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Communications in Theoretical Physics

Padé approximant approach to singular properties of quantum gases: the ideal cases

Yuan-Hong Tian, Wen-Du Li, Yao Shen, Wu-Sheng Dai

Published Apr 3, 2021• 4 citations
Publisher

Abstract

Abstract In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padé approximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.

DOI
10.1088/1572-9494/abf4b6

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Abstract In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padé approximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.
We study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length, and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the local density approximation (LDA), analytical results are obtained, and the special case of the isotropic trap is discussed in detail.
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.
ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~
By developing a quantum virial expansion theory, we quantitatively calculate the dynamic density response function of a trapped strongly interacting Fermi gas at high temperatures near unitarity. A clear transition from atomic to molecular responses is identified in the spectra when crossing from the BCS to BEC regimes, in qualitative agreement with recent Bragg spectroscopy observations. Our virial expansion method provides a promising way to solve the challenging strong-coupling problems and is applicable to other dynamical properties of strongly correlated Fermi gases.
Abstract Knowledge of the chemical potential is essential in application of the Fermi–Dirac and the Bose–Einstein distribution functions for the calculation of properties of quantum gases. We give expressions for the chemical potential of ideal Fermi and Bose gases in 1, 2 and 3 dimensions in terms of inverse polylogarithm functions. We provide Mathematica functions for these chemical potentials together with low- and high-temperature series expansions. In the 3d Bose case we give also expansions about TBT_{{{{\mathrm {B}}}}}TB​ T B . The Mathematica routines for the series allow calculation to arbitrary order.
The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality ddd. In particular, the convergence radius ρc(d)ρ_c(d)ρc​(d) of the expansion is particulary large at {\em exactly\/} d=3d=3d=3 dimensions, ρc(3)=7.1068…×lim⁡d→3ρc(d)ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)ρc​(3)=7.1068…×limd→3​ρc​(d). The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.