Physical Review A

Efimov states in He4 trimers by two-body effective-range and scattering-length analysis: A comparison with Faddeev calculations

Stephen Huber

Published Jun 1, 1985• 27 citations

Abstract

A comparison is made between the number of Efimov States present in $^{4}\mathrm{He}$ trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/${r}_{0}$) using realistic phenomenological He-He potentials. It is found that there is good agreement for systems supporting few states and that agreement remains when the Efimov states are forced to disappear by mathematically altering the potentials.

DOI: 10.1103/physreva.31.3981

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Universality in Efimov-associated tetramers in He4

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Boson-Faddeev in the Unitary Limit and Efimov States

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Physical Review A

Efimov states in He4 trimers by two-body effective-range and scattering-length analysis: A comparison with Faddeev calculations

Stephen Huber

Published Jun 1, 1985• 27 citations

Publisher

Abstract

A comparison is made between the number of Efimov States present in

^{4}\mathrm{He}

trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/

{r}_{0}

) using realistic phenomenological He-He potentials. It is found that there is good agreement for systems supporting few states and that agreement remains when the Efimov states are forced to disappear by mathematically altering the potentials.

DOI: 10.1103/physreva.31.3981

Related papers

A study of the Efimov states and binding energies of the helium trimer through the Faddeev-coordinate–momentum approach

H. S. Huber, T. K. Lim

The Journal of Chemical PhysicsFeb 1, 197854 citations

Zero-energy scattering and bound states in the He4 trimer and tetramer

S. Nakaichi-Maeda, T. K. Lim

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$ -wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$ -wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$-wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

Physical Review AAug 1, 198346 citations

Analytic representation of the Efimov effect in the helium trimer

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

Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$ , are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$ -wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.

Physical Review AJun 2, 20045 citations

Efimov State in the He4 Trimer

T. K. Lim, Sister Kathleen Duffy, William C. Damer

On the basis of a Faddeev calculation, an Efimov state is predicted to exist in $^{4}\mathrm{He}_{3}$ . This discovery represents the first manifestation of the Efimov effect and may have farreaching consequences in the statistical mechanics of $^{4}\mathrm{He}$ gas at low temperatures.On the basis of a Faddeev calculation, an Efimov state is predicted to exist in $^{4}\mathrm{He}_{3}$. This discovery represents the first manifestation of the Efimov effect and may have farreaching consequences in the statistical mechanics of $^{4}\mathrm{He}$ gas at low temperatures.

Physical Review LettersFeb 14, 1977104 citations

Universality in Efimov-associated tetramers in He4

Emiko Hiyama, M. Kamimura

We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the $^{4}\mathrm{He}$ trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the $^{4}\mathrm{He}$ trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length ${r}_{\mathrm{vdW}}$ , and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the $^{4}\mathrm{He}$ trimer with the realistic potentials.We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the $^{4}\mathrm{He}$ trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the $^{4}\mathrm{He}$ trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length ${r}_{\mathrm{vdW}}$, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the $^{4}\mathrm{He}$ trimer with the realistic potentials.

arXiv:1409.2501Physical Review ANov 18, 201430 citations

Boson-Faddeev in the Unitary Limit and Efimov States

H. S. K"øhler

A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges $r_0 > 0$ and rank-1 potentials the total energy $E_T$ is found to converge with momentum cut-off $Λ$ for $Λ> \sim 10/r_0$ . In the Unitary limit ( $1/a=r_0= 0$ ) the energy does however diverge. It is shown (analytically) that in this case $E_T=E_uΛ^2$ . Calculations give $E_u=-0.108$ for the ground state and $E_u=-1.\times10^{-4}$ for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges $r_0 > 0$ and rank-1 potentials the total energy $E_T$ is found to converge with momentum cut-off $Λ$ for $Λ> \sim 10/r_0$ . In the Unitary limit ($1/a=r_0= 0$) the energy does however diverge. It is shown (analytically) that in this case $E_T=E_uΛ^2$. Calculations give $E_u=-0.108$ for the ground state and $E_u=-1.\times10^{-4}$ for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.

arXiv:1012.2926Dec 14, 20100 citations

A comparison is made between the number of Efimov States present in

^{4}\mathrm{He}

trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/

{r}_{0}

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the

^{4}\mathrm{He}

atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the

s

-wave two-body

t

matrix to a separable form through the unitary-pole approximation and did the same for the

s

-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

Exact solutions for the low-temperature helium dimer and trimer,

^{4}\mathrm{He}_{2}

and

^{4}\mathrm{He}_{3}

, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the

S

-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and

{r}_{0}

are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.

On the basis of a Faddeev calculation, an Efimov state is predicted to exist in

^{4}\mathrm{He}_{3}

. This discovery represents the first manifestation of the Efimov effect and may have farreaching consequences in the statistical mechanics of

^{4}\mathrm{He}

gas at low temperatures.

We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the

^{4}\mathrm{He}

trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the

^{4}\mathrm{He}

trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length

{r}_{\mathrm{vdW}}

, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the

^{4}\mathrm{He}

trimer with the realistic potentials.

r_0 > 0

and rank-1 potentials the total energy

E_T

is found to converge with momentum cut-off

Λ

for

Λ> \sim 10/r_0

. In the Unitary limit (

1/a=r_0= 0

) the energy does however diverge. It is shown (analytically) that in this case

E_T=E_uΛ^2

. Calculations give

E_u=-0.108

for the ground state and

E_u=-1.\times10^{-4}

for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.