Effective action for superfluid Fermi systems in the strong-coupling limit
N. Dupuis
We derive the low-energy effective action for three-dimensional superfluid Fermi systems in the strong-coupling limit, where superfluidity originates from Bose-Einstein condensation of composite bosons. Taking into account density and pairing fluctuations on the same footing, we show that the effective action involves only the fermion density $\rho_{r}$ and its conjugate variable, the phase $\theta_{r}$ of the pairing order parameter $\Delta_{r}$. We recover the standard action of a Bose superfluid of density $\rho_{r}/2$, where the bosons have a mass $m_B=2m$ and interact {via} a repulsive contact potential with amplitude $g_B=4\pi a_B/m_B$, $a_B=2a$ ($a$ the s-wave scattering length associated to the fermion-fermion interaction in vacuum). For lattice models, the derivation of the effective action is based on the mapping of the attractive Hubbard model onto the Heisenberg model in a uniform magnetic field, and a coherent state path integral representation of the partition function. The effective description of the Fermi superfluid in the strong-coupling limit is a Bose-Hubbard model with an intersite hopping amplitude $t_B=J/2$ and an on-site repulsive interaction $U_B=2Jz$, where $J=4t^2/U$ ($t$ and $-U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, $z$ the number of nearest-neighbor sites).
cond-mat0412563.tex
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%\def\Deltaqthsd{{{\Delta_{q}^{\rm HS}}^*}} 
%\def\Deltaqtmhsd{{{\Delta_{-q}^{\rm HS}}^*}}
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%\def\betaq{{\beta_{\bf q}}}
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%\def\Xkq{{X_{{\bf k},{\bf q}}}} 
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%\def\sumr{{\sum_{\bf r}}}
%\def\nrup{{n_{{\bf r}\uparrow}}}
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%\def\Omegab{{\bf\Omega}}
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\title{Effective action for superfluid Fermi systems in the strong-coupling
  limit} 
\author{N. Dupuis}
\affiliation{Department of Mathematics, Imperial College, 
180 Queen's Gate, London SW7 2AZ, UK \\ and \\ 
Laboratoire de Physique des Solides, CNRS UMR 8502, \\
  Universit\'e Paris-Sud, 91405 Orsay, France }

\date{December 20, 2004}

%--------------------------------------------------------------------------

\begin{abstract}
We derive the low-energy effective action for three-dimensional superfluid
Fermi systems in the strong-coupling limit, where 
superfluidity originates from Bose-Einstein condensation of composite
bosons. Taking into account density and pairing fluctuations on the same
footing, we show that the effective action involves only the fermion density
$\rho_{\bf r}$ and its conjugate 
variable, the phase $\theta_{\bf r}$ of the pairing order parameter
$\Delta_{\bf r}$. We recover the standard
action of a Bose superfluid of density $\rho_{\bf r}/2$, where the bosons have
a mass 
$m_B=2m$ and interact {\it via} a repulsive contact potential with amplitude
$g_B=4\pi a_B/m_B$,  
$a_B=2a$ ($a$ the s-wave scattering length associated to the
fermion-fermion interaction in vacuum). For lattice models, the derivation of
the effective action is based on the mapping of the attractive Hubbard model
onto the Heisenberg model in a uniform magnetic field, and a coherent state
path integral representation of the partition function. The effective
description of the Fermi superfluid in the strong-coupling limit is a
Bose-Hubbard model with an intersite hopping amplitude $t_B=J/2$ and an
on-site repulsive interaction $U_B=2Jz$, where $J=4t^2/U$ ($t$ and $-U$ are 
the intersite hopping amplitude and the on-site attraction in the (fermionic)
Hubbard model, $z$ the number of nearest-neighbor sites). 
\end{abstract}

\pacs{74.20.Fg, 71.10.Fd, 05.30.Jp}

\maketitle

\section{Introduction}

Recent progress in the experimental control of ultracold atomic Fermi 
\cite{Kinast04,Bartenstein04,Zwierlein04,Regal04,Bourdel04}
gases has revived the interest in the crossover from the weak-coupling BCS
limit of superfluid fermions to the strong-coupling limit of condensed
composite bosons. \cite{Randeria95,BCS-BEC} In this paper, we derive the
low-energy effective action for a superfluid Fermi system in the
strong-coupling limit, both in continuum and lattice models. The latter may
be relevant for high-$T_c$ superconductors or ultracold Fermi gases in an
optical lattice.  

A Bose superfluid is described by a complex field $\psi_{B\bf r}=
\sqrt{\rho_{B\bf r}} e^{i\theta_{B\bf r}}$ where $\rho_{B\bf r}$ is the boson
density at position ${\bf r}$ in space. The equation of motion derived from the
standard action of a Bose system leads to the Gross-Pitaevskii
equation,\cite{Gross61,Pitaevskii61} i.e. a non-linear Schr\"odinger equation
for the $\psi_B$ field. The Gross-Pitaevskii equation yields a simple
description of quantum macroscopic phenomena like the Josephson effect or the
flux quantization,\cite{Feynman72,Ao95} and has proven to be a tool of choice
for the understanding of many phenomena in ultracold atomic Bose
gases. \cite{Pitaevskii03} In Fermi systems, there is in general
no simple relation between the amplitude of the superfluid (pairing) order
parameter $\Delta_{\bf r}$ and the fermion density $\rho_{\bf r}$. This
suggests that 
a minimal description, aiming at making contact with the standard description
of a Bose superfluid, should at least include the superfluid order parameter
$\Delta_{\bf r}$ and the density $\rho_{\bf r}$ from the outset. In the
strong-coupling 
limit, where superfluidity originates from Bose-Einstein condensation (BEC) of
composite bosons, we expect the description in terms of $\rho_{\bf r}$ and
$\Delta_{\bf r}=|\Delta_{\bf r}|e^{i\theta_{\bf r}}$ to be redundant and the
superfluid to be 
described by a single complex field $\psi_{\bf r}=\sqrt{\rho_{\bf r}/2}
e^{i\theta_{\bf r}}$ 
($\rho_{\bf r}/2$ being the density of composite bosons). 

Previous studies of the BCS-BEC crossover in superfluid Fermi systems can be
divided into two categories. In the first type of approach, 
\cite{Randeria95,Drechsler92,Haussmann93,deMelo93,Pistolesi96,Piery2000,Pieri03}
the density $\rho_{\bf r}$ is not 
considered explicitely and a pairing field $\Delta_{\bf r}^{\rm HS}$ is
introduced by means 
of a Hubbard-Stratonovich transformation of the fermion-fermion
interaction. In the BEC limit, the standard action $S[\psi^*,\psi]$ of a Bose
superfluid is recovered if one identifies $\psi_{\bf r}$ to $\Delta_{\bf
  r}^{\rm HS}$ (after 
a proper rescaling). For a continuum model, the bosons have a mass $m_B=2m$ and
interact {\it via} a repulsive contact potential with amplitude $g_B=4\pi
a_B/m_B$, $a_B=2a$ ($a$ is the s-wave scattering length associated to the
fermion-fermion interaction in vacuum). The main (conceptual) difficulty of
this approach 
is that the Hubbard-Stratonovich field $\Delta_{\bf r}^{\rm HS}$ is {\it not}
the physical 
pairing field $\Delta_{\bf r}=|\Delta_{\bf r}|e^{i\theta_{\bf r}}$ but rather
its conjugate 
field. \cite{Depalo99} Although both fields coincide at the mean-field level,
they differ when fluctuations are taken into account. As a result,
${\psi_{\bf r}}\propto\Delta_{\bf r}^{\rm HS}$ does not correspond to
$\sqrt{\rho_{\bf r}/2}e^{i\theta_{\bf r}}$ as expected.  

In the second type of approach to the BCS-BEC crossover,
\cite{Depalo99,Dupuis04,Stone95} the {\it physical} density and pairing
fields, $\rho_{\bf r}$ and $\Delta_{\bf r}$, are introduced from the
outset. For continuum models, only the weak-coupling limit has been
considered. \cite{Depalo99,Stone95}  
For lattice (Hubbard) models in the strong-coupling low-density limit, one
finds that the order parameter amplitude and the density are tied by the
relation $|\Delta_{\bf r}|=\sqrt{\rho_{\bf r}/2}$, so that the low-energy
effective action  
can be written in terms of a single complex field, $\Delta_{\bf r} =
\sqrt{\rho_{\bf r}/2} 
e^{i\theta_{\bf r}} \equiv \psi_{\bf r}$. \cite{Depalo99,Dupuis04} In the
continuum 
limit, one finds that the (composite) bosons have a mass $m_B=1/J$ and interact
{\it via} a repulsive contact potential with amplitude $g_B=8J$ (in two
dimensions), where $J=4t^2/U$ ($t$ being the intersite hopping amplitude and
$-U$ ($U\geq 0$) the on-site attractive interaction). \cite{Dupuis04}

Most of the theoretical works on the BCS-BEC crossover in ultracold atomic
Fermi gases have been formulated within a fermion-boson model,
\cite{fermion-boson} aiming at incorporating the molecular states involved in
the Feshbach resonance which drives the crossover. While the
equivalence of the fermion-boson 
model to an effective single-channel model in the crossover region may be
questionable, \cite{Falco04,Simonucci04} both models are equivalent in the
strong-coupling limit. 

The outline of the paper is as follows. In Sec.~\ref{sec:cm}, we extend the
approach of Ref.~\onlinecite{Depalo99} to the strong-coupling limit of a
continuum model. The particle-particle and particle-hole channels
are considered on the same footing, and the (physical) density
($\rho_{\bf r}$) and pairing ($\Delta_{\bf r}$) fields are 
introduced from the outset. The low-energy effective action is derived by
assuming small fluctuations of the collective fields about their mean-field
values. We find that fluctuations of $\rho_{\bf r}$ and $\Delta_{\bf r}$ are
not independent, 
so that the low-energy action can be written in term of a single complex field
${\psi_{\bf r}}=\sqrt{\rho_{\bf r}/2}e^{i\theta_{\bf r}}$. We recover the
standard action of a Bose 
superfluid with $m_B=2m$ and $g_B=4\pi a_B/m_B$, $a_B=2a$. For a lattice
model (Sec.~\ref{sec:lm}), we follow the approach introduced in
Ref.~\onlinecite{Dupuis04}. We map the attractive Hubbard model onto
the half-filled repulsive Hubbard model in a uniform magnetic field coupled to
the fermion spins. In the strong-coupling limit, the latter reduces to the
Heisenberg model in a uniform field. The low-energy effective action of the
attractive model is finally deduced from the coherent state path integral
representation of the Heisenberg model. In the low-density limit, where the
Pauli principle (which prevents two composite bosons to occupy the same site)
should not matter, $|\Delta_{\bf r}|\simeq \sqrt{\rho_{\bf r}/2}$ and the
superfluid Fermi 
system can be described by the complex field ${\psi_{\bf r}}= \sqrt{\rho_{\bf
    r}/2} 
e^{i\theta_{\bf r}}$. We find that the effective description of the Fermi
superfluid 
is a Bose-Hubbard model with intersite hopping amplitude $t_B=J/2$ and an
on-site repulsive interaction $U_B=2Jz$ (where $z$ is the number of
nearest-neighbor sites). 

\section{Continuum model}
\label{sec:cm} 

We consider a three-dimensional superfluid fermion system with the action
$S=S_0+S_{\rm int}$,  
\begin{eqnarray}
S_0 &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,}  c^\dagger_{\bf r}
\left(\partial_\tau - \mu - 
\frac{{\boldsymbol{\nabla}^2}}{2m} \right)  c_{\bf r} , \nonumber \\ 
S_{\rm int} &=& -g {\int_0^\beta d\tau} {\int d^3{\bf r}\,} c^*_{{\bf
    r}\uparrow}c^*_{{\bf r}\downarrow} 
c_{{\bf r}\downarrow} c_{{\bf r}\uparrow} ,
\label{action1} 
\end{eqnarray} 
where $c^{(*)}_{{\bf r}\sigma}$ are Grassmann variables, $ c_{\bf r}=(c_{{\bf
r}\uparrow},c_{{\bf r}\downarrow})^T$, $\tau$ an imaginary time and
$\beta=1/T$ the inverse temperature. $-g$ is the attractive interaction
between fermions ($g\geq 0$). The chemical potential $\mu$ fixes the average
fermion density $\rho_0$. To suppress ultraviolet divergences appearing in
the perturbation theory, one regularizes\cite{deMelo93} the local
fermion-fermion interaction 
with a cutoff $\Lambda$ acting on the fermion dispersion: $\epsilon_{\bf
  k}=|{\bf k}|^2/2m \leq \Lambda$.  $g$ and $\Lambda$ determine the s-wave
scattering length $a$ defined by the low-energy limit of the two-body problem
in vacuum,  
\begin{equation}
\frac{m}{4\pi a} = - \frac{1}{g} + \int_{\epsilon_{\bf k}\leq \Lambda}
\frac{d^3{\bf k}}{(2\pi)^3} \frac{1}{2\epsilon_{\bf k}} .
\label{scattering}
\end{equation}
$a$ is negative for small $g$ and diverges when $g=2\pi^2/m\Lambda$. For $g
>2\pi^2/m\Lambda$, there is a two-body bound-state (composite boson) with
energy $E_B=-1/ma^2$ and the scattering length $a$ is positive. The latter
also determines the extension of the bound-state. Low-energy
properties depend solely on $a$ (and not $g$ or $\Lambda$); we shall therefore
take the limit $g\to 0$ and $\Lambda\to\infty$ with $a$ fixed. In the
following, we consider the BEC limit defined by ${\rho_0 a^3}\ll 1$ ($a>0$),
where superfluidity originates from BEC of composite bosons. 

The (real) density and (complex) pairing fields,
\begin{eqnarray}
\rho_{\bf r} &=&  c^\dagger_{\bf r}  c_{\bf r} , \nonumber \\
S^z_{\bf r} &=&  c^\dagger_{\bf r} \sigma^z  c_{\bf r} , \nonumber \\ 
\Delta_{\bf r} &=&  c_{{\bf r}\downarrow} c_{{\bf r}\uparrow} , \nonumber \\
\Delta^*_{\bf r} &=& c^*_{{\bf r}\uparrow} c^*_{{\bf r}\downarrow} ,
\label{constraints}
\end{eqnarray}
can be introduced in the action by means of real ($ \rho^{\rm HS}_{\bf
  r},\tilde\rho^{\rm HS}_{\bf r}$) and 
complex ($\Delta_{\bf r}^{\rm HS}$) Lagrange multipliers:
\begin{eqnarray}
S &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl\lbrace  c^\dagger_{\bf
  r} \biggl( \partial_\tau - \mu - 
\frac{{\boldsymbol{\nabla}^2}}{2m} \biggr)  c_{\bf r}  \nonumber \\ &&  
-g\gamma |\Delta_{\bf r}|^2 - \frac{g\alpha}{4}(\rho^2_{\bf r}-{S^z_{\bf r}}^2)
\nonumber \\ && 
+ i  \rho^{\rm HS}_{\bf r}(\rho_{\bf r}- c^\dagger_{\bf r} c_{\bf r})
+ i \tilde\rho^{\rm HS}_{\bf r} (S^z_{\bf r}- c^\dagger_{\bf r}\sigma^z c_{\bf
  r}) \nonumber \\ && 
+ i [\Delta_{\bf r}^{\rm HS}(\Delta^*_{\bf r}-c^*_{{\bf r}\uparrow}c^*_{{\bf
  r}\downarrow})  
+ {\rm c.c.}]  \biggr\rbrace .
\label{action2}
\end{eqnarray}
$(\sigma^x,\sigma^y,\sigma^z)$ denotes the Pauli matrices. 
Integrating over $ \rho^{\rm HS}_{\bf r},\tilde\rho^{\rm HS}_{\bf
  r},\Delta_{\bf r}^{\rm HS}$ and $\rho_{\bf r},S^z_{\bf 
  r},\Delta_{\bf r}$, we recover the original action (\ref{action1}) if we
choose $\alpha+\gamma=1$.\cite{note4} The
relative weights $\alpha$ and $\gamma$ of the particle-hole and
particle-particle channels are arbitrary. All the resulting effective actions
are equivalent when treated exactly. However, to recover the mean-field
results from a saddle-point approximation, we take $\alpha=\gamma=1$. When
only low-energy long-wavelength fluctuations about the mean-field state are
considered, there is no overlapping of the two channels and therefore no
overcounting. \cite{Depalo99} Note that by integrating out the 
physical fields $S^z_{\bf r}$, $\rho_{\bf r}$ and $\Delta_{\bf r}$, one
recovers the 
action $S[c, \rho^{\rm HS}_{\bf r},\tilde\rho^{\rm HS}_{\bf r},\Delta_{\bf
    r}^{\rm HS}]$ which is generally obtained by 
means of a Hubbard-Stratonovich decoupling of the interaction term. Thus the
Lagrange multipliers $ \rho^{\rm HS}_{\bf r}$, $\tilde\rho^{\rm HS}_{\bf r}$
and $\Delta_{\bf r}^{\rm HS}$ enforcing the 
constraints (\ref{constraints}) can also be seen as Hubbard-Stratonovich
fields.\cite{Depalo99}  In the following, we neglect spin fluctuations
($\tilde\rho^{\rm HS}_{\bf r}$ and $S^z_{\bf r}$) since they do not play an
important role when the interaction is attractive. 


\subsection{Mean-field theory}

The mean-field theory is obtained from a saddle-point approximation where
the fields $\rho_{\bf r}$, $\Delta_{\bf r}$, $ \rho^{\rm HS}_{\bf r}$ and
$\Delta_{\bf r}^{\rm HS}$ are taken space and 
time independent. The saddle-point equations read
\begin{eqnarray}
\rho_0 = \langle  c^\dagger_{\bf r}  c_{\bf r} \rangle , && \,\,\,\, 
i\rho_0^{\rm HS} = \frac{g}{2} \rho_0 , \nonumber \\
\Delta_0 = \langle  c_{{\bf r}\downarrow} c_{{\bf r}\uparrow} \rangle , &&
\,\,\,\, 
i\Delta_0^{\rm HS} = g \Delta_0 , \nonumber \\ 
\Delta_0^* = \langle c^*_{{\bf r}\uparrow} c^*_{{\bf r}\downarrow} \rangle ,
&& \,\,\,\, 
i{\Delta_0^{\rm HS}}^* = g \Delta_0^* .
\label{speq}
\end{eqnarray} 
With no loss of generality, we can take $\Delta_0=\Delta_0^*$
real. $i\Delta_0^{\rm HS}=i{\Delta_0^{\rm HS}}^*$ is then real at the saddle
point. It is convenient to redefine 
$i\Delta_0^{\rm HS}\to \Delta_0^{\rm HS}$ and $i{\Delta_0^{\rm HS}}^*\to
{\Delta_0^{\rm HS}}^*$ (so that 
$\Delta_0^{\rm HS}={\Delta_0^{\rm HS}}^*$ is real) and absorb $i\rho_0^{\rm
  HS}$ in the definition of 
the chemical potential. The mean-field action is then (up to an additive
constant)
\begin{eqnarray}
S_{\rm MF} &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[
  c^\dagger_{\bf r} \biggl(\partial_\tau-\mu- 
  \frac{{\boldsymbol{\nabla}^2}}{2m} \biggr) c_{\bf r} \nonumber \\ && 
- \Delta_0^{\rm HS}(c^*_{{\bf r}\uparrow} c^*_{{\bf r}\downarrow}+{\rm c.c.})
  \biggr] . 
\label{actionhf}
\end{eqnarray}
From (\ref{actionhf}), we readily obtain the normal and anomalous Green
functions,
\begin{eqnarray}
G({\bf k},i\omega) &=& - \langle c_\sigma({\bf k},i\omega) c^*_\sigma({\bf
  k},i\omega) 
\rangle = \frac{-i\omega-{\xi_{\bf k}}}{\omega^2+{E^2_{\bf k}}} , \nonumber \\
F({\bf k},i\omega) &=&  - \langle c_\sigma({\bf k},i\omega)
  c_{\bar\sigma}(-{\bf k},-i\omega) 
\rangle = \frac{\Delta_0^{\rm HS}}{\omega^2+{E^2_{\bf k}}} , \nonumber \\
F^*({\bf k},i\omega) &=&  - \langle c^*_{\bar\sigma}(-{\bf k},-i\omega)
c^*_\sigma({\bf k},i\omega) \rangle = F({\bf k},i\omega) ,
\label{hfgf}
\end{eqnarray} 
where ${E_{\bf k}}=(\xi^2_{\bf k}+{\Delta_0^{\rm HS}}^2)^{1/2}$, ${\xi_{\bf
  k}}=\epsilon_{\bf 
  k}-\mu$, and 
$\bar\sigma=-\sigma$. $c_\sigma({\bf k},i\omega)$ is the Fourier transformed
field of $c_{{\bf r}\sigma}$ and $\omega$ a fermionic Matsubara frequency. 
Using (\ref{scattering}) and (\ref{hfgf}), we can rewrite the saddle-point
equations (\ref{speq}) as  
\begin{eqnarray}
\frac{m}{4\pi a} &=& {\int_{\bf k}} \biggl( \frac{1}{2\epsilon_{\bf k}} -
\frac{1}{2{E_{\bf k}}} 
\biggr) ,  \nonumber \\
\rho_0 &=& {\int_{\bf k}} \biggl( 1 - \frac{{\xi_{\bf k}}}{{E_{\bf k}}}
\biggr) , 
\label{speq1} 
\end{eqnarray}
where ${\int_{\bf k}}\equiv\int d^3{\bf k}/(2\pi)^3$. Eqs.~(\ref{speq1})
determine the 
chemical potential $\mu$ and the order parameter $\Delta_0^{\rm
  HS}=g\Delta_0$.  
In the strong-coupling limit ${\rho_0 a^3}\ll 1$, one obtains (see Appendix
\ref{app:hfcorr})  
\begin{eqnarray}
\mu &=& - \frac{1}{2ma^2}(1-2\pi {\rho_0 a^3}) , \nonumber \\
\Delta_0^{\rm HS} &=& \biggl(\frac{4\pi \rho_0}{m^2a}\biggr)^{1/2}
\biggl(1+\frac{\pi}{4}{\rho_0 a^3} \biggr) .  
\label{muhf} 
\end{eqnarray} 


\subsection{Low-energy effective action}
\label{subsec:lea}

In this section, we derive the low-energy effective action for the physical
fields $\rho_{\bf r}$ and $\Delta_{\bf r}$. Since our derivation partially
follows 
Ref.~\onlinecite{Depalo99}, we describe only the main steps (technical details
are given in Appendix \ref{app:lea}).  
The main assumption is that the collective bosonic fields $\rho_{\bf r}$, $
\rho^{\rm HS}_{\bf r}$, 
$\Delta_{\bf r}$ and $\Delta_{\bf r}^{\rm HS}$ weakly fluctuate about their
mean-field values.  

Starting from the action (\ref{action2}) (with $\alpha=\gamma=1$), where 
\begin{equation}
\Delta_{\bf r} = |\Delta_{\bf r}| e^{i\theta_{\bf r}} ,
\end{equation}
we perform the change of variables 
\begin{equation}
 c_{\bf r} \to  c_{\bf r} e^{\frac{i}{2}\theta_{\bf r}} , \,\,\,\,
\Delta_{\bf r}^{\rm HS} \to \Delta_{\bf r}^{\rm HS} e^{i\theta_{\bf r}}  .
\end{equation}
We then consider the shift $ \rho^{\rm HS}_{\bf r} \to \rho_0^{\rm HS}+
\rho^{\rm HS}_{\bf r}$, $i\Delta_{\bf r}^{\rm HS} \to 
\Delta_0^{\rm HS}+i\Delta_{\bf r}^{\rm HS}$ and $i{\Delta_{\bf r}^{\rm HS}}^*
\to 
\Delta_0^{\rm HS}+i{\Delta_{\bf r}^{\rm HS}}^*$ (recall that 
a factor $i$ has been included in $\Delta_0^{\rm HS}$ and ${\Delta_0^{\rm
    HS}}^*$), so that the 
Hubbard-Stratonovich fields $ \rho^{\rm HS}_{\bf r}$ and $\Delta_{\bf r}^{\rm
  HS}$ now describe (small) 
fluctuations about the mean-field values. This leads to the action
\begin{eqnarray}
S &=& S_{\rm MF} + {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[
  c^\dagger_{\bf r} \biggl( \frac{i}{2} 
  \dot\theta_{\bf r} - 
  \frac{i}{4m}{\boldsymbol{\nabla}\theta_{\bf r}} \cdot
  \tensor{\boldsymbol{\nabla}}  \nonumber \\ &&
+ \frac{({\boldsymbol{\nabla}\theta_{\bf r}})^2}{8m} 
  -\frac{\mu_B}{2} -i \rho^{\rm HS}_{\bf r} \biggr)  c_{\bf r}
- i (\Delta_{\bf r}^{\rm HS} c^*_{{\bf r}\uparrow}c^*_{{\bf r}\downarrow} +
  {\rm c.c.}) \nonumber \\ && 
+ (i{\Delta_{\bf r}^{\rm HS}}^*+i\Delta_{\bf r}^{\rm
  HS}+2\Delta_0^{\rm HS})|\Delta_{\bf r}| \nonumber \\ && 
-g |\Delta_{\bf r}|^2 - \frac{g}{4} \rho_{\bf r}^2 +(i\rho_0^{\rm HS}+i
  \rho^{\rm HS}_{\bf r})\rho_{\bf r} \biggr] , 
\end{eqnarray} 
where
$\tensor{\boldsymbol{\nabla}} = \roarrow{\boldsymbol{\nabla}} -
\loarrow{\boldsymbol{\nabla}}$.  
Here we write the chemical potential as $\mu=\mu_{\rm MF}+\mu_B/2$ where
$\mu_{\rm MF}$ is the chemical potential in the mean-field
approximation. The next step is to shift $ \rho^{\rm HS}_{\bf r}$, $i
\rho^{\rm HS}_{\bf r}\to i \rho^{\rm HS}_{\bf r}+ 
i\dot\theta_{\bf r}/2+({\boldsymbol{\nabla}\theta_{\bf r}})^2/8m-\mu_B/2$, and
to introduce Nambu spinors 
$\phi_{\bf r}=(c_{{\bf r}\uparrow},c^*_{{\bf r}\downarrow})^T$. This gives
\begin{eqnarray}
S &=& S_{\rm MF} + S' + {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[
  -g|\Delta_{\bf r}|^2-\frac{g}{4} \rho_{\bf r}^2 
  \nonumber \\ && + (2\Delta_0^{\rm HS}+i\Delta_{\bf r}^{\rm HS}+i{\Delta_{\bf
      r}^{\rm HS}}^*)|\Delta_{\bf r}| \nonumber \\ 
  &&
+\rho_{\bf r} \Bigl(i\rho_0^{\rm HS}+i \rho^{\rm HS}_{\bf
  r}+\frac{i}{2}\dot\theta_{\bf r}+\frac{({\boldsymbol{\nabla}\theta_{\bf
  r}})^2}{8m} - 
\frac{\mu_B}{2}\Bigr) \biggr] , 
\end{eqnarray}
where 
\begin{eqnarray}
S' &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \Bigl(-i \rho^{\rm HS}_{\bf r}
{j^z_{0{\bf r}}} + \frac{1}{2} 
{\boldsymbol{\nabla}\theta_{\bf r}}  \cdot {\bf j}^0_{\bf r} 
\nonumber \\ && -
i\Delta_{\bf r}^{\rm HS} {j^+_{0{\bf r}}} - i {\Delta_{\bf r}^{\rm HS}}^*
{j^-_{0{\bf r}}} \Bigr)  ,  
\label{Sprime} \\ 
{j^z_{0{\bf r}}} &=& \phi^\dagger_{\bf r} \tau^z \phi_{\bf r} =
c^\dagger_{\bf r} 
c_{\bf r} , \nonumber \\  
{j^+_{0{\bf r}}} &=& \phi^\dagger_{\bf r} \tau^+ \phi_{\bf r} = c^*_{{\bf
    r}\uparrow} 
c^*_{{\bf r}\downarrow} , \nonumber 
\\  
{j^-_{0{\bf r}}} &=& \phi^\dagger_{\bf r} \tau^- \phi_{\bf r} =  c_{{\bf
    r}\downarrow} 
c_{{\bf r}\uparrow} ,  
\nonumber \\ 
{\bf j}^0_{\bf r} &=& -\frac{i}{2m} \phi^\dagger_{\bf r} \tensor
{\boldsymbol{\nabla}} 
\phi_{\bf r} = -\frac{i}{2m}  c^\dagger_{\bf r} \tensor {\boldsymbol{\nabla}}
c_{\bf r} . 
\label{jdef}  
\end{eqnarray} 
$(\tau_x,\tau_y,\tau_z)$ are Pauli
matrices acting in Nambu space. The effective action
$S[\rho, \rho^{\rm HS},\Delta,\Delta^{\rm HS}]$ is obtained by integrating out
the 
fermions. To quadratic order in the bosonic fields and their gradient
($\partial_\tau$ or $\boldsymbol{\nabla}$),  it is sufficient to retain the
first and second order cumulants of $S'$ with respect to the mean-field
action:   
\begin{multline}
S[\rho, \rho^{\rm HS},\Delta,\Delta^{\rm HS}] = \biggl\langle
  S'-\frac{{S'}^2}{2} 
\biggr\rangle_c \\
+ {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[ -g|\Delta_{\bf r}|^2 -
  \frac{g}{4} \rho_{\bf r}^2  \\ 
+ (2\Delta_0^{\rm HS}+i{\Delta_{\bf r}^{\rm HS}}^*+i\Delta_{\bf r}^{\rm
  HS})|\Delta_{\bf r}| \\ 
+\rho_{\bf r} \Bigl(i\rho_0^{\rm HS}+i \rho^{\rm HS}_{\bf
  r}+\frac{i}{2}\dot\theta_{\bf r}+\frac{({\boldsymbol{\nabla}\theta_{\bf
  r}})^2}{8m} - 
\frac{\mu_B}{2}\Bigr) \biggr] ,
\label{action_cf}
\end{multline}
where the averages $\langle \cdots\rangle_c$ are calculated with respect to 
the mean-field action $S_{\rm MF}$. 
Calculating the first and second order cumulants and integrating out the
Hubbard-Stratonovich fields $ \rho^{\rm HS}_{\bf r}$ and $\Delta_{\bf r}^{\rm
  HS}$ (Appendix \ref{app:lea}), we obtain  
\begin{multline} 
S[\rho,\Delta] = {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \rho_{\bf r}
\biggl(\frac{i}{2}\dot\theta_{\bf r}+\frac{({\boldsymbol{\nabla}\theta_{\bf
      r}})^2}{8m}  
-\frac{\mu_B}{2} \biggr) \\
+ \sum_q (\delta\rho_{-q},\delta|\Delta|_{-q})
\left(
\begin{array}{ll}
{\beta_{\bf q}} & -g{\alpha_{\bf q}} \\
-g{\alpha_{\bf q}} & g^2 {\gamma_{\bf q}} 
\end{array}
\right) \left(
\begin{array}{l}
\delta\rho_q \\ \delta|\Delta|_q
\end{array}
\right) , 
\label{action3} 
\end{multline}
where $\delta\rho_q$ and $\delta|\Delta|_q$ are the Fourier transforms of
$\delta\rho_{\bf r}=\rho_{\bf r}-\rho_0$ and $\delta|\Delta_{\bf
  r}|=|\Delta_{\bf r}|-\Delta_0$, and  
\begin{eqnarray}
{\beta_{\bf q}} &=&  \frac{1}{2}{\Pi^{zz}_{00}({\bf q})}^{-1}-\frac{g}{4} -
    \frac{1}{{C_{\bf q}}} [{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf
    q})}]^2 , \nonumber \\ 
-g {\alpha_{\bf q}} &=& 
\frac{1}{C_q} {\Pi^{zz}_{00}({\bf q})}^{-1} {\Pi^{z+}_{00}({\bf q})}  ,
\nonumber \\ 
g^2 {\gamma_{\bf q}} &=& -\frac{1}{{C_{\bf q}}}-g , \nonumber \\ 
{C_{\bf q}} &=& - {\Pi^{+-}_{00}({\bf q})} - {\Pi^{++}_{00}({\bf q})} +
    2{\Pi^{zz}_{00}({\bf q})}^{-1} {\Pi^{z+}_{00}({\bf q})}^2 . 
\nonumber \\ && 
\label{action3a}
\end{eqnarray} 
We use the notation $q=({\bf q},i\omega_\nu)$ and $\sum_q=\sum_{{\bf
q},\omega_\nu}$ where $\omega_\nu$ is a bosonic Matsubara frequency. 
The mean-field correlation function $\Pi^{\nu\nu'}_{00}(q)= \langle
j^\nu_0(q) j^{\nu'}_0(-q)\rangle_c$ is calculated in Appendix
\ref{app:hfcorr} and $\Pi^{\nu\nu'}_{00}({\bf q})=\Pi^{\nu\nu'}_{00}({\bf
  q},\omega_\nu=0)$. $j^\nu_0(q)$ is the Fourier transformed field of
$j^\nu_{0{\bf r}}$ [Eq.~(\ref{jdef})]. Eq.~(\ref{action3}) shows that half the
fermion density is the conjugate variable of the phase $\theta_{\bf r}$ of the
pairing field. Eqs.~(\ref{action3}-\ref{action3a}) agree with Eq.~(2.3) of 
Ref.~\onlinecite{Depalo99} except for the coefficient of $\delta\rho_{-q}
\delta|\Delta|_q$ which is found to have opposite sign.\cite{note5} 

We now discuss the strong-coupling limit (not considered in
Ref.~\onlinecite{Depalo99}). To leading order in ${\rho_0 a^3}$
and $|{\bf q}|a$, we have (Appendix \ref{app:hfcorr}) \cite{Marini98} 
\begin{eqnarray}
{\alpha_{\bf q}} &=& \biggl(\frac{1}{4\pi{\rho_0 a^3}}\biggr)^{1/2} \biggl( 1 +
\frac{9}{4}\pi{\rho_0 a^3}+\frac{1}{6}|{\bf q}|^2a^2 \biggr) , \nonumber \\
{\beta_{\bf q}} &=& \frac{1}{2\rho_0 ma^2} \biggl(1+4\pi{\rho_0
  a^3}+\frac{1}{4}|{\bf q}|^2a^2 
\biggr) , \nonumber \\ 
{\gamma_{\bf q}} &=& \frac{m}{2\pi a}  \biggl( 1 + \frac{3}{2}\pi{\rho_0
  a^3}+\frac{7}{48} 
|{\bf q}|^2a^2 \biggr)  . 
\label{abg} 
\end{eqnarray} 
Denoting by $\lambda^+_{\bf q}$ and  $\lambda^-_{\bf q}$  the two eigenvalues
of the fluctuation matrix appearing in (\ref{action3}), we have 
\begin{eqnarray}
\lambda^+_{\bf q} &=& {\beta_{\bf q}} + g^2 \frac{{\alpha_{\bf
      q}}^2}{{\beta_{\bf q}}} , \nonumber \\  
\lambda^-_{\bf q} &=& g^2 \biggl( {\gamma_{\bf q}} - \frac{{\alpha_{\bf
      q}}^2}{{\beta_{\bf q}}} \biggr) ,
\end{eqnarray}
to order ${\cal O}(g^2)$. For $g\to 0$ (at fixed $a$), the mode corresponding
to the eigenvalue $\lambda^+_{\bf q}$ is frozen, which leads to 
\begin{equation}
\frac{\delta|\Delta_{\bf r}|}{\delta\rho_{\bf r}} = \frac{1}{g} \biggl(
  \frac{\pi}{\rho_0 m^2 
  a} \biggr)^{1/2}.
\label{tied} 
\end{equation}
Density ($\delta\rho_{\bf r}$) and modulus ($\delta|\Delta_{\bf r}|$)
fluctuations do not 
fluctuate independently in the low-energy limit but are tied by the relation
(\ref{tied}). From (\ref{action3},\ref{abg},\ref{tied}), we deduce that 
the dynamics of the Fermi superfluid is determined by the effective action
\begin{eqnarray}
S[\rho,\theta] &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[ \rho_{\bf
  r} \biggl( 
  \frac{i}{2}\dot\theta_{\bf r} + 
  \frac{({\boldsymbol{\nabla}\theta_{\bf r}})^2}{8m} - \frac{\mu_B}{2} \biggr)
  \nonumber \\  
&& + \frac{\pi a}{2m} (\delta\rho_{\bf r})^2 +
  \frac{(\boldsymbol{\nabla}\delta\rho_{\bf 
  r})^2}{32\rho_0 m} \biggr] .
\label{action4}
\end{eqnarray} 
Introducing the bosonic field
\begin{equation}
{\psi_{\bf r}} = \sqrt{\frac{\rho_{\bf r}}{2}} e^{i\theta_{\bf r}} ,
\label{psidef}
\end{equation}
we recover the standard action of a Bose superfluid,
\begin{eqnarray}
S[\psi^*,\psi] &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[
  {\psi^*_{\bf r}} \biggl(\partial_\tau - \mu_B - 
  \frac{{\boldsymbol{\nabla}^2}}{2m_B} \biggr) {\psi_{\bf r}} \nonumber \\ 
&& + \frac{2\pi a_B}{m_B} ({\psi^*_{\bf r}}{\psi_{\bf r}}-\rho_0/2)^2 \biggr], 
\label{action5}
\end{eqnarray}
where $m_B = 2m$ and $a_B = 2a$ are the mass and the scattering length of the
bosons. The result $a_B=2a$ corresponds to the Born approximation for the
boson-boson scattering, while the exact result is $a_B=0.6a$. \cite{Petrov04}
Eqs.~(\ref{action5}) and  
(\ref{action4}) are equivalent in the hydrodynamic regime where
$(\boldsymbol{\nabla} 
\rho_{\bf r})^2/\rho_0\simeq (\boldsymbol{\nabla}\rho_{\bf r})^2/\rho_{\bf
  r}$. \cite{note2}  

Thus, we have shown how, by introducing the physical fields $\rho_{\bf r}$ and
$\Delta_{\bf r}$ from the outset and expanding about the mean-field state in
the 
strong-coupling limit, one obtains the standard action of a Bose
superfluid. Our approach should be contrasted with a number of previous works 
\cite{Randeria95,Drechsler92,Haussmann93,deMelo93,Pistolesi96,Piery2000,Pieri03}  
where only the pairing Hubbard-Stratonovich field $\Delta_{\bf r}^{\rm HS}$ is
considered and the expansion is carried out about the non-interacting
state, which gives the action (\ref{action5}) but for the field
$\sqrt{\rho_{\bf r}/2}e^{i\theta^{\rm HS}_{\bf r}}$ instead of the $\psi$
field defined in (\ref{psidef}). \cite{note3} 


\section{Lattice model}
\label{sec:lm}

In this section, we consider the attractive Hubbard model on a bipartite
lattice, with Hamiltonian 
\begin{equation}
H = -t \sum_{\langle{\bf r},{\bf r}'\rangle}  (c^\dagger_{\bf r} c_{{\bf r}'}
+{\rm h.c.}) 
- \mu {\sum_{\bf r}}  c^\dagger_{\bf r}  c_{\bf r} - U {\sum_{\bf r}} {n_{{\bf
      r}\uparrow}} {n_{{\bf r}\downarrow}} . 
\label{ham1}
\end{equation}
The operator $c^\dagger_{{\bf r}\sigma}$ ($c_{{\bf r}\sigma}$) creates
(annihilates) a fermion with spin $\sigma$ at the lattice site ${\bf r}$,
$ c_{\bf r}=(c_{{\bf r}\uparrow},c_{{\bf r}\downarrow})^T$, and $n_{{\bf
    r}\sigma}=c^\dagger_{{\bf r}\sigma}c_{{\bf r}\sigma}$. $\langle{\bf
  r},{\bf r}'\rangle$ denotes nearest-neighbor sites. The chemical potential
$\mu$ fixes the average density $\rho_0$ (i.e. the average number of fermions
per site) and $-U$ ($U\geq 0$) is the on-site attractive interaction. 
%At half-filling, particle-hole symmetry implies that $\mu=-U/2$. 

We are interested in the strong-coupling limit $U\gg t$ where fermions form
tightly bound composite bosons which behave as local pairs. The latter Bose
condense at low temperature giving rise to superfluidity. In order to derive
the low-energy effective action, we could follow the procedure used in 
Sec.~\ref{sec:cm}. Here, we shall use a different method, based on the
mapping of the attractive Hubbard model in the strong-coupling limit onto the
Heisenberg model in a uniform magnetic field. \cite{Dupuis04} Thus this
approach is based on a $t/U$ expansion about the $t=0$ limit rather than
on an expansion about the mean-field state. \cite{note1}

Under the canonical particle-hole transformation \cite{Micnas90}
\begin{equation}
c_{{\bf r}\downarrow} \to (-1)^{\bf r} c^\dagger_{{\bf r}\downarrow} , \,\,\, 
c^\dagger_{{\bf r}\downarrow} \to (-1)^{\bf r} c_{{\bf r}\downarrow} ,
\end{equation}
the Hamiltonian becomes (omitting a constant term)
\begin{eqnarray}
H &=& -t \sum_{\langle{\bf r},{\bf r}'\rangle} ( c^\dagger_{\bf r} c_{{\bf
    r}'}  +{\rm h.c.})
- {\sum_{\bf r}}  c^\dagger_{\bf r} \biggl(\frac{U}{2} + h_0 \sigma^z \biggr)
c_{\bf r} \nonumber \\ && 
+ U {\sum_{\bf r}} {n_{{\bf r}\uparrow}} {n_{{\bf r}\downarrow}} ,
\label{ham2}
\end{eqnarray}
and corresponds now to the repulsive Hubbard model in a magnetic
field ${\bf h}_0=h_0 \hat {\bf z}$ along the $z$ axis, 
\begin{equation}
h_0 = \mu + \frac{U}{2},
\end{equation}
coupled to the fermion spins. The chemical potential $U/2$ in (\ref{ham2}),
together with particle-hole symmetry, implies that the system is half-filled. 
The density and pairing operators transform into the three components of the
spin density operator: 
\begin{eqnarray}
\rho_{\bf r} =  c^\dagger_{\bf r}  c_{\bf r} &\to&  c^\dagger_{\bf r} \sigma^z
c_{\bf r} +1 , \nonumber \\   
\Delta_{\bf r} =  c_{{\bf r}\downarrow} c_{{\bf r}\uparrow} &\to& (-1)^{\bf r}
c^\dagger_{{\bf r}\downarrow} c_{{\bf r}\uparrow} ,  
\nonumber \\  
\Delta^\dagger_{\bf r} = c^\dagger_{{\bf r}\uparrow} c^\dagger_{{\bf r}
  \downarrow} &\to& (-1)^{\bf r} c^\dagger_{{\bf r}\uparrow}  c_{{\bf
    r}\downarrow}  . 
\label{correspondence}
\end{eqnarray}
The equation fixing $\mu$, $\langle  c^\dagger_{\bf r} c_{\bf r}
\rangle=\rho_0$, becomes an equation 
fixing the magnetic field: $\langle  c^\dagger_{\bf r} \sigma^z c_{\bf
  r}\rangle=\rho_0-1$. 

In the strong-coupling limit $U\gg t$, the Hamiltonian (\ref{ham2}) simplifies
into \cite{Dupuis04}
\begin{equation}
H = J \sum_{\langle {\bf r},{\bf r}'\rangle} {\bf S_r}\cdot {\bf S}_{{\bf r}'}
- 2 {\bf h}_0 \cdot {\sum_{\bf r}} {\bf S_r} ,
\label{ham3}
\end{equation}
where $J=4t^2/U$ and ${\bf S_r}$ is a spin-$\frac{1}{2}$ operator. Using
spin-$\frac{1}{2}$ coherent states $|{\bf\Omega_r}\rangle$ (${\bf\Omega}^2_{\bf
  r}=1$), \cite{Auerbach94} the action of the Heisenberg model (\ref{ham3})
can be written as 
\begin{equation}
S[{\bf\Omega}] = {\int_0^\beta d\tau} \biggl[ {\sum_{\bf r}}
  [\langle{\bf\Omega_r}|\dot{\bf\Omega}_{\bf r} 
  \rangle - {\bf h}_0 \cdot {\bf\Omega_r}] + J \sum_{\langle{\bf r},{\bf
   r}'\rangle} \frac{{\bf\Omega_r} \cdot {\bf\Omega}_{{\bf r}'}}{4} \biggr] ,
\label{action_heis}
\end{equation} 
where $|\dot{\bf\Omega}_{\bf r}\rangle=\partial_\tau|{\bf\Omega_r}\rangle$. 

The effective action $S[\rho,\Delta]$ of the superfluid system is obtained by
rewriting the action (\ref{action_heis}) in terms of the density and pairing
fields of the attractive model. In the strong-coupling limit,
Eqs.~(\ref{correspondence}) (written now for fields rather than operators)
become \cite{Dupuis04}  
\begin{eqnarray}
\rho_{\bf r} &=& \Omega^z_{\bf r} + 1 , \nonumber \\ 
\Delta_{\bf r} &=& \frac{(-1)^{\bf r}}{2} \Omega^-_{\bf r} , \nonumber \\
\Delta^*_{\bf r} &=& \frac{(-1)^{\bf r}}{2} \Omega^+_{\bf r} ,
\label{correspondence2}
\end{eqnarray}
where $\Omega^\pm_{\bf r}=\Omega^x_{\bf r}\pm i\Omega^y_{\bf r}$. The
condition ${\bf\Omega}^2_{\bf r}=1$ implies that $\rho_{\bf r}$ and
$\Delta_{\bf r}$ do not 
fluctuate independently but are tied by the relation
\begin{equation}   
|\Delta_{\bf r}| = \frac{1}{2} [\rho_{\bf r}(2-\rho_{\bf r})]^{1/2} .
\label{tied1}
\end{equation}  

In the low-density limit ($\rho_{\bf r}\ll 1$), where the Pauli principle
(which 
prevents two composite bosons to occupy the same site) should not matter,
we expect to recover the standard action of a Bose superfluid. In that limit,
$|\Delta_{\bf r}|\simeq\sqrt{\rho_{\bf r}/2}$; the pair density $|\Delta_{\bf
  r}|^2$ equals half 
the fermion density $\rho_{\bf r}$, and $\Delta_{\bf r}=|\Delta_{\bf
  r}|e^{i\theta_{\bf r}}$ coincides   
with the bosonic field ${\psi_{\bf r}}=\sqrt{\rho_{\bf r}/2}e^{i\theta_{\bf
    r}}$.  
To order ${\cal O}(\rho^2_{\bf r})$, we deduce from
(\ref{action_heis}-\ref{tied1}) 
\begin{eqnarray}
S[\rho,\theta] &=& {\int_0^\beta d\tau} \biggl\lbrace  
{\sum_{\bf r}} \biggl[ \frac{i}{2} \rho_{\bf r}\dot\theta_{\bf r} -
  \biggl(h_0+\frac{Jz}{4}\biggr)\rho_{\bf r}  \biggr] \nonumber \\ && 
+ \frac{J}{4} \sum_{\langle {\bf r},{\bf r}'\rangle} [\rho_{\bf r}\rho_{{\bf
      r}'}  
\nonumber \\ && 
- (\rho_{\bf r}\rho_{{\bf r}'})^{1/2}(2-\rho_{\bf r})\cos(\theta_{\bf
  r}-\theta_{{\bf r}'}) ]  
\biggr\rbrace .
\end{eqnarray}
The term
$(i/2)\rho_{\bf r}\dot\theta_{\bf r}$ comes from the Berry phase term
$\langle{\bf\Omega_r}|\dot{\bf\Omega}_{\bf r} \rangle$ of the action
$S[{\bf\Omega}]$ [Eq.~(\ref{action_heis})] with a proper gauge
choice. \cite{Dupuis04} If we further assume that $\rho_{\bf r}$ and
$\theta_{\bf r}$ are slowly varying in space, we obtain
\begin{eqnarray}
S[\rho,\theta] &=& {\int_0^\beta d\tau} \biggl\lbrace 
{\sum_{\bf r}} \biggl[ \frac{i}{2} \rho_{\bf r}\dot\theta_{\bf r} -
  \biggl(h_0+\frac{Jz}{4} \biggr) \rho_{\bf r} 
+ \frac{Jz}{4} \rho^2_{\bf r} \biggr] \nonumber \\ &&
- \frac{J}{2} \sum_{\langle {\bf r},{\bf r}'\rangle} (\rho_{\bf r}\rho_{{\bf
    r}'})^{1/2} \cos(\theta_{\bf r}-\theta_{{\bf r}'}) \biggr\rbrace .
\end{eqnarray}
or, equivalently,
\begin{eqnarray}
S[\psi^*,\psi] &=& {\int_0^\beta d\tau} \biggl\lbrace
{\sum_{\bf r}}  \biggl[{\psi^*_{\bf r}}(\partial_\tau - \mu_B) {\psi_{\bf r}}
  + \frac{U_B}{2} 
  |\psi^4_{\bf r}| \biggr] \nonumber \\ &&
- t_B \sum_{\langle {\bf r},{\bf r}'\rangle} ({\psi^*_{\bf r}}  \psi_{{\bf
    r}'} + {\rm c.c.}) \biggr\rbrace ,
\end{eqnarray}
where $t_B=J/2$, $U_B=2Jz$, $\mu_B=2h_0+Jz/2$, and $z$ is the number of
nearest-neighbor sites. We therefore obtain the
action of the Bose-Hubbard model with on-site repulsive interaction $U_B$ and
nearest-neighbor hopping amplitude $t_B$. In the continuum limit and for a
cubic lattice, the latter gives a boson mass $m_B=1/J$ as obtained in
Ref.~\onlinecite{Dupuis04}.  


\section{Concluding remarks}

In this paper, we have shown that a Fermi superfluid in the strong-coupling
limit, where superfluidity originates from BEC of composite bosons, can be
described by the complex field $\psi_{\bf r}=\sqrt{\rho_{\bf
    r}/2}e^{i\theta_{\bf r}}$, 
where $\rho_{\bf r}$ is the fermion density and $\theta_{\bf r}$ the phase of
the pairing 
field $\Delta_{\bf r}$. Such is description is made possible by the fact that
density 
($\rho_{\bf r}$) and amplitude ($|\Delta_{\bf r}|$) fluctuations are not
independent in the 
strong-coupling limit. The effective action $S[\rho,\theta]$ is derived by
introducing the physical fields $\rho_{\bf r}$ and $\Delta_{\bf r}$ from the
outset by means 
of Lagrange multiplier fields $ \rho^{\rm HS}_{\bf r}$ and $\Delta_{\bf
  r}^{\rm HS}$. The latter play the 
role of the Hubbard-Stratonovich fields usually introduced {\it via} a
Hubbard-Stratonovich transformation of the fermion-fermion interaction. 

For continuum models, the effective action is derived from an expansion about
the mean-field state. It corresponds to the usual action of a Bose superfluid
of density $\rho_{\bf r}/2$ where the bosons have a mass $m_B=2m$ and interact
{\it via} a contact potential with amplitude $g_B=4\pi a_B/m_B$, $a_B=2a$. 

For lattice (Hubbard) models, the effective action is obtained from an
expansion about the $t=0$ limit, using the mapping of the attractive Hubbard
model in the strong-coupling limit onto the Heisenberg model in a uniform
magnetic field. The effective model is a Bose-Hubbard model with an on-site
repulsion $U_B=2Jz$ (with $z$ the number of nearest-neighbor sites) and a
nearest-neighbor intersite hopping amplitude $t_B=J/2$, where $J=4t^2/U$. 



%------------------------------------------------------------------------

\appendix

\begin{widetext} 

\section{Low-energy effective action $S[\rho,\Delta]$}
\label{app:lea}

In this appendix, we derive the effective action (\ref{action3}) starting from
Eq.~(\ref{action_cf}). The first and second order cumulants are given by
\begin{eqnarray}
\langle S' \rangle &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} [-i\rho_0
  \rho^{\rm HS}_{\bf r} -i \Delta_0 
  (\Delta_{\bf r}^{\rm HS}+{\Delta_{\bf r}^{\rm HS}}^*) ] , \nonumber \\ 
\langle {S'}^2 \rangle_c &=& \sum_q \biggl[ -\rho_{-q}^{\rm HS}
  {\Pi^{zz}_{00}(q)} \rho_{q}^{\rm HS} - 2 \rho_{-q}^{\rm HS}
  {\Pi^{z+}_{00}(q)} {\Delta_{q}^{\rm HS}}  
- 2 \rho_{-q}^{\rm HS} {\Pi^{z-}_{00}(q)} {{\Delta_{-q}^{\rm HS}}^*} 
- {\Delta_{-q}^{\rm HS}} {\Pi^{++}_{00}(q)}  {\Delta_{q}^{\rm HS}} \nonumber
  \\ && 
 - {{\Delta_{q}^{\rm HS}}^*} {\Pi^{--}_{00}(q)} {{\Delta_{-q}^{\rm HS}}^*} - 2
  {\Delta_{-q}^{\rm HS}} 
{\Pi^{+-}_{00}(q)} {{\Delta_{-q}^{\rm HS}}^*} \biggr] .
\label{cumulant}
\end{eqnarray} 
The second order cumulant is written in Fourier space. The mean-field
correlation function $\Pi^{\nu\nu'}_{00}(q) = \langle j^\nu_0(q)
j^{\nu'}_0(-q) \rangle_c$ is defined in Sec.~\ref{subsec:lea}. To obtain
(\ref{cumulant}), 
we have used the fact that mean-field correlation functions involving the
current ${\bf j}^0_{\bf r}$ vanish (Appendix \ref{app:hfcorr}). In the
low-energy limit, we 
can approximate $\Pi^{\nu\nu'}_{00}(q)$ by its static limit
$\Pi^{\nu\nu'}_{00}({\bf q})=\Pi^{\nu\nu'}_{00}({\bf q},\omega_\nu=0)$. 

Integrating out the  Hubbard-Stratonovich field $ \rho^{\rm HS}_{\bf r}$, we
obtain 
\begin{eqnarray}
S &=& {\int_0^\beta d\tau} {\int d^3{\bf r}\,} \biggl[ - g\delta|\Delta_{\bf
  r}|^2 - 
  \frac{g}{4} \delta\rho_{\bf r}^2
+ \rho_{\bf r} \Bigl( \frac{i}{2} \dot\theta_{\bf r} +
  \frac{({\boldsymbol{\nabla}\theta_{\bf r}})^2}{8m}  
- \frac{\mu_B}{2} \Bigr) 
+i ({\Delta_{\bf r}^{\rm HS}}^* +\Delta_{\bf r}^{\rm HS})\delta|\Delta_{\bf
  r}| \biggr] \nonumber  \\ && 
+ \sum_q \biggl\lbrace \frac{1}{2} {\Pi^{++}_{00}({\bf q})} ({\Delta_{-q}^{\rm
  HS}} {\Delta_{q}^{\rm HS}} + 
{\rm c.c.}) +{{\Delta_{q}^{\rm HS}}^*} {\Pi^{+-}_{00}({\bf q})}
  {\Delta_{q}^{\rm HS}} \nonumber \\ && 
+ \frac{1}{2}{\Pi^{zz}_{00}({\bf q})}^{-1} \Bigl[ \delta\rho_{-q} \delta\rho_q
  - 2i {\Pi^{z+}_{00}(q)} 
  \delta\rho_{-q}({\Delta_{q}^{\rm HS}}+{{\Delta_{-q}^{\rm HS}}^*})
- {\Pi^{z+}_{00}({\bf q})}^2({\Delta_{-q}^{\rm HS}}{\Delta_{q}^{\rm
  HS}}+{{\Delta_{-q}^{\rm HS}}^*} {\Delta_{q}^{\rm HS}} + 2 
|{\Delta_{q}^{\rm HS}}|^2) \Bigr] \biggr\rbrace ,
\label{appA1}
\end{eqnarray} 
where $\delta\rho_{\bf r} = \rho_{\bf r}-\rho_0$ and $\delta|\Delta_{\bf
  r}|=\Delta_{\bf r}-\Delta_0$. Here
we have neglected constant terms and use the saddle-point equations
(\ref{speq}).  

To obtain the action $S[\rho,\Delta]$ in terms of the physical fields only,
one has then to integrate out the Hubbard-Stratonovich field $\Delta^{\rm HS}$:
\begin{eqnarray}
&& \int {\cal D}[\Delta^{\rm HS}] \exp\Bigl\lbrace-S_0[\Delta^{\rm HS}] -i
  \sum_q 
  ({\Delta_{q}^{\rm HS}}+{{\Delta_{-q}^{\rm HS}}^*})
  [\delta|\Delta|_{-q}-{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf q})} 
    \delta\rho_{-q}]\Bigr\rbrace \nonumber \\ 
&=& 
\exp \Bigl\lbrace -\frac{1}{2} \sum_q \langle ({\Delta_{q}^{\rm
  HS}}+{{\Delta_{-q}^{\rm 
  HS}}^*}) 
({\Delta_{-q}^{\rm HS}}+{{\Delta_{q}^{\rm HS}}^*}) \rangle_0
[\delta|\Delta|_{q}-{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf q})}
  \delta\rho_q] 
[\delta|\Delta|_{-q}-{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf q})}
  \delta\rho_{-q}] 
\Bigr\rbrace \nonumber \\ 
&=&
\exp \Bigl\lbrace \frac{1}{2} \sum_q [M_{11}({\bf q})+M_{22}({\bf
  q})+2M_{12}({\bf q})] [\delta|\Delta|_{q}-{\Pi^{zz}_{00}({\bf
  q})}^{-1}{\Pi^{z+}_{00}({\bf q})} \delta\rho_q] 
[\delta|\Delta|_{-q}-{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf q})}
  \delta\rho_{-q}] \Bigr\rbrace, 
\label{appA2}
\end{eqnarray}
where averages $\langle \cdots \rangle_0$ are taken with the Gaussian action 
\begin{eqnarray}
S_0[\Delta^{\rm HS}] &=& - \frac{1}{2} \sum_q ({{\Delta_{q}^{\rm
  HS}}^*},{\Delta_{-q}^{\rm HS}}) 
  M^{-1}({\bf q})  
\left(
\begin{array}{l}
{\Delta_{q}^{\rm HS}}  \\ {{\Delta_{-q}^{\rm HS}}^*}  
\end{array}
\right) , \nonumber \\ 
M^{-1}({\bf q}) &=& \left(
\begin{array}{lr}
-{\Pi^{+-}_{00}({\bf q})} + {\Pi^{zz}_{00}({\bf q})}^{-1} {\Pi^{z+}_{00}({\bf
    q})}^2 & -{\Pi^{++}_{00}({\bf q})}+{\Pi^{zz}_{00}({\bf
    q})}^{-1}{\Pi^{z+}_{00}({\bf q})}^2 \\ 
 -{\Pi^{++}_{00}({\bf q})}+{\Pi^{zz}_{00}({\bf q})}^{-1}{\Pi^{z+}_{00}({\bf
    q})}^2 & -{\Pi^{+-}_{00}({\bf q})} + {\Pi^{zz}_{00}({\bf q})}^{-1}
    {\Pi^{z+}_{00}({\bf q})}^2  
\end{array}
\right) .
\end{eqnarray} 
In the following, we denote by ${A_{\bf q}}$ and ${B_{\bf q}}$ the diagonal and
off-diagonal 
components of $M^{-1}({\bf q})$, and ${C_{\bf q}}={A_{\bf q}}+{B_{\bf q}}$. The
effective action 
$S[\rho,\Delta]$ deduced from Eqs.~(\ref{appA1},\ref{appA2}) is given by
Eq.~(\ref{action3}).  
 
\end{widetext}


\section{Mean-field correlation function} 
\label{app:hfcorr}

In this appendix, we calculate the mean-field correlation function
$\Pi^{\nu\nu'}_{\mu\mu'}(q) = \langle j^\nu_\mu(q) j^{\nu'}_{\mu'}(-q)
\rangle_c$ ($\nu,\nu'=x,y,z$; $\mu,\mu'=0,x,y,z$) in the strong-coupling limit
${\rho_0 a^3}\ll 1$ and for $|{\bf q}|a\ll 1$. 

\subsection{General expression}

$j^\nu_\mu(q)$ is the Fourier transformed field of 
$j^\nu_{\mu{\bf r}}$ [Eq.~(\ref{jdef})]:
\begin{eqnarray}
j^\nu_0(q) &=& \frac{1}{\sqrt{\beta V}} \sum_k \phi^\dagger_k \sigma^\nu
  \phi_{k+q} ,  \nonumber \\ 
j^0_\mu(q) &=& \frac{1}{\sqrt{\beta V}} \sum_k \frac{1}{m} \biggl(k_\mu +
  \frac{q_\mu}{2} \biggr) \phi^\dagger_k \phi_{k+q} \,\,\, (\mu\neq 0) ,
\end{eqnarray}
where $V$ is the volume of the system and $V^{-1}\sum_{\bf k}={\int_{\bf k}}$
for 
$V\to\infty$. $k=({\bf k},i\omega)$ and $\sum_k = \sum_{{\bf k},\omega}$ where
$\omega$ is a fermionic Matsubara frequency. We have 
\begin{eqnarray}
{\Pi^{zz}_{00}(q)}  &=& - \frac{2}{\beta V} \sum_k [ G(k)G(k+q)-F(k)F(k+q)] ,
\nonumber \\
{\Pi^{z+}_{00}(q)}  &=& {\Pi^{z-}_{00}(-q)} = - \frac{2}{\beta V} \sum_k
G(k+q)F(k) , 
\nonumber \\ 
{\Pi^{++}_{00}(q)}  &=& {\Pi^{--}_{00}(q)} = - \frac{1}{\beta V} \sum_k
F(k)F(k+q) , 
\nonumber \\
{\Pi^{+-}_{00}(q)} &=&  \frac{1}{\beta V} \sum_k G(k)G(-k-q) ,
\nonumber \\ 
\Pi^{00}_{\mu\mu'}(q) &=& - \frac{2}{\beta V} \sum_k
\frac{1}{m^2} \biggl(k_\mu + \frac{q_\mu}{2} \biggr) \biggl(k_{\mu'} +
\frac{q_{\mu'}}{2} \biggr) \nonumber \\ && \times [G(k+q) G(k)+F(k+q) F(k)] ,
\label{pi1} 
\end{eqnarray} 
where $G$ and $F$ are the mean-field propagators [Eq.~(\ref{hfgf})]. The
correlation function $\langle j^0_\mu(q) j^\nu_0(-q) \rangle$ ($\mu\neq 
0$) vanishes. 

In the following, we consider the static limit $\Pi^{\nu\nu'}_{00}({\bf q}) = 
\Pi^{\nu\nu'}_{00}({\bf q},\omega_\nu=0)$. Performing the sum over Matsubara
frequency in (\ref{pi1}) in the $T=0$ limit, we obtain 
\begin{eqnarray}
{\Pi^{zz}_{00}({\bf q})} &=& {\int_{\bf k}} \frac{1}{{E_{\bf k}}+{E_{{\bf
	k}+{\bf q}}}} 
\biggl(1 - \frac{{\xi_{\bf k}}{\xi_{{\bf k}+{\bf q}}}}{{E_{\bf k}}{E_{{\bf
	k}+{\bf 
	q}}}} 
+ \frac{{\Delta_0^{\rm HS}}^2}{{E_{\bf k}}{E_{{\bf k}+{\bf q}}}} \biggr) , 
\nonumber \\  
{\Pi^{z+}_{00}({\bf q})}  &=& {\Pi^{z-}_{00}({\bf q})} = {\int_{\bf k}}
	\frac{\Delta_0^{\rm HS}{\xi_{\bf k}}}{({E_{\bf 
      k}}+{E_{{\bf k}+{\bf q}}}){E_{\bf k}}{E_{{\bf k}+{\bf q}}}} ,  
\nonumber \\
{\Pi^{++}_{00}({\bf q})} &=& {\Pi^{--}_{00}({\bf q})} = - {\int_{\bf k}}
	\frac{{\Delta_0^{\rm 
      HS}}^2}{2({E_{\bf k}}+{E_{{\bf k}+{\bf q}}}){E_{\bf k}}{E_{{\bf k}+{\bf
	q}}}} ,  
\nonumber \\
{\Pi^{+-}_{00}({\bf q})} &=& {\int_{\bf k}} \frac{1}{2({E_{\bf k}}+{E_{{\bf
	k}+{\bf q}}})} 
\biggl( 1 + 
\frac{{\xi_{\bf k}}{\xi_{{\bf k}+{\bf q}}}}{{E_{\bf k}}{E_{{\bf k}+{\bf q}}}}
\biggr) . 
\end{eqnarray} 
The correlation function $\Pi^{00}_{\mu\mu'}({\bf q})$ vanishes for ${\bf
q}=0$. Since $j^0_{\mu{\bf r}}$ multiplies $\partial_\mu\theta_{\bf r}$ in the
action $S'$, it is sufficient to consider $\Pi^{00}_{\mu\mu'}({\bf q}=0)$ to
obtain the effective action $S[\rho,\Delta]$ to order $(\partial_\mu
\theta_{\bf r})^2$.  

We next expand the correlations to order ${\cal O}(|{\bf q}|^2)$. Writing
${\xi_{{\bf k}+{\bf q}}}={\xi_{\bf k}} + {X_{{\bf k},{\bf q}}}$ with ${X_{{\bf
      k},{\bf q}}}={\bf k}\cdot{\bf q}/m+|{\bf 
  q}|^2/2m$, we 
obtain
\begin{eqnarray}
{\Pi^{zz}_{00}({\bf q})} &=& {\int_{\bf k}} \biggl[ \frac{{\Delta_0^{\rm
	HS}}^2}{{E^3_{\bf 
	k}}} + \biggl( 
  -\frac{3{\xi_{\bf k}}}{2{E^3_{\bf k}}} + \frac{3\xi^3_{\bf k}}{2E^5_{\bf k}}
	\biggr) 
  {X_{{\bf k},{\bf q}}} 
\nonumber \\ &&  
+ \biggl(-\frac{1}{2{E^3_{\bf k}}} + \frac{3{\xi^2_{\bf k}}}{E^5_{\bf k}} -
\frac{5\xi^4_{\bf k}}{2E^7_{\bf k}} \biggr) {X^2_{{\bf k},{\bf q}}} \biggr] ,
\nonumber \\ 
\Pi^{z\pm}_{00}({\bf q}) &=& {\int_{\bf k}} \frac{\Delta_0^{\rm
    HS}{\xi_{\bf k}}}{2E^3_{\bf k}} 
\biggl( 
  1 -\frac{3{\xi_{\bf k}}}{2{E^2_{\bf k}}} {X_{{\bf k},{\bf q}}} \nonumber \\
	&& 
- \frac{3{E^2_{\bf k}}-10{\xi^2_{\bf k}}}{4E^4_{\bf k}} {X^2_{{\bf k},{\bf
	q}}} \biggl) ,  
\nonumber \\ 
{\Pi^{++}_{00}({\bf q})} &=& {\Pi^{--}_{00}({\bf q})} \nonumber \\ 
&=& - {\int_{\bf k}} \frac{{\Delta_0^{\rm HS}}^2}{4{E^3_{\bf k}}} \biggl( 1 -
\frac{3{\xi_{\bf k}}}{2{E^2_{\bf k}}} {X_{{\bf k},{\bf q}}} \nonumber \\ &&
- \frac{3{E^2_{\bf k}}-10{\xi^2_{\bf k}}}{4{E^2_{\bf k}}} {X^2_{{\bf k},{\bf
	q}}} \biggr)  
\nonumber \\ 
{\Pi^{+-}_{00}({\bf q})} &=& \frac{1}{2} {\int_{\bf k}} \biggl[
	\frac{1}{2{E_{\bf k}}} + 
  \frac{{\xi^2_{\bf k}}}{2{E^3_{\bf k}}} + 
  \biggl( \frac{{\xi_{\bf k}}}{4{E^3_{\bf k}}} - \frac{3{\xi^3_{\bf
	k}}}{4E^5_{\bf k}} 
  \biggr) 
  {X_{{\bf k},{\bf q}}}    
\nonumber \\
&& + \biggl( - \frac{1}{8{E^3_{\bf k}}} - \frac{7{\xi^2_{\bf k}}}{8E^5_{\bf
    k}} + 
    \frac{5\xi^4_{\bf k}}{4E^7_{\bf k}} \biggr) {X^2_{{\bf k},{\bf q}}}
	\biggr] . 
\label{pi2}
\end{eqnarray} 

\subsection{Strong-coupling limit ${\rho_0 a^3}\ll 1$} 

In the strong-coupling limit, the chemical potential $\mu$ is negative. We
then have
\begin{eqnarray}
&& {\int_{\bf k}} \frac{1}{\xi_{\bf k}} = 
\frac{m\Lambda}{\pi^2} - \frac{m^{3/2}|\mu|^{1/2}}{\sqrt{2}\pi} , 
\nonumber \\ &&
\int_0^\infty dk \frac{k^4}{\xi^3_{\bf k}} = 
\frac{3\pi m^{5/2}}{2\sqrt{2}|\mu|^{1/2}} .
\label{B5}
\end{eqnarray}
Other useful relations are obtained by differentiating Eqs.~(\ref{B5})  with
respect to $\mu$. Note that $\Lambda$ is sent to infinity whenever the
integral over ${\bf k}$ converges. In the strong-coupling limit, the
small parameter expansion is ${\Delta_0^{\rm HS}}^2/|\mu|^2\sim
{\rho_0 a^3}$. Approximate 
expressions of the mean-field correlation functions can be obtained by
expanding Eqs.~(\ref{speq1}) and (\ref{pi2}) in power of ${\Delta_0^{\rm
HS}}^2$ and using Eqs.~(\ref{B5}) (as well as those obtained from (\ref{B5})
by differentiating 
with respect to $\mu$). A straightforward (but somewhat lengthly) calculation
then gives Eq.~(\ref{muhf}) and 
\begin{eqnarray}  
{\Pi^{zz}_{00}({\bf q})} &=& \rho_0 m a^2 \biggr(1-4\pi{\rho_0 a^3} -
\frac{{|{\bf q}|}^2 a^2}{4} 
\biggl) , 
\nonumber \\ 
\Pi^{z\pm}_{00}({\bf q}) &=& \biggl( \frac{\rho_0 m^2a}{4\pi} \biggr)^{1/2}
\biggl( 1 - \frac{7}{4} \pi{\rho_0 a^3} - \frac{{|{\bf q}|}^2 a^2}{12} \biggl)
, 
\nonumber \\
{\Pi^{++}_{00}({\bf q})} &=& - \frac{\rho_0 ma^2}{4} \biggl( 1 - 4\pi{\rho_0
  a^3} - 
\frac{5}{16}{|{\bf q}|}^2 
a^2 \biggr) , 
\nonumber \\
{\Pi^{+-}_{00}({\bf q})}  &=& \frac{1}{g} - \frac{\rho_0 ma^2}{4} ( 1 -
4\pi{\rho_0 a^3}) - \frac{ma{|{\bf q}|}^2}{32\pi} .
\end{eqnarray}  


\begin{thebibliography}{99}

\bibitem{Kinast04} J. Kinast, S.L. Hemmer, M.E. Gehm, A. Turpalov, and
  J.E. Thomas, Phys. Rev. Lett. {\bf 92}, 150402 (2004).

\bibitem{Bartenstein04} M. Bartenstein, A. Altmeyer, S. Riedl, S. Joachim,
  C. Chin, J.H. Denschlag, and R. Grimm, Phys. Rev. Lett. {\bf 92}, 120401
  (2004).  

\bibitem{Zwierlein04}  M.W. Zwierlein, C.A. Stan,
  C.H. Schunk, S.M.F. Raupach, A.J. Kerman, and
  W. Ketterle, Phys. Rev. Lett. {\bf 92}, 120403 (2004). 

\bibitem{Regal04} C.A. Regal, M. Greiner, and D.S. Jin, Phys. Rev. Lett. {\bf
  92}, 040403 (2004). 

\bibitem{Bourdel04} T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang,
  F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and
  C. Salomon, Phys. Rev. Lett. {\bf 93}, 050401 (2004). 


\bibitem{Randeria95} For a review on the BCS-Bose crossover in Fermi
  superfluids, see M. Randeria in {\it Bose-Einstein condensation},
  edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge
  University Press, 1995). 

\bibitem{BCS-BEC} For earlier works on the BCS-Bose crossover in
  Fermi superfluids, see D.M. Eagles, Phys. Rev. {\bf 186}, 456
  (1969); A.J. Leggett in {\it Modern Trends in the
  Theory of Condensed Matter}, edited by A. Peralski and J. Przystawa
  (Springer-Verlag, Berlin, 1980); P. Nozi\`eres and S. Schmitt-Rink,
  J. Low Temp. Phys. {\bf 59}, 195 (1985). 

\bibitem{Gross61} E.P. Gross, Nuovo Cimento {\bf 20}, 454 (1961).

\bibitem{Pitaevskii61} L.P. Pitaevskii, Sov. Phys. JETP {\bf 12}, 155
  (1961). 

\bibitem{Feynman72} R.P. Feynman, {\it Statistical Mechanics}
  (Benjamin, 1972).

\bibitem{Ao95} P. Ao, D.J. Thouless, and X.M. Zhu, Mod. Phys. Lett. B
  {\bf 9}, 755 (1995).

\bibitem{Pitaevskii03} See, for instance, L. Pitaevskii and S. Stringari,
  {\it Bose-Einstein condensation} (Oxford University Press, 2003). 

\bibitem{Drechsler92} M. Drechsler and W. Zwerger, Ann. Physik {\bf
  1}, 15 (1992). 

\bibitem{Haussmann93} R. Haussmann, Z. Phys. {\bf 91}, 291 (1993). 

\bibitem{deMelo93} C. A. R. S\'a de Melo, M. Randeria, and
  J. R. Engelbrecht, Phys. Rev. Lett. {\bf 71}, 3202 (1993);
  J. R. Engelbrecht, M. Randeria, and C. A. R. S\'a de Melo,
  Phys. Rev. B {\bf 55}, 15153 (1997). 

\bibitem{Pistolesi96} F. Pistolesi and G.C. Strinati, Phys. Rev. B
  {\bf 53}, 15168 (1996). 

\bibitem{Piery2000}  P. Pieri and G.C. Strinati, Phys. Rev. B 61, 15370
  (2000). 

\bibitem{Pieri03} P. Pieri and G.C. Strinati, Phys. Rev. Lett. {\bf
  91}, 030401 (2003).

\bibitem{Depalo99} S. De Palo, C. Castellani, C. Di Castro, and
  B.K. Chakraverty, Phys. Rev. B {\bf 60}, 564 (1999). 

\bibitem{Dupuis04} N. Dupuis, Phys. Rev. B {\bf 70}, 134502 (2004). 

\bibitem{Stone95} See also M. Stone, Int. J. Mod. Phys. B {\bf 9}, 1359
  (1995); I.J.R Aitchison, P. Ao, D.J. Thouless, and
  X.M. Zhu, Phys. Rev. B {\bf 51}, 6531 (1995). 

\bibitem{fermion-boson} See, for instance, M. Holland, S.J.J.M.F. Kokkelmans,
  M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. {\bf 87}, 120406 (2001); 
  Y. Ohashi and A. Griffin, Phys. Rev. Lett. {\bf 89}, 130402 (2002). 

\bibitem{Falco04} G.M. Falco and H.T.C. Stoof, Phys. Rev. Lett. {\bf 92},
  130401 (2004). 

\bibitem{Simonucci04} S. Simonucci, P. Pieri, and G.C. Strinati,
   Europhys. Lett. {\bf 69}, 713 (2005). 

\bibitem{note4} Here we use the identity
  $c^*_{{\bf r}\uparrow}c^*_{{\bf
  r}\downarrow} c_{{\bf r}\downarrow} c_{{\bf
  r}\uparrow}=\frac{1}{4}[( c^\dagger_{\bf r} c_{\bf r})^2-( c^\dagger_{\bf
  r}\sigma^z c_{\bf r})^2]$.    

\bibitem{note5} An opposite sign for the coefficient of the $\delta\rho_{-q}
  \delta|\Delta|_q$ would give a minus sign in Eq.~(\ref{tied}), i.e. $\delta
  |\Delta_{\bf r}|\delta\rho_{\bf r}<0$. Since $|\Delta_{\bf r}|^2$ is related
  to the density of Cooper pairs, we expect $|\Delta_{\bf r}|$ to
  increase with the fermion density, i.e. $\delta|\Delta_{\bf
  r}|\delta\rho_{\bf r}>0$, in agreement with Eq.~(\ref{tied}). Note also that
  a minus sign in Eq.~(\ref{tied}) would not allow us to recover the bosonic
  action (\ref{action5}).  

\bibitem{Marini98} Mean-field correlation functions and their strong-coupling
  limit have also been computed in M. Marini, F. Pistolesi, and G.C. Strinati,
  Eur. Phys. J. B {\bf 1}, 151 (1998). 

\bibitem{Petrov04}  D.S. Petrov, C. Salomon, and G.V. Shlyapnikov,
  Phys. Rev. Lett. {\bf 93}, 090404 (2004).

\bibitem{note2} In Ref.~\onlinecite{Depalo99}, it was (incorrectly) stated
  that the coefficient $1/32$ in the term $(\boldsymbol{\nabla}\rho_{\bf
  r})^2/32\rho_0 m$ is not 
  obtained from the expansion about the mean-field state, so that the
  action (\ref{action5}) is recovered only in the strict hydrodynamic limit
  where only the gradient term $\rho_{\bf r}({\boldsymbol{\nabla}\theta_{\bf
  r}})^2/8m$ contained in
  $|\boldsymbol{\nabla}{\psi_{\bf r}}|^2/4m$ is retained, while the term
  $(\boldsymbol{\nabla}\rho_{\bf r})^2/32m\rho_{\bf r}$ is ignored.

\bibitem{note3} In our formalism, this approach amounts to integrating out the
  physical field $\Delta_{\bf r}$ from the outset and then carrying out the
  expansion 
  about the non-interacting state. Integrating out the fermions and $
  \rho^{\rm HS}_{\bf r}$, one obtains 
  $|\Delta_{\bf r}^{\rm HS}|=(4\pi\rho_{\bf r}/m^2a)^{1/2}$ and an effective
  action    
  $S[\rho_{\bf r},\theta^{\rm HS}_{\bf r}]$, where $\theta^{\rm HS}_{\bf r}$
  is the phase of the 
  Hubbard-Stratonovich field $\Delta_{\bf r}^{\rm HS}$. When written in terms
  of the field 
  $\sqrt{\rho_{\bf r}/2}e^{i\theta^{\rm HS}_{\bf r}}$, the action takes the
  form (\ref{action5}).   

\bibitem{note1} The derivation of the effective action based on the mapping of
  the attractive Hubbard model onto the repulsive Hubbard model
  presents several interesting features. First, in the repulsive model, the
  difficulties with dealing with both the particle-particle and particle-hole
  channels do not arise since all the important fluctuations pertain to the
  particle-hole channel. Second, the repulsive model allows one to take into
  account on the same footing ${\bf 
  q}=(\pi,\pi)$ density and ${\bf q}=0$ pairing fluctuations near
  half-filling. At half-filling, both types of
  fluctuations are degenerate so that the symmetry of the order parameter is
  enlarged to SO(3). See Ref.~\onlinecite{Dupuis04} for a detailed discussion.
  
\bibitem{Micnas90} R. Micnas, J. Ranninger, and S. Robaszkiewicz,
  Rev. Mod. Phys. {\bf 62}, 113 (1990). 
         
\bibitem{Auerbach94} See, for instance, A. Auerbach, {\it Interacting
  Electrons and Quantum Magnetism} (Springer-Verlag, New York, 1994).  

\end{thebibliography} 


\end{document}


cond-mat0412563.extracted_bib
\begin{thebibliography}{99}

\bibitem{Kinast04} J. Kinast, S.L. Hemmer, M.E. Gehm, A. Turpalov, and
  J.E. Thomas, Phys. Rev. Lett. {\bf 92}, 150402 (2004).

\bibitem{Bartenstein04} M. Bartenstein, A. Altmeyer, S. Riedl, S. Joachim,
  C. Chin, J.H. Denschlag, and R. Grimm, Phys. Rev. Lett. {\bf 92}, 120401
  (2004).  

\bibitem{Zwierlein04}  M.W. Zwierlein, C.A. Stan,
  C.H. Schunk, S.M.F. Raupach, A.J. Kerman, and
  W. Ketterle, Phys. Rev. Lett. {\bf 92}, 120403 (2004). 

\bibitem{Regal04} C.A. Regal, M. Greiner, and D.S. Jin, Phys. Rev. Lett. {\bf
  92}, 040403 (2004). 

\bibitem{Bourdel04} T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang,
  F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and
  C. Salomon, Phys. Rev. Lett. {\bf 93}, 050401 (2004). 


\bibitem{Randeria95} For a review on the BCS-Bose crossover in Fermi
  superfluids, see M. Randeria in {\it Bose-Einstein condensation},
  edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge
  University Press, 1995). 

\bibitem{BCS-BEC} For earlier works on the BCS-Bose crossover in
  Fermi superfluids, see D.M. Eagles, Phys. Rev. {\bf 186}, 456
  (1969); A.J. Leggett in {\it Modern Trends in the
  Theory of Condensed Matter}, edited by A. Peralski and J. Przystawa
  (Springer-Verlag, Berlin, 1980); P. Nozi\`eres and S. Schmitt-Rink,
  J. Low Temp. Phys. {\bf 59}, 195 (1985). 

\bibitem{Gross61} E.P. Gross, Nuovo Cimento {\bf 20}, 454 (1961).

\bibitem{Pitaevskii61} L.P. Pitaevskii, Sov. Phys. JETP {\bf 12}, 155
  (1961). 

\bibitem{Feynman72} R.P. Feynman, {\it Statistical Mechanics}
  (Benjamin, 1972).

\bibitem{Ao95} P. Ao, D.J. Thouless, and X.M. Zhu, Mod. Phys. Lett. B
  {\bf 9}, 755 (1995).

\bibitem{Pitaevskii03} See, for instance, L. Pitaevskii and S. Stringari,
  {\it Bose-Einstein condensation} (Oxford University Press, 2003). 

\bibitem{Drechsler92} M. Drechsler and W. Zwerger, Ann. Physik {\bf
  1}, 15 (1992). 

\bibitem{Haussmann93} R. Haussmann, Z. Phys. {\bf 91}, 291 (1993). 

\bibitem{deMelo93} C. A. R. S\'a de Melo, M. Randeria, and
  J. R. Engelbrecht, Phys. Rev. Lett. {\bf 71}, 3202 (1993);
  J. R. Engelbrecht, M. Randeria, and C. A. R. S\'a de Melo,
  Phys. Rev. B {\bf 55}, 15153 (1997). 

\bibitem{Pistolesi96} F. Pistolesi and G.C. Strinati, Phys. Rev. B
  {\bf 53}, 15168 (1996). 

\bibitem{Piery2000}  P. Pieri and G.C. Strinati, Phys. Rev. B 61, 15370
  (2000). 

\bibitem{Pieri03} P. Pieri and G.C. Strinati, Phys. Rev. Lett. {\bf
  91}, 030401 (2003).

\bibitem{Depalo99} S. De Palo, C. Castellani, C. Di Castro, and
  B.K. Chakraverty, Phys. Rev. B {\bf 60}, 564 (1999). 

\bibitem{Dupuis04} N. Dupuis, Phys. Rev. B {\bf 70}, 134502 (2004). 

\bibitem{Stone95} See also M. Stone, Int. J. Mod. Phys. B {\bf 9}, 1359
  (1995); I.J.R Aitchison, P. Ao, D.J. Thouless, and
  X.M. Zhu, Phys. Rev. B {\bf 51}, 6531 (1995). 

\bibitem{fermion-boson} See, for instance, M. Holland, S.J.J.M.F. Kokkelmans,
  M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. {\bf 87}, 120406 (2001); 
  Y. Ohashi and A. Griffin, Phys. Rev. Lett. {\bf 89}, 130402 (2002). 

\bibitem{Falco04} G.M. Falco and H.T.C. Stoof, Phys. Rev. Lett. {\bf 92},
  130401 (2004). 

\bibitem{Simonucci04} S. Simonucci, P. Pieri, and G.C. Strinati,
   Europhys. Lett. {\bf 69}, 713 (2005). 

\bibitem{note4} Here we use the identity
  $c^*_{{\bf r}\uparrow}c^*_{{\bf
  r}\downarrow} c_{{\bf r}\downarrow} c_{{\bf
  r}\uparrow}=\frac{1}{4}[( c^\dagger_{\bf r} c_{\bf r})^2-( c^\dagger_{\bf
  r}\sigma^z c_{\bf r})^2]$.    

\bibitem{note5} An opposite sign for the coefficient of the $\delta\rho_{-q}
  \delta|\Delta|_q$ would give a minus sign in Eq.~(\ref{tied}), i.e. $\delta
  |\Delta_{\bf r}|\delta\rho_{\bf r}<0$. Since $|\Delta_{\bf r}|^2$ is related
  to the density of Cooper pairs, we expect $|\Delta_{\bf r}|$ to
  increase with the fermion density, i.e. $\delta|\Delta_{\bf
  r}|\delta\rho_{\bf r}>0$, in agreement with Eq.~(\ref{tied}). Note also that
  a minus sign in Eq.~(\ref{tied}) would not allow us to recover the bosonic
  action (\ref{action5}).  

\bibitem{Marini98} Mean-field correlation functions and their strong-coupling
  limit have also been computed in M. Marini, F. Pistolesi, and G.C. Strinati,
  Eur. Phys. J. B {\bf 1}, 151 (1998). 

\bibitem{Petrov04}  D.S. Petrov, C. Salomon, and G.V. Shlyapnikov,
  Phys. Rev. Lett. {\bf 93}, 090404 (2004).

\bibitem{note2} In Ref.~\onlinecite{Depalo99}, it was (incorrectly) stated
  that the coefficient $1/32$ in the term $(\boldsymbol{\nabla}\rho_{\bf
  r})^2/32\rho_0 m$ is not 
  obtained from the expansion about the mean-field state, so that the
  action (\ref{action5}) is recovered only in the strict hydrodynamic limit
  where only the gradient term $\rho_{\bf r}({\boldsymbol{\nabla}\theta_{\bf
  r}})^2/8m$ contained in
  $|\boldsymbol{\nabla}{\psi_{\bf r}}|^2/4m$ is retained, while the term
  $(\boldsymbol{\nabla}\rho_{\bf r})^2/32m\rho_{\bf r}$ is ignored.

\bibitem{note3} In our formalism, this approach amounts to integrating out the
  physical field $\Delta_{\bf r}$ from the outset and then carrying out the
  expansion 
  about the non-interacting state. Integrating out the fermions and $
  \rho^{\rm HS}_{\bf r}$, one obtains 
  $|\Delta_{\bf r}^{\rm HS}|=(4\pi\rho_{\bf r}/m^2a)^{1/2}$ and an effective
  action    
  $S[\rho_{\bf r},\theta^{\rm HS}_{\bf r}]$, where $\theta^{\rm HS}_{\bf r}$
  is the phase of the 
  Hubbard-Stratonovich field $\Delta_{\bf r}^{\rm HS}$. When written in terms
  of the field 
  $\sqrt{\rho_{\bf r}/2}e^{i\theta^{\rm HS}_{\bf r}}$, the action takes the
  form (\ref{action5}).   

\bibitem{note1} The derivation of the effective action based on the mapping of
  the attractive Hubbard model onto the repulsive Hubbard model
  presents several interesting features. First, in the repulsive model, the
  difficulties with dealing with both the particle-particle and particle-hole
  channels do not arise since all the important fluctuations pertain to the
  particle-hole channel. Second, the repulsive model allows one to take into
  account on the same footing ${\bf 
  q}=(\pi,\pi)$ density and ${\bf q}=0$ pairing fluctuations near
  half-filling. At half-filling, both types of
  fluctuations are degenerate so that the symmetry of the order parameter is
  enlarged to SO(3). See Ref.~\onlinecite{Dupuis04} for a detailed discussion.
  
\bibitem{Micnas90} R. Micnas, J. Ranninger, and S. Robaszkiewicz,
  Rev. Mod. Phys. {\bf 62}, 113 (1990). 
         
\bibitem{Auerbach94} See, for instance, A. Auerbach, {\it Interacting
  Electrons and Quantum Magnetism} (Springer-Verlag, New York, 1994).  

\end{thebibliography}