Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/0209025 · Physical Review A

Scaling limit of virtual states of triatomic systems

Marcelo Takeshi Yamashita, T. Frederico, A. Delfino, Lauro Tomio

Published Nov 8, 2002• 53 citations
arXivHTMLPDF

Abstract

For a system with three identical atoms, the dependence of the s-wave virtual state energy on the weakly bound dimer and trimer binding energies is calculated in the form of a universal scaling function. The scaling function is obtained from a renormalizable three-body model with a pairwise Dirac-$\ensuremath{\delta}$ interaction. The threshold condition for the appearance of the trimer virtual state was also discussed.

DOI
10.1103/physreva.66.052702

Related papers

Scaling predictions for radii of weakly bound triatomic molecules

Marcelo Takeshi Yamashita, R.S. Marques de Carvalho, Lauro Tomio +1 more

The mean-square radii of the molecules ${}^{4}{\mathrm{He}}_{3},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{6}\mathrm{Li},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{7}\mathrm{Li},$ and ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{23}\mathrm{Na}$ are calculated using a three-body model with contact interactions. They are obtained from a universal scaling function calculated within a renormalized scheme for three particles interacting through pairwise Dirac-$\ensuremath{\delta}$ interaction. The root-mean-square distance between two atoms of mass ${m}_{A}$ in a triatomic molecule are estimated to be of the order of $\mathcal{C}\sqrt{{\ensuremath{\Elzxh}}^{2}/[{m}_{A}{(E}_{3}\ensuremath{-}{E}_{2})]},$ where ${E}_{2}$ is the dimer and ${E}_{3}$ is the trimer binding energies, and $\mathcal{C}$ is a constant (varying from $\ensuremath{\sim}0.6$ to $\ensuremath{\sim}1),$ which depends on the ratio between ${E}_{2}$ and ${E}_{3}.$ Considering previous estimates for the trimer energies, we also predict the sizes of rubidium and sodium trimers in atomic traps.

arXiv:physics/0303031•Physical Review A•Jul 16, 2003•21 citations

Universality and scaling limit of weakly-bound tetramers

M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio +2 more

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.

AIP conference proceedings•Jan 1, 2012•6 citations

Binding and structure of tetramers in the scaling limit

M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio +2 more

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 ${B}_{3}$, 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 $N$th tetramer energy ${B}_{4}^{(N)}$ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer ${B}_{4}^{(N+1)}$ 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 $N$. We show that both channels of the FY decomposition [atom-trimer ($K$ type) and dimer-dimer ($H$ type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the $H$ channel is favored over the $K$ channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.

arXiv:1110.5214•Physical Review A•Feb 8, 2012•33 citations

Scaling limit of weakly bound triatomic states

T. Frederico, Lauro Tomio, A. Delfino +1 more

The usefulness of a scale-independent approach to identify Efimov states in three-body systems is shown by comparing such an approach with a realistic calculation in the case of three helium atoms. We show that the scaling limit is realized in practice in this case, and suggest its application to study other similar systems, including the case where two kinds of atoms are mixed. We also consider the observed large scattering length of $\mathrm{the}{ }^{87}\mathrm{Rb}$ dimer to estimate the critical value of the ground-state energy of the corresponding trimer $(>~1.5 \mathrm{mK})$, in order to allow for one Efimov state above the ground state.

Physical Review A•Jul 1, 1999•71 citations

Triatomic molecules in two-dimensional layers

F. F. Bellotti

We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.

•Jan 1, 2012•0 citations

Delta function model for the helium trimer: origin of three-body forces

S. M. Blinder, LAWRENCE L. JUN. LOHR

The Dirac bubble potential, previously used to model the helium dimer, is applied to the trimer. It is shown that the quantum mechanical kinetic energy operator for a three-body system contains terms over and above the analogues of classical pairwise contributions. The additional terms, the ‘Borromean couplings’, are responsible for a dramatic increase in the binding energy of the trimer. For example, 4He3 is bounded by two orders of magnitude more strongly than 4He2, 279 mK versus 1.31 mK, respectively, according to our calculations. Moreover, the trimer is considerably more compact than the dimer, with (r12) decreasing to 9.01 Å from 51.9 Å.

Molecular Physics•Jan 10, 2001•6 citations
For a system with three identical atoms, the dependence of the s-wave virtual state energy on the weakly bound dimer and trimer binding energies is calculated in the form of a universal scaling function. The scaling function is obtained from a renormalizable three-body model with a pairwise Dirac-\ensuremathδ\ensuremath{\delta}\ensuremathδ interaction. The threshold condition for the appearance of the trimer virtual state was also discussed.
The mean-square radii of the molecules 4He3,{}^{4}{\mathrm{He}}_{3},4He3​, ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{6}\mathrm{Li},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{7}\mathrm{Li},$ and ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{23}\mathrm{Na}$ are calculated using a three-body model with contact interactions. They are obtained from a universal scaling function calculated within a renormalized scheme for three particles interacting through pairwise Dirac-$\ensuremath{\delta}$ interaction. The root-mean-square distance between two atoms of mass mA{m}_{A}mA​ in a triatomic molecule are estimated to be of the order of $\mathcal{C}\sqrt{{\ensuremath{\Elzxh}}^{2}/[{m}_{A}{(E}_{3}\ensuremath{-}{E}_{2})]},$ where E2{E}_{2}E2​ is the dimer and E3{E}_{3}E3​ is the trimer binding energies, and C\mathcal{C}C is a constant (varying from $\ensuremath{\sim}0.6$ to $\ensuremath{\sim}1),$ which depends on the ratio between E2{E}_{2}E2​ and E3.{E}_{3}.E3​. Considering previous estimates for the trimer energies, we also predict the sizes of rubidium and sodium trimers in atomic traps.
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.
The usefulness of a scale-independent approach to identify Efimov states in three-body systems is shown by comparing such an approach with a realistic calculation in the case of three helium atoms. We show that the scaling limit is realized in practice in this case, and suggest its application to study other similar systems, including the case where two kinds of atoms are mixed. We also consider the observed large scattering length of the87Rb\mathrm{the}{ }^{87}\mathrm{Rb}the87Rb dimer to estimate the critical value of the ground-state energy of the corresponding trimer (> 1.5mK)(>~1.5 \mathrm{mK})(> 1.5mK), in order to allow for one Efimov state above the ground state.
We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.
The Dirac bubble potential, previously used to model the helium dimer, is applied to the trimer. It is shown that the quantum mechanical kinetic energy operator for a three-body system contains terms over and above the analogues of classical pairwise contributions. The additional terms, the ‘Borromean couplings’, are responsible for a dramatic increase in the binding energy of the trimer. For example, 4He3 is bounded by two orders of magnitude more strongly than 4He2, 279 mK versus 1.31 mK, respectively, according to our calculations. Moreover, the trimer is considerably more compact than the dimer, with (r12) decreasing to 9.01 Å from 51.9 Å.