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AIP conference proceedings

Universality and scaling limit of weakly-bound tetramers

M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio, A. Delfino, T. Frederico

Published Jan 1, 2012• 6 citations
Publisher

Abstract

The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.

DOI
10.1063/1.3688794

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The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.
The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy B3{B}_{3}B3​, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited NNNth tetramer energy B4(N){B}_{4}^{(N)}B4(N)​ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer B4(N+1){B}_{4}^{(N+1)}B4(N+1)​ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on NNN. We show that both channels of the FY decomposition [atom-trimer (KKK type) and dimer-dimer (HHH type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the HHH channel is favored over the KKK channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.
We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths (a), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for |a|→∞ when the ratio between the tetramer and trimer energies is ≃4.6 and a new tetramer is formed. The new scale is also revealed for a < 0 by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.
Four-boson Efimov physics is well known in the negative energy regime but far less above the four-body breakup threshold. The part of this region with negative two-boson scattering length is studied solving rigorous four-particle scattering equations for transition operators in the momentum space. Moving away from the unitary limit, the Efimov tetramers evolve from unstable bound states into resonances. Their energies and widths are studied as functions of the two-boson scattering length; a universal behavior is established and given in a dimensionless representation. The Efimov tetramers have finite width in the whole regime; they broaden rapidly in the resonance regime but remain narrower than the associated trimer. The resonant behavior is most clearly seen in the four-particle recombination rate.
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
We investigate the appearance of a four-boson limit cycle in Hamiltonian systems at the unitarity limit. The model interaction incorporates two-, three-, and four-body short-range potentials, which allow us to disentangle the interwoven dynamics of three- and four-boson energy levels. Through numerical evidence, we observe a correlation between the energies of two successive universal tetramer levels for a fixed weakly bound trimer, which is found to be largely model independent. Interestingly, this correlation cycle is consistent with the findings of Hadizadeh et al. [Phys. Rev. Lett. 107, 135304 (2011)] using a zero-range model.
The description of unitary few-boson systems is conceptually simple: only one parameter – the three-body binding energy – is required to predict the binding energies of clusters with an arbitrary number of bosons. Whether this correlation between the three- and many-boson systems still holds for two species of bosons for which only the inter-species interaction is resonant depends on how many particles of each species are in the system. For few-body clusters with species A and B and a resonant AB interaction, it is known that the emergent AAB and ABB three-body scales are correlated to the ground-state binding energies of the AAAB and ABBB systems, respectively. We find that this link between three and four bodies is broken for the AABB tetramer whose binding energy is neither constrained by the AAB nor by the ABB trimer. From this de-correlation, we predict the existence of a scale unique to the AABB tetramer. In our explanation of this phenomenon, we understand the AABB and AAAB/ABBB tetramers as representatives of two different universal classes of N-body systems with distinct renormalization-group and discrete-scaling properties.