arXiv:0809.1892 · Physical Review A

Universal properties of the ultracold Fermi gas

Shizhong Zhang, Anthony J. Leggett

Published Feb 2, 2009• 184 citations

Abstract

We present some general considerations on the properties of a two-component ultracold Fermi gas along the BEC-BCS crossover. It is shown that the interaction energy and the free energy can be written in terms of a single dimensionless function $h(\ensuremath{\xi},\ensuremath{\tau})$, where $\ensuremath{\xi}=\ensuremath{-}{({k}_{F}{a}_{s})}^{\ensuremath{-}1}$ and $\ensuremath{\tau}=T∕{T}_{F}$. The function $h(\ensuremath{\xi},\ensuremath{\tau})$ incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the average rf-spectroscopy shift $\overline{\ensuremath{\delta}\ensuremath{\omega}}(\ensuremath{\xi},\ensuremath{\tau})$ and the molecular fraction ${f}_{c}(\ensuremath{\xi},\ensuremath{\tau})$ in the closed channel can be expressed in terms of $h(\ensuremath{\xi},\ensuremath{\tau})$ and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.

DOI: 10.1103/physreva.79.023601

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arXiv:0809.1892 · Physical Review A

Universal properties of the ultracold Fermi gas

Shizhong Zhang, Anthony J. Leggett

Published Feb 2, 2009• 184 citations

arXiv HTML PDF

Abstract

h(\ensuremath{\xi},\ensuremath{\tau})

, where

\ensuremath{\xi}=\ensuremath{-}{({k}_{F}{a}_{s})}^{\ensuremath{-}1}

and

\ensuremath{\tau}=T∕{T}_{F}

. The function

h(\ensuremath{\xi},\ensuremath{\tau})

incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the average rf-spectroscopy shift

\overline{\ensuremath{\delta}\ensuremath{\omega}}(\ensuremath{\xi},\ensuremath{\tau})

and the molecular fraction

{f}_{c}(\ensuremath{\xi},\ensuremath{\tau})

in the closed channel can be expressed in terms of

h(\ensuremath{\xi},\ensuremath{\tau})

and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.

DOI: 10.1103/physreva.79.023601

Related papers

Strongly interacting Fermi gases, radio frequency spectroscopy and universality

Shizhong Zhang

In Chapter 5, we consider the BEC-BCS crossover problem in the dilute atomic gases frm a general point of view. We show that as a result of the specific properties of the ultra-cold Fermi gases, there exists a universal function which incorporates all the many-body information of the system. Although we do not yet know how to compute the function analytically, we show that many physical quantities can be expressed in terms of this function. It gives an intuitive way of understanding of how universality arises in the cold atomic gases.In Chapter 5, we consider the BEC-BCS crossover problem in the dilute atomic gases frm a general point of view. We show that as a result of the specific properties of the ultra-cold Fermi gases, there exists a universal function which incorporates all the many-body information of the system. Although we do not yet know how to compute the function analytically, we show that many physical quantities can be expressed in terms of this function. It gives an intuitive way of understanding of how universality arises in the cold atomic gases.

Jan 1, 20091 citation

Non-universal BCS-BEC crossover in resonantly interacting Fermi gases

Lars Melwyn Jensen

We investigate the non-universal behavior of the BCS-BEC crossover model at the normal to superfluid transition. By using a finite temperature quantum field theoretical approach due to Nozieres and Schmitt-Rink and by making the effective range expansion of the effective two-body interaction we numerically calculate the crossover transition temperature as a function of the scattering length, $a_{F},$ and the effective range parameter, $r_{\mathrm{e}}.$ In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of $-r_{\mathrm{e}}$ we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of $T_{c}$ at fixed coupling for increasing $-r_{\mathrm{e}}$ in the crossover region.We investigate the non-universal behavior of the BCS-BEC crossover model at the normal to superfluid transition. By using a finite temperature quantum field theoretical approach due to Nozieres and Schmitt-Rink and by making the effective range expansion of the effective two-body interaction we numerically calculate the crossover transition temperature as a function of the scattering length, $a_{F},$ and the effective range parameter, $r_{\mathrm{e}}.$ In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of $-r_{\mathrm{e}}$ we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of $T_{c}$ at fixed coupling for increasing $-r_{\mathrm{e}}$ in the crossover region.

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We review the functional renormalization group (RG) approach to the BCS-BEC crossover for an ultracold gas of fermionic atoms. Formulated in terms of a scale-dependent effective action, the functional RG interpolates continuously between the atomic or molecular microphysics and the macroscopic physics on large length scales. We concentrate on the discussion of the phase diagram as a function of the scattering length and the temperature which is a paradigm example for the non-perturbative power of the functional RG. A systematic derivative expansion provides for both a description of the many-body physics and its expected universal features as well as an accurate account of the few-body physics and the associated BEC and BCS limits.We review the functional renormalization group (RG) approach to the BCS-BEC crossover for an ultracold gas of fermionic atoms. Formulated in terms of a scale-dependent effective action, the functional RG interpolates continuously between the atomic or molecular microphysics and the macroscopic physics on large length scales. We concentrate on the discussion of the phase diagram as a function of the scattering length and the temperature which is a paradigm example for the non-perturbative power of the functional RG. A systematic derivative expansion provides for both a description of the many-body physics and its expected universal features as well as an accurate account of the few-body physics and the associated BEC and BCS limits.

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arXiv:cond-mat/0502534Physical Review AMar 17, 200651 citations

h(\ensuremath{\xi},\ensuremath{\tau})

, where

\ensuremath{\xi}=\ensuremath{-}{({k}_{F}{a}_{s})}^{\ensuremath{-}1}

and

\ensuremath{\tau}=T∕{T}_{F}

. The function

h(\ensuremath{\xi},\ensuremath{\tau})

incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the average rf-spectroscopy shift

\overline{\ensuremath{\delta}\ensuremath{\omega}}(\ensuremath{\xi},\ensuremath{\tau})

and the molecular fraction

{f}_{c}(\ensuremath{\xi},\ensuremath{\tau})

in the closed channel can be expressed in terms of

h(\ensuremath{\xi},\ensuremath{\tau})

and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.

a_{F},

and the effective range parameter,

r_{\mathrm{e}}.

In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of

-r_{\mathrm{e}}

we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of

T_{c}

at fixed coupling for increasing

-r_{\mathrm{e}}

in the crossover region.

^{6}\mathrm{Li}

and qualitative agreement with the condensate fraction in

^{6}\mathrm{Li}

and

^{40}\mathrm{K}

. In addition to the phase diagram and condensate fraction we compute the correlation length for molecules, the in-medium scattering length for molecules and atoms, and the sound velocity.