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arXiv:2206.13185 · Physics of Particles and Nuclei

Conserving Approximations to Dilute Equilibrium Systems. Pair Interaction Potential

Evgeni E. Kolomeitsev, P. D. Lukianov, Dmitry N. Voskresensky

Published Oct 1, 2024• 1 citation
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Abstract

Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems we study a system of non-relativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit $$n{{\lambda }^{3}} \ll 1$$ , where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon-nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional including the tadpole and sandwich diagrams. The classical high-temperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order $${{\hbar }^{3}}$$ ), and purely quantum (Beth-Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, $$T \lesssim 20$$ MeV. Then, we study the problem of the anomalously large empirical value of the nucleon-nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon-nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low temperature dependence as one obtained within the Φ derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ derivable and virial approaches.

DOI
10.1134/s1063779624700928

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Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems we study a system of non-relativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit nλ3≪1n{{\lambda }^{3}} \ll 1nλ3≪1 , where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon-nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional including the tadpole and sandwich diagrams. The classical high-temperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order ℏ3{{\hbar }^{3}}ℏ3 ), and purely quantum (Beth-Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, T≲20T \lesssim 20T≲20 MeV. Then, we study the problem of the anomalously large empirical value of the nucleon-nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon-nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low temperature dependence as one obtained within the Φ derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ derivable and virial approaches.
The quantum theory of the third virial coefficient CCC 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 CBE{C}_{\mathrm{BE}}CBE​ 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 He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB 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 BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
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The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth order of Fermi gases in one, two, and three dimensions, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes $\mathrm{\ensuremath{\Delta}}{b}_{3}, \mathrm{\ensuremath{\Delta}}{b}_{4}$, and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{2}$, the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for $\mathrm{\ensuremath{\Delta}}{b}_{3}$ with previous calculations in all dimensions and we formulate predictions for $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ in one and two dimensions. We also provide, for a range of couplings, the subspace contributions $\mathrm{\ensuremath{\Delta}}{b}_{31}, \mathrm{\ensuremath{\Delta}}{b}_{22}, \mathrm{\ensuremath{\Delta}}{b}_{41}$, and $\mathrm{\ensuremath{\Delta}}{b}_{32}$, which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Pad\'e and Pad\'e-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.
We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range R∗{R}^{*}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 aaa. 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{R}^{*}=0R∗=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.
Thermodynamic properties of a Fermi system close to the unitarity limit, where the 2-body scattering length a approaches +/-infinity, are studied in the high temperature Boltzmann regime. For dilute systems the virial expansion coefficients in the Boltzmann regime are expected, from general arguments, to be universal. A model independent finite temperature T calculation of the third virial coefficient b3(T) is presented. At the unitarity limit, b3infinity approximately 1.11 is a universal number. The energy density up to the third virial expansion is derived. These calculations are of interest in dilute neutron matter and could be tested in current atomic experiments on dilute Fermi gases near the Feshbach resonance.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.