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Phase diagram of a strongly interacting polarized fermi gas in one dimension

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This work identifies the polarized superfluid as having an Fulde-Ferrell-Larkin-Ovchinnikov structure, and predicts the resulting mode frequency as a function of the spin polarization, and identifies the ground state of a one-dimensional trapped Fermi gas with imbalanced spin population.
Abstract
Based on the integrable Gaudin model and local density approximation, we discuss the ground state of a one-dimensional trapped Fermi gas with imbalanced spin population, for an arbitrary attractive interaction. A phase separation state, with a polarized superfluid core immersed in an unpolarized superfluid shell, emerges below a critical spin polarization. Above it, coexistence of polarized superfluid matter and a fully polarized normal gas is favored. These two exotic states could be realized experimentally in highly elongated atomic traps, and diagnosed by measuring the lowest density compressional mode. We identify the polarized superfluid as having an Fulde-Ferrell-Larkin-Ovchinnikov structure, and predict the resulting mode frequency as a function of the spin polarization.

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Phase Diagram of a Strongly Interacting Polarized Fermi Gas in One Dimension
Hui Hu,
1,2
Xia-Ji Liu,
2
and Peter D. Drummond
2
1
Department of Physics, Renmin University of China, Beijing 100872, China
2
ARC Centre of Excellence for Quantum-Atom Optics, Department of Physics, University of Queensland, Brisbane,
Queensland 4072, Australia
(Received 17 October 2006; published 14 February 2007)
Based on the integrable Gaudin model and local density approximation, we discuss the ground state of a
one-dimensional trapped Fermi gas with imbalanced spin population, for an arbitrary attractive interac-
tion. A phase separation state, with a polarized superfluid core immersed in an unpolarized superfluid
shell, emerges below a critical spin polarization. Above it, coexistence of polarized superfluid matter and a
fully polarized normal gas is favored. These two exotic states could be realized experimentally in highly
elongated atomic traps, and diagnosed by measuring the lowest density compressional mode. We identify
the polarized superfluid as having an Fulde-Ferrell-Larkin-Ovchinnikov structure, and predict the
resulting mode frequency as a function of the spin polarization.
DOI: 10.1103/PhysRevLett.98.070403 PACS numbers: 05.30.Fk, 03.75.Ss, 71.10.Pm, 74.20.Fg
Strongly attractive Fermi gases with imbalanced spin
components are common in different branches of physics
[18], so spin-polarized ultracold atomic gases are an
atomic analog of many other exotic forms of matter. The
true ground state of an attractive polarized Fermi gas
remains elusive, because the standard Bardeen-Cooper-
Schrieffer (BCS) mechanism requires the pairing of two
fermions with opposite spin. Polarized Fermi gases cannot
be explained by the BCS theory, due to mismatched Fermi
surfaces. Various exotic forms of pairing have been pro-
posed, including the deformed Fermi surface [3], interior
gap pairing [4] or Sarma superfluidity [5], phase sepa-
ration [6], and the inhomogeneous Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) state [7].
Recent measurements on polarized
6
Li gases [1,2] near a
Feshbach resonance provide a route towards testing these
theories in experiment. However, the presence of a har-
monic trap in these experiments makes it difficult to iden-
tify which if any of the different pairing schemes occurs.
Theoretically, it is desirable to have an exactly soluble
mode of polarized uniform Fermi gases to identify various
pairing schemes, and clarify the issue of the trap.
In the present Letter we report on the exact ground state
of a homogeneous 1D polarized Fermi gas [911] with
attractive intercomponent interactions at zero temperature.
We then study the phase diagram of an inhomogeneous
Fermi gas under harmonic confinement, using the local
density approximation (LDA). We complement this with a
mean-field Bogoliubovde Gennes (BdG) theory in the
weak-coupling limit, where the phase fluctuations are
small, in order to clarify the physical meaning of the
solutions. Collective mode frequencies are also calculated
as an experimental diagnostic, thus extending previous
work on the unpolarized case [12 14].
As well as being a theoretical test bed for the ground-
state issue, 1D Fermi gases in traps can be realized using
two-dimensional optical lattices, where the radial motion
of atoms is frozen out. Thus, one can experimentally check
these quantum many-body predictions, which has also
been recently carried out for the 1D Bose gas [15].
The following remarkable features are found: (i) In the
ground state of a uniform system, we find three distinct
phases with increasing chemical potential difference be-
tween species: an unpolarized BCS superfluid, a polarized
superfluid, and a fully polarized normal state. The polar-
ized superfluid is most widespread, and reduces to the
FFLO-type for weak interactions. Therefore, it is relatively
easy to observe the FFLO physics in one dimension, as
anticipated in previous approximate studies [16]. In earlier
work the phase diagram was not conclusive, as the nature
of the transition from BCS to FFLO states was under
debate [16]. (ii) Within the LDA, we consider the inhomo-
geneous phase diagram of the trapped gas. This leads to a
phase separation, as the inhomogeneous cloud separates
into either a mixture of a polarized superfluid and an
unpolarized superfluid, or a polarized superfluid and a fully
polarized normal gas. (iii) We calculate the longitudinal
size of the two spin components and the frequency of the
lowest density compressional mode. These quantities give
a measurable fingerprint of the whole phase diagram.
We describe a 1D polarized Fermi gas with N N
"
N
#
fermions each of mass m and spin polarization P
N
"
N
#
=N > 0 in a harmonic trap, by
H H
0
X
N
i1
1
2
m!
2
ho
x
2
i
; (1)
where
H
0

@
2
2m
X
N
i1
@
2
@x
2
i
g
1D
X
N
"
i1
X
N
#
j1
x
i
x
j
(2)
is the Hamiltonian of Gaudin model of a spin 1=2 Fermi
gas attracting via a short range potential g
1D
x [9]. The
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coupling constant g
1D
(<0) can be expressed through the
1D scattering length a
1D
, g
1D
2@
2
=ma
1D
. A two-
fermion bound state arises once N
#
> 0, with binding
energy
b
@
2
=ma
2
1D
.
In the absence of the harmonic trap, the integrable
Gaudin model, Eq. [9], can be solved exactly using
Bethe’s ansatz [9,10]. Introducing linear number densities,
n N=L and n
N
=L ( " , # ), where L is the size
of the system, the uniform gas is characterized by the
polarization p n
"
n
#
=n > 0 and a dimensionless pa-
rameter mg
1D
=@
2
n2=na
1D
> 0. The ground
state is obtained by numerically solving the Gaudin inte-
gral equations [9,10]. We have clarified the physical nature
of the resulting solutions by also solving the weak-
coupling mean-field BdG equations.
Figure 1 shows the energy per particle, E
hom
=N, the
chemical potential of spin up fermions,
hom;"
@E
hom
=@N
"
, and the chemical potential difference,

hom
@E
hom
=@N
"
N
#
, as a function of polarization
p at two intermediate interaction strengths 0:5 and
2:0. The mean-field calculations lead to the same
general behavior. Their asymptotic behavior in the weak
and strong coupling limits may be understood analytically.
Of particular interest is the chemical potential difference.
In the weakly interacting limit of 1 , 
hom
is given by
( maxfp; 1 pg)

hom
@
2
n
2
2m
2
2
p
@
2
n
2
2m
p ; (3)
where the first term on the right-hand side is the Fermi
energy difference of an ideal polarized gas, and the second
term arises from the mean-field Hartree-Fock interactions.
The chemical potential difference increases with increas-
ing , and reaches a half of the binding energy of bound
states in the strongly attracting limit of !1,

hom
b
2
@
2
n
2
2m
2
1p
2
16
@
2
n
2
2m
2
p
2
: (4)
The first two terms in magnitude coincide with the chemi-
cal potential of a Tonks-Girardeau bosonic gas of paired N
#
dimers [12,13], which is fermionized due to strong attrac-
tions, while the third term is equal to the chemical potential
of residual unpaired N
"
N
#
noninteracting fermions.
Therefore, in the strong coupling regime the system be-
haves like a coherent mixture of a molecular Bose gas and
fully polarized single-species Fermi sea.
We analyze the phase structure of the ground state of
uniform polarized Fermi gases (Fig. 2). Given an interac-
tion strength the chemical potential difference at finite
polarization p is bounded by two critical values, 
c;p0
and 
c;p1
, as indicated by arrows in Fig. 1(b) for
2:0. Below 
c;p0
, the gas remains in the BCS-like
superfluid state with zero polarization (SF), while above

c;p1
, a fully polarized normal state is favored (N). In
between, a superfluid state with finite polarization (SF
P
)
dominates. The mean-field calculation of a uniform gas
shows that the SF
P
is of FFLO character, as the exactly
soluble ground-state energy corresponds precisely with the
FFLO solution in the weak-coupling limit. Physically

c;p0
is the energy cost required to break pairs in un-
polarized superfluid, i.e., the spin gap [13], while 
c;p1
is also associated with the pair-breaking (for the last pair),
but is promoted upwards due to the Pauli repulsion from
existing fermions. The dependence of 
c;p0
and

c;p1
on the parameter is reported in Fig. 2, resulting
in a homogeneous phase diagram.
In units of @
2
n
2
=2m, these have the following limiting
behavior: 
c;p0
2


p
exp
2
=2 and 
c;p1
2
=2 as ! 0, while 
c;p0
b
=2
2
=16 and

c;p1
b
=2
2
as !1. Both critical chemical
potential differences saturate to the half of the binding
energy in the strong coupling limit. Converting into the
chemical potential
hom
@E
hom
=@N, we can obtain the
phase diagram in the
hom
-
hom
plane.
FIG. 1 (color online). The energy per particle, chemical po-
tential for spin up atoms (a), and chemical potential difference
(b) in units of @
2
n
2
=2m as a function of the spin polarization at
two coupling parameters 0:5 and 2:0.
FIG. 2 (color online). Uniform phase diagram, displaying
‘normal’ (N), unpolarized superfluid (SF), and polarized super-
fluid states (SF
P
). At weak coupling or high density, the pre-
dominant SF
P
phase corresponds to the mean-field FFLO
solution. The dashed and dotted lines are asymptotic behavior
as described in the text.
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We now turn to describe the 1D polarized gas in a
harmonic trap. We partition the system into cells that can
be treated locally as being uniform, with a local chemical
potential. This LDA is applicable provided that the number
of fermions in a cell is much greater than unitary, and the
variation of the trap potential across the cell is small
compared with the local Fermi energy and hence the inter-
face effects are negligible [17]. Overall it requires a suffi-
cient large local density, which implying N 1,a
condition readily satisfied in the 1D experiment. Note
that the breakdown of LDA has been observed in the
elongated 3D trap [17], when the linear density in the
transverse axis becomes small. The LDA amounts to de-
termining the chemical potential 
"
#
=2 and the
chemical potential difference  
"
#
=2 of the
inhomogeneous gas from the local equilibrium condition,
hom;
nx;px
1
2
m!
2
ho
x
2
: (5)
The normalization conditions are N
R
nxdx and NP
R
nxpxdx, where nxis the total linear number density
and px the local spin polarization. By rescaling the
chemical potentials, coordinate and linear density into
dimensionless form [ ~
=
b
,
~
x x=a
2
ho
=a
1D
and
~
n na
1D
], the normalization equations can be rewritten as
Na
2
1D
=a
2
ho
R
~
n
~
xd
~
x and Na
2
1D
=a
2
ho
P
R
~
n
~
xp
~
xd
~
x.
These expressions emphasize that the dimensionless cou-
pling constant in a trap is controlled by Na
2
1D
=a
2
ho
, where
Na
2
1D
=a
2
ho
1 corresponds to weak coupling while
Na
2
1D
=a
2
ho
1 corresponds to the strongly interacting
regime.
The qualitative feature of density profiles n
x is sim-
ple to understand. Within the LDA, the local chemical
potential x decreases parabolically away from the cen-
ter of the trap while the local chemical potential difference
x stays constant. It is then clear from Fig. 2 that with a
nonzero P we always have a polarized superfluid in the
center where the local chemical potential (interaction pa-
rameter) is large (small). Away from the center with de-
creasing local chemical potential, the Fermi gas goes into
either an unpolarized superfluid ( <
b
=2) or a fully
polarized normal cloud ( >
b
=2). Thus, there is a
critical chemical potential difference 
c
b
=2 that
separates the inhomogeneous system into two phase sepa-
ration states: a mixture of a polarized superfluid core and
an unpolarized superfluid shell (SF
P
-SF), or a coexistence
of a polarized superfluid at the center and a fully polarized
normal gas outside (SF
P
-N).
The former phase is exotic, as the BCS-like superfluid
state occurs at the edge of the trap, in marked contrast to
the 3D case [1]. This is caused by the peculiar effect of low
dimensionality, for which the gas becomes more nonideal
with decreasing 1D density towards the edge of the trap,
and hence the energy required to break the pairs ap-
proaches
b
=2 from below. As  <
b
=2, there should
be a fully paired region once the local critical chemical
potential 
c;p0
>, i.e., the BCS-like superfluid.
We show the density profile of each component in Fig. 3
with a typical coupling parameter Na
2
1D
=a
2
ho
1. In addi-
tion, we have performed a BdG calculation for the trapped
gas and observe a FFLO-N phase. The resulting density
profiles on the weak-coupling side are in perfect agreement
with the LDA calculation, indicating unambiguously that
the SF
P
phase is a FFLO state.
We determine the phase diagram of the inhomogeneous
polarized 1D Fermi gas as a function of the interaction
strength and spin polarization by calculating the critical
polarization P
c
that corresponds to 
c
as a function
of the coupling constant, and plot it in Fig. 4(a). The
asymptotic behavior of P
c
can be computed analytically
in the weak and strong coupling limits. We find that P
c
1=Na
2
1D
=a
2
ho
as Na
2
1D
=a
2
ho
!1, and P
c
1=5
256=225
2
0:4Na
2
1D
=a
2
ho
1=2
as Na
2
1D
=a
2
ho
! 0.
We consider the experimental relevance of the two phase
separation states, by calculating the size of the cloud and
the lowest density compressional mode. These are readily
detectable via absorption imaging. Figure 4(b) reports the
evolution of the Thomas-Fermi radius of two spin compo-
nents as a function of polarization at three different inter-
action couplings. The radius for spin up and down fermions
is the same in the SF
P
-SF phase, but diverges in opposite
directions in the SF
P
-N phase.
Using a sum-rule approach, the frequency ! of the
lowest density and spin density compressional (breathing)
modes of 1D trapped gases can be calculated from the
identity @
2
!
2
m
1
=m
1
, where m
1
hF
; H ;Fi=2
and m
1
F=2 are two energy-weighted moments
of the breathing operator F cos
P
i
x
2
i"
sin
P
j
x
2
j#
,
with a mixing angle =2 <<=2. The linear static
response of the system can be calculated in terms of the
susceptibility matrix @
2
E
hom
=@n
@n
0
. Basically these two
different breathing modes correspond to the in-phase and
out-of-phase oscillations of the two imbalanced spin pop-
FIG. 3 (color online). Density profiles of each spin component
and their difference at a typical interaction coupling of Na
2
1D
=
a
2
ho
1 for the SF
P
-SF phase (a) and the SF
P
-N phase (b).
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ulations. They are decoupled for a mixing angle
in
> 0 or
out
< 0 that minimizes the mode frequency, analogous to
the spin-charge separation in 1D. Then, the operators F
with these angles are anticipated to exhaust all the weights
in the density and spin density channels, respectively.
Thus, the sum-rule approach is well applicable, providing
only an upper bound on the mode frequency. A stringent
test of this single mode approximation merits further study,
e.g., by using the random-phase approximation theory.
Figure 5 shows the density breathing mode frequency as
a function of polarization at several interaction strengths.
With increasing polarization the frequency initially rises in
the SF
P
-SF state and then gradually decreases to the ideal
gas result 2!
ho
in the SF
P
-N phase. A peak structure is
found that gives an independent means of identifying the
FFLO phase, which dominates the phase diagram in the
vicinity of this peak.
We emphasize that the FFLO superfluid reported here
can be detected via a Josephson junction that is formed by
confining two 1D polarized gases in a double-well trap.
There is also a signature present in the density correlation
function, which has an oscillatory behavior in this phase.
Further details will be provided elsewhere.
In conclusion, we have investigated a 1D polarized
Fermi gas in a harmonic trap, and have shown that the
trap generally gives rise to phase separation: with at least
one FFLO-type phase present. Two distinct exotic phase-
separated structures can occur, and are detectable via ab-
sorption imaging and collective mode experiments.
We acknowledge fruitful discussions with Dr. X.-W.
Guan. This work is supported by the Australian Research
Council Center of Excellence and by the National Science
Foundation of China under Grant No. NSFC-10574080 and
the National Fundamental Research Program under Grant
No. 2006CB921404.
Note added.After this work was completed, we be-
came aware of a related work of G. Orso [18].
[1] M. W. Zwierlein et al., Science 311, 492 (2006); Nature
(London) 442, 54 (2006); Y. Shin et al., Phys. Rev. Lett.
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¨
ther and A. Sedrakian, Phys. Rev. Lett. 88, 252503
(2002).
[4] W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002
(2003).
[5] G. Sarma, J. Phys. Chem. Solids 24, 1029 (1963); C.-H.
Pao, S.-T. Wu, and S.-K. Yip, Phys. Rev. B 73, 132506
(2006); H. Hu and X.-J. Liu, Phys. Rev. A 73, 051603(R)
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47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].
[8] For reviews, see, for example, R. Casalbuoni and
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[13] J. N. Fuchs, A. Recati, and W. Zwerger, Phys. Rev. Lett.
93, 090408 (2004).
[14] X.-J. Liu, P. D. Drummond, and H. Hu, Phys. Rev. Lett.
94, 136406 (2005).
[15] H. Moritz et al., Phys. Rev. Lett. 91, 250402 (2003); K. V.
Kheruntsyan et al., ibid. 91, 040403 (2003); P. D.
Drummond et al., ibid. 92, 040405 (2004).
[16] K. Yang, Phys. Rev. B 63, 140511(R) (2001);
H. Shimahara, Phys. Rev. B 50, 12 760 (1994).
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070402 (2006); A. Imambekov et al., Phys. Rev. A
74,
053626 (2006).
[18] G. Orso, Phys. Rev. Lett. 98, 070402 (2007).
FIG. 5 (color online). Square of the lowest density breathing
mode frequency, !
2
, as a function of the spin polarization for
interaction parameters as indicated. The inset shows the mixing
angle that minimizes the sum-rule mode frequency at
Na
2
1D
=a
2
ho
1:0.
FIG. 4 (color online). (a) Phase diagram of an inhomogeneous
1D polarized Fermi gas. The dashed and dotted lines are asymp-
totic limits described in the text. (b) The Thomas-Fermi radius of
each spin component as a function of the polarization at
Na
2
1D
=a
2
ho
0:01, 1, and 100.
PRL 98, 070403 (2007)
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Q1. What contributions have the authors mentioned in the paper "Phase diagram of a strongly interacting polarized fermi gas in one dimension" ?

Based on the integrable Gaudin model and local density approximation, the authors discuss the ground state of a one-dimensional trapped Fermi gas with imbalanced spin population, for an arbitrary attractive interaction.