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arXiv:1705.09511 · Physical Review A

Thermodynamic equivalence of two-dimensional imperfect attractive Fermi and repulsive Bose gases

Marek Napiórkowski, J. Piasecki

Published Jun 29, 2017• 9 citations
arXivHTMLPDF

Abstract

We consider two-dimensional imperfect attractive Fermi and repulsive Bose gases consisting of spinless point particles whose total interparticle interaction energy is represented by $a{N}^{2}/2V$ with $a=\ensuremath{-}{a}_{F}\ensuremath{\le}0$ for fermions and $a={a}_{B}\ensuremath{\ge}0$ for bosons. We show that, in spite of the attraction, the thermodynamics of a $d=2$ imperfect Fermi gas remains well defined for $0\ensuremath{\le}{a}_{F}\ensuremath{\le}{a}_{0}={h}^{2}/2\ensuremath{\pi}m$, and is exactly the same as the one of the repulsive imperfect Bose gas with ${a}_{B}={a}_{0}\ensuremath{-}{a}_{F}$. In particular, for ${a}_{F}={a}_{0}$ one observes the thermodynamic equivalence of the attractive imperfect Fermi gas and the ideal Bose gas.

DOI
10.1103/physreva.95.063627

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A uniform ground-state three-dimensional Fermi gas with short-range repulsive pairwise interaction is under consideration. Its kinetic and interaction energies are calculated up to the second order of the expansion in the gas parameter. Similar to recent results for an interacting Bose gas, the quantities in question are found to depend on the pairwise interaction through two characteristic lengths: the former, aaa, is the sss-wave scattering length, and the latter, bbb, is related to aaa by $b=a\ensuremath{-}m(\ensuremath{\partial}a∕\ensuremath{\partial}m)$, where mmm stands for the fermion mass. To control the results, we proceed in two independent ways. The first involves the Hellmann-Feynman theorem applied to derive the kinetic and interaction energies from the total-energy expansion in the gas parameter first found by Huang and Yang. The second way operates with in-medium pair wave functions and allows one to calculate the quantities of interest ``from scratch.'' The results of the present investigation, taken together with those of the recent consideration of a dilute Bose gas, make it possible to conclude that the pairwise interaction in a quantum gas has an essential and nontrivial effect on the kinetic energy, which is not the case for a classical many-particle system.