Induced interactions in the BCS-BEC crossover of two-dimensional Fermi gases with Rashba spin-orbit coupling

0.57 MB·

by  Juhee Lee

Induced interactions in the BCS-BEC crossover of two-dimensional Fermi gases with
Rashba spin-orbit coupling
Juhee Lee and Dong-Hee Kim∗
Department of Physics and Photon Science, School of Physics and Chemistry,
Gwangju Institute of Science and Technology, Gwangju 61005, Korea
WeinvestigatetheGorkov–Melik-Barkhudarov(GM)correctiontosuperfluidtransitiontempera-
tureintwo-dimensionalFermigaseswithRashbaspin-orbitcoupling(SOC)acrosstheSOC-driven
BCS-BECcrossover. Inthecalculationoftheinducedinteraction,wefindthatthespin-component
mixingduetoSOCcaninducebothoftheconventionalscreeningandadditionalantiscreeningcon-
7
tributionsthatinterplaysignificantlyinthestrongSOCregime. WhiletheGMcorrectiongenerally
1
lowers the estimate of transition temperature, it turns out that at a fixed weak interaction, the
0
2 correction effect exhibits a crossover behavior where the ratio between the estimates without and
withthecorrectionfirstdecreaseswithSOCandthenbecomesinsensitivetoSOCwhenitgoesinto
r
the strong SOC regime. We demonstrate the applicability of the GM correction by comparing the
a
M zero-temperature condensate fraction with the recent quantum Monte Carlo results.

8
I. INTRODUCTION dimensions[49–52], andveryrecentlybyusingthequan-

] tumMonteCarlo(QMC)methodforthegroundstatein
s two dimensions [53].
a The synthetic spin-orbit coupling (SOC) in ultracold
g gas systems has attracted a lot of attention because of Inthispaper,weinvestigateamany-bodycorrectionto
- its importance in realizing a new tunable platform for theestimatesofsuperfluidtransitiontemperatureintwo-
t
n nontrivial condensed matter phenomena [1–6]. For in- dimensionalFermigaseswithRashbaSOCbyemploying
a stance, the presence of SOC is an essential part of topo- the Gorkov–Melik-Barkhudarov (GM) approach [54, 55].
u
logicalinsulatorsandsuperconductors[7,8]andsystems The GM correction incorporates the induced interaction
q
. for quantum anomalous Hall effect [9] and topological due to the second-order particle-hole processes in the
t
a quantum computation [10]. The realization of SOC in Fermisea,providingasimpleextensionoftheBCSmean-
m ultracold atomic gases has rapidly progressed in recent field framework. In two-component Fermi gases with at-
years. TheequalmixingofRashbaandDresselhausSOC tractive s-wave interaction, without SOC, the screening
-
d has been realized firstly with bosons [11] and then with effectoftheinducedinteractionreducescriticaltempera-
n fermions [12–14]. Many proposals have been suggested ture,forinstance,byaratioofabout2.22inthreedimen-
o
forexperimentalrealizationofRashba-onlySOC[15–21]. sions [54, 55] and about 2.72 in two dimensions [56, 57].
c
Very recently, the realization of two-dimensional SOC The GM correction has been also extended to systems
[
which can be transformed into either Rashba or Dres- with mass imbalance [58], optical lattices [59], BEC-

2 selhaus SOC has been reported [22–24]. BCS crossover [60], spin-density polarization [61], three-
v
In attractively interacting Fermi gases, spin-orbit cou- component gases [62], polar molecules [63–65], and spin-
6
0 pling can largely affect the formation of fermion pairs orbit coupling [66, 67].
7 in transition to superfluid (for reviews, see, e.g., [4–6]). HereweaddresstwoimportantpropertiesofSOCthat
2 In particular, tuning the strength of Rashba SOC can make the calculations of the induced interaction essen-
0 produce another type of crossover between a Bardeen- tially different from the conventional case without SOC.
1. Cooper-Schrieffer (BCS) superfluid and a Bose-Einstein First, in the presence of SOC, the spin components are
0 condensate (BEC) of tightly bound molecules. In the mixed, forming the two helicity branches of energy dis-
7 FermigaseswithoutSOC,theBCS-BECcrossoveristyp- persion. These mixed spin states can mediate an an-
1 icallyimplementedbycontrollinginterparticlescattering tiscreening particle-hole polarization in addition to the
:
v length throughout the Feshbach resonance [25–29]. In conventional screening one, which is analogous to the
i the presence of the SOC, it has been suggested that a three-component gases where the third component in-
X
BEC can be realized even at a fixed weak interaction ducessucheffect[62]. Second,themixed-spin-statechar-
ar where very strong SOC can lead to the formation of the acter of the Fermi sea abruptly changes across the SOC-
bosonic bound state called rashbon [30–32]. The BCS- drivenBCS-BECcrossover[31],whichmaysubstantially
BECcrossoverinspin-orbit-coupledFermigaseshasbeen affect the medium contributions to the induced interac-
studiedwithvarioussettingsofSOC,interaction,anddi- tion. However, these properties are not included in the
mensionalityinthemean-fieldtheory[33–43],alsointhe earlierconsiderationoftheinducedinteractionforaspin-
beyond-mean-field approaches for three [44–48] and two orbit-coupled system [66].
In the calculation of superfluid transition tempera-
ture in the weakly interacting limit, we find that the
correction effect with the induced interaction exhibits a
∗ [email protected] crossover behavior between the weak and strong SOC
2
regimes which is attributed to the change of the Fermi k) k)
seacharacter. AtzeroSOC,thetransitiontemperatureis (⇓ (⇓
⇑/ ⇑/
confirmedtobereducedwiththecorrectionbyaratioof ξ ξ
(a) (b)
about2.7. AstheSOCstrengthincreases,thiscorrection k k>
F,⇑ F,⇑
ratiodecreasesintheweakSOCregimebutthenbecomes
insensitive to SOC in the strong SOC regime. The ratio kF,⇓ k<F,⇑
in the strong SOC limit is estimated to be about 1.5 for ky ky
the mean-field transition temperature, however the cor-
rection effect turns out to be much more suppressed in
the estimate of the Berezinskii-Kosterlitz-Thouless tran- weak SOC strong SOC
sitiontemperature. Inaddition,wecalculatetheconden-
k k
x x
satefractionforcomparisonwiththerecentground-state
QMC results [53], showing significant improvement with
FIG. 1. Fermi surfaces of two-dimensional Fermi gases with
the GM correction to the mean-field estimate. Rashba SOC in noninteracting limit. The solid (dotted) line
This paper is organized as follows. In Sec. II, we de- presents the ⇑(⇓) helicity branch of the energy dispersion.
scribe the weak and strong SOC regimes in association The horizontal dashed lines indicate the Fermi energies for
with the change in the noninteracting Fermi sea charac- the SOC strengths vR =1 (a) and 1.7 (b) that belong to the
ter in the two-dimensional Fermi gas with Rashba SOC. weak and strong SOC regimes, respectively.
AlsoinSec.II,themean-fieldapproximationtoestimate
the superfluid transition temperature is briefly reviewed.
In Sec. III, we provide the detailed procedures of our in- where eiϕk =hR(k)/hR(k). The resulting noninteract-
| |
duced interaction calculation and the correction to the ing energy dispersions with SOC are obtained as
mean-field equations. In Sec. IV, we present the correc-
tiontotheestimatesofsuperfluidtransitiontemperature ξk,⇑ =ξk vR k, (3)
− | |
in the weakly interacting limit as a function of the SOC ξ =ξ +v k, (4)
k,⇓ k R
strength. The comparison with the QMC results for the | |
condensate fraction is given in this section. In Sec. V, which are referred to as the - and -helicity branches,
summary and conclusions are given. respectively. Forthesimplicit⇑yofcalc⇓ulations,weusethe
naturalunit(cid:126)=2m=1,andtheBoltzmannconstantk
B
is set to be unity. The particle density n is fixed at 1/2π
II. RASHBA SOC AND HELICITY BRANCH throughout our calculations, which also sets the Fermi
momentum in the absence of SOC (v = 0) to be unity
R
A. Helicity basis transformation as k =√2πn=1. This is equivalent to setting the unit
F
of momentum and energy to be k and (cid:15) (cid:126)2k2/2m.
F F ≡ F
The Hamiltonian of the Rashba spin-orbit-coupled TheSOCstrengthvR isaccordinglyexpressedintheunit
Fermi gases in two dimensions can be written as of (cid:15)F/kF.
(cid:88) (cid:88)(cid:104) (cid:105)
= ξ c† c h (k)c† c +h.c.
H k k,σ k,σ− R k,↓ k,↑ B. The weak and strong SOC regime
k,σ=↑,↓ k
(cid:88)
+g c†k,↑c†−k,↓c−k(cid:48),↓ck(cid:48),↑ (1) At a given particle density, the character of the non-
k,k(cid:48)
interacting Fermi sea abruptly changes as the SOC
strength v increases from a small to large value as il-
where c† (c ) is the creation (annihilation) operator R
k,σ k,σ lustrated in Fig. 1. At a small v , both of the - and
R
of a σ-species fermion with momentum k (k ,k ), and ⇑
x y -helicity branches can be occupied, and thus the Fermi
≡
the dispersion ξk = (cid:126)22mk2 −µ for chemical potential µ. ⇓sea is composed of two different helicity branches. In
Thevolumeissettobeunityforsimplicity. Weconsider contrast, at a large v the Fermi sea transforms into
R
an attractive s-wave interaction between the - and - a doughnutlike shape with two surfaces of the same -
↑ ↓ ⇑
spin species, and thus a negative value is given to the helicitybranch. Fork =√2πn=1,onecaneasilyverify
F
interaction strength g. The Rashba SOC is given by the that this transition occurs at v = √2. For v < √2,
R R
spin-off-diagonaltermwithhR(k) vR( ky+ikx)where the radii of the - and -helicity Fermi seas are
≡ − ⇑ ⇓
v indicates the SOC strength.
R
(cid:112)
The noninteracting part of the Hamiltonian is diago- k =v /2+ 1 (v /2)2, (5)
F,⇑ R R
nalized in the ( , )-helicity basis defined by the unitary (cid:112)−
⇑ ⇓ k = v /2+ 1 (v /2)2. (6)
transformation written as F,⇓ − R − R
(cid:18)cckk,,↓↑(cid:19)= √12(cid:18)ei1ϕk −e−1iϕk(cid:19)(cid:18)aakk,,⇑⇓(cid:19), (2) Oronuntdheedotbhyerthheantdw,ofosrurvfRace>s √wi2t,hthraedFiierkmF>,i⇑seaandisksFu<,r⇑-
3
which are obtained as (a) (b)
kF>,⇑ =vR/2+1/vR, (7) p," q �k,# p," k,"
kF<,⇑ =vR/2−1/vR. (8) p+k+q q p�k+q
In order to distinguish these two different characters
of the noninteracting Fermi sea, we refer to the SOC p, k, p, k,
� # " � # � #
strengths of v < √2 and v > √2 as the weak and
R R
strong SOC regimes, respectively.
(c) + +
'
C. Mean-field approach to superfluid transition
The GM correction procedures are based on a per- FIG.2. Second-orderdiagramsofthemedium-inducedinter-
turbative approach that preserves the structure of the actionbetweenthe↑-and↓-spincomponents. Thesolidlines
mean-field theory. The many-body effect induced by the arepropagators,andthesingledottedlineindicatesthebare
Fermi sea is considered as an effective interaction. With two-body interaction. The diagram in (a) is conventional in
thesuperfluidorderparameter∆ g(cid:80) c c ,the usualtwo-componentsystems,andtheonein(b)ismediated
≡ k(cid:104) −k,↓ k,↑(cid:105) bythespin-componentmixingduetoSOC.(c)Theschematic
mean-field Hamiltonian is written as
diagram of the effective interaction (double dotted line) with
HMF = 21(cid:88)Ψ†kH(cid:101)kΨk+(cid:88)ξk− |∆g|2, (9) the induced interaction correction.
k k
where Ψ† =(c† ,c† ,c ,c ), and the matrix
k k,↑ k,↓ −k,↑ −k,↓ III. INDUCED INTERACTIONS
 ξ h∗(k) 0 ∆ 
k − R
H(cid:101)k =−hR0(k) ξ∆k∗ −ξ∆k hR0(k). (10) graFmigsuoreft2heilliunsdturacteedsitnhteertawctoiornelienvatnhtesperceosnedn-coerodferSOdiCa-.
∆∗ −0 −h−∗R(k) −−ξk While diagram 1 shown in Fig. 2(a) is typical in a two-
component system with an s-wave interaction, diagram
Diagonalizing (cid:101)(k), onecan obtainthe quasiparticleen-
H (cid:112) 2 shown in Fig. 2(b) is allowed by the spin-offdiagonal
ergies as E = (ξ v k)2+ ∆2, and accordingly
k,± k± R | | propagatorsavailableinthemixedspinstatesduetothe
one can write down the thermodynamic potential as
SOC. The particle-hole polarizations presented in dia-
Ω= (cid:88) (cid:20)ξk Ek,s 1 ln(cid:0)1+e−βEk,s(cid:1)(cid:21) |∆|2. grams 1 and 2 contribute to the induced interaction cor-
2 − 2 − β − g rection. In the GM correction scheme, the effective in-
k,s=± teraction g¯ is written with the correction up to the
eff
From the saddle-point condition 1 ∂Ω = 0, the mean- second-order of interaction as
∆∂∆∗
field gap equation is derived as
(cid:90) d2k (cid:34) 1 (cid:88) tanhβEk,s(cid:35) g¯eff =g+g¯ind (cid:39)g−g2(cid:0)Π1+Π2(cid:1), (14)
2 =0, (11)
(2π)2 2(cid:15) +(cid:15) − 4E
k B s=± k,s where the induced interaction g¯ g2(cid:0)Π +Π (cid:1) is
ind 1 2
where (cid:15) = (cid:126)2k2/2m. Note that the bare interaction to be averaged over the Fermi surfac≡es t−hat are expected
k
strength g is replaced by the two-body binding energy to dominantly contribute to the fermion pairing.
(cid:15)B through the relation for a two-dimensional gas The contributing polarization functions Π1 and Π2 of
1 (cid:90) d2k 1 diagrams 1 and 2 are expressed by using propagators in
= . (12) an imaginary-frequency domain as
− g (2π)22(cid:15) +(cid:15)
k B
The number equation n=−∂∂Ωµ is also written as Π (p˜,k˜)= 1 (cid:88) (0)(p˜+k˜+q˜) (0)(q˜), (15)
(cid:90) d2k (cid:34) (cid:88) tanhβEk,s(cid:35) 1 β q˜ G↑↑ G↓↓
n= 1 (ξ +sv k) 2 . (13)
(2π)2 −s=± k R 2Ek,s Π2(p˜,k˜)=−β1 (cid:88)G↑(0↓)(p˜−k˜+q˜)G↓(0↑)(q˜), (16)
Foragivenparticledensity,thesuperfluidtransitiontem- q˜
perature is determined by self-consistently solving the
gap and number equations for the vanishing order pa- where the four-momentum z˜ (z,iz ) is fermionic,
n
rameter. ThemodificationbytheGMcorrectionistobe and the noninteracting Matsu≡bara Green’s function
(cid:68) (cid:16) (cid:17)(cid:69)
binrgoutghhetboanrleyiinntteorathcteiomneagnw-fiiethldtghaepeeffqeucattivioeninbtyerraecptliaocn- Gσ(0σ)(cid:48)(z˜) = −(cid:82)0β Tτ czσ(τ)c†zσ(cid:48)(0) 0eiznτdτ. The
(0)
g¯ that includes the induced interaction correction. spin-diagonal and offdiagonal parts of can be eval-
eff Gσσ(cid:48)
4
uated through the helicity basis transformation as The GM correction to the mean-field gap equation is
then readily done by replacing bare interaction g with
1(cid:104) (cid:105)
Gσ(0σ)(k˜)= 2 G0⇑(k˜)+G0⇓(k˜) , (17) effective interaction g¯eff. In the weakly interacting limit,
the inverse of the effective interaction becomes
1 (cid:104) (cid:105)
Gσ(0σ¯)(k˜)= 2esσσ¯iϕk G0⇑(k˜)−G0⇓(k˜) , (18) 1 1 1
= +Π +Π , (24)
(cid:0) (cid:1) 1 2
where s = s = 1, and G (k˜) = 1/(ik ξ ) for g¯eff g−g2 Π1+Π2 (cid:39) g
↑↓ ↓↑ 0a n k,a
− −
a , . Then, Π and Π can be decomposed with
1 2 which rewrites the mean-field gap equation (11) as
∈ {⇑ ⇓}
the inter- and intra-helicity-branch components as
(cid:88) (cid:90) d2k (cid:34) 1 (cid:88) tanhβEk,s(cid:35)
Π (p,k)= [Π (p,k)] (19) 2 =Π +Π . (25)
1,2 1,2 ab (2π)2 2(cid:15) +(cid:15) − 4E 1 2
k B k,s
a,b∈{⇑,⇓} s=±
where The applicability of the GM-corrected gap equation
mainly depends on the interaction strength since the in-
1(cid:90) d2q
[Π (p,k)] = χ (p+k,q), (20) ducedinteractionisbasicallyasecond-orderperturbation
1 ab 4 (2π)2 ab expansionforaweakinteraction. Inaddition,wehaveas-
c (cid:90) d2q sumedthatallintra-andinter-surfacepairsoftheFermi
[Π (p,k)] = ab ei∆ϕχ (p k,q).(21)
2 ab − 4 (2π)2 ab − momenta equally participate in the Fermi surface aver-
age for the induced interaction. These approximations
The coefficient cab is +1 for a = b or 1 for a = b, and may need to be examined in more rigorous approaches.
− (cid:54)
∆ϕ = ϕq ϕp−k+q. The function χab is calculated by Wewillexaminetheimprovementwiththecorrectionby
−
performing imaginary-frequency summation as comparingwiththerecentground-stateQMCresultsfor
the condensate fraction [53] in the following section.
1 (cid:88)
χ (p k,q)= G (p˜ k˜+q˜)G (q˜) (22)
ab 0a 0b
± β ±
iqn
IV. RESULTS AND DISCUSSIONS
n (ξ ) n (ξ )
F p±k+q,a F q,b
= −
iΩ+ξ ξ
p±k+q,a q,b
− A. Induced interactions
where n (ξ) = 1/(eβξ +1), and the bosonic frequency
F
Ω is taken to be zero in the low temperature limit in Figure 3 presents the induced interaction calculated
the evaluation of the effective interaction. Our formu- fromdiagrams1and2asafunctionoftheSOCstrength
lation of the induced interaction extends the previous v . The total induced interaction g¯ = g2(Π +Π )
R ind 1 2
−
approach [66] where only the interbranch χ⇑⇓ was con- is positive for all vR and thus reduces the bare attrac-
sidered within diagram 1. Later, we will show that the tion g in the effective interaction g¯ g +g¯ . This
eff ind
≡
contribution of χ⇑⇓ is rather small at strong SOC. would lower the estimation of superfluid transition tem-
In the standard procedures of the GM correction, the perature. However, we find that the induced interaction
inducedinteractionisevaluatedasΠbeingaveragedover and its components exhibit a nontrivial SOC-strength
the Fermi surface momenta [55], which can be justified dependence. This appears as a crossover from the weak
in the weakly interacting limit where the scattering pro- to strong SOC regime that is attributed to the abrupt
cessesforpairingwoulddominantlyoccurneartheFermi changeinthemixed-spin-statecharacterofthenoninter-
surface. Indeed, the recent QMC calculations for two- acting Fermi sea.
dimensional Fermi gases with Rashba SOC showed that In the weak SOC regime, the induced interaction is
the condensate wave functions are peaked at the two mainly determined by the screening effect of Π from
1
Fermi surfaces in the weak and strong SOC regimes [53]. the contribution of diagram 1. In the limit of v 0,
R
→
Therefore, we evaluate the induced interaction as all components of the polarization function, χ given in
ab
g¯ g2(cid:2)Π +Π (cid:3) Eq. (23), become identical, leading to the complete can-
ind ≡− 1 2 cellationinΠ2,whileinturnΠ1recoversthecasewithout
= g2[ Π (p,k) + Π (p,k) ], (23) SOC. The negative sign of Π can be easily understood
− (cid:104) 1 (cid:105)FS (cid:104) 2 (cid:105)FS 1
fromtheformof χ whichissimilartotheconventional
where Π = 1 (cid:80) (cid:80) Π(p,k),and de- ab
(cid:104) (cid:105)FS ||SF||2 p∈SF k∈SF SF two-component systems without SOC. As vR increases,
notes a set of the Fermi surface momenta. We treat the Π becomes finite with the opposite sign indicating an
2
twoFermisurfacesonanequalfootingforaveraging,and antiscreening effect, although it remains much smaller in
tchoumspSoFseidncolfutdheesaolnlemwoimthenktFa,⇑re(skidF>i,n⇑g) aonndthtehetwoothseurrfwacitehs maOgnnitthuedeotthhearnhΠan1dfo,rinsmthaellsvtrRo.ng SOC regime, the an-
k (k< )intheweak(strong)SOCregime. Allmomen- tiscreeningcontributionofΠ isnolongersmallbutcon-
F,⇓ F,⇑ 2
tum integrations are done numerically for the evaluation siderably weakens the screening effect of Π in the total
1
of the induced interaction. induced interaction. At v > √2, the noninteracting
R
5
(a) 1 (a) 0.3 2.5 (b) 0.4 2
Tc(0) Tc(0)
D 0 Π−1+ΠΠ−−2 F 0.2 Tc(GM) F 0.3 Tc(GM) T(c0) / T(cGM) 1.8
−ΠN/2 −1 Π−12 εT / Bc Tc(0) / Tc(GM) 2.4 εT / Bc 0.2 1.6
k 0.1 k
0.1 1.4
−2
v k = 0.5ε v k = 2.0ε
R F F R F F
0 1 2 3 4 5 0 2.3 0 1.2
0 0.04 0.08 0 0.04 0.08
(b) 0.5 diagram 1 (c) 1 diagram 2 εB / εF εB / εF
0.8
0
FIG.4. Induced-interactioncorrectiontothemean-fieldtran-
D ⇑⇑ D 0.6
−ΠN() / 1ab2−−10−..155 ⇑⇓(⇓⇓⇑⇓) −ΠN() / 2ab2 00..24 ⇑⇓(⇓⇑⇓⇑⇑⇓) sT(Tithci(eo0)nr)attaeinomdopfwetrihatehtuttrrheaenascsiotairornefucttneicomtnipo(enTraco(tGfuMtrhe))ewiiisnthtceoormuatcpttaihoreendcsobtrrreeetncwgteitoehnn.
the weak and strong SOC regimes at (a) v = 0.5 and (b)
−2 0 R
v =2.0, respectively.
R
−2.5 −0.2
0 1 2 3 4 5 0 1 2 3 4 5
v k /ε v k /ε
R F F R F F
tween the estimates of the transition temperature with-
FIG.3. InducedinteractionasafunctionoftheSOCstrength. out and with the correction.
The values are scaled with the density of states N2D of two- WefirstexaminetheinteractiondependenceoftheGM
dimensional Fermi gases without SOC. (a) Total induced in- correctiontothesuperfluidtransitiontemperatureinthe
teraction g¯ /(−g2) ≡ Π +Π , and the screening and an-
ind 1 2 weak and strong SOC regimes. Figure 4 shows the com-
tiscreening contributions Π and Π from diagrams 1 and 2
1 2 parison between the mean-field transition temperature
given in Fig. 2, respectively. The inter- and intra-helicity-
(0) (GM)
branch components (Π) are presented for diagrams 1 (b) estimates Tc without the correction and Tc with
ab √ (0) (GM)
and 2 (c). The vertical dotted lines indicate vR = 2 where the correction and the reduction ratio Tc /Tc as a
the character of the Fermi sea abruptly changes. function of (cid:15) . In both of the values of v that we have
B R
examined,thereductionratiogoesbelowtheknownzero-
SOC value of about 2.7 in the weakly interacting limit
Fermi sea is available only with the -helicity branch for and also shows the weaker influence of the correction at
⇑
particles to reside, which leads to the dominant contri- the stronger SOC.
bution of the χ⇑⇑ component in both Π1 and Π2 while The variation of the correction effect with the SOC
theothercomponentsarelargelysuppressed. Akinkwas strength is more systematically shown in Figs. 5(a) and
foundatvR =√2wherethe⇓-helicitybranchdisappears 5(b)atthefixedvaluesof(cid:15)B =0.001and(cid:15)B =0.01that
in the Fermi sea. The induced interaction is determined are close to the weakly interacting limit. We find that
byaneteffectoftheoppositecontributionsofΠ1andΠ2, the reduction ratio Tc(0)/Tc(GM) first decreases monoton-
and their proper consideration becomes very important ically from the zero-SOC value of about 2.7 as v in-
R
for the evaluation of the GM correction in spin-orbit- creases within the weak SOC regime, but then the ratio
coupled systems. becomesinsensitivetoSOCwhenv goesintothestrong
R
SOCregime. Akinkappearsatv =√2wherethenon-
R
interacting Fermi sea abruptly changes. While the effect
B. Correction to superfluid transition temperature oftheinducedinteractiondecreaseswithincreasingSOC
in terms of the reduction ratio, it turns out that the ra-
tio at a large v approaches a constant value about 1.5
We calculate the correction to the estimates of su- R
(0)
which still gives a sizable reduction to T .
perfluidtransitiontemperatureintheweaklyinteracting c
limit. First,wepresenttheeffectsoftheinducedinterac- On the other hand, in strictly two dimensions, the
tion correction on the mean-field calculations which can true long-ranger order is absent because of strong quan-
be more relevant in a quasi-two-dimensional system sup- tum fluctuations. Although, it is well known that the
porting the long-range order. Additionally, for a strictly superfluid transition at finite temperature is still possi-
two-dimensional system where the long-range order is ble by the Berezinskii-Kosterlitz-Thouless (BKT) mech-
absent, we provide a rough estimate of the Berezinskii- anism of the vortex-antivortex pairing [68, 69]. In the
Kosterlitz-Thoulesstransitiontemperatureundertheas- spin-orbit-coupled Fermi gases, the BKT transition tem-
sumptionthattheinducedinteractioneffectislimitedto perature and superfluid properties have been estimated
the mean-field gap amplitude. The influence of the in- with consideration of the phase fluctuations in the order
duced interaction correction varies with the strength of parameterbeyondthemean-fieldapproximation[49–52].
SOC, which we discuss below in terms of the ratio be- Regarding the GM correction, the following question
6
(a) 0.2 5 (c) 5 (e) 1
T(GM) T(GM) GM total
εkT / BcF 00 0..01.155 Tcc(0) 234 ccT / T(0)(GM) εkT / BFBKT 00..0048 TTBB(c(0GKK)MTT) 234 BKTBKTT / T(0)(GM) ndensate fraction 000...468 QMGQMMMMFFα C Ctt=torr i1iptttpaorlilelpteatltel t
1 1 o 0.2
ε =0.001ε ε =0.001ε c
B F B F
0 0 0 0 0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
(b) 0.3 5 (d) 5 (f) 1
T(GM) T(GM) GM total
εkT / BcF 00..12 Tcc(0) 234 ccT / T(0)(GM) εkT / BFBKT 0 0.0.15 TTB(c(B0GKK)MTT) 234 BKTBKTT / T(0)(GM) ndensate fraction 000...468 QMGQMMMMFFα C Ctt=torr i7iptttpaorlilelpteatltel t
1 1 o 0.2
ε =0.01ε ε =0.01ε c
B F B F
0 0 0 0 0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
v k /ε v k /ε ln(k a)
R F F R F F F
FIG.5. SuperfluidtransitiontemperaturewiththeGMcorrectionandcomparisonwiththeQMCdataofcondensatefraction.
The mean-field transition temperature T (a) and (b), and the BKT transition temperature T (c) and (d), are calculated
c BKT
asafunctionoftheSOCstrengthv atweakattractionsof(cid:15) =0.001and(cid:15) =0.01. Thesuperscripts(GM)and(0)indicate
R B B
ones with and without the GM correction, respectively. The condensate fraction at zero temperature is compared with the
recent QMC results [53] for the weak and strong SOC regimes at (e) α ≡ 2v2/(cid:15) = 1 and (f) α = 7. For direct comparison,
R F
the binding energy (cid:15) is converted into the scattering length a through the relation (cid:15) ≡ 4(cid:126)2/ma2e2γ where γ is the Euler’s
B B
constant. The lines marked by MF indicate the mean-field values without the GM correction.
naturally arises for the BKT transition temperature. At ated as ρ (T) = n ρ (T) ρ (T) [50] as a function of
s 1 2
− −
weakinteractions,thepreviousresult[50]showsthatthe temperature T, where ρ and ρ in our unit are given as
1 2
BKT transition temperature recovers the mean-field es-
timate when the Rashba SOC strength is small, while
in the strong SOC limit, the BKT temperature largely ρ (T)= vR (cid:88)(cid:90) ∞dks(cid:18)ξ + ∆2(cid:19)tanh[βE2k,s],
1 k,s
deviates from the mean-field estimate. However, we find 16π s=± 0 ξk Ek,s
tmheaatnf-ofirelidnsttaranncsei,tiaotnatveemrypesrmataullrevRw,itthhetehsetiGmMatecoofrrtehce- ρ2(T)= β (cid:88)(cid:90) ∞kdk(cid:16)k+svR(cid:17)2sech2[βEk,s],
8π 2 2
tion is reduced by a factor of about 2.7, and therefore s=± 0
it happens that the mean-field estimate with the correc-
tion actually goes much below the BKT temperature if
for ξ ξ . With the GM-corrected gap equa-
k,+/− k,⇓/⇑
estimated without such correction. ≡
tionemployedfor∆, onecanfindthecorrespondingcor-
Here we suggest that in the weakly interacting limit, rection in the estimate of the BKT transition tempera-
the GM correction may be applicable to the estimate of ture T(GM) by solving the NK relation.
BKT
the BKT temperature by restricting the influence of the
The correction in the BKT transition temperature is
inducedinteractionwithinthemean-fieldorsaddle-point
shown in Figs. 5(c) and 5(d) for the same choices of (cid:15) .
B
gap equation. We follow the description of the phase
(GM)
With the correction, it turns out that T becomes
fluctuations given in Ref. [50]. In the procedures, when BKT
(GM)
the amplitude fluctuations of the order parameter is ne- very close to the mean-field estimate Tc at small vR,
glected, the amplitude is still determined from the same recovering the expectation of Ref. [50]. In the reduction
mean-fieldgapequation. Thus,asaroughapproximation ratio between T(0) , the one without the correction, and
BKT
intheweaklyinteractinglimit,byignoringanyeffectson (GM)
T , the behavior of its SOC dependence exhibits the
BKT
the anomalous propagators, one may include the correc-
similar crossover observed in the mean-field result: The
tion in the same way as for the mean-field calculations
ratio decreases with v at first and then becomes insen-
R
with the GM-corrected gap equation.
sitive to v in the strong SOC regime. Interestingly, the
R
The BKT transition temperature (T ) can be de- asymptoticvalueoftheratioapproachesunityatalarge
BKT
termined through the universal Nelson-Kosterlitz (NK) v , implying that the effect of the induced interaction
R
relation[70]whichcanbewrittenasT = πρ (T ) on the BKT transition temperature may diminish in the
BKT 4 s BKT
in our natural unit. The superfluid density ρ is evalu- Rashbon limit.
s
7
C. Comparison with the QMC results responding correction to the superfluid transition tem-
perature largely depends on the SOC strength, which is
Finally, we demonstrate the applicability of the GM- attributed to the abrupt change of the Fermi sea charac-
corrected gap equation by the comparison with the re- ter across the SOC-driven BCS-BEC crossover.
cent ground-state QMC calculations [53] for the conden- In the estimate of the mean-field transition tempera-
sate fraction. In the mean-field approximation [49, 50], tureintheweaklyinteractinglimit,thecorrectiondueto
the condensed density of fermion pairs n is expressed theinducedinteractionshowsacrossoverbehavioracross
0
as n = (cid:80) φ (k)2 + φ (k)2 where the spin-singlet the weak and strong SOC regimes. At strong SOC, it is
0 k| ↑↓ | | ↑↑ | (cid:80)
pairing field φ (k) c c = ∆ 1/4E found that the mean-field temperature is reduced by a
↑↓ ≡ (cid:104) k,↑ −k,↓(cid:105) − s=± k,s
and the SOC-induced spin-triplet pairing field φ (k) ratio of about 1.5 with the GM correction which is in
↑↑
(cid:104)ck,↑c−k,↑(cid:105) = −∆e−iϕk(cid:80)s=±s/4Ek,s at T = 0. Th≡e contrast to the known value of about 2.7 for the case
total condensate fraction n 2n /n is then written as without SOC. This ratio decreases with increasing SOC
c 0
≡
in the weak SOC regime, but then it becomes largely in-
(cid:32) (cid:33)
∆2 (cid:90) ∞ 1 1 sensitive to SOC in the strong SOC regime. The similar
nc = | 4| kdk E2 + E2 . (26) crossoverisalsofoundinourestimationoftheBKTtran-
0 k,+ k,− sition temperature at weak interactions. However, our
calculation indicates that the correction becomes quan-
Thisonlydependsonthezero-temperatureorderparam-
titatively very small in the strong SOC limit in the case
eter which is obtained from Eq. (25) when the GM cor-
of the BKT transition temperature.
rection is included.
In Figs. 5(e) and 5(f), we provide the condensate frac- The essence of the GM correction procedures is in
tion calculated at the two chosen values of the SOC the modification of the mean-field gap equation with
strength for direct comparison with the QMC data in the induced interaction. Our comparison with the re-
theweakandstrongSOCregimes. Thetotalcondensate cent ground-state QMC result for the condensate frac-
fractions indicate good agreement, showing a significant tion [53] shows remarkable improvement to the use of
improvement with the GM correction to the mean-field theusualmean-fieldorderparameter,demonstratingthe
estimate. The agreement is better in the stronger SOC importance of the GM correction in a spin-orbit-coupled
even at relatively strong interactions, which is notable system. While exact numerics with SOC for the tran-
sincetheGMcorrectionisgenerallyexpectedtobemost sition temperature have not been available yet, our es-
reliable at weak interactions. In the spin-triplet compo- timates with the GM correction would motivate appli-
nent, the perturbative nature of the GM correction in- cations of other beyond-mean-field approaches, for in-
deed becomes apparent at strong interactions where the stance, with the functional renormalizaton group [71],
QMCdatadeviatefromtheGM-correctedone. Suchsys- particle-hole fluctuations with self-energy renormaliza-
tematic deviation with increasing interaction is not seen tion [72], non-Gaussian fluctuation theory [45], and t-
inthetotalcondensatefractionatthestrongerSOC,im- matrix approaches [46, 47, 73–76], to two-dimensional
plyingthattheunderestimationofthesingletcomponent Fermi gases with SOC. In addition, we expect that it is
compensates the difference. alsopossibletoextendourformulationoftheGMcorrec-
tion to systems in three dimensions or with other types
ofSOC,whichwouldprovidemoreimprovedmean-field-
V. SUMMARY AND CONCLUSIONS based predictions of experimental relevance.
We have investigated the Gorkov–Melik-Barkhudarov
correctiontosuperfluidtransitiontemperatureinattrac-
ACKNOWLEDGMENTS
tively interacting Fermi gases with Rashba SOC in two
dimensions. The main differences due to the presence
of SOC are summarized as follows. First, the spin- We thank Peter Rosenberg and Shiwei Zhang for
component mixing allows another second-order process providing their quantum Monte Carlo data and Jildou
that additionally leads to the antiscreening effect that Baarsma for valuable discussions. This work was sup-
is prohibited in the usual two-component case without ported from Basic Science Research Program through
SOC. The induced interaction is thus determined by a the National Research Foundation of Korea funded by
neteffectoftheantiscreeningandconventionalscreening the Ministry of Science, ICT & Future Planning (Grant
contributions. Second, the induced interaction and cor- No. NRF-2014R1A1A1002682).
[1] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg (2013).
Rev. Mod. Phys. 83, 1523 (2011). [3] J. Dalibard, arXiv:1504.05520.
[2] V.GalitskiandI.B.Spielman,Nature(London)494,49 [4] J.Zhang,H.Hu,X.-J.Liu,andH.Pu,Annu.Rev.Cold
8
At. Mol. 2, 81 (2014). Rev. B 84, 014512 (2011).
[5] W. Yi, W. Zhang and X.-L. Cui, Sci. Chin. Phys. Mech. [32] J. P. Vyasanakere and V. B. Shenoy, New J. Phys. 14,
Astron. 58, 1 (2015). 043041 (2012).
[6] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [33] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett.
[7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 107, 195304 (2011); L. Jiang, X.-J. Liu, H. Hu, and H.
(2010). Pu, Phys. Rev. A 84, 063618 (2011).
[8] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 [34] M.Gong,S.Tewari,andC.Zhang,Phys.Rev.Lett.107,
(2011). 195303 (2011).
[9] N.Nagaosa,J.Sinova,S.Onoda,A.H.MacDonald,and [35] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305
N. P. Ong, Rev. Mod. Phys. 82, 1539 (2011). (2011).
[10] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das [36] M.IskinandA.L.Suba¸sı,Phys.Rev.Lett.107,050402
Sarma, Rev. Mod. Phys. 80, 1083 (2008). (2011); Phys. Rev. A 84, 043621 (2011).
[11] Y.-J.Lin,K.Jim´enez-Garc´ıa,andI.B.Spielman,Nature [37] W. Yi and G.-C. Guo, Phys. Rev. A 84, 031608(R)
(London) 471, 83 (2011). (2011).
[12] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, [38] J.Zhou,W.Zhang,andW.Yi,Phys.Rev.A84,063603
W.S.Bakr,andM.W.Zwierlein,Phys.Rev.Lett.109, (2011).
095302 (2012). [39] L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys.
[13] P.Wang,Z.-Q.Yu,Z.Fu,J.Miao,L.Huang,S.Chai,H. Rev. A 84, 033633 (2011); 86, 053632 (2012).
Zhai,andJ.Zhang,Phys.Rev.Lett.109,095301(2012). [40] G. Chen, M. Gong, and C. Zhang, Phys. Rev. A 85,
[14] R.A.Williams,M.C.Beeler,L.J.LeBlanc,K.Jim´enez- 013601 (2012).
Garc´ıa,andI.B.Spielman,Phys.Rev.Lett.111,095301 [41] L.HeandX.-G.Huang,Phys.Rev.B86,014511(2012).
(2013). [42] L. Han and C. A. R. Sa´ de Melo, Phys. Rev. A 85,
[15] J. Ruseckas, G. Juzeliu¯nas, P. O¨hberg, and M. Fleis- 011606(R) (2012); K. Seo, L. Han, and C. A. R. Sa´ de
chhauer, Phys. Rev. Lett. 95, 010404 (2005). Melo, Phys. Rev. Lett. 109, 105303 (2012).
[16] G. Juzeliu¯nas, J. Ruseckas, and J. Dalibard, Phys. Rev. [43] Y. Yi-Xiang, J. Ye, and W.-M. Liu, Phys. Rev. A 90,
A 81, 053403 (2010). 053603 (2014).
[17] D.L.Campbell,G.Juzeliu¯nas,andI.B.Spielman,Phys. [44] L.He,X.-G.Huang,H.Hu,andX.-J.Liu,Phys.Rev.A
Rev. A 84, 025602 (2011). 87, 053616 (2013).
[18] D. L. Campbell and I. B. Spielman, New J. Phys. 18, [45] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 92,
033035 (2016). 121111(R) (2015).
[19] B.M.Anderson,I.B.Spielman,andG.Juzeliu¯nas,Phys. [46] C.-T. Wu, B. M. Anderson, R. Boyack, and K. Levin,
Rev. Lett. 111, 125301 (2013). Phys. Rev. B 91, 220504(R) (2015).
[20] Z.-F.Xu,L.You,andM.Ueda,Phys.Rev.A87,063634 [47] B. M. Anderson, C.-T. Wu, R. Boyack, and K. Levin,
(2013). Phys. Rev. B 92, 134523 (2015).
[21] X.-J. Liu, K. T. Law, and T. K. Ng, Phys. Rev. Lett. [48] C.-Y. Wang and Y. He, Phys. Lett. A 379, 3089 (2015).
112, 086401 (2014); 113, 059901 (2014) [49] K. Zhou and Z. Zhang, Phys. Rev. Lett. 108, 025301
[22] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. (2012).
Chen, D. Li, Q. Zhou, and J. Zhang, Nature Phys. 12, [50] L. He and X.-G. Huang, Phys. Rev. Lett. 108, 145302
540 (2016). (2012).
[23] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C. [51] M. Gong, G. Chen, S. Jia, and C. Zhang, Phys. Rev.
Zhang, P. Wang, and J. Zhang, Phys. Rev. Lett. 117, Lett. 109, 105302 (2012).
235304 (2016). [52] J.P.A.Devreese,J.Tempere,andC.A.R.Sa´deMelo,
[24] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Phys. Rev. Lett. 113, 165304 (2014); Phys. Rev. A 92,
Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 043618 (2015).
354, 83 (2016). [53] H. Shi, P. Rosenberg, S. Chiesa, and S. Zhang, Phys.
[25] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Rev. Lett. 117, 040401 (2016).
Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. [54] L. P. Gorkov and T. K. Melik-Barkhudarov, Zh. Eksp.
92, 120401 (2004). Teor. Fiz. 40, 1452 (1961) [Sov. Phys. JETP 13, 1018
[26] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. (1961)].
Chevy,M.Teichmann,L.Tarruell,S.J.J.M.F.Kokkel- [55] H. Heiselberg, C. J. Pethick, H. Smith, and L. Viverit,
mans, and C. Salomon, Phys. Rev. Lett. 93, 050401 Phys. Rev. Lett. 85, 2418 (2000).
(2004). [56] D. S. Petrov, M. A. Baranov, and G. V. Shlyapnikov,
[27] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Phys. Rev. A 67, 031601(R) (2003).
Jochim, J. Hecker-Denschlag, and R. Grimm, Science [57] M.A.Resende,A.L.Mota, R.L.S.Farias, andH.Cal-
305, 1128 (2004). das, Phys. Rev. A 86, 033603 (2012).
[28] G.B.Partridge,K.E.Strecker,R.I.Kamar,M.W.Jack, [58] M. A. Baranov, C. Lobo, and G. V. Shlyapnikov, Phys.
and R. G. Hulet, Phys. Rev. Lett. 95, 020404 (2005). Rev. A 78, 033620 (2008).
[29] M.W.Zwierlein,J.R.Abo-Shaeer,A.Schirotzek,C.H. [59] D.-H.Kim,P.To¨rma¨,andJ.-P.Martikainen,Phys.Rev.
Schunck, and W. Ketterle, Nature (London) 435, 1047 Lett. 102, 245301 (2009).
(2005). [60] Z.-Q.Yu,K.Huang,andL.Yin,Phys.Rev.A79,053636
[30] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83, (2009).
094515 (2011). [61] Z.-Q. Yu and L. Yin, Phys. Rev. A 82, 013605 (2010).
[31] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. [62] J.-P. Martikainen, J. J. Kinnunen, P. To¨rma¨, and C. J.
Pethick, Phys. Rev. Lett. 103, 260403 (2009).
9
[63] M. A. Baranov, M. S. Marenko, V. S. Rychkov, and G. 1201 (1977).
V. Shlyapnikov, Phys. Rev. A 66, 013606 (2002). [71] S. Floerchinger, M. Scherer, S. Diehl, and C. Wetterich,
[64] J.Levinsen,N.R.Cooper,andG.V.Shlyapnikov,Phys. Phys. Rev. B 78, 174528 (2008).
Rev. A 84, 013603 (2011). [72] K.B.Gubbels,J.E.Baarsma,H.T.C.Stoof,Phys.Rev.
[65] A.K.Fedorov,S.I.Matveenko,V.I.Yudson,andG.V. Lett.103,195301(2009);J.E.Baarsma,K.B.Gubbels,
Shlyapnikov, Sci. Rep. 6, 27448 (2016). H. T. C. Stoof, Phys. Rev. A 82, 013624 (2010).
[66] H. Caldas, R. L. S. Farias, and M. Continentino, Phys. [73] Q.Chen,I.Kosztin,B.Janko´,andK.Levin,Phys.Rev.
Rev. A 88, 023615 (2013). Lett. 81, 4708 (1998).
[67] J. Lee and D.-H. Kim, Phys. Rev. A 94, 067601 (2016). [74] R.Haussmann,W.Rantner,S.Cerrito,andW.Zwerger
[68] V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 59, 907 (1971) Phys. Rev. A 75, 023610 (2007).
[Sov. Phys. JETP 32, 493 (1971)]. [75] M. Bauer, M. M. Parish, and T. Enss, Phys. Rev. Lett.
[69] J. M. Kosterlitz and D. Thouless, J. Phys. C 5, L214 112, 135302 (2014).
(1972). [76] Q. Chen, Sci. Rep. 6, 25772 (2016).
[70] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39,