arXiv:cond-mat/0404313 · Physical Review A

Four-boson system with short-range interactions

Lucas Platter, H.‐W. Hammer, Ulf-G. Meißner

Published Nov 5, 2004• 201 citations

Abstract

We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to $^{4}\mathrm{H}\mathrm{e}$ atoms and calculate the binding energy of the $^{4}\mathrm{H}\mathrm{e}$ tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to $^{4}\mathrm{H}\mathrm{e}$ atoms, the correlation is approximately linear.

DOI: 10.1103/physreva.70.052101

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arXiv:cond-mat/0404313 · Physical Review A

Four-boson system with short-range interactions

Lucas Platter, H.‐W. Hammer, Ulf-G. Meißner

Published Nov 5, 2004• 201 citations

arXiv HTML PDF

Abstract

^{4}\mathrm{H}\mathrm{e}

atoms and calculate the binding energy of the

^{4}\mathrm{H}\mathrm{e}

tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to

^{4}\mathrm{H}\mathrm{e}

atoms, the correlation is approximately linear.

DOI: 10.1103/physreva.70.052101

Related papers

An effective theory for the four-body system

Lucas Platter

arXiv:cond-mat/0410016AIP conference proceedingsJan 1, 20050 citations

Three- and four-boson systems expanded around the unitarity limit: Application to $^4$ He moleculesThree- and four-boson systems expanded around the unitarity limit: Application to $^4$He molecules

Feng Wu, Xincheng Lin, U. van Kolck +1 more

The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of $^4$ He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of $^4$ He atomic clusters is governed by only small deviations from discrete scale invariance.The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of $^4$He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of $^4$He atomic clusters is governed by only small deviations from discrete scale invariance.

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Xincheng Lin

Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold $^{4}\mathrm{He}$ atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy ${B}_{3}^{(0)}$ . The binding energy and decay width for the associated tetramer ground state are ${E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, ${E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold $^{4}\mathrm{He}$ atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy ${B}_{3}^{(0)}$. The binding energy and decay width for the associated tetramer ground state are ${E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, ${E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.

arXiv:2304.06172Physical Review CFeb 20, 20243 citations

^{4}\mathrm{H}\mathrm{e}

atoms and calculate the binding energy of the

^{4}\mathrm{H}\mathrm{e}

tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to

^{4}\mathrm{H}\mathrm{e}

atoms, the correlation is approximately linear.

Three- and four-boson systems expanded around the unitarity limit: Application to

^4

He molecules

^4

He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of

^4

He atomic clusters is governed by only small deviations from discrete scale invariance.

^{4}\mathrm{He}

atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.

^{4}\mathrm{He}

atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy

{B}_{3}^{(0)}

. The binding energy and decay width for the associated tetramer ground state are

{E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}

and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state,

{E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}

and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.