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arXiv:2503.01511

Confinement-induced unatomic trimer states in mass-imbalanced systems

Rafael M. Francisco, D. S. Rosa, T. Frederico, M. T. Yamashita, G. Krein

cond-mat.quant-gasquant-phSubmitted Mar 3, 2025• Announced Mar 4, 2025• 0 citations
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Abstract

As resonantly interacting trimers of the type AAB are progressively squeezed from $D=3$ to $D=2$, 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, $D_c$. This continuous scale invariance starts to dominate the behavior of the system at the dimension $\overline{D}<D_c$, 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 $^{7}$Li$-^{23}$Na$_{2}$, $^{7}$Li$-^{87}$Rb$_{2}$ and $^{7}$Li$-^{133}$Cs$_{2}$. The increase in the mass-imbalance of the trimers reduces the interval between $D_c$ and $\overline{D}$. 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.

Journal reference
Phys. Rev. A 112, 033315 2025
DOI
10.1103/h9d1-bvs4

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•Mar 24, 2014•0 citations
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.
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.
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.
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 studied the single-particle momentum distribution of mass-imbalanced Efimov states embedded in noninteger dimensions. The contact parameters, which can be related to the thermodynamic properties of the gas, were extracted from the tail of the single-particle momentum densities. We studied the dependence of the contact parameters on the progressive change in the noninteger dimension, ranging from three (D=3D=3D=3) to two (D=2D=2D=2) dimensions. Within this interval, we move from the D=3D=3D=3 regime where the Efimov discrete-scale symmetry drives the physics to close to the critical dimension, which depends on the mass imbalance, where the continuum-scale symmetry takes place. We found that the two- and three-body contacts grow significantly in magnitude with the decrease of the noninteger dimension towards the critical dimension, impacting observables of resonantly interacting trapped Bose gases.
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.
The Efimov states are three-body bound states, which appear universally when the s-wave scattering length between the particles is resonantly large.Since the Efimov states and their characteristic discrete scale invariance have recently been observed in ultracold atoms, the Efimov states and their associated few-body phenomena have attracted widening interest both theoretically and experimentally.In this thesis, I theoretically study the Efimov physics and related phenomena, focusing on the following three topics: (i) Universal three-body physics for a mass-imbalanced Fermi system (Chapter 3).For a system of two identical spinless (i.e., spin-polarized) fermions and one distinguishable particle which interact via a short-range potential with a large s-wave scattering length, two classes of universal three-body bound states have been known to appear in different regimes of the mass ratio: the Efimov trimers and the Kartavtsev-Malykh trimers, which feature the discrete and continuous scale invariance, respectively.I have found the third class of universal three-body bound states, which I call the "crossover trimers".The crossover trimers show neither the discrete nor continuous scale invariance.I have identified the regions of these three classes of trimers as a function of the mass ratio and the s-wave scattering length, and shown that the Kartavtsev-Malykh trimers and the Efimov trimers can continuously transform into each other via the crossover trimers as the mass ratio and the s-wave scattering length are varied.Owing to the presence of the crossover trimers, the Kartavtsev-Malykh trimers dissociate into a particle and a dimer when the s-wave scattering length is varied, inducing resonances in the particle-dimer scattering.I have calculated the elastic particle-dimer scattering lengths in arbitrary angular-momentum channels, and shown that the particle-dimer resonances indeed occur.From the resonance positions, I have found accurate values of the critical mass ratios at which the Kartavtsev-Malykh trimers in the higher angular-momentum channels appear. (ii) Universal three-body parameter (Chapter 4).In the Efimov physics, the three-body parameter fixes the short-range three-body phase and hence the scale of the energy spectrum.Until quite recently, it has been widely held that the threebody parameter should depend on the short-range part of the inter-atomic potential, and hence vary almost randomly between different atomic species and hyperfine states.However, mounting evidence in recent experiments suggests otherwise.While some theoretical studies have succeeded in reproducing the universal behavior in the three-body parameter, the underlying physical mechanism has remained unclear and controversial.In this thesis, I elucidate the physical origin of the universal three-body parameter.I propose that a non-adiabatic deformation of the three-body wave function induced by a universal two-body correlation results in a universal three-body repulsion, which prevents three particles from coming close and renders the three-body parameter universal.This mechanism is verified by reproducing the universal three-body repulsion with a simple model wave function.The mechanism found here suggests that the three-body parameter would also be universal for non-van der Waals types of potentials.For various classes of deep two-body potentials, it is shown that the effective range is the relevant length scale to characterize the two-body correlation and sets the three-body parameter.Depending on the shape of the two-body correlation, there