Contact Intensity and Extended Hydrodynamics in the BCS-BEC Crossover
In the first part of this chapter we analyze the contact intensity C, which has been introduced by Tan (Ann Phys 323:2952 (2008)) and appears in several physical observables of the strongly correlated two-component Fermi gas. We calculate the contact C in
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Abstract In the first part of this chapter we analyze the contact intensity C , which has been introduced by Tan (Ann Phys 323:2952 (2008)) and appears in several physical observables of the strongly correlated two-component Fermi gas. We calculate the contact C in the full BCS-BEC crossover for a uniform superfluid Fermi gas by using an efficient parametetrization of the ground-state energy. In the case of harmonic confinement, within the Thomas-Fermi approximation, we derive analytical formulas of C in the three relevant limits of the crossover. In the second part of this chapter we discuss the extended superfluid hydrodynamics we have recently proposed to describe static and dynamical collective properties of the Fermi gas in the BCS-BEC crossover. In particular we show the relation with the effective theory for the Goldstone field derived by Son and Wingate (Ann Phys 321, 197 (2006)) on the basis of conformal invariance. By using our equations of extended hydrodynamics we determine nonlinear sound waves, static response function and structure factor of a generic superfluid at zero temperature.
1 Contact Intensity It has been proved by Tan [1] that the momentum distribution .k/ in an arbitrary system consisting of fermions in two spin states ( D"; #) with a large scattering length has a tail that falls off as .k/
C k4
(1)
L. Salasnich () Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, Via Marzolo 8, 35122 Padova, Italy e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__6, © Springer International Publishing Switzerland 2014
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for k ! 1, where C is the so-called contact intensity [1]. Here large s-wave scattering length a means that jaj r0 , where r0 is the effective interaction radius. Under this condition Tan [1] has shown that C is related to the total energy E of the Fermi system by the rigorous expression C D
4 ma2 dE ; „2 da
(2)
where the derivative is taken under constant entropy and, in general, C depends on the number N of fermions, the scattering length a and the parameters of the trapping potential [2,3]. Remarkably, Eqs. (1) and (2) work also at finite temperature and in this case C will be a function of T [2, 4]. Tan has also derived, for finite scattering lengths, a generalized virial theorem and a generalized pressure relation where the contact C appears [3]. The contact intensity C appears also in other physical observables of the strongly correlated Fermi system. For instance, the radio-frequency spectroscopy shift is proportional to C [5–8], and the same happens to the photoassociation rate [9]. Very recently, it has been shown that the contact C gives the asymptotic tail behavior of the shear viscosity as a function of the frequency [10]. Using the methods of quantum field theory, Braaten and Platter have rederived [11, 12] the Tan’s universal relations [1–3]. In addition, they
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