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arXiv:2303.14952 · Physical Review A

Universal tetramer limit cycle at the unitarity limit

T. Frederico, M. Gattobigio

Published Sep 5, 2023• 5 citations
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Abstract

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.

DOI
10.1103/physreva.108.033302

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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 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.
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Abstract The independence between few-body scales beyond the van der Waals universality is demonstrated for the extreme mass-imbalanced case of a specific many-boson system. This finding generalizes the scaling properties of universal tetramers to a broader class of heterogeneous few-boson systems. We assume two heavy atoms interacting with ( N − 2)-lighter ones at the unitary limit, using a particular case where no interactions are active between identical particles, by investigating the interwoven spectra of this many-body system for an arbitrary number of light bosons. A large mass-ratio between the particles allows us to treat this N -body system analytically, by solving an effective inverse-squared long-range interaction which is stablished for the two heavy bosons. For a cluster with N − 2 light bosons ( N ⩾ 4), we discuss the implications of the corresponding long-range potentials associated with different subsystem thresholds, implying in independent interwoven limit cycles for the correlation between the energies of excited N -body system. Our study with extreme mass-imbalanced few-boson bound states provides a fundamental understanding of the scaling behavior of their interwoven spectra. The novel insights enlarge the well-known Efimov physics paradigm and show the existence of different limit cycles, which could be probed by new experiments.
Three- and four-boson systems expanded around the unitarity limit: Application to 4^44He molecules
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The independence between few-body scales beyond the van der Waals universality is demonstrated for the extreme mass-imbalanced case of a specific many-boson system. This finding generalizes the scaling properties of universal tetramers to a broader class of heterogeneous few-boson systems. We assume two heavy atoms interacting with (N−2)−(N-2)-(N−2)−lighter ones at the unitary limit, using a particular case where no interactions are active between identical particles, by investigating the interwoven spectra of this many-body system for an arbitrary number of light bosons. A large mass-ratio between the particles allows us to treat this N−N-N−body system analytically, by solving an effective inverse-squared long-range interaction which is stablished for the two heavy bosons. For a cluster with N−2N-2N−2 light bosons (N≥4N\ge 4N≥4), we discuss the implications of the corresponding long-range potentials associated with different subsystem thresholds, implying in independent interwoven limit cycles for the correlation between the energies of excited N−N-N−body system. Our study with extreme mass-imbalanced few-boson bound states provides a fundamental understanding of the scaling behavior of their interwoven spectra. The novel insights enlarge the well-known Efimov physics paradigm and show the existence of different limit cycles, which could be probed by new experiments.