Mixed triplet and singlet pairing in multicomponent ultracold fermion systems with dipolar interactions

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by  Congjun Wu

Mixed triplet and singlet pairing in ultracold multicomponent fermion systems with
dipolar interactions
Congjun Wu and J. E. Hirsch
Department of Physics, University of California, San Diego, CA 92093
The symmetry properties of the Cooper pairing problem for multi-component ultra-cold dipolar
molecularsystemsareinvestigated. Thedipolaranisotropyprovidesanaturalandrobustmechanism
for both triplet and singlet Cooper pairing to first orderin theinteraction strength. With a purely
0
1 dipolar interaction, thetriplet pz-likepolar pairing is the most dominant. A short-range attractive
interaction can enhance the singlet pairing to be nearly degenerate with the triplet pairing. We
0
point out that these two pairing channels can mix by developing a relative phase of ±π, thus
2 2
spontaneously breaking time-reversal symmetry. We also suggest the possibility of such mixing of

n triplet and singlet pairing in other systems.
a
J
PACSnumbers: 03.75.Ss,74.20.Rp,67.30.H-,05.30

1
1
The study of ultracold dipolar molecules has recently teraction. Itcanmix withthe singlets+d -pairing
r2 3z2
] become a research focus of cold atom physics [1–4]. The whosepairingstrengthistunablethroughthe−shortrange
s
a prominentfeatureofthedipolarinteractionisitsdr2 3z2- non-dipolar s-wave scattering. The mixing occurs with
g type anisotropy when the dipolar moments are ali−gned a relative phase of ±π which breaks time-reversal (TR)
- 2
by an external electric field. Considerable progress has symmetry spontaneously. Pairing in dipolar Fermi gases
t
n been made instudying anisotropiccondensationof dipo- with more than two components is also discussed.
a lar bosons [5–8]. Furthermore, dipolar fermionic sys- Samokhinet al. studied non-uniformmixed parity su-
u
tems provide an exciting opportunity to study exotic perfluidstatesinthepresenceofdipolarinteractions[22].
q
. anisotropic many-body physics of fermions. Experi- They considered coupling between singlet and triplet
t
a mentally, a near quantum-degenerate gas of the dipo- channelswithzerorelativephaseonly. Kabanov[23]also
m lar fermion 40K-87Rb has been achieved [1]. A number considered recently mixture of singlet and triplet pair-
- of theoretical works have been done for the anisotropic ing with zero relative phase. While this work was being
d Fermi liquid properties of dipolar Fermi gases [9–13], in- completed, a study of the competition between triplet
n
cluding both singlet particle and collective excitations. andsingletpairingindipolarfermionicsystemsappeared
o
that analyzed some of the cases considered here [24].
c The dipolar interaction also has important effects in
[ the Cooper pairing symmetry as studied in Refs [14–18]. We begin with the two-component dipolar fermionic
5 In the single component case, the only possible pairing system with the electric dipole moments aligned along
the z-axis. The dipolar interaction reads V (~r −~r )=
v channels are of odd parity. Assuming dipole moments 3D 1 2
40 awliotnhgstlihgehtz-hayxbisr,idtihzeatpioaniriwngithsymotmheertroydids mpaaritnilayl owfapvze t−w|e~r1e2−nd~r2(2~r|3P−2(~rco)sθa~rn1d−~rt2h)e, ewlehcetrreicθfi~re1l−d~r2E~;isdtihsethaengelleectbreic-
9 1 2
components. Dipolar molecules can have an internal de- dipole moment. The anisotropy is manifested in the an-
4
greeoffreedomarisingfromthe hyperfineconfigurations gulardependence ofthe V withthe formofthe second
. 3D
0 of the constituent atoms. The electric dipolar interac- Legendre polynomial. The Fourier transform of the 3D
1 tion is independent of these internal components which dipolarinteraction,V(~k)= 8πd2P (cosθ ),onlydepends
9 will be denoted as spin below. The inter-component in- 3 2 k
0 on the polar angle of~k. The Hamiltonian is written as
teractionopensupthepossibilityofbothspinsingletand
:
Xiv twroipulledtbpeaiirnitnegrsesftoirngthteossitmudpylesetvetnwori-cchoemrpCoonoepnetrcpaaseir.inIgt H = k,α ǫ(~k)−µ c†α(~k)cα(~k)+ 21V k,k′,qV(~k−~k′)
patterns and the competition among them. X(cid:2) (cid:3) X
ar In this article, we show that the dipolar interaction × Pβ†α(~k;~q)Pαβ(~k′;~q) (1)
favorsCooper pairingin the triplet channel overthe sin- where ǫ(~k) = ~k2/(2m); µ is the chemical potential;
glet channel. This is an effect directly arising from the P (~k;~q) = c (−~k + ~q)c (~k + ~q) is the pairing opera-
αβ α β
anisotropyofthedipolarinteraction,anditoccurstofirst
tor; α,β refer to ↑ and ↓. Please note that V(~k −~k )
orderinthe interactionstrength. Incontrast,itdoesnot ′
depends on the polar angle of the vector~k−~k , not the
appearintheusualcondensedmattertripletpairingsys- ′
tems suchassuperfluid3He [19–21]: the spinfluctuation relative angle between~k and~k′. We define a dimension-
mechanism based on the strong ferromagnetic tendency lessparametertodescribetheinteractionstrengthasthe
in 3He arises from the repulsive part of the 3He-3He in- ratio between the characteristic interaction energy and
teraction at second order. For a two-component dipo- the Fermi energy: λ≡E /E = 2d2mkf.
int F 3 π2~2
lar fermion system, we find the dominate pairing in the We only consider uniform pairing states at the mean-
spin triplet p -like channel with the purely dipolar in- field level, thus set ~q = 0 in the pairing interaction in
z
2
Eq. 1. We define P (~k) = P (~k;~q = 0) which satis- -1
αβ αβ ws+d
fies P (~k) = −P (−~k). The pairing operators can be si
αβ βα
decomposed into the spin singlet P and triplet chan-
si
an√1ne2lds[PPP↑t↓ytrx(ri~,ki(y~k),z)+:=PP−↓si↑(√(~ki~k2))[]P,=↑↑P(t~xk√r1i)2([+~kP)↑P↓=↓(↓~k()−~k−)√]1.2PP[P↓s↑↑i((↑~k~k()~k)]),a−nPdtzPrPi↓(t↓~kµr()i~k(~)k=]), λ/Eigenvalues--23
are even and odd functions of ~k respectively; Pµ de-
tri
scribes the triplet pairing operators whose total spin is wz
the eigenstate of eˆ ·S~ with zero eigenvalue. Using tri
µ pair
-4
these operators, the pairing interaction of Eq. 1 with
-3 -2 -1 0 1 2 3
~q =0 can be rewritten as V /λ
00;0
1
Hpair = 2V k,k′, Vtri(~k;~k′) [ µ=x,y,zPt†r,iµ(~k)Ptµri(~k′) ] FφzIG-c.h1a:nn(Celoalonrdownlssii+nde)foTrhteheeigseinngvlaeltueφss+wdtzr-cihfoarnntheel.sTpihnetlraipttleert
+ Vsi(~kX;~k′)(cid:8)Ps†i(~k)Psi(~k′) ,X (2) dependson the short-range s-waveinteraction V00;0.
whereVtri,si(~k;~k′)= 12 V(~k−~k′)(cid:9)∓V(~k+~k′) . Vsi(~k;~k′) procedureexplainedinRef. [14,25]byreplacingthebare
is an even function of both arguments ~k and ~k , while interactionV withthe renormalizedzeroenergy ver-
(cid:8) (cid:9) ′ tri,si
Vtri(~k;~k′) is an odd function of both. texfunctionsΓtri,si. AttheleveloftheBornapproxima-
The decoupled mean-field Hamiltonian reads tion, this regularizationis equivalent to just introducing
an energy cutoff of ±ω¯ for ξ in Eq. 5, where ω¯ is at the
~k′
H = ′ Ψ (~k) ξ(~k)I ∆αβ(~k) Ψ(~k), (3) scale of the Fermi energy.
mf X~k † ∆∗βα(~k) −ξ(~k)I ! EqT.o5ainsalliynzeeartihzeedd.omCionnasnidteprainirgintghaintstthaebislittryonagroeustndpaTirc-,
ing occurs at the Fermi surface and following the stan-
where ′k means summation over half of mo- dardprocedureinRef. [14],wedefinetheeigen-gapfunc-
(mce(n~ktu),mcP(~ks)p,acc†e(;−~k)ξ,(c~k†)(−~k)=)T. Tǫ(h~ke)m−eanµ-fi;eldΨg(a~kp)func=- tions φjtri,si(~k) satisfying
∆tio↑αnβiscadn↓efibneedd↑eacsom∆pαβo↓s(~ekd) =intV1o sin~kg′lVet(~ka−nd~k′t)rhi|pPleαtβ(c~kh′a)|ni-. N0 d4Ωπk′Vtri,si(~k;~k′)φjtri,si(~k′)=wtjri,siφjtri,si(~k) (6)
nels as ∆ (~k) = ∆ (~k)iσyP+ ∆ (~k)(iσµσy) . Z
The Bogoliαuβbov quasipasriticle sαpβectra btreic,µome E (~k)αβ= where N0 = πm2k~f2 is the density of state at the Fermi
1,2 surface; wj are dimensionless eigenvalues;~k,~k are at
ξ2+λ2 (~k) where λ2 (~k) are the eigenvalues of the tri,si ′
k 1,2 1,2 the Fermi surface. We neglectthe effectof Fermisurface
qpositive-definite Hermitian matrix ∆ (~k)∆(~k). Its trace distortiononpairingwhichis ahigherordereffect inthe
†
satisfiesλ2(~k)+λ2(~k)=|∆ (~k)|2+ |∆ (~k)|2. The interaction strength λ. Then Eq. 5 is linearized into a
1 2 si µ tri,µ
free energy becomes set of decoupled equations
P
2 βE~k,i 1 φjtri,si{1+wtri,siln[(2eγω¯)/(πkBT)]}=0. (7)
F = − ln 2cosh −
β 2 2V
k,Xi=1,2 (cid:2) (cid:3) ~k,~k′,a=Xsi,(tri,µ) ThesphericalharmonicsdecompositionofVtri,si(~k;~k′)
× ∆∗a(~k)Va−1(~k;~k′)∆a(~k) , (4) reads
whereV(cid:8)s−i,1tri(~k;~k′)istheinverse(cid:9)oftheinteractionmatrix N4π0Vtri,si(~k;~k′)= Vll′;mYl∗m(Ωk)Yl′m(Ω~k′), (8)
defined as V1 k′Vsi,tri(~k,~k′)Vs−i,1tri(~k′,~k′′)=δ~k,~k′′. lX,l′;m
The gap equations are expressed as
P where Vll′;m remains diagonal for m but couples partial
d3k wave channels with l′ = l,l ±2. Vtri,si only have ma-
∆tri,µ(~k) = − (2π)′3Vtri(~k;~k′)K(~k′)∆tri,µ(~k′), trix elements in odd and even partial wave channels, re-
Z spectively. Vll;m has the same expressions of the Landau
∆ (~k) = − d3k′ V (~k;~k )K(~k )∆ (~k ), (5) parameters of the dipolar Fermi gases given by Fregoso
si (2π)3 si ′ ′ si ′ et al. [13] except for an overall minus sign due to pair-
Z
ing and a trivial overall numerical factor difference. For
where K(~k ) = tanh[βE (~k )]/[2E (~k )]. Eq. 5 formally l = l = m = 0, V = 0 because the average value of
′ 2 i ′ i ′ ′ 00;0
diverges. It can be regularized following the standard the dipolar interaction is zero.
3
We diagonalize the matrix Vll′;m to find the dom- follow the method presented in Ref. [28] to define the
inant negative eigenvalues which determine the domi- function f(x) = −2 ln[2coshβ ξ2+x], which satisfies
β 2 k
nant pairing channels. Two eigenvectors are found with d2 f(x)>0, thus f(λ2)+f(λ2p)≥2f(λ21+λ22). Then the
eigenvalues much more negative than other channels. dx2 1 2 2
first term in Eq. 4 is minimized by the unitary pairing,
One lies in the triplet odd parity sector with dominant
and the second term is degenerate for unitary and non-
p -wave character with slight hybridization with other
z unitary pairings. Therefore, φz +iφs+d is favored. This
odd parity channels as same as in the single compo-
nent case [14]: φz(Ω ) with the most negative eigen- is an exotic fully gapped TR breaking pairing state be-
value wz = −3.820λk, whose eigenvector is φz(Ω ) ≈ cause the nodes of φs+d and φs do not coincide. It also
tri k breaksparity but is invariantunder the combinedparity
0.993Y −0.120Y . Theotheroneliesintheevenparity
10 30
and TR operation. Its Goldstone manifold for the con-
spin singlet channel. For the purely dipolar interaction,
its eigenvalue is ws+d = −1.935λ and the eigenvector tinuous symmetry breaking is the same as in the purely
lies in the mixed chsiannel of s+d as φs+d(Ω ) ≈ triplet-φz pairing phase.
k2 3k2 k
0.6Y −0.8Y withnodes. However−,thzischannelissen- TheaboveanalysiscanberecapturedintheGinzburg-
00 20
Landau (GL) framework. The bulk pairing order pa-
sitivetothestrengthoftheshortranges-wavescattering,
which contributes only to the matrix element of V00;0 as rameters are defined as ∆zµ = kφz(~k)∆tri,µ(~k) and
depictedinFig. 1. Experimentally,thisscatteringcanbe ∆s+d = φs+d(~k)∆ (~k). The GL free energy is con-
µ k si P
tuned to the scale of the FermienergythroughFeshbach structed as
resonance, i.e., V can become of order 1. As for λ, as P
00;0
estimatedinRef. [13],itcouldreach0.1∼0.2. Thusthe F = αz(T) |∆zµ|2+αs+d(T)|∆s+d|2+βz|∆zµ|4
competition between φs+d and φs can be studied experi- µ
X
mentallyinthefuture. Whentheybecomedegenerateat + β |∆s+d|4+γ |∆z|2|∆s+d|2
V /λ≈−3.15,φs+d becomesmostly ofs-wavecharac- s+d 1 µ
00;0 µ
ter as φs+d(Ω )≈0.901Y −0.434Y . X
k 00 20
+ γ {∆z ∆z ∆s+d∆s+d+cc}, (9)
We firstconsiderthecaseofthedominanttripletpair- 2 µ∗ µ∗
µ
ing, whose critical temperature is determined from Eq. X
7 as T ≈(2eγω¯/π)exp(−1/|wz |). Its order parameter
configupzration is ∆tri,µ(~k) = ∆ttrrii,φz(Ωk)dˆµ. dˆis a spin iwshdeerfienαezd=asNT0ln(≈TTpz()2,eαγω¯s+/dπ)=exNp(0−ln1(/T|wsT+sd+)d,|)a;nβd Tasn+dd
space unit vector. Without losing generality, it is taken s+d si z
along the z-axis as dˆ=eˆ . ∆ (Ω )=∆ eiγφz(Ω ) βs+d termsarethe standardquartic termsfor the triplet
z tri,φz k tri k φz and singlet φs+d channels, respectively; γ describe
where γ is the pairing phase. This phase has line-nodes 1,2
the coupling between the φz and φs+d-channels. Fol-
on the equator. It breaks the U (1) gauge, and the spin
c lowing the analysis on a similar problem in Ref. [29],
SU (2) symmetries, but maintains TR, parity, and a
s we consider the situation where the two channels are
Z -symmetry of the combined operation of dˆ → −dˆ
2 µ µ nearly degenerate and T is slightly larger than T ,
and γ → γ + π [26, 27]. Its Goldstone manifold is pz s+d
thenthe tripletpairingdevelopsfirstatT =T . Defin-
G = [U (1)⊗SU (2)]/[U (1)⊗Z ] = [U (1)⊗S2]/Z . c pz
c s s 2 c s 2 ing r = (γ −2|γ |)/(2β ), the condition for the second
1 2 z
The corresponding low energy excitations include the
instability to occur is that there exists a lower temper-
phononandthe spin-wavemodes. Itsupportstwodiffer-
ature T below which |α (T )| > r|α (T )|. This can
eflnutidciltays,saensdofthveorhtaiclef-si:nttehgeerusquuaalnitnutmegevrorvtoerxteoxfosufpsuerpfleur-- Tbehesa±tiπs′fi-pedhafsoerdriff<er1en,cwesh+bicdehtwre′eesnultthseintprziTTpc′le=′t a(nTdTs+csdi)n1g−1lert.
idity combined with a π-disclination of the d-vector. 2
channel pairing requires that γ > 0. In Refs. [22, 23],
2
Next we consider the coexistence of the singlet φs+d couplingbetweenthesingletandtripletpairingsthrough
pairing and the triplet φz pairing, when they be- a linear spatial derivative is considered, which leads to
come nearly degenerate. The order parameter can be spatially non-uniform states. Due to spin conservation,
chosen as ∆si(Ωk) = ∆0φs+d(Ωk) and ∆tri,µ(Ωk) = such a term is not allowed here.
∆zφz(Ωk)δµ,z. An important observation is that a rel- Itis naturaltofurtherconsidercompetingpairingsfor
ative ±π2-phase difference between ∆si and ∆tri is fa- even larger number of components represented by the
vored, thus spontaneously breaking TR symmetry. This internal hyperfine spin degrees of freedom, which is an
can be proved as follows: the quasi-particle spectra even number. The dipolar interaction is independent
reads Ei = ξ2+λ2i, with λ21,2 = |∆0φs+d(Ωk)|2 + on them, thus the system has an SU(2N) symmetry.
|∆zφz(Ωk)|2 ±p2Re(∆∗z∆0)φs+d(Ωk)φz(Ωk). The last The 2N ×2N pairing matrix ∆αβ(~k) can be classified
term vanishes for relative phase ±π2 between ∆0 and as N(2N +1)-component symmetric (odd parity) pair-
∆z. In other words, φz +iφs+d is unitary pairing, i.e., ingandN(2N−1)-componentantisymmetric (evenpar-
∆ (~k)∆(~k) is an identity matrix up to a factor, and ity) pairing, which are generalizations of the triplet and
†
λ2 = λ2 = |∆ φs+d(Ω )|2 + |∆ φz(Ω )|2|. All other singlet channel pairings, respectively. They can be ex-
1 2 0 k z k
phase differences give λ 6= λ , thus are non-unitary plicitlyconstructedasfollows. Wedefinethechargecon-
1 2
pairing. To show that unitary pairing is optimal, we jugation matrix R as R =(−)iδ and the time-
ij i,2N i+1
−
4
reversal operators T = RC where C is complex conju- and φz (spin symmetric) channels, a relative phase ±π
2
gation. For 2N = 2, R reduces to the familiar −iσ . R is needed to maintain the unitary pairing.
y
satisfies RT = −R and R2 = −1. On the other hand,
In summary,we haveinvestigatedthe competing pair-
any 2N ×2N Hermitian matrix can be expanded in the
ing symmetries in ultracold multicomponent dipolar
basis of the identity matrix and 4N2 −1 generators of
molecular systems, which provides a wonderful oppor-
the SU(2N) group. They can be classified as even and
tunity to investigated exotic pairings. We predict that
odd under TR transformation. N(2N +1) of them are
the anisotropyof the dipolar interactionprovides a well-
TR odd which can be constructed as spin-tensor matri-
defined pairing mechanism to the spin triplet, or, more
cesP withoddranknumbers(e.g. spin,spin-octupoles,
µ generally, the spin symmetric channel Cooper pairing.
etc.), and N(2N−1) of them are TR even which can be
The spin singlet even parity channel pairing in the φs+d
constructed as spin-tensor matrices Q with even rank
ν channel is tunable by the short range s-wave scattering.
numbers(e.g. theidentitymatrix,spinquadrupole,etc.).
It mixes with the spin triplet odd parity channel pair-
Using TP T 1 = −P , TQ T 1 = Q , and RT = −R,
µ − µ ν − ν ing by developing a relative phase ±π to maintain the
it can be shown that the matrices of P R are sym- 2
µ unitary pairing. This is another type of unconventional
metric and Q R are antisymmetric, respectively. Thus
ν Cooper pairing breaking TR symmetry.
we can decompose ∆ (Ω ) = ∆ (Ω )(Q R) +
αβ k asy,µ k µ αβ
∆ (Ω )(P R) , where ∆ (Ω ) (µ = 1 ∼ WepointoutthatourmechanismofTRbreakingmix-
sym,ν k ν αβ asy,µ k
N(2N −1)) and ∆ (Ω ) (ν = 1 ∼ N(2N +1)) are ing between triplet and singlet pairings is very general.
sym,ν k
even and odd functions of Ω , respectively. Forexample,insuperfluid3Heitwasoriginallyproposed
k
Theeigenvalueanalysisforcompetingpairingchannels that pairing would occur in the singlet d-wave chan-
is the same as in Eq. 8 by replacing the triplet (singlet) nel [31], induced by the attractive part of the Van der
pairing with the spin-symmetric (antisymmetric) pair- Waals interaction. Later, attention focused exclusively
ing, respectively. We next consider the unitary pairing on pairing in the triplet p-wave channel induced by the
in both spin-symmetric and spin asymmetric channels, short-range repulsive interaction [31, 32]. It is natural
respectively. A convenient choice for the matrix kernels to expect that both channels could contribute to pair-
is that P = σ ⊗I for the spin symmetric channel, ing at sufficiently low temperatures, leading to coupled
z z N
andQ=I whereI andI aretheidentitymatrices Balian-Werthamer(BW)tripletpairingandsingletchan-
2N N 2N
withN and2N-dimensions. Thefirstonecorrespondsto nel pairing with a relative phase ±π. In metallic super-
2
thepairingof i=1 2Nc†i(~k)c†2N i(−~k),whilethesecond conductors, coupling of an isotropic s-wave state with a
corresponds tPo i=∼1 2N(−)i−1−c†i(~k)c†2N i(−~k). In the BfoWr thtreippleatrtsictautlaerwciallseleawdhetroeathsiengrelelaitsiovteropphiacsgeaips o±nπly.
φz-channelpairing sta∼tes, the spin-SU(2N−)symmetry is 2
P These possibilities will be discussed separately.
broken down into SU(N)×SU(N)×U(1), thus it has
2N2branchesofspin-waveGoldstonemodes. Thevortex C. W. thanks helpful discussions with E. Fradkin and
configuration is similar to the N = 1 case including the S.Kivelson. C.W.issupportedbyNSFunderNo. DMR-
usual integer vortex and the half-quantum vortex com- 0804775,andSloanResearchFoundation. Thisworkwas
binedwithaπ-disclinationofspin-texture. Againforthe also supported in part by NSF under No. PHY05-51164
mixingbetweenpairingintheφs+d (spinantisymmetric) at the KITP.
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