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arXiv:2408.02362 · Physical Review A

Confinement-induced unatomic trimer states

D. S. Rosa, R.M. Francisco, T. Frederico, Gastão I. Krein, Marcelo Takeshi Yamashita

Published Oct 7, 2024• 1 citation
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Abstract

The signature of an unatomic system is revealed by a continuous scale invariance that appears during a progressive dimensional squeezing of a resonantly interacting trimer. The unatomic regime is reached at the dimension $\overline{D}$, which for three identical atoms is found to be $\overline{D}=2.292$---below this value, the trimer wave function at short distances displays a power-law behavior. The fingerprint of this crossover is a sharp evolution of the contacts that characterizes the trimer momentum distribution tail.

DOI
10.1103/physreva.110.042803

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The signature of an unatomic system is revealed by a continuous scale invariance that appears during a progressive dimensional squeezing of a resonantly interacting trimer. The unatomic regime is reached at the dimension D‾\overline{D}D, which for three identical atoms is found to be D‾=2.292\overline{D}=2.292D=2.292---below this value, the trimer wave function at short distances displays a power-law behavior. The fingerprint of this crossover is a sharp evolution of the contacts that characterizes the trimer momentum distribution tail.
The dimensionality of a system can fundamentally impact the behavior of interacting quantum particles. Classic examples range from the fractional quantum Hall effect to high-temperature superconductivity. As a general rule, one expects confinement to favor the binding of particles. However, attractively interacting bosons apparently defy this expectation: While three identical bosons in three dimensions can support an infinite tower of Efimov trimers, only two universal trimers exist in the two-dimensional case. Here, we reveal how these two limits are connected by investigating the problem of three identical bosons confined by a harmonic potential along one direction. We show that the confinement breaks the discrete Efimov scaling symmetry and successively destroys the weakest bound trimers. However, the deepest bound trimers persist even under strong confinement. In particular, the ground-state Efimov trimer hybridizes with the two-dimensional trimers, yielding a superposition of trimer configurations that effectively involves tunneling through a short-range repulsive barrier. Our results suggest a way to use strong confinement to engineer more stable Efimov-like trimers, which have so far proved elusive.
Experimental progress in the study of strongly interacting ultracold atoms has recently allowed the observation of Efimov trimers. We study theoretically a nonconventional observable for these trimer states, which may be accessed experimentally, the momentum distribution n(k)n(\mathbf{k})n(k) of the constitutive bosonic particles. The large momentum part of the distribution is particularly intriguing: In addition to the expected 1/k41/{k}^{4}1/k4 tail associated with contact interactions, it exhibits a subleading tail 1/k51/{k}^{5}1/k5 which is a hallmark of Efimov physics and leads to a breakdown of a previously proposed expression of the energy as a functional of the momentum distribution.
As resonantly interacting trimers of the type AAB are progressively squeezed from D=3D=3D=3 to D=2D=2D=2, unatomic states emerge. We calculated the contacts from the high momentum tail of the single particle densities. The sharp increase of the contacts serves as a signature of the transition between the Efimov and unatomic regimes, characterized by the emergence of continuous scale invariance when the system reaches a critical dimension, DcD_cDc​. This continuous scale invariance starts to dominate the behavior of the system at the dimension D‾<Dc\overline{D}<D_cD<Dc​, below which the trimers momentum distribution tails exhibit a power-law behavior signaling the unatomic regime. To illustrate our findings, we studied compounds of the forms 7^{7}7Li−23-^{23}−23Na2_{2}2​, 7^{7}7Li−87-^{87}−87Rb2_{2}2​ and 7^{7}7Li−133-^{133}−133Cs2_{2}2​. The increase in the mass-imbalance of the trimers reduces the interval between DcD_cDc​ and D‾\overline{D}D. The emergence of unatomic states can be experimentally verified by observing the two-body contact parameter, which is a quantity directly related to the thermodynamic properties of the gas.
We investigate the single-particle momentum distribution, the two-body contact and three-body tail coefficients of mass-imbalanced three-body systems at the critical dimension DcD_cDc​, where the transition between discrete and continuous scale invariance takes place as the spatial dimension is tuned between three and two dimensions. We show that the asymptotic momentum distribution at DcD_cDc​ is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime. This structure contains an additional quadratic-logarithmic contribution, characterized by the asymptotic coefficient C3′′C_3''C3′′​, which leads to a finite and well-defined description of the momentum tail at the transition. The coefficient C3′′C_3''C3′′​ depends sensitively on the mass imbalance, changing sign across different mass configurations and vanishing for identical particles. For heavy-light-light configurations, the subleading three-body term vanishes at a characteristic momentum, leaving the leading two-body contribution at the order retained in the asymptotic expansion. The physical accessibility of this cancellation depends on the separation between the trimer size and the interaction range. We further analyze the narrow intermediate region connecting the Efimov and unatomic regimes, here identified as an intermediate scaling regime, whose extent and properties are strongly controlled by the mass ratio. These results establish the critical dimension as a regime with emergent scaling properties and provide quantitative momentum-space signatures for probing the transition between discrete and continuous scale invariance in few-body quantum systems.
We investigate the single-particle momentum distribution and contact parameters of mass-imbalanced three-body systems at the critical dimension Dc, where the transition between discrete and continuous scale invariance takes place as the spatial dimension is tuned between three and two dimensions. We show that the asymptotic momentum distribution at Dc is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime. This structure requires the introduction of an additional three-body contact parameter associated with a quadratic logarithmic contribution, leading to a finite and well-defined description of the momentum tail at the transition. This additional three-body parameter depends sensitively on the mass imbalance, changing sign across different mass configurations and vanishing for identical particles. As a consequence, the three-body contribution to the momentum distribution can be suppressed at a characteristic momentum scale, leaving the asymptotic tail entirely determined by the two-body contact. We further analyze the narrow intermediate region connecting the Efimov and unatomic regimes, here identified as an intermediate scaling regime, whose extent and properties are strongly controlled by the mass ratio. These results establish the critical dimension as a regime with emergent scaling properties and provide experimentally accessible signatures for probing the transition between discrete and continuous scale invariance in few-body quantum systems.