Breached Superfluidity via P-Wave Coupling

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by  E. Gubankova

MIT-CTP # 3528
Breached superfluidity via p-wave coupling
E. Gubankova,1 E.G. Mishchenko,2,3 and F. Wilczek1
1Center for Theoretical Physics, Department of Physics, MIT, Cambridge, Massachusetts 02139
5 2Department of Physics, University of Utah, Salt Lake City, UT 84112
0 3Lyman Laboratory, Department of Physics, Harvard University, MA 02138
0
2 Anisotropicpairingbetweenfermionspecieswithdifferentfermimomentaopenstwo-dimensional
n areasofgaplessexcitations,thusproducingaspatiallyhomogeneousstatewithcoexistingsuperfluid
a and normal fluids. This breached pairing state is stable and robust for arbitrarily small mismatch
J and weak p-wavecoupling.

7
2
Introduction. Recently there has been considerablein- as one integrates out high-energy modes near the Fermi

] terestinthepossibilityofnewformsofsuperfluiditythat surface. Thus the effective Hamiltonian will come to re-
n
could arisewhen one has attractive interactionsbetween sembletheformweassumeiftheinterspeciesinteractions
o
species with different fermi surfaces. This is stimulated are repulsive in s-wave but attractive in p-wave. Fermi
c
- by experimental developments in cold atom systems [1] statistics forbids diagonal (intraspecies) s-wave interac-
r
p and by considerations in high-density QCD [2]. Possible tions; if the higher partialwavesare repulsive,orweakly
u coexistenceofsuperfluiditywithgaplessexcitationsisan attractive, the model discussed here will apply. In the
s especiallyimportantqualitativeissue. Spatiallyhomoge- context of cold atom systems, tuning to an appropriate
.
t neous superfluid states that coexist with gapless modes p-wave Feshbach resonance, as recently reported in [16],
a
m at isolated points or lines in momentum space arise in can insure interspecies p-wave dominance.
a straightforward way when BCS theory is generalized A crucial difference between the model we consider
-
d to higher partial waves [3]. Gapless states also are well and the conventionalp-wave superfluid system, 3He, lies
n known to occur in the presence of magnetic impurities in the distinguishability of the paired species. So al-
o
[4] and, theoretically, in states with spontaneous break- though there are two components, there is no approxi-
c
[ ing of translationsymmetry [5], where the gapless states mate quasispin symmetry, and no analogue of the fully
span a two-dimensional fermi surface. Strong coupling gapped B phase [17]. In the absence of a magnetic field
3
between different bands also may lead to zeros in quasi- 3He has an approximate SO(3) SO(3) U(1) sym-
v L× S ×
8 particle excitations and gapless states [6]. For spheri- metry under separate spatial, rotations, spin rotations,
8 cally symmetric (s-wave) interactions a spherically sym- and number, which is spontaneously broken to the di-
0 metric ansatz of this type naturally suggests itself when agonal SO(3) in the B phase. The residual sym-
L+S
9 one attempts to pair fermions of two different species metry enforces a gap of uniform magnitude in all di-
0
with distinct fermi surfaces, and a pairing solution can rections in momentum space. The systems we consider
4
0 be found [7, 8, 9, 10, 11]. The stability of the result- havequitedifferentsymmetryandbreakingpatterns,for
t/ ing state against phase separation [12] or appearance of example U(1)Lz × U(1)A × U(1)B → U(1)Lz+A+B for
a a tachyon in the gauge field (negative squared Meissner two spin-polarized species A, B in a magnetic field, or
m mass) [13] is delicate, however [14, 15]. It appears to re- SO(3) U(1) U(1) U(1) if the mag-
L × A × B → Lz+A+B
- quire some combination of unequal masses, momentum- netic field can be neglected. The reduced residual sym-
d
dependent pairing interactions, and long-range neutral- metry allows for interesting direction-dependent struc-
n
ityconstraints. Herewedemonstrateanotherpossibility: ture in momentum space. (In the A phases 3He pairs
o
c direction-dependent interactions, specifically p-wave in- effectively as two separate single-species systems, which
: teractions. In this case, stability appears to be quite ro- again is quite different from our set-up.)
v
i bust. Itseemsquitereasonable,intuitively,thatexpand- Experimentalrealizationsofp-waveinteractionincold
X
ing an existing (lower-dimensional) locus of zeros into a atom systems have been reported recently in Ref. [18].
r two-dimensional zone should be significantly easier than Feshbachresonanceinp-waveoccursbetween 40K atoms
a
producingasphereofgaplessexcitations“fromscratch”. in f =9/2, m = 7/2 hyperfine states. This is in con-
f
−
We shall show that it even occurs for arbitrarily small trasttothes-waveresonance,whichoccursbetweennon-
coupling and small Fermi surface mismatch. identical f =9/2, m = 9/2 and f =9/2, m = 7/2
f f
− −
Interactionsrelevanttopairingcanbedominatedbyp- states [19]. A promising system for possible observa-
wave(orhigher)harmonics under severalcircumstances. tion of the p-wave breached pairing superconductivity
If the s-wave interaction is repulsive, it will not be sub- is a mixture of f = 9/2, m = 9/2 and f = 9/2,
f
−
jecttotheCooperinstability,andwillnotinducepairing. m = 7/2 atoms 40K tuned into the repulsive side
f
−
The Cooper instability can be regarded as an enhance- of the s-wave Feshbach resonance. Different densities
ment of the effective interaction for attractive channels (Fermi momenta) of m = 9/2 and m = 7/2 par-
f f
− −
2
ticles can be prepared using different magnitudes of an where θ∗ = arccosz, for z = I/∆ < 1, and θ∗ = 0 for
initial additional magnetic field, which is then removed. z >1. Performingthe integrations(detailedcalculations
Large atomic relaxation times ensure that the created will be given elsewhere [20]) we obtain the algebraic gap
(metastable) states will exist long enough to allow for- equation,
mation of a superfluid phase.
Model: Having in mind cold atoms in a magnetic trap π
ln(1/y)=z3 , (z <1)
with atomic spins fully polarizedby a magnetic field, we 4
consideramodelsystemwiththetwospeciesoffermions 1/y z z3
ln = z2 1+ arcsin[z−1] ,
having the same Fermi velocity vF, but different Fermi z+√z2 1 −2 − 2
momenta p I/v . The effective Hamiltonian is (cid:18) − (cid:19) p
F ± F (z >1) (5)
H = [ǫAa†a +ǫBb† b ∆∗a†b† ∆ b a ] (1)
p p p p −p −p− p p −p− p −p p where y = ∆/∆ is the relative magnitude of the gap
p 0
X
compared to its value at I =0,
with ǫA = ξ + I, ǫB = ξ I, ξ = v(p p ),
p p p p − p − F
i∆anrtpoeur=nadcPtitokhnVepis−Fek−rhmaV†kpib−†−skukriwf.aicHteheirn(eωtthheea“ttDrIea)bc,tyieva”endienntteehrreg-sypine2tcωrieaDs- ∆p0ol =exp(cid:18)−ν3g + 13(cid:19)≈1.40 exp(cid:18)−ν3g(cid:19). (6)
D
≫
species interaction is assumed to be either repulsive or
for weak coupling. There is a factor 3 in the exponent
negligibly small. Excitations of the Hamiltonian (1),
with anisotropicinteractioninsteadof 1 as in the s-wave
Ep± = ± ξp2+∆2p +I, are gapless provided that there BCS. For small values of z the solution to the gap equa-
are areasqon the Fermi surface where I > ∆p. The gap tion is y = 1− π4x3 with x = I/∆0. We depict the solu-
equation at zero temperature, tion of the polar phase gap equation in Fig. 1, with the
following numerical values of the characteristic points:
1 ∆
∆ = V k θ ξ2+∆2 I , (2) x = (4/3πe)1/3 = 0.538, y = e−1/3 = 0.717 (at the
p 2Xk p−k ξk2+∆2k (cid:18)q k k− (cid:19) pAointAy′(x)→∞),yC =e−A1/3/2=0.358(atthe point
can be simplified byptaking the integral over dξ , C y(x)=0).
p
∆n = νZ d4øπn′V(n,n′)∆n′(cid:18)ln|∆1n′| y
1
∆n′
+ Θ(I ∆n′ )ln | | . (3)
−| | I + I2−∆2n′! 0.8 m=0 m=1
whereν =p2/(2π2v )isthedensitpyofstates. Inthelast
F F
expression it is assumed that I and ∆ are dimension-
n
0.6
less, and scaled with the “Debye” energy: I 2ω I,
D
→
∆ 2ω ∆ . In deriving Eq. (3) we neglect de-
n D n
→
pendence of V on the absolute values of p and k;
p−k 0.4
this is valid for ω E . At weak coupling we
D F
≪
may linearize in the partial wave expansion, V(n,n′) =
VY (n)Y⋆ (n′). Assumingp-wavedominance,we 0.2
l,m l lm lm
parameterizeV(n,n′)=g(n n′)withg >0. P-wavegap
P ·
parameterscan arise in the forms Y andY , describ-
10 1±1 x
ing polar and planar phases respectively. 0.2 0.4 0.6 0.8
Polar phase: ∆ Y (n). We look for a solution in
n 10
∼
the form ∆ = ∆cos(n,z) where z is a fixed but arbi-
n FIG.1: Solutionsy(x)ofthegap equation in thepolar, m=
trarydirection(rotationalsymmetryisbroken). Thegap
0, and planar, m=1, phases. The lower branch corresponds
equation becomes, totheunstablestate. Thebranchesmergeatthepointswhere
y′(x)→∞, beyond which there are no non-zero solutions of
π/2
1 thegapequation. Thebrokenline∆=I isincludedtoguide
= dθsinθcos2θln(∆cosθ) (4) theeye.
− νg
Z
0
π/2 Planar phase: ∆ Y (n), ∆ Y (n). We now
+ dθsinθcos2θln z+√z2−cos2θ look for a solutionni∼n th1e1 form n∆∼ =1−∆1 sin(n,z)eiφ,
n
cosθ
θZ∗ ! where φ is the polar angle in the plane perpendicular
3
to z. The gap equation becomes, For the polar phase, assuming z >1 we find,
π/2 κ 1 3z3/4π,
2 = dθsin3θln(∆sinθ) κzz = 1−3πz/4+3πz3/8, (14)
− νg (cid:20) xx (cid:21) −
Z
0
θ∗ z+ z2 sin2θ aTthehicgoheeffir cviaelnutesκxoxf zbecom0.e7s52n)eginatdiivceataintgza≥n0in.4s8ta0b(iκlitzyz
+ dθsin3θln − (7) ≥
sinθ with respect to a transition into some inhomogeneous
p !
Z
0 state(probablysimilartoaLOFFstate). Fortheplanar
where θ∗ = arcsinz, for z < 1, and θ = π/2, for z > 1. phase,
0
Performing the integration we obtain the algebraic gap
equation, κzz =1 3z2 3z(1 z2)ln 1+z . (15)
z2 z 1+z 1 (cid:20)κxx (cid:21) ∓ 4 − 8 ∓ (cid:18)1−z(cid:19)
ln(1/y)= + (3+z2)ln + ln 1 z2 ,(8)
− 4 8 1 z 2 | − | The coefficients κ and κ remain always positive for
(cid:12) − (cid:12) xx zz
(cid:12) (cid:12) the whole range of z <0.623where the gapequation (8)
where again y is the relative m(cid:12)agnitu(cid:12)de of the gap com-
(cid:12) (cid:12) hasstablesolutions. Thus,wefindthattheplanarphase
pared to its value for a zero mismatch. For the planar
haslowerenergyandhigherdensityofCooperpairsthan
phase,
the polar phase and is therefore more stable.
1 3 5 3
∆pl = exp + 1.15 exp . (9) SpecificHeat: TheimportantmanifestationoftheBCS
0 2 −νg 6 ≈ −νg states with gapless excitations is the appearance of the
(cid:18) (cid:19) (cid:18) (cid:19)
Forsmallvaluesofz thesolutiontothegapequationhas term linear in temperature in the specific heat, which
the form y = 1 3x4 and x is defined as before. Note is characteristic for a normal Fermi liquid. The specific
− 4 heat is given by
that the planar phase gap is more robust than the polar
phase,beingperturbedbythefourthpowerinsteadofthe
∂n(E+) ∂n(E−)
third. Solution of the gap equation for the planar phase C = E+ p +E− p , (16)
p ∂T p ∂T
is depicted in Fig. 1, with the following numericalvalues p (cid:18) (cid:19)
X
of the characteristic points: x = 0.674, y = 0.787,
A A
zAS=tabxiAli/tyy:AT=he0.c8o5n6d;exnCsa=tioen−5e/n6er=gy0.i4s3g5i.venby(atT = where Ep± =± ξp2+∆2n+I. At lowtemperatures T ≪
I the first termqin Eq. (16) gives an exponentially small
0),
contribution. The second term, with E−, in Eq.(16) is,
dø ∆ 2
Ω Ω = ν n | n| +I2 2
s− n 4π − 2 ∞ ξ2+ ∆ 2 I
Z (cid:18) ν døn | n| −
C = dξ . (17)
− I I2−|∆n|2 Θ(I−|∆n|) . (10) 4T2−Z∞ Z 4π c(cid:16)opsh2 √ξ2+2|∆Tn|2−(cid:17)I
Evaluating this exprepssion for z = I/∆ < 1, w(cid:17)e obtain (cid:20) (cid:21)
for polar phase Performingtheintegration,wecalculatethecontribution
Ω Ω =ν∆2 1 πz3 +z2 , (11) ofthe gaplessmodes to the specific heatat T ≪I to be
s n
− (cid:18)−6 − 4 (cid:19) π πz, polar phase,
C =νT (18)
which is negative for z <0.537,and for planar phase 6 4z2, planar phase.
(cid:26)
1 z2 z(1 z2) 1+z
Ω Ω =ν∆2 + + − ln , (12) As expected, the “normal” contribution to the specific
s n
− −3 2 4 1 z heat, is proportionalto the area occupied by the gapless
(cid:18) − (cid:19)
which is negative for z < 0.623. For our specific model modes, i.e. the I/∆ strip around the equator for the
Hamiltonian,at weakcoupling,the planar phaseis more polar phase and the I2/∆2 islands around the poles for
stable. For I > ∆ the condensation energy is always the planar phase.
positive,indicatingthatthelowerbranchesareunstable. Conclusion and Comments: We have presented sub-
Followingthestandardmethodsinthetheoryofsuper- stantial evidence that our simple model supports the
conductivity [21] we calculate the super-currents in our planar phase gapless superfluidity in the ground state.
systemunderthe influence ofhomogeneousinspacevec- For I ∆ the gapless modes contribute high powers in
≪
tor potential A. The super-current is anisotropic, j = terms of mismatch, I4 for the solution and I2 for
i ∼ ∼
e2Nκ A with the components given by (κ =κ ), theheatcapacity,i.e. theyrepresentsmallperturbations.
m ik k xx yy The residual continuous symmetry of this state, and its
κzz =1 3I døn cos2θ Θ(I−|∆n|). (13) favorable energy relative to plausible competitors (nor-
(cid:20) κxx (cid:21) − 2 Z 4π (cid:20)sin2θ (cid:21) I2−|∆n|2 mal state, polar phase) suggest that it is a true ground
p
4
state in this model. The planar phase is symmetric un- for useful discussions. This work is supported in part
der simultaneous axial rotation and gauge (i.e., phase) by funds provided by the U. S. Department of En-
transformation. Also, we obtaina positive density of su- ergy(D.O.E.)undercooperativeresearchagreementDF-
perconducting electrons, suggesting that inhomogeneous FC02-94ER40818,and by NSF grant DMR-02-33773.
LOFF phases are disfavored at small I.
In some respects the same qualitative behavior we
find here in the p-wave resembles what arose in s-wave
[14]. Namely,isotropics-wavesuperconductivityhastwo
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