. Although all the considered systems are floppy, trimers with all bound dimer subsystems are less spread and have significantly lower percentage of quasi-linear configurations than those which have at least one unbound dimer subsystem.Postprint (author's final draft

Boronat Medico, Jordi

Stipanovic, P.

Vranješ Markic, Leandra

Few-Body Systems

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We report a diffusion Monte Carlo study of A4He2 and A4He3He trimers’ structural properties, were A is one of the alkali atoms 6,7Li, 23Na, 39K, 85Rb or 133Cs. Some of them are in a pure halo state, characterized by large spatial extent and universality, while some are close to the halo limit. The theoretical analysis of these trimers enables insight on how structural properties of weakly bound systems change when approaching the halo edge. For that purpose, two-variable distributions of inter-particle separations and angles were calculated. Extreme spatial extensions of some trimers with 3He confirm their halo nature. Although all the considered systems are floppy, trimers with all bound dimer subsystems are less spread and have significantly lower percentage of quasi-linear configurations than those which have at least one unbound dimer subsystem

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Structure of halo and quasi-halo helium–helium–alkali trimers

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Few-Body Systems manuscript No.(will be inserted by the editor)Structure of Halo and Quasi-Halo Helium-Helium-Alkali TrimersPetar Stipanovic´ · Leandra Vranjesˇ Markic´ · Jordi BoronatReceived: date / Accepted: dateAbstract We report a diffusion Monte Carlo study of A4He2and A4He3He trimers’ structural properties, were A is oneof the alkali atoms 6,7Li, 23Na, 39K, 85Rb or 133Cs. Someof them are in a pure halo state, characterized by large spa-tial extent and universality, while some are close to the halolimit. The theoretical analysis of these trimers enables in-sight on how structural properties of weakly bound systemschange when approaching the halo edge. For that purpose,two-variable distributions of inter-particle separations andangles were calculated. Extreme spatial extensions of sometrimers with 3He confirm their halo nature. Although all theconsidered systems are floppy, trimers with all bound dimersubsystems are less spread and have significantly lower per-centage of quasi-linear configurations than those which haveat least one unbound dimer subsystem.Keywords halo states · three-body · helium-alkali clusters ·structure · quantum Monte CarloPACS 02.70.Ss · 36.40.-c · 67.90.+z1 IntroductionWeakly bound and spatially extended few-body systems havebeen attracting considerable interest since they share uni-versal features. This key characteristic enables comparisonPetar Stipanovic´University of Split, Faculty of Science, R. Bosˇkovic´a 33, HR-21000Split, CroatiaTel.: +385 21 619 234E-mail: pero@pmfst.hrLeandra Vranjesˇ Markic´University of Split, Faculty of Science, R. Bosˇkovic´a 33, HR-21000Split, CroatiaJordi BoronatDepartament de Fı´sica, Campus Nord B4-B5, Universitat Polite`cnicade Catalunya, E-08034 Barcelona, Spainof systems from different fields of physics, from ultracoldgases to nuclear physics. Specifically, in quantum three-bodyphysics the most famous property is the Efimov effect [1].An infinite series of three-body geometrically spaced andstable excited states was predicted [1], and latter experimen-tally confirmed [2], to appear when a third particle is addedto a pair of bosons that are on the edge of binding. Universal-ity is also related to the two- and three-body systems, whichmostly occupy classically forbidden regions of space, the socalled halo states [3–5]. Such systems are not sensitive todetails of the short-range interactions. Thus, their propertiescan be predicted with any potential that has the same inte-gral properties, i.e., the scattering length and the effectiverange. The scale invariance noticed in nuclei [3] was con-firmed also for small atomic clusters [3,5]. Those works pro-posed a universal relation between scaled size and energy fortwo- and three-body systems as well as separation of linescorresponding to different types of trimers. Namely, trimerscan be classified as Borromean [3], tango [6], samba [7]and all-bound type if they have zero, one, two and all threetwo-body subsystems bound, respectively. Universal rela-tions between scaled size and energy of weakly bound three-body systems were calculated in three [8] and two [9] spatialdimensions.Due to their extremely weak interaction, both heliumisotopes, 4He and 3He played a significant role in the studyof molecular halos [3,5]. Furthermore, it was experimentallyconfirmed that the excited state of 4He3 is an Efimov state,while the ground state of 4He23He was suggested to be aquantum halo state [10,11]. Recently, helium trimer prop-erties have been thoroughly investigated not only theoret-ically [12–14], but also experimentally [12,13]. Since He-alkali potentials are even shallower, one expects that swap-ping a He isotope by an alkali could produce a halo stateor a diffuse system close to the halo border (quasi-halo).Very weak binding of mixtures HeHeA, where A is one of2 Petar Stipanovic´ et al.the alkali atoms, was predicted by different methods [15–20,22,21]. In addition, the first excited states of 4He26Liand 4He27Li were predicted to be Efimov states. [15–19,21].Also recently, weak binding of 4He2A has been confirmed[23–25]. Swapping a 4He by 3He generates even more dif-fuse clusters, showing a pure halo characteristics [5,25].However, structural properties of helium-alkali clustershave not been thoroughly investigated so far. In particular,two-dimensional angular and radial distribution functions,which would give better insight into the different triangu-lar configurations, are lacking. Since helium-alkali trimersoffer a huge set of weakly bound systems under natural con-ditions, we analyzed their structure in more detail to testhow mass and binding strength affect their diffuseness indifferent directions. In particular, we notice the difference intrimer configuration belonging to different types, all-boundand tango/samba trimers.Since very weakly bound system are computationallyquite demanding, we chose the diffusion Monte Carlo method(DMC) [26] and pure estimators [27] to extract exact ground-state properties. The method is discussed in Sec. 2. Sec. 3reports the distribution functions. Finally, Sec. 4 comprisesa summary of the work and an account of the main conclu-sions.2 MethodClusters A4He2 and A4He3He were studied at zero tem-perature using the DMC method. The DMC method solvesstochastically the Schro¨dinger equation written in imaginarytime τ = it/h¯,−∂Ψ(R,τ)∂τ = (H−Er)Ψ (R,τ) , (1)applying reasonable approximations for the Green’s func-tion when imaginary time step ∆τ → 0. In Eq. (1), Er is areference energy and the walker R≡ (r1,r2,r3) collectivelydenotes the positions of the trimer’s constituents.The Hamiltonian H for the trimer isH =−3∑i=1h¯22mi∇2i +3∑i,j=1i<jV(rij) , (2)where V denotes the interatomic potentials between the threepairs of the trimer. Explicit three-body potentials are not in-cluded because their contribution has proved to be negligiblein the case of pure He trimers [13,28]. For He-He interac-tions we used the semi-empirical HFDB potential [29] as inRefs. [5,13] and for He-A the KTTY [30] model as in Ref.[5]. The best available constants from the NIST databasewere taken,0.5h¯2m−1/(mK A˚2)for 3He, 4He, 6Li, 7Li, 23Na, 39K, 85Rb, and 133Cs, respec-tively equal to 8041.811058, 6059.640786, 4032.226566,3457.001455,1055.005713, 622.4853844, 285.6415818, and182.4931752.In order to solve Eq.(1) we applied the same procedureas in Refs. [13,25] In particular, the details of the calcula-tions are given in the work that reports the stability and one-dimensional distribution functions of 4He2A, 4He3HeA and3He2A, where A is one of the alkali atoms Li, Na, K, Rb orCs. [25]For the calculation of the atom pair-pair distributionsP(ri j,rkl) and angular-angular distributions P(θ1,θ3) it wasnecessary to use pure estimators [27], because the corre-sponding operators do not commute with the HamiltonianH. In order to ensure exact ground-state properties, it is cru-cial in this algorithm to verify that the chosen block size islarge enough. We used 105 steps per block for all studiedstructural properties, although for some trimers they con-verged even for 3 times smaller block sizes. For the calcu-lations of the distribution functions we have chosen a timestep ∆τ = 10−3 K−1 that we checked to be small enough toreduce any finite time-step bias.3 ResultsMotivated by the recent experimentally measured [10,11,31] distributions of helium trimers 4He3 and 4He23He, wedetermined the structural properties of the ground state forall A4He2 and A4He3He trimers. The structure is visualizedby two-variable distribution functions, inter-particle separa-tions P(He−He,4 He−A) and angles P(θ1,θ3) where θ1and θ3 denote respectively α and γ angle of the triangleABC formed by atoms.Results for the distribution functions are shown in fig-ures 1, and 2. Top rows present A4He3He trimers, whileA4He2 are in a bottom rows. In Fig. 1, the separation be-tween He-He atoms is in the x-axis, while the 4He-A dis-tances are in the y-axis. The repulsive barrier of the He-Apotential is wider than the He-He one, producing a widerzero probability area extending next to the x-axis than nextto the y-axis. Color-boxes for top and bottom row are differ-ent for clarity reasons because trimers A4He3He are muchmore diffuse than A4He2, which are not in a halo state [25].Our previous study has shown that universal scaling of en-ergy and size [5] extends also somewhat below the halodomain. Similar distributions for A4He2 suggest that theyshould be concentrated very close in the scaling plot fromRef. [5]. Higher peaks mean higher localization which isin agreement with the predicted binding energies [14,25].Swapping the heavier He isotope by the lighter, the effectbecomes more pronounced. All A4He3He trimers are tangostates, except the one with Na, which is a samba state be-cause the 3He-Na dimer is bound with the energy of -1.242Structure of Halo and Quasi-Halo Helium-Helium-Alkali Trimers 3 0 10 20 306Li4He3He 7Li4He3He 23Na4He3He 39K4He3He 85Rb4He3He01234133Cs4He3Her (4He - A) / Å 0  10  20  30 0 10 20 306Li4He2 0  10  20  307Li4He2 0  10  20  3023Na4He2r (He - He) / Å 0  10  20  3039K4He2 0  10  20  3085Rb4He2 0  10  20  3002468133Cs4He2103 P(He - He,4He - A) / Å-2Fig. 1 Distributions of He-He and 4He-A separations in: A4He3He (top) and A4He2 (bottom), where A is an alkali metal;∫Pd2r = 1θ3θ1θ23He6Li4He 0 0.2 0.4 0.6 0.8 7Li4He3He 23Na4He3He 39K4He3He 85Rb4He3He 133Cs4He3He6Li4He2θ3 / pi 0  0.2  0.4  0.6  0.8 0 0.2 0.4 0.6 0.8 7Li4He2 0  0.2  0.4  0.6  0.823Na4He2 0  0.2  0.4  0.6  0.839K4He2θ1 / pi 0  0.2  0.4  0.6  0.885Rb4He2 0  0.2  0.4  0.6  0.8133Cs4He2 0  0.2  0.4  0.6  0.80123456pi2 P (θ1 , θ3 )Fig. 2 Distributions of angles θ1 = ∡(3,4He−A− 4He) and θ3 = ∡(4He− 3,4He−A) in helium-alkali trimers: A4He3He (top) and A4He2 (bot-tom), normalized to ∫ P(θ1,θ3)dθ1dθ3 = 1mK. Larger differences in distributions are in agreement withtrimers’ spread along the universal tango line [5]. E. g. scaledsizes of 3.3 and 1.7 were obtained [5] for 3He4He7Li and3He4He39K, respectively, what classifies them as halo andquasi-halo close to the halo limit 2. All clusters are squeezedin the direction of the axis belonging to the heavier pair, y-axis. The effect is the more pronounced the stronger bind-ing of the He-A pair is with respect to the the He-He bind-ing (-1.69 mK). Namely, 4He-A binding energies in mK are-1.515, -5.622, -28.98, -11.2, -10.27, -4.954 for 6Li, 7Li,23Na, 39K, 85Rb and 133Cs, respectively. Manifolded sys-tem’s configuration arrangements are noticeable in all trimers,i.e., for a particular ri j value a wide range of distances rkl hashigh probability of occurrence.Angle-angle distribution functions shown in Fig. 2 pro-vide additional insight into the possible types of triangu-lar configurations. Angle θ1 =∡(3,4He−A− 4He) is shownon x-axis, while y-axis stands for θ3 = ∡(4He− 3,4He−A).Angles are marked for 6Li4He3He in upper right corner ofthe first subfigure in the top row. All of the trimers are spreadamong different possible triangular shapes. As the mass ofthe alkali atom is increased a more defined maximum startsto form. The overall symmetry is significantly different inthe cases of A4He3He trimers, presented in the upper rowand A4He2 ones in the bottom row. In A4He3He trimers,which have at least one unbound subsystem, when the Amass in increased, a maximum forms with the small valueof the angle at the 3He atom and the large angle at the Aatom, suggesting a significant proportion of nearly linearconfigurations. On the other hand, in the case of A4He2 suchconfigurations have very low probability. The most probabletriangular configuration are given in Table 1. Isosceles con-figurations are preferred by A4He2, with approximately twotimes smaller angle in the A corner than in the He one. Incontrast to that, in the A4He3He trimer, the angle in the 4Hevertex is two times larger than in the 3He one while the angleθ1 increases with the A mass from 97°to 165°.4 Petar Stipanovic´ et al.Table 1 Most probable triangular configuration of atoms in trimersA−4H−4He and A−4He−3He, given with the uncertainty of 5°θ1−θ2−θ3 [°] in A−B−CA A−4H−4He A−4He−3He6Li 36−71−73 097−59−247Li 37−74−69 103−48−2923Na 43−70−67 150−19−1139K 34−76−70 159−14−0785Rb 33−74−73 159−14−07133Cs 34−80−66 165−11−044 ConclusionsUsing accurate diffusion Monte Carlo simulations we ob-tained the ground-state structural properties for all trimersA4He2 and A4He3He, where A is an alkali metal 6,7Li, 23Na,39K, 85Rb or 133Cs. Structural properties were calculated us-ing pure estimators in order to completely eliminate the biasof the trial wavefunction used for guiding the diffusion pro-cess within the DMC method.Our results for the distribution of He-He and 4He-A dis-tances show a noticeable squeezing of the HeHeA cluster inthe He-A direction. The effect is the more pronounced thestronger binding of the He-A pair is in comparison to the He-He binding. Samba and tango type trimers are more spatiallystretched than all-bound trimers. From angular-angular distri-butions it is noticeable that samba 23Na4He3He and A4He3Hetango-type trimers show with large probability triangular con-figurations close to the stretched triangle with A in the mid-dle. The unbound He-He pair enables their appearance. Incontrast to that, all-bound trimers barely experience theseconfigurations since all bound pairs prevent stretching. In-creasing the binding energy, the distributions become moreclassical, although the high localizations of angles around aparticular most probable configuration, as in Ne3 [14] is stillnot observed.The one-variable distributions of inter-particle distancesand angles in 4He3 and 4He23He trimers have been recentlymeasured. [10,11] This opens up the possibility that many-variable distributions could be also extracted. Since they givemore information about the system, comparison of theoreti-cal and experimental results would help to know if theoret-ical models need additional refinements. Particularly, thereare significantly different He-A model potentials in the lit-erature. We chose the KTTY model [30] since it was re-cently used in the construction of a universal potential model[32] and was mostly used in previous few-body calculations.Anyway, any experimental confirmation of its quality wouldbe very welcome from the theoretical side.Acknowledgements This work has been supported in part by the Croa-tian Science Foundation under the project number IP-2014-09-2452. J.B. acknowledge additional support by the MICINN-Spain, Grant No.FIS2014-56257-C2-1-P. The computational resources of the Isabellacluster at Zagreb University Computing Center (Srce), the HYBRIDcluster at the University of Split, Faculty of Science and Croatian Na-tional Grid Infrastructure (CRO NGI) were used.References1. V. Efimov, Phys. Lett. 33B, 563 (1970)2. T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. En-geser, A. D. Lange, K. Pilch, A. Jaakkola, H.C. Na¨gerl, R. Grimm,Nature 440, 315 (2006)3. A. S. Jensen, K. Riisager, D. V. Fedorov, E. Garrido, Rev. 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Structure of halo and quasi-halo helium–helium–alkali trimers

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