1 /27

2022.11.2 CLUSHIQ2022

Feasibility of the non-uniform FFLO superfluid Fermi atomic gas
Taira Kawamura and Yoji Ohashi, Phys. Rev. A 106, 033320 (2022)

Taira Kawamura (Keio University)
Yoji Ohashi (Keio University)

Contents
1. Introduction

2. Formalism

3. Results

4. Summary

2 /27

Contents
1. Introduction

2. Formalism

3. Results

4. Summary

3 /27

Ultracold Fermi Gas

4 /27

Powerful tool to experimentally study strongly correlated quantum many-body systems

https://ultracold.phys.virginia.edu/public_html/

Tunable interaction associated with a Feshbach resonance
第 1 章 序論

C. A. Regal, et al., PRL 90, 230404 (2003)

pseudospin
σ=↑
擬スピン↑

Feshbach molecule
Feshbach分子40↑

H. Tajima (doctor thesis)

Temperature T

Fermi liquid

−U
↑

K

Superfluid
(BCS)

擬スピン↓
σ=↓

hbach 共鳴を表した模式図。2 つの異なる擬スピン状態 (超微細構造) の Fermi 原子が鳴分子ボソン (Feshbach 分子) を形成する。このときに生じる原子間相互作用を、外部

実験的に制御することができる。

External magnetic field

Molecular
Bose gas(BEC)

weak-couplingstrong-coupling
Attractive interaction strength
BCS regime
BEC regime

Superconductivity in Different Hierarchies
Ultracold Fermi Gas

Nuclear Matter

Temperature T

Neutron Star

5 /27

Quark Matter
Rep. Prog. Phys. 74 (2011) 014001
K Fukushima and T Hatsuda
K. Fukushima and T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011)

Molecular
Bose gas

Fermi liquid

2SC(2-flavor superconductor)

Tc
Superfluid

Attractive Interaction Strength

https://astronomy.com/news/2020/03/
how-big-are-neutron-stars
Figure 1. Conjectured QCD phase diagram with boundaries that define various states of QCD matter based on Sχ B patterns.

pp

pseudospin
σ=↑

−p

−p
σ=↓

p

Fermi surfaceσ=↓

Neutronand sets a natural scale for the critical temperature of light flavours it is appropriate to address more on the chiral
σchiral=restoration.
↑
In the chiral perturbation theory (χ PT) phase transition. Recent analyses on the basis of the staggeredd-quarkp

the chiral condensate for two massless quark flavours at
low temperature is known to behave as !ψ̄ψ"T /!ψ̄ψ" =1 − T 2 /(8fπ2 ) − T 4 /(384fπ4 ) − · · · with the pion decay
constant fπ $ 93 MeV [29]. Although the validity ofχ PT is limited to low temperature, this is clear evidenceof the melting of chiral condensate at a finite temperature.
At low baryon density, likewise, the chiral condensate
decreases as !ψ̄ψ"nB /!ψ̄ψ" = 1 − σπ N nB /(fπ2 m2π ) − · · ·[30–32] where σπ N ∼ 40 MeV is the π–N sigma term.
(For higher order corrections, see [33, 34].)
The chiral transition is a notion independent of the(low density)
deconfinement transition. In section 3.2 we classify thechiral transition according to the Sχ B pattern.

u-quark−p

fermion and Wilson fermion indicate a crossover from thehadronic phase to the quark–gluon plasma for realistic u, dand s quark masses [38, 39]. The pseudo-critical temperature
Tpc , which characterizes the crossover location, is likely to bewithin the range 150–200 MeV as summarized in section 4.2.
0
Even for the temperature above Tpc the system may bestrongly correlated and Bardeen,show non-perturbative
phenomena
Cooper,Schrieffer (1957)
such as the existence of hadronic modes or pre-formed
hadrons in the quark–gluon plasma at µB = 0 [28, 40]
as well as at µB (= 0 [41–43]. Similar phenomena canbe seen in other strong-coupling systems such as the hightemperature superconductivity and in the BEC regime of(2SC)
ultracold fermionic atoms [44].

Δ(x) = Δ

Uniform Fermi superfluid (BCS state)

QCD critical points.

In the density region beyond µ ∼ TNon-uniform Superconductivity in Different Hierarchies
Ultracold Fermi Gas

Nuclear Matter

Temperature T

Neutron Star

6 /27

Quark Matter
Rep. Prog. Phys. 74 (2011) 014001
K Fukushima and T Hatsuda
K. Fukushima and T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011)

Molecular
Bose gas

Fermi liquid

2SC(2-flavor superconductor)

Tc
Superfluid

Attractive Interaction Strength

https://astronomy.com/news/2020/03/
how-big-are-neutron-stars
Figure 1. Conjectured QCD phase diagram with boundaries that define various states of QCD matter based on Sχ B patterns.

pσ=↑neutron

Q≠0

Q≠0

σ=↓

protonSpin imbalanceand sets a natural scale for the critical temperature ofchiral restoration. In the chiral perturbation theory (χ PT)
the chiral condensate for two massless quark flavours at
low temperature is known to behave as !ψ̄ψ"T /!ψ̄ψ" =1 − T 2 /(8fπ2 ) − T 4 /(384fπ4 ) −Q
· · ·≠
with0 the pion decay
constant fπ $ 93 MeV [29]. Although the validity ofχ PT is limited to low temperature, this is clear evidenceof the melting of chiral condensate at a finite temperature.
At low baryon density, likewise, the chiral condensate
decreases as !ψ̄ψ"nB /!ψ̄ψ" = 1 − σπ N nB /(fπ2 m2π ) − · · ·[30–32] where σπ N ∼ 40 MeV is the π–N sigma term.
(For higher order corrections, see [33, 34].)
The chiral transition is a notion independent of thedeconfinement transition. In section 3.2 we classify thechiral transition according to the Sχ B pattern.

Isospin asymmetryu-quark

light flavours it is appropriate to address more on the chiralphase transition. Recent analyses on the basis of the staggered
fermion and Wilson fermion indicate a crossover from thehadronic phase to the quark–gluon plasma for realistic iQxu, dand s quark masses [38, 39]. The pseudo-critical temperature0
Tpc , which characterizes the crossover location, is likely to be
Fulde,
Ferrell(1964)
within the range 150–200 MeV assummarized
in section
4.2.
Even for the temperature above Tpc the system may bestrongly correlated and show non-perturbative phenomena
such as the existence of hadronic modes or0pre-formed
hadrons in the quark–gluon
plasma Ovchinnikov
at µB = 0 [28, 40]
Larkin,(1965)
as well as at µB (= 0 [41–43]. Similar phenomena canbe seen in other strong-coupling systems such as the hightemperature superconductivity and in the BEC regime ofultracold fermionic atoms [44].

d-quark

Mismatched Fermi surfaces
Δ(x) = Δ e

Δ(x) = Δ cos(Qx)Electrical and color neutrality

Non-uniform Fermi superfluid (FFLO state)

QCD critical points.

In the density region beyond µ ∼ T

FFLO State in a Spin-imbalanced Ultracold Fermi Gas
Spin-imbalanced Fermi gas (N↑ ≠ N↓)

Theory (Mean-field approximation)
M. M. Parish et. al., Nature Physics 3, 124 (2007)
G. C. Strinati et. al., Phys. Rep. 738, 1 (2018)

polarization(N↑ > N↓)

Temperature T

Fermi liquid

7 /27

⋮

P = [N↑ − N↓]/[N↑ + N↓]

Molecular
Bose gas

Tc
Superfluid

Attractive Interaction Strengthpolarization P

The FFLO state has not been observed experimentally in a 2D and 3D Fermi gas.

Difficulty (I) : Destruction of the FFLO state by pairing fluctuations

8 /27

In isotropic systems, the FFLO-type non-uniform superfluid state is unstable against pairing fluctuations at nonzero temperature- H. Shimahara, J. Phys. Soc. Jpn. 67, 1872 (1998)
even in a 3D system.
- Y. Ohashi, J. Phys. Soc. Jpn. 71, 2625 (2002)- J. Wang et. al., Phys. Rev. B 97, 134513 (2018)- P. Zdybel et. al., Phys. Rev. A 104, 063317 (2021)

gas system(continuous rotational symmetry)⋮

This mean-field phase diagram is completely wrong !
(as pF)−1 = − 1

Q↑infinite degeneracy

Large fluctuations caused by an infinite degeneracy
with respect to the direction of the FFLO Q vector
destroy the FFLO-type long-range order.

0.12

Temperature T/TF↑

NormalUniform Superfluid
(BCS)

0.08

0.04

Phase Separation
0

0

0.1
0.2
Polarization P = [N↑ − N↓]/[N↑ + N↓]

Non-uniform Superfluid(FFLO)

Difficulty (I) : Destruction of the FFLO state by pairing fluctuations

9 /27

In isotropic systems, the FFLO-type non-uniform superfluid state is unstable against pairing fluctuations at nonzero temperature- H. Shimahara, J. Phys. Soc. Jpn. 67, 1872 (1998)
even in a 3D system.
- Y. Ohashi, J. Phys. Soc. Jpn. 71, 2625 (2002)- J. Wang et. al., Phys. Rev. B 97, 134513 (2018)- P. Zdybel et. al., Phys. Rev. A 104, 063317 (2021)

gas system(continuous rotational symmetry)

Q

Q

Laser

↑

↑

⋮

Optical lattice system(discrete rotational symmetry)

Explicitly breaking
the continuous rotational symmetry

Atom
Periodic potential made by optical interference

infinite degeneracy

finite degeneracy

http://www.kozuma.phys.titech.ac.jp/
research_category/entry7.html

depletion in the central region. The size of this inner corethe ‘‘visibility’’ " of the core region (Fig. 5). The sudden
decreases for increasing imbalance, and the core shows adrop of " around ! # !c results from the sudden jump of
distinctive boundary until it disappears for large imbalnd0 . The comparison between Rp and Rc shows that the
10 /27ance. We observe this shell structure even for very small
boundary layer between the superfluid and the normal
imbalances, down to 5!3"%, which excludes a homogeneregion is rather thin. It has been suggested that the detailed
ous superfluid state at this low imbalance, contrary to theshape of profiles in the intermediate region could be usedconclusions
in Ref. into
[6]. the BCS uniform superfluidtoand
In a spin-imbalanced Fermi gas, phaseseparation
theexotic
polarized
normalcan occur.
identify
states suchas the gas
Fulde-Ferrell-LarkinThe reconstructed 3D profile of the density difference
Ovchinnikov
state[23–25].
This
will 91,
be a247002
subject (2003)
of futureP.
F.
Bedaque
et.
al.,
PRL
shows that the two components in the core region haveresearch.
⋮equal densities. We reconstruct 3D profiles from the 2D
Experiments↑
density difference usdistributions n~d !x; z" of the column
ing the inverse Abel transformation (Fig. 3) [20]. The‣onMIT(2006)of the local
reconstructiondoes
notdependthevalidity↑
density approximation (LDA) or Y.
a harmonic
Shin et.al.,approximaPRL 97,
030401 (2006)⇒

Difficulty (II) : Competition with phase separation (BCS+Normal)

Phase separation (BCS + Normal)
↑

Q=0
↑

BCS-type Cooper pairs+

FIG. 3. Reconstruction of 3D distributions from their inte‣ Rice (2006)
grated 2D distributions. (a) An integrated 2D distribution n~dwith! & 58% at B & 834 G. (b) A less noisy distribution
was
n
G.
B.
Partridge
et.al.,
Science311,↑↑ FIG. 2. In situ direct imaging of trapped Fermi gases for obtained by averaging four quadrants with respect to the central
5760 (2006)
dashed lines in (a). (d) The 1D profile obtained by integrating thevarious population imbalance !. The integrated
2D distributions
Raveraged distribution along the z direction shows then ‘‘flattop’’of the density difference n~d !x; z" % nd !~r"dy were measured↓
feature. (c) The 2D cut nd !x; y & 0; z" of the 3D distribution
using phase-contrast imaging at B & 834 G for total atom numnd !~r" was reconstructed by applying the inverse Abel transber Nt # 1 ' 107 . For ! ( 75%, a distinctive core was observed
formation to (b). (e) The radial profile of the central section of
Polarized
normalgas
showing
the shell
structure of the cloud. The field of view forδn dashed
the reconstructed 3D distribution in the xy plane. The
each image is 160 #m ' 800 #m. The three leftmost images arelines in (d) and (e) are fits to the profiles’ wings using a TF
displayed with different contrast levels for clarity. The image
distribution.
with ! & 5!3"% was taken for Nt # 1:7 ' 107 .

Thus, even if the destruction of the FFLO state by pairing fluctuations can be removed in an optical lattice, we still need to overcome the030401-2
competition between the FFLO state and the unwanted phase separation.

This work

11 /27(1) Is it possible to stabilize the FFLO state against pairing fluctuations by introducing an optical lattice ?
Q

gas system↑

lattice system

Q↑
Explicitly breaking
the continuous rotational symmetry

infinite degeneracy

finite degeneracy(2) Can the FFLO state be stabilized against the phase separation (BCS + Normal) in an optical lattice ?
FFLO statepPhase separation (BCS + Normal)

σ=↑
Q≠0

σ=↓

Q=0

vs.

⇒

+
BCS-type Cooper pairs

polarized normal gas

Contents
1. Introduction

2. Formalism

3. Results

4. Summary

12 /27

Nozières and Schmitt-Rink (NSR) Theory

13 /27

Hamiltonian (3D attractive Hubbard model)⇒
Mean-field Hamiltonian HFL‣ Superfluid order parameter

Fluctuations in the Cooper channel HFL‣ Hartree energy ・effective chemical potential

Fulde-Ferrell-type order parameter‣ Nambu field
τi : Pauli matrix in Nambu space‣ Single-particle energy(nearest-neighbor hopping)Nozières and Schmitt-Rink (NSR) Theory

14 /27

Hamiltonian (3D attractive Hubbard model)

Mean-field hamiltonian HMF

Fluctuations in the Cooper channel HFL

Thermodynamic potential

ER 2. STRONG COUPLING FORMALISM FOR ATTRACTIVELYINTERACTING FERMI SYSTEMS

Phase transition
ΩMF

T > TcT = Tc2nd order phase transition (Thouless criterion)
BCS

T < TcΩFL!U= +

=
τ

^

τj

Π̂

+

+

−U

ΩMF

h > hcNumber equation

NSR corrections ΩNSR to the thermodynamic potential Ω in the superbelow T c . Π̂i j is the (i j) component of the particle-particle correlation
Eq.(2.51).
Fluctuation correction

FFLO

Δ

h = hcΔh < hc1st order phase transition

Contents
1. Introduction

2. Formalism

3. Results

4. Summary

15 /27

FFLO State in a Gas System (without an optical lattice)1(neglecting pairing fluctuations)temperature T/Tc0

Mean-field approximation

normalBCS

0.8(as pF)−1 = − 1(weak-coupling BCS regime)

0.6

0.4

0.2

0
0

FFLO
0.05

0.1

0.15

0.2

polarization P = [N↑ − N↓]/[N↑ + N↓]

0.25

16 /27

FFLO State in a Gas System (without an optical lattice)1(neglecting pairing fluctuations)temperature T/Tc0

Mean-field approximation

normalBCS

0.8(as pF)−1 = − 1(weak-coupling BCS regime)

0.6

0.4

0.2

0
0

FFLO
0.05

0.1

0.15

0.2

0.25

1

NSR theory(including pairing fluctuations)

BCS

temperature T/Tc0

0.8

normal
0.6

0.4

0.2

0
0

0.02

0.04

0.06

polarization P = [N↑ − N↓]/[N↑ + N↓]

0.08

17 /27

FFLO State in a Gas System (without an optical lattice)1(neglecting pairing fluctuations)temperature T/Tc0

Mean-field approximation(as pF)−1 = − 1

normalBCS

0.8(weak-coupling BCS regime)

0.6

0.4

0.2

0
0

FFLO
0.05

0.1

0.15

0.2

0.25

1

NSR theory(including pairing fluctuations)

Intensity of pairing fluctuations
BCS

0.8

temperature T/Tc0

18 /270.2

normal

0.1

0.6

0.4

qy 0

0.2

-0.1

0
0

0.02

0.04

0.06

polarization P = [N↑ − N↓]/[N↑ + N↓]

0.08

-0.2
-0.2

Q

-0.1

0

qx

0.1

0.2

infinite degeneracy

FFLO State in a Lattice System (Stabilization of the FFLO state)1(continuous
rotational symmetry)(as pF)−1 = − 1 (BCS regime)

temperature T/Tc0

Gas system

0.8

0.6

0.4

qy 0

0.2

-0.1

0.04

0.06

0.08

-0.2
-0.2

-0.1

0

qx

0.1

0.2

0.1

0.2

infinite degeneracy

0.2

0.8

temperature T/Tc0

n = n↑ + n↓ = 0.3

0.02

Q

1

Optical lattice system

U/6t = 0.4 (BCS regime)

0.2

0.1

0
0

(discrete fourfold
rotational symmetry)

Intensity of pairing fluctuations

normalBCS

19 /27

BCS

normal

0.1

0.6

qy

0.4

Q
0

Q

-0.1

0.2

FFLO
0

0

0.01

0.02

0.03

0.04

polarization P = [N↑ − N↓]/[N↑ + N↓]

-0.2
-0.2

-0.1

0

qx

finite degeneracy

FFLO State in a Lattice System (Fermi Surface shape)
Filling n dependence

temperature T/Tc0

BCS

FFLO

polarization P

filling n

20 /27

FFLO State in a Lattice System (Fermi Surface shape)

21 /27Filling n dependence
BCS

weak
5.1 熱平衡 FFLO 状態

temperature T/Tc0

(a)

FFLO

polarization P

filling n(b)Anisotropy of the Fermi surface

strongCompetition between the FFLO and the Phase Separation (BCS+Normal)

n = 0.3(t′ = 0)
BCS (2nd order)
temperature T/Tc0

normal phase

phase separation(1st order)

FFLO (2nd order)￼polarization P = [N↑ − N↓]/[N↑ + N↓]

22 /27

Competition between the FFLO and the Phase Separation (BCS+Normal)
Anisotropy of the Fermi surface

weak
n = 0.3(t′ = 0)temperature T/Tc0

BCS (2nd order)

BCS (2nd order)
normal phasenormal phase

phase separation(1st order)

phase separation(1st order)

FFLO (2nd order)￼polarization P = [N↑ − N↓]/[N↑ + N↓]

￼n = 0.305(t′ = 0)FFLO (2nd order)

polarization P = [N↑ − N↓]/[N↑ + N↓]

strong23 /27

Competition between the FFLO and the Phase Separation (BCS+Normal)
Anisotropy of the Fermi surface

weak
n = 0.3(t′ = 0)temperature T/Tc0

BCS (2nd order)

normal phase

phase separation(1st order)

phase separation(1st order)

￼

￼

￼

polarization P = [N↑ − N↓]/[N↑ + N↓]

n = 0.31(t′ = 0)
BCS (2nd order)

normal phase

FFLO (2nd order)

strongn = 0.305(t′ = 0)
BCS (2nd order)

24 /27FFLO (2nd order)

polarization P = [N↑ − N↓]/[N↑ + N↓]

normal phase

FFLO (2nd order)

polarization P = [N↑ − N↓]/[N↑ + N↓]

Contents
1. Introduction

2. Formalism

3. Results

4. Summary

25 /27

Summary

26 /27

Taira Kawamura and Yoji Ohashi, Phys. Rev. A 106, 033320 (2022).

‣ We theoretically explored a promising route to achieve the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in a spin-imbalanced
ultracold Fermi gas.
‣ We have found that the anisotropy of the Fermi surface is crucial to the realization of the FFLO state. As the Fermi surface deviatesfrom spherical with increasing the filling fraction, the FFLO pairing fluctuations become weak, which promotes the stabilizationof the FFLO-type long-range order.

Future work
‣ Fluctuations in other channels‣ Crystalline (multiple-Q) FFLO state/27

where ξp,σ = p /(2m) − µ and Ep =
ξp + ∆ ∓ δµ. The stationary condition of ΩMF with respect to ∆0
and x reads the gap equation and the mechanical equilibrium condition,

Phase
diagram
! 1 #
∂Ω
mix
MF

∂∆0

= 0 ⇔ ∆0 = U

k

2Ẽk





&

1 − f (Ẽk+ ) − f (Ẽk− ) ,Chemical potential is fixed

Phase diagram [µ = εF , (as pF)−1 = −1.5]N= 0 ⇔ ΩBCS
=
Ω
MF
MF .









(91)

Total particle number is fixed

28 /27n = n↑ + n↓ = 0.5
U/(6t) = 0.4





 



∂x



Phase diagram(μ = εF)




∂Ωmix
MF

























  

 
























 



























The 1st-order transition line (green dotted line) corresponds to the Chandrasekhar-Clogston limit. The super• nlimit,= n↑ which+ n↓ =
0.5, U/(6t)= 0.4
fluid fraction jumps from x = 0 to x = 1 at the CC
means
that there
is no phase separation.

• If we only need the 1st-order phase transition line between normal and PS, we can set x = 0 beforeDestruction of the FFLO long-range order by FFLO pairing fluctuations/27

In isotropic systems, the FFLO-type non-uniform superfluid state is unstable against pairing fluctuations at nonzero temperature- H. Shimahara, J. Phys. Soc. Jpn. 67, 1872 (1998)
even in a 3D system.
- Y. Ohashi, J. Phys. Soc. Jpn. 71, 2625 (2002)- J. Wang et. al., Phys. Rev. B 97, 134513 (2018)- P. Zdybel et. al., Phys. Rev. A 104, 063317 (2021)

gas system(continuous rotational symmetry)

⋮

Q
↑

↑

Large fluctuations caused by an infinite degeneracy with respect to the direction of the FFLO Q vector
destroy the FFLO-type long-range order.

Long-range order can be stabilized!

(cf.) Hohenberg-Mermin-Wagner theorem⋯⋯

2D XY modelO(2) symmetry
continuous symmetry⋯⋯

2D Ising modelZ2 symmetry
discrete symmetry

FFLO State in a Spin-imbalanced Ultracold Fermi Gas
Spin-imbalanced Fermi gas (N↑ ≠ N↓)

Theory (Mean-field approximation)
M. M. Parish et. al., Nature Physics 3, 124 (2007)
G. C. Strinati et. al., Phys. Rep. 738, 1 (2018)

polarization(N↑ > N↓)⋮P = [N↑ − N↓]/[N↑ + N↓]

Temperature T

Fermi liquid

Tc

Temperature T/TF

0.12

Molecular
Bose gas

Uniform Superfluid
(BCS)

0.08

0.04

Phase Separation0

polarization P(as pF)−1 = − 1

NormalSuperfluid

Attractive Interaction Strength

30 /27

Non-uniform Superfluid(FFLO)

00.1
0.2
Polarization P = [N↑ − N↓]/[N↑ + N↓]

The FFLO state has not been observed experimentally in a 2D and 3D Fermi gas.

Destruction of the FFLO long-range order by FFLO pairing fluctuations/27

In isotropic systems, the FFLO-type non-uniform superfluid state is unstable against pairing fluctuations at nonzero temperature- H. Shimahara, J. Phys. Soc. Jpn. 67, 1872 (1998)
even in a 3D system.
gas system(continuous rotational symmetry)

Optical lattice system(discrete rotational symmetry)

Q

⋮

Laser

↑

↑

Q

- Y. Ohashi, J. Phys. Soc. Jpn. 71, 2625 (2002)- J. Wang et. al., Phys. Rev. B 97, 134513 (2018)- P. Zdybel et. al., Phys. Rev. A 104, 063317 (2021)

Explicitly breaking
the continuous rotational symmetry

Atom
Periodic potential made by optical interference

infinite degeneracy

finite degeneracy

http://www.kozuma.phys.titech.ac.jp/
research_category/entry7.html(cf.) Hohenberg-Mermin-Wagner theorem⋯⋯

2D XY modelO(2) symmetry
continuous symmetry⋯⋯

2D Ising modelZ2 symmetry
discrete symmetry-SU(2) caseFlavor-SU(2) casea general chiral orderparameter:φ(z)≡"ψψ#(x)+i"ψiγ
5 τ3 ψ#(x) 5 τ3 ψ#(x)a generalchiral orderparameter: φ(z)≡"ψψ#(x)+i"ψiγ

iqz
Landau-Peierls
instability
FF-type=∆e)groundʲNakano-Tatsumi(2005);
akin to Dautry-Nyman(1979)ʳ!(φFF-type
FF(φFF =∆eiqzstate:) ground state:ʲNakano-Tatsumi(2005);
akin to Dautry-Nyman(1979)ʳ!ψψ"(z) =!ψψ"(z)∆ cos(qz),!ψiγ5 τ3 ψ"(z)sin(qz)= ∆ cos(qz),!ψiγ== ∆ sin(qz)5 τ∆
3 ψ"(z)QCD (chiral condensate)TGLexpansionforthe(3+1)-dimensional
NJL
model
in
the
chirallimit[Nickel(2009)]
2nd/crossover
LO-type!(φLO-typegroundstate:
ʲ
Nickel(2009);
cf.
Thies(2006)ʳ
LO =∆(z))(φʲNickel(2009); cf. Thies(2006)ʳ
LO =∆(z)) ground state:

Dual Chiral Density Wave√√!ψψ"(z) =∆ ν sn(∆z|ν)!ψψ"(z)= ∆ ν sn(∆z|ν)Fulde-Ferrell typechiral restoration⟨ϕ⟩ = 0

2nd

chiral condensed⟨ϕ⟩ = 0, q = 0(∆: amplitude,
q: wavenumber,
ν: elliptic modulus)(∆: amplitude,
q: wavenumber,
ν:Larkin-Ovchinikov
typeelliptic modulus)
real kink crystal

Dual chiral
waves(DCDW)" density
Dual chiraldensity
waves (DCDW)
DCDW

/27

inhomogeneous chiral condensed⟨ϕ⟩ = 0, q ≠ 0

μ" Real kink"crystals(RKC)
Real kink
crystals (RKC)

DCDW: 1st, RKC: 2nd

RKC

Z

Z

(chiral spirals in (chiral
3+1D systems)
spirals in 3+1D systems)(periodic domain(periodic
walls in 3+1D
domainsystems)
walls in 3+1D systems)
.

.

.

.

.

.

.

.

.

.

.

.

Chemical

Highm is the mass of a 6Li
P energy, andm, er 5 B2k2/2m is the recoiltion). We perform an inverse Abel transform to obtain the number ofP
There are several requirements to be met for the system to be 1D. particles and polarization in the central tube. Following ref. 6, we plot
/27
nly the lowest transverse modethese radii as a function of the central tube polarization (Fig. 3),
Pc in each tube may be populated.
Fermi normalizing the radii by (N )1/2a , where N is the total number of
equires that both the thermal energy kBT and the 1DFully0z0KeF 5 N1Bvz be small compared to the transverse confinement
polarized
particlesinthecentral
tube
and
a
5
(B/mv
)
is
the
harmonic
oscilzz
VacuumN
isthenumberof atoms
pergas
1D tubein state
BvH. Herej1æ,Nature
1
Y. Liao,
et. al.,(London)1D spin-imbalanced
Fermi
lator
length 467,
along567
the(2010).
central tube. The critical polarization Pc correand vH are the axial and transverse confinement frequencies of sponds to the crossing of these two radii where the entire cloud is
Effective magnetic
fieldvidual tube.
Second,thesingle-particle
tunnellingt should
Theory: exactly solvable Gaudin–Yang rate
model
ll compared to both eF and T. The condition eF . t is equivalentbifying that the Fermi surface is 1D, and the condition T . t
1.2
he inter-tube coupling incoherent. All conditions are well satisfied
experiment: the tube aspectVacuum
ratio vH/vz 5 1,000 is larger than
1.0
20 for the central
1.0 tube; and t/kB < 17 nK is much smaller than/kB < 1.2 mK and T < 175 nK.
0.8
Pctune an external magnetic field to the Bardeen–Cooper–
Fully polarizedfer (BCS) side (890 G) of the broad 3D Feshbach resonance in
0.6
s 25 and 26), where the 1D interactions are strongly attractive27,28.
0.4
easure the in 0.5
situ density of the two spin species by sequential
g with two probe laser beams, choosing their intensity and fre0.2
to maximize the signal-to-noise ratio of the density difference
Partially polarized (FFLO)
ethods). Assuming hydrostatic equilibrium, the 1D spatial density
0.0
n1,2(z) can be0.0expressed in terms of m 5 m0 2 V(z), and h 5 h0,
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.6
0.8
1.0
m0 and h0 are the chemical potential and chemical potential
Central tube polarization
Centraltube
polarization
nce at the centre of the tube, set by the total number of particles
5 (N1 2(adapted
N2)/N; V(z)the 6). Figure 3 | Experimental phase diagram as a function of polarization in the
ube N 51 N
1 1 N2 and polarization
Figure| Theoretical
T 5 0 phase Pdiagram
fromisref.
1 particular, the phase
1 boundary between central tube. The scaled radii of the axial density difference (red diamonds)
onfinement potential. In
density
profilepopulation
imbalanced
1D Fermi
gas
wasfoundstate
to qualitatively
agree
with
a theoreticalprediction,
exhibiting the FFLO state.
a, Schematic
with
m~ (m
) versus
h~ (m
three
phases:
‣ The
|
2æ)
axial
density(blue
circles)compared
with
a
175
nK
and
the
minority(
1 zm2of1 {m2 ) , showing
y paired and partially polarized
regions occurs
2
2 where the density
Bethe ansatz calculation (solid lines). The dimensionless scaled axial radius
fully
paired(green),fully
polarized(blue),
andpartiallypolarized(yellow),
0, and theboundary
betweenthe fully associatedand
nce n1(z)‣2Inn2a(z)1/2 the Cooper pair’s momentum.
1D5system,
there
is no infinite
degeneracy
with
N
R/(a
z 0 ) is plotted, where R is the position along the bundle of tubes wherewhichispredicted
to
be
FFLO.
In
a
trap,
m
decreasesfrom
thecentretotheedge,
y polarized phases corresponds to n2(z) 5 0, as shown in Fig. 1b. the respective density vanishes, N is the total number of particles in the central
0
while
h
is
constant
throughout
the
tube.
The
verticalarrowsshow
twopossibleor 6 0.03, bothre 2 shows axial density profiles of state j1æ, state j2æ, and their tube, and az is the axial harmonic confinement length. At P < 0.13
pathsforfrom
the trap
centre to edge:These
The partiallypolarized thecentrences
a rangeof polarizations.
images represent
sumis surroundedradii intersect, indicating that the entire cloud is partially polarized. The data
eithera fullyinpaired
superfluid
at lowa fully polarizedphase atinear bydensity
all tubes
in ourphasesystem,andh orarebyproduced
by arein reasonable agreement with the theoretical crossing at slightly higher
high
h. At
a criticalvalueimagesof h, corresponding
to a polarization
Pc, thewholepolarization
P < 0.17.
ting ourcolumndensityacross the remaining
transverse
Radius/(azN01/2)

Radius/(azN01/2)

FFLO state in a 1D spin-imbalanced ultracold Fermi gas

shown in Fig. 4, including the steep enhancementof H abc2 at a low temperature and the first-order phase$transitionatalargelyT
independentH
bears a strikingfield)(superconductor under an external magnetic
resemblance to that expected for the FFLO transition [3,5].

of the field slope and increases steeply with H above H $ .

Observation of the FFLO state ?
‣ Heavy fermion superconductor

FeSe

CeCoIn5- Bianchi et.al., PRL 89, 137002 (2002)

CeCu2Si2- Kitagawa et.al., PRL 121, 157004 (2018)

magnetic field- Kumagi et.al., PRL 97, 227002 (2006)‣ Iron-based superconductor

FeSe
KFe2As2- Ok et.al., PRB 101, 224509 (2020)- Kasahara et.al., PRL 124, 107001 (2020)

Kasahara et.al., PRL(2020)- Kumagi et.al., PRL 119, 217002 (2017)

FIG. 3. (a) Magnetic-field dependence of the thermal conductivity in FeSe for Hkab. The inset shows a schematic illustration‣ Organicofsuperconductor
the experimental setup of the thermal-conductivity measurements. Orange arrows indicate the magnetic field H k at which a
κ − (BEDT− TTF)
kink-likeminimumof κðHÞ2 appears. (b) The same data below
2Cu(NCS)
T ¼ 2 K plotted for the high-field regime above μ0 H ¼ 16 T. A
- Wright et.al., PRL 107, 087002 (2011)
discontinuous downward jump at μ0 H ¼ 24 T appears inside the- Mayaffreet.al., Nat.
Phys.
928 (2014)superconducting
state
as 10,
indicatedby the black arrows. Green
arrows indicate the field Hp determined by our resistivity
measurements.
β′′ − (BEDT− TTF) SF CH CF SO
2

5

2

2

3

- Beyer et.al., PRL 109, 027003 (2012)￼- Koutroulakis et.al., PRL 116, 067003 (2016)￼/27

temperature
FIG. 4. High-field phase diagram of FeSe for Hkab plane.
Blue circles and green crosses show Hirr and Hp determinedby resistivity measurements. Orange and yellow circles show Hkand Conditions
H$ determinedthermal-conductivity
forbyrealizing
the FFLO measurements,
state
respectively. Above the first-order phase transition field H$ ,a distinct
field-induced
superconducting
at low effect(1) Paulipair-breaking
effect > phaseorbitalemerges
pair-breaking
temperatures.

107001-3(2) ultra-clean system

Boundary between the normal and phase-separated
phase
(32)

nMF
↑,BCS =nMF↓,BCS =

#!

"

k

#!

"

ũ2k f (Ẽk− ) + ṽk2 f (−Ẽk+ ) ,(33)

Phase separation (BCS superfluid + Normal)MFnσ,N =k#

We neglect the interface energy between
the normal and superfluid components

with the coherence factorsf (ξ˜k,σ ).

122ũk,Q = 1 − ṽk,Q =

k

2

!

1+

s
ξ˜k,Q

Ẽk,Q

" (34),∆0ũk,Q ṽk,Q =.
2Ẽk,Q(14)
Equilibrium conditions- thermal equilibrium TSF = TNect to ∆0 gives the gap equation in Eq. (19);The Bogoliubov excitation energy is given by

⇔

∂ΩBCS
MF
∂∆0

/27

ũ2k f (Ẽk+ ) + ṽk2 f (−Ẽk− ) ,

=0

⇔
with

- chemical equilibrium μSF,σ = μN,σ

#$
(
# 1 %
%2
±+
a
s
2 ± ξ˜a ,
˜
˜
−
Ẽ
=
Ẽ
±
ξ
=
ξ
+
∆
k,Q
∆0 = U
1 − f (Ẽk,Q.
(35) 0 k,Qk,Qk ) − f (Ẽk )k,Q
2Ẽk
k

(15)

- mechanical equilibrium PSF = PN

superfluid fraction x' 1&' U & MF
'
1&˜
s/a
MF
˜
˜
S = PNmix
ξk,Q respect= condition
ξk+Q/2,↑±, ξμ=), xξk+Q/2,↑ ± ξ−k+Q/2,↓ ∓n↑ ± n↓ .
−k+Q/2,↓ect toWex reads
theminimize
mechanicalequilibrium
(P
FMF
μ
Δ
need to
with
to
,
,
and
2
2
2
↑ ↓
0

(16)



MF andItisnowpossibletoderiveasetofcoupledself-consistentequationswithrespecttoorderparameters
n
σ
mix
∂FMF
mix
SF
BCS
N
F
∆MF
given by mechanical equilibrium condition(36)0==Ω− Ωare0 . They
MF
MF .

⇔

∂x

n↑MF =(k#ĉ†k,↑ ĉk,↑ $ =

(&
k



'
†
†
2
2
ũk,Q #γ̂k,+ γ̂k,+ $ + ṽk,Q #γ̂k,− γ̂k,− $ =


(&
'
5
−
+
MF
2
2
- Hartree density equationsn↓ =ũk,Q f (Ẽk,Q ) + ṽk,Q f (−Ẽk,Q ) ,

(&
k

'

+ 2−ũ2k,Q f (Ẽk,Q) + ṽk,Q f (−Ẽk,Q) ,


(17)
(18)

k

- gap equation∆0 = U

(
k

1.2

)



*

1
+
−
1 − f (Ẽk,Q) − f (Ẽk,Q) .
2Ẽk,Q


(19)





Phase diagram (fixed
particlenumber condition)SFN- mechanical equilibrium condition

ΩMF = ΩMFIn this study, with experiments in ultracold
atomic systems in mind, we consider the following situation:• We fix the total filling number n = nMF + nMF .







(BCS + N)

FFLO State in a Lattice System (Fermi Surface shape)
Filling n

5.1 熱平衡 FFLO 状態

Fermi surface

BCS(t′ = 0)temperature T/Tc0(a)t′ = 05.1 熱平衡 FFLO 状態

polarization P

NNN hopping t′

￼

￼

￼

temperature T/Tc0(n = 0.4)

￼

36 /27FFLO(a)filling n

(b)

BCS

Anisotropy of the Fermi surface(b)n = 0.4

FFLO
弱polarization P

Fermi面の異方性

強

!
図
5.6.
Fermi
面の
(a)
フィリング
n
と
(b)
次最近接サイトへの飛び移り積分tに対する依存next-nearest-neighbor

hopping t′n(t! ) が大きく (小さく弱 ) なるほど、Fermi 面の異方性が強くなる。
Fermi面の異方性

強