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arXiv:0706.3774 · Physical Review Letters

Superfluid Equation of State of Dilute Composite Bosons

X. Leyronas, R. Combescot

Published Oct 23, 2007• 31 citations
arXivHTMLPDF

Abstract

We present an exact theory of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime, which treats explicitly dimers as made of two fermions. We apply our framework, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n(3/2), the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a(M), the composite nature of the dimers appearing only in the next order term proportional to n(2).

DOI
10.1103/physrevlett.99.170402

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We present an exact theory of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime, which treats explicitly dimers as made of two fermions. We apply our framework, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n(3/2), the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a(M), the composite nature of the dimers appearing only in the next order term proportional to n(2).
We show how the 3 and 4-body problems emerge in the BEC limit of the BEC-BCS crossover, where we treat explicitly dimers as made of two fermions. We give the argument leading, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n 3/2, the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a M , the composite nature of the dimers appearing only in the next order term proportional to n 2.
We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density nnn. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aM{a}_{M}aM​. We go next in the expansion to the Lee-Huang-Yang order, proportional to n3∕2{n}^{3∕2}n3∕2. We find the far less obvious result that it retains also the same expression in terms of aM{a}_{M}aM​ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n2{n}^{2}n2.
We present a theoretical study of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime in the case of an unequal number of fermions of two species. We take full account of the composite nature of the dimers made of fermions. In the limit of low densities, we calculate the ground-state energy of the system, or equivalently the chemical potentials of each species, as well as the one-particle gap and the energy of an ``impurity'' immersed in a Fermi sea. For the chemical potentials we go up to order (density)4/3(\mathrm{density}){}^{4/3}(density)4/3. The results found involve the exact atom-dimer aAD{a}_{\mathrm{AD}}aAD​ and dimer-dimer aDD{a}_{\mathrm{DD}}aDD​ scattering lengths and therefore include the three- and four-body problems in the many-body problem. We briefly comment on the importance of the different mean-field corrections for recent experiments.
We study the zero-temperature grand potential of a three-dimensional superfluid made of ultracold fermionic alkali-metal atoms in the BCS-BEC crossover. In particular, we analyze the zero-point energy of both fermionic single-particle excitations and bosonic collective excitations. The bosonic elementary excitations, which are crucial to obtain a reliable equation of state in the Bose-Einstein condensate regime, are obtained with a low-momentum expansion up to the forth order of the quadratic (Gaussian) action of the fluctuating pairing field. By performing a cutoff regularization and renormalization of Gaussian fluctuations, we find that the scattering length aB{a}_{B}aB​ of composite bosons, bound states of fermionic pairs, is given by aB=(2/3)aF{a}_{B}=(2/3)\phantom{\rule{4pt}{0ex}}{a}_{F}aB​=(2/3)aF​, where aF{a}_{F}aF​ is the scattering length of fermions.
We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the two-dimensional mean-field theory indicates that the quantum fluctuations are much more pronounced than those in three dimensions. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of two-dimensional Bose gases and determine the ratio of the composite boson scattering length aB{a}_{B}aB​ to the fermion scattering length a2D{a}_{2\mathrm{D}}a2D​. We find ${a}_{B}\ensuremath{\simeq}0.56{a}_{2\mathrm{D}}$, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.
We study Gaussian fluctuations of the zero-temperature attractive Fermi gas in the two-dimensional (2D) BCS-BEC crossover showing that they are crucial to get a reliable equation of state in the Bose-Einstein condensation (BEC) regime of composite bosons, bound states of fermionic pairs. A low-momentum expansion up to the fourth order of the quadratic action of the fluctuating pairing field gives an ultraviolent divergent contribution of the Gaussian fluctuations to the grand potential. Performing dimensional regularization we evaluate the effective coupling constant in the beyond-mean-field grand potential. Remarkably, in the BEC regime our grand potential gives exactly the Popov's equation of state of 2D interacting bosons, and allows us to identify the scattering length aB{a}_{B}aB​ of the interaction between composite bosons as aB=aF/(21/2e1/4)=0.551...aF{a}_{B}={a}_{F}/({2}^{1/2}{e}^{1/4})=0.551...{a}_{F}aB​=aF​/(21/2e1/4)=0.551...aF​, with aF{a}_{F}aF​ is the scattering length of fermions. Remarkably, the value from our analytical relationship between the two scattering lengths is in full agreement with that obtained by recent Monte Carlo calculations.