Efimov Physics in small bosonic clusters
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Efimov Physics in small bosonic clusters
M. Gattobigio · A. Kievsky · M. Viviani
Received: date / Accepted: date
Abstract
We study small clusters of bosons, A = 2 , , , ,
6, characterized by a reso-nant interaction. Firstly, we use a soft-gaussian interaction that reproduces the values ofthe dimer binding energy and the atom-atom scattering length obtained with LM2M2potential, a widely used He- He interaction. We change the intensity of the potentialto explore the clusters’ spectra in different regions with large positive and large neg-ative values of the two-body scattering length and we report the clusters’ energies onEfimov plot, which makes the scale invariance explicit. Secondly, we repeat our calcu-lation adding a repulsive three-body force to reproduce the trimer binding energy. Inall the region explored, we have found that these systems present two states, one deepand one shallow close to the A − Keywords
Efimov Physics · Bosonic Clusters · Hyperspherical Harmonics
PACS
Systems composed by few particles having large value of the two-body scattering length, a , with respect to the natural length, ℓ , fixed by the inter-particle potential, havebeen the object of an intense investigation both from a theoretical and experimentalpoint of view (for recent reviews see Refs. [1,2,3]). The interest is driven by their Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20- 25 August, 2012, Fukuoka, JapanM. GattobigioUniversit´e de Nice-Sophia Antipolis, Institut Non-Lin´eaire de Nice, CNRS, 1361 route desLucioles, 06560 Valbonne, FranceE-mail: [email protected]. Kievsky - M. VivianiIstituto Nazionale di Fisica Nucleare, Largo Pontecorvo 2, 56127 Pisa, Italy universal properties; the behavior of observables do not depend on the microscopicalcharacteristics, namely the inter-particle potential, but only on symmetries.In the limit a/ℓ → ∞ , known either as resonant a → ∞ or as scaling ℓ → λ = e π/s , andfor three-identical bosons s = 1 . λ ≈ .
69. This symmetry implies thatall observables can be written as an universal log-period function of the dimensionlessvariable aκ ∗ , where κ ∗ is a three-body parameter encoding the high-energy (shortdistance) physics, which enters, at leading order, only through this parameter. Thefact that the observables’ behavior is governed by discrete-scale symmetry is knowas Efimov physics. For A = 3, these properties have been studied for large positiveand large negative values of the scattering length in the ( a − , k ) plane, with k =sign( E )[ | E | / ( ~ /m )] / , constructing the Efimov plot [5]. This plot is useful in theidentification of discrete-scale symmetry; in fact, if one introduce the radial, H , andangular, ξ Efimov variables by k = H sin ξ and a − = H cos ξ , the scale invariancereads H → λ − H , and ξ → ξ .In the present work we extend our previous analysis of the A = 4 − a − , k ) plane. We have modified the strength of the LM2M2 [7]potential in order to cover the region of negative values of a up to a − , with this valueindicating the threshold of having a three-body system bound. We have also increasedthe intensity of the interaction in order to extend the analysis to positive values of a in which the universal character of the system starts to be questionable, i.e, when theground-state E approaches the natural energy E ℓ = − ~ /mℓ , which delimits theEfimov window.We used the LM2M2 potential to fix the two-body soft-core potential as in discussedin Refs. [8,6]; this has been possible because of the scale separation between the He- He scattering length, a = 189 .
41 a.u., and the natural length ℓ = 10 . A ≥
3, by meansof the non-symmetrized hyperspherical harmonic (NSHH) expansion method with thetechnique recently developed by the authors in Refs. [9,10,11,12]. In this approach,the authors have used the Hyperspherical Harmonic (HH) basis, without a previoussymmetrization procedure, and on the representation of the Hamiltonian matrix, as asum of products of sparse matrices, well suited for a numerical implementation.As a result, we have observed that in all the region explored the A = 4 , , E A − threshold. This analysisconfirms, at least in one zone of the Efimov plot, previous observations that each Efimovstate in the A = 3 system produces two bound states in the A = 4 system, and extendsthis observation to the A = 5 , In our calculation we used ~ /m = 43 . K as mass parameter. The LM2M2interaction has been modified in the following way V λ ( r ) = λ · V LM2M2 ( r ) , (1) and we have varied λ from λ = 0 . a = a − = − .
84 a.u., up to λ = 1 . a = 44 .
79 a.u. The unitary limit is produced for λ ≈ . V ( r ) = V e − r /R , (2)with range R = 10 a.u., and we have varied the strength V in order to reproduce thevalues of a given by V λ ( r ). In the three-body sector we need an hypercentral-three-body (H3B) interaction to better describe the A = 3 system obtained with the modifiedLM2M2 potential W ( ρ ) = W e − ρ /ρ , (3)with the strength W tuned to reproduce the trimer energy E obtained using V λ ( r ).Here ρ = ( r + r + r ) is the hyperradius of three particles and the range ofthe three-body force ρ = R We have solved the A = 3 problem for bound states using the modified LM2M2 po-tential, and then we used the resulting energies to fix the strength of the H3B force.Than we have diagonalized the Hamiltonian for A = 3 , , , a − , k ) plots, whichhave been scaled to shrink the scale factor to √ λ ≈ . A = 4 , , a studied. These calculations confirms the prediction for A = 4of a pair of tetramers attached to a trimer [13,14,15], and extends the observation to A = 5 and A = 6 systems.When the repulsive three-body force is included, lower panel of Fig. 1, the spectrummoves up and we can observe that the excited state E A disappears for A = 5 , a approaches a − . This behaviour is sensitiveto the range of the three-body force ρ , and it has been deeply investigated in Ref. [16].In Fig. 2 we investigate the scale invariance using Efimov-polar coordinates: for afixed value of ξ we report the ratio ( E A /E A ) / calculated with TBG potential. Fornegative values of a , the ratios tend to be constants in agreement with discrete-scaleinvariance; we note that, even if the ratio for A = 3 tends to be constant, the value islower than the expected λ . For a >
0, which corresponds to the shaded zone in Fig. 2,the non-universal behaviour becomes stronger, and the ratios are no more constants.To sum up, we have shown that Efimov physics for
A > A = 4 , ,
6, and that the ratio between each pair tends tobe constant as a function of the Efimov angle ξ . References
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