arXiv:cond-mat/0206527 · Physical review. B, Condensed matter

Dilute resonating gases and the third virial coefficient

Paulo F. Bedaque, Gautam Rupak

Published May 19, 2003• 28 citations

Abstract

We study dilute gases with short-range interactions and large two-body scattering lengths. At temperatures between the condensation temperature and the scale set by the range of the potential there is a high degree of universality. The first two terms in the expansion of thermodynamic functions in powers of the fugacity z, which measures the diluteness of the system, are determined by the scattering length only. The term proportional to ${z}^{3}$ depends only on one new parameter describing the three-body physics. We compute the third term of the expansion and show that, for many values of this new parameter, the ${z}^{3}$ term may be the dominant one.

DOI: 10.1103/physrevb.67.174513

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arXiv:cond-mat/0206527 · Physical review. B, Condensed matter

Dilute resonating gases and the third virial coefficient

Paulo F. Bedaque, Gautam Rupak

Published May 19, 2003• 28 citations

arXiv HTML PDF

Abstract

{z}^{3}

depends only on one new parameter describing the three-body physics. We compute the third term of the expansion and show that, for many values of this new parameter, the

{z}^{3}

term may be the dominant one.

DOI: 10.1103/physrevb.67.174513

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The equation of state of dilute Bose gases, in which the energy only depends on the s -wave scattering length, is rather unknown beyond the universal limit. We have carried out a set of diffusion Monte Carlo calculations up to gas parameters of 10 − 2 to explore how the departure from the universality emerges. Using different model potentials, we calculate the energies of the gas in an exact way, within some statistical noise, and report the results as a function of the three relevant scattering parameters: the s -wave scattering length a 0 , the s -wave effective range r 0 , and the p -wave scattering length a 1 . If the effective range is not large, we observe universality in terms of a 0 and r 0 up to gas parameters of 10 − 2 . If r 0 grows, the regime of universality in these two parameters is reduced and effects of a 1 start to be observed. In the ( a 0 , r 0 ) universal regime we propose an analytical law that reproduces fairly well the exact energies. Published by the American Physical Society 2024The equation of state of dilute Bose gases, in which the energy only depends on the s -wave scattering length, is rather unknown beyond the universal limit. We have carried out a set of diffusion Monte Carlo calculations up to gas parameters of 10 − 2 to explore how the departure from the universality emerges. Using different model potentials, we calculate the energies of the gas in an exact way, within some statistical noise, and report the results as a function of the three relevant scattering parameters: the s -wave scattering length a 0 , the s -wave effective range r 0 , and the p -wave scattering length a 1 . If the effective range is not large, we observe universality in terms of a 0 and r 0 up to gas parameters of 10 − 2 . If r 0 grows, the regime of universality in these two parameters is reduced and effects of a 1 start to be observed. In the ( a 0 , r 0 ) universal regime we propose an analytical law that reproduces fairly well the exact energies. Published by the American Physical Society 2024

arXiv:2407.18059Physical Review ResearchDec 30, 20241 citation

{z}^{3}

depends only on one new parameter describing the three-body physics. We compute the third term of the expansion and show that, for many values of this new parameter, the

{z}^{3}

term may be the dominant one.

The quantum theory of the third virial coefficient

C

is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of

{C}_{\mathrm{BE}}

are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for

{\mathrm{He}}^{4}

and

{\mathrm{He}}^{3}

, by means of a comparison of the corresponding expansion for the second virial coefficient

B

with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to

B

. Then the leading term of

C

is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for

C

is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.

{R}^{*}

in the interaction. In the limit where the thermal wavelength $\ensuremath{\lambda}\ensuremath{\ll}{R}^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths

a

. By interpolating from the perturbative limits $\ensuremath{\lambda}/|a|\ensuremath{\gg}1$ and ${R}^{*}/\ensuremath{\lambda}\ensuremath{\gg}1$, we estimate the value of the fourth virial coefficient at unitarity for

{R}^{*}=0

and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.