Energy of Stable Half-Quantum Vortex in Equal-Spin-Pairing

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by  Shokouh Haghdani

Energy ofStableHalf-Quantum Vortex inEqual-Spin-Pairing
Shokouh Haghdani1,∗ and Mohammad Ali Shahzamanian1
1Department of Physics, Faculty of Sciences, University of Isfahan, 81744 Isfahan, Iran
Inthetripletequal-spin-pairingstatesofboth3He−AphaseandSr2RuO4 superconductor, existenceof
Half-QuantumVortices(HQVs)arepossible. Thevorticescarryhalf-integermultiplesofmagneticquantum
fluxΦ0 =hc/2e.Toobtainequilibriumconditionforsuchsystems,onehastotakeintoaccountnotonlyweak
interactionenergybutalsoeffectsofLandauFermiliquid.OurmethodisbasedontheexplanationoftheHQV
intermsofaBCS−likewavefunctionwithaspin-dependentboots. Wehaveconsideredℓ = 2ordereffects
2
oftheLandauFermiliquid. WehaveshownthattheeffectsofLandauFermiliquidinteractionwithℓ =2are
1
negligible. InstableHQV,aneffectiveZeemanfieldexists. Inthethermodynamicstabilitystate,theeffective
0
Zeemanfieldproducesanon-zerospinpolarizationinadditiontothepolarizationofexternalmagneticfield.
2

n PACSnumbers:
a
J

4 1. INTRODUCTION

]
n One view to the liquid phase of 3He containsin normaland superfluidpartswhich at 3×10−3 Kelvin the superfluidpart
o startstobeoccurredmostlyintripletpairing1,2. ThespintripletpairinginthesuperconductorcompoundSr2RuO4isobserved
c experimentallybellow1.5Kelvin3.
-
r Thetripletpairingcontainsparticleswiththesamespindirectionsthatleadstospincurrent.Thespincurrentleadstointeresting
p
andimportantphenomenalikeHQVsinequal-spin-paring(ESP). Unlikecommonvortices,thehalf-quantumvorticescontain
su half-integer multiplications of the flux quantum Φ0 = hc/2e . The origins of vortices in type-II superconductors and 3He
. or 4He superfluidare different. In theformercase, the vorticesare appearedin the presenceofexternalmagneticfield while
at appearanceof thevorticesin the lattercases iscausedbythe rotationofthe vesselwhich3Heor4Heis contained. External
m magneticfieldinfluencesthevorticesin3Hetooandcangeneratehalf-quantumvortices.
- VakaryukandLeggett4 haveshownthatinequal-spin-pairingstate,thestabilityconditionofhalf-quantumvorticesisobtained
d whenstronginteractionsalsotakenintoaccount.Thegeneralmethodforcalculatingthestronginteractionsarepresentedbythe
n
LandauFermiliquidtheory.Theyhaveconsideredonlyℓ=1terminthisinteraction.Inthiswork,ℓ=2termisaccountedand
o
itisfoundthattermwithℓ=2isverysmallcomparedtoℓ=0,1terms.
c
[

1
2. THEORETICALAPPROACH
v
1
2 In the ESP state of a spin triplet condensate, the spin of particles in the Cooper pair is either aligned (up) or antialigned
8 (down)withacommondirectioninthespacewhichiscalledESP axis1,2. ThereforeCooperpairsmaybedescribedviaalinear
0 superpositionof states| ↑↑iand| ↓↓i. Thepairsarecondensedinthe sameorbitalstates, weshow thembyϕ andϕ . The
↑ ↓
1. many-bodywavefunctionwhichdescribesasystemofN/2pairsbyϕ↑ andϕ↓ canbewrittenas
0
2 ΨESP =A{[ϕ↑(r1,r2)|↑↑i+ϕ↓(r1,r2)|↓↓i]...[ϕ↑(rN−1,rN)|↑↑i+ϕ↓(rN−1,rN)|↓↓i]}, (1)
1
: where A is the antisymmetrization operator with respect to particles coordinates ri and spins. Here, for simplicity it will be
v
assumed that the ESP state is neutral. Annular geometry with the radius R and the wall thickness d with d/R ≪ 1 is
i
X considered,topreventcomplicationsconnectedtothepresenceofthecoreofvortex. Thereforeatzerotemperature,theHQV
r stateofthecondensatecanbedescribedvia4
a
iℓ iℓ
↑ ↓
ΨHQV =exp{ Xθi+ Xθi}ΨESP, (2)
2 2
i=↑ i=↓
hereθ istheazimuthalcoordinateoftheithparticleontheannulusandthespinaxisisalongthesymmetryaxisoftheannulus.
i
Theintegerℓ denotestheangularmomentumofσ,thecomponentofpairwavefunction.
σ
The thermal stability of system is obtained via minimization of energy of the system by using the wave function in Eq. (2).
In fact one needsfullversion of originalHamiltonianessentially. In the simplest case, one may considerBCS Hamiltonian,
H , along with spin triplet pairing term; however the mentioned Hamiltonian H is unable to provide the thermodynamic
BCS
stability of the HQV. In such systems, the stability condition is obtained when strong interactions taken into account. The
generalmethodforcalculationthe stronginteractionsare presentedby the LandauFermiliquidtheory. Althoughthe method
was used for investigatingthe normal metals, the generalized version of the method is used for studying superconductorand
superfluid1. ThusHamiltonianofthesysteminvolvestwoparts;1)BCSHamiltonianand2)LandauFermiliquideffects,H ,
FL
2
thenH = H +H . First,theexpectationvalueoftheweakcouplingHamiltonianwiththestateEq. (2)iscalculated. In
BCS FL
thiscase,onecanwriteitasasumofthreetermswithdifferentphysicalsources:
EBCS =E0+ES +T. (3)
whereE0istheenergycontributionoriginatedfromthefreedominternaldegreesofCooperpairs.Thiscontributionisindepen-
dentonthecenterofmassmotionoftheCooperpairsi.e.,onquantumnumbersℓ andℓ andthemagneticfieldmagnitudefor
↑ ↓
theannulusradiusRwhereismuchlargerthantheBCS coherencelengthξ05. Inthisworkweassumealargeenoughannuls
sothatthistermisignorable.
ThesecondtermintheEq. (3)istheenergyofspinpolarizationofthesystem. WeassumeN astheparticlesnumberwithspin
σ
projectionσ. onecandefineS asaprojectionofthetotalnumberspinpolarizationonthesymmetryaxisasS ≡(N −N )/2.
↑ ↓
Therefor,spinpolarizationenergyisobtainedas
2
(g µ S)
E = S B −g µ B.S, (4)
S S B
2χ
ESP
where g is the gyromagneticratio for particles and χ is the ESP state spin susceptibility. It is importantthat the spin
S ESP
polarizationS,isavariationalparameterandtheactualvalueofS isobtainedbyminimizationoftheenergy.
ThethirdtermintheEq.(3)isthekineticenergyofthecurrentscirculatinginthesystem. Weintroducethenewparametersas4:
ℓ +ℓ Φ ℓ −ℓ
↑ ↓ ↑ ↓
ℓsΦ ≡ − ,ℓsp ≡ , (5)
2 Φ0 2
whereΦisthetotalfluxthroughtheannulus.Byusingtheaboveintroducedparameters,T takesthefollowingform:
~2
2 2
T = 8m∗R2{(ℓsΦ+ℓsp)N +4ℓspℓsΦ}, (6)
where m∗ is the effective mass of particlescontainingthe Fermi liquid corrections. It is related to the bare mass of particles
m bythe usualrelationofFermiliquidtheoryasm∗ = m(1+F1/3)1,2. InEq. (6)the firsttermin thebracketsis constant.
Because it is proportionalto the total numberof particles N ≡ N↑ +N↓ and given valuesof ℓsΦ and ℓsp. The second term
generatesaneffectiveZeemanfieldintheHQV stateduetoproportionaltothespinpolarizationS.Themagnitudeofthisfield
andhencethevalueofthethermalequilibriumspinpolarizationshouldbeobtainbyenergyminimization4.
Formakingan HQV stable, thestrongcouplingeffectsshouldbeconsidered1,6. Also itneedsto be accountedthe changeof
FermiliquidenergyE duetothepresenceofspinandmomentumcurrentsintheHQV state. Thesecurrentsarecreatedby
FL
thespindependentboostofEq. (2). WeusethestandardformalismofFermiliquidtheoryandinvestigateℓ=2ordereffectsof
LandauFermiliquid.Byconsideringthementionedassumptions,wefinallyreachatthelengthlyexpressionasfollow:
1 dn
EFL = ( )−1Z0S2+
2 dε
N−1~2 1 2 2 Z1 2 2 2 Z1 2 Z1
8m∗R23{(ℓsΦF1+ℓsp 4 )N +4(ℓspF1+ℓsΦ 4 )S +4ℓsΦℓsp(F1+ 4 )SN}
N−1~4 Z2 2 2 2 Z2 2 2 2 2 2
+64m∗p2R4{[(F2+ 4 )(ℓsΦ+ℓsp) + 2 ℓsΦℓsp]N +[(16F2+2Z2)ℓsΦℓsp]×
F
N−1p2
S2+[(8F2+2Z2)ℓsΦℓsp(ℓ2sΦ+ℓ2sp)]SN}− 12m∗F(F2N2+Z2S2), (7)
where (dn/dε) is the density of states at the Fermi surface and Z0,Z1,Z2,F0 and F1 are Landau parameters. The first term
intheaboveequation,proportionaltoZ0,istheenergycostgeneratedbyaspinpolarizationandtherestexplainsFermiliquid
correctionscausedbythepresenceofcurrents.
NowweobtaintheequilibriumspinpolarizationintheHQV statebyminimizingthetotalenergyE +E withrespect
BCS FL
toS. ThenS canbewrittenas
S =(g µ )−1χB′, (8)
s B
whereχisthespinsusceptibilityofthesystem. For3He−Athevalueofχisapproximately0.37timessmallerthanthenormal
statesusceptibilityatlowtemperatures6. Wealsofindχas:
χ−1 = χ 1 +(ddnε)−1Z0(gSµB)−2+ m∗RN2−(g1~µ2 )2(ℓ2spF31 +ℓ2sΦZ121)
ESP S B
+16m∗pN2R−14~(g4 µ )2ℓ2spℓ2sΦ(8F2+Z2)− 6mN∗−(1gp2FµZ2)2. (9)
F S B S B
3
′
TheZeemanfieldB isinvolvedtwoparts:1)theexternalZeemanfieldB and2)theeffectiveZeemanfieldB :
eff
′
B =B+B , (10)
eff
TheeffectiveZeemanfieldB isduetothepresentofspincurrents:
eff
Beff =−~2(2gmSµ∗RB2)−1ℓspℓsΦ{1+ F31 + Z121 + 16p~22R2(4F2+Z2)(ℓ2sΦ+ℓ2sp)}. (11)
F
TheeffectiveZeemanfieldisaperiodicfunctionofthetotalexternalmagneticfluxΦwithaperiodequaltoΦ0. Thesignofthe
effectivefieldisalteredwhenthetotalmagneticfieldisequaltohalf-integervaluesofthefluxquantum4.
Finally,theenergyofthesystemE ≡ hHicanbeobtainedbyinsertingthevalueofS,accordingtoEq. (8)-Eq. (11),tothe
relationofthetotalenergyandignoringtheinternalenergycontribution.Thenonecanwrite:
E =−1χB′ + ~2N {ℓ2 +ℓ2 1+ Z121}+ ~4N {(ℓ2 +ℓ2 )2F2+ Z42 +ℓ2 ℓ2 Z2 }. (12)
2 8mR2 sΦ sp1+ F1 64mp2R4 sΦ sp 1+ F1 sΦ sp1+ F1
3 F 3 3
The contribution of the spin polarizationto the energy( first term of Eq. (12)) is small for reasonable values of the external
magneticfield. Thethirdtermoftheequationisrelatedtoℓ = 2whichcomparetothesecondtermisorderof~2/2mR2ε .
F
Evidently,one canignorethem foranalyzingthe stability ofHQVs. The stability regionof the HQVs dependson the (1+
Z1/12)/(1+F1/3)thatis,theratioofsuperfluidspindensitytosuperfluiddensityρsp/ρs1. Thecriteriaofstabilityisdirectly
obtainedbyminimizingofEq. (12)andleadstoρ /ρ ≺ 17. Inthe3He−A,theratioislessthanunityforalltemperatures
sp s
bellowcriticaltemperatureandthenHQVsarepossibleforalargelimitofthephasesdiagram7.
3. CONCLUSIONS
In the equal-spin pairing condensation of HQV an effective Zeeman field B exists. In the thermodynamic stability
eff
state,theeffectiveZeemanfieldproducesanonzerospinpolarizationinadditiontothepolarizationofexternalmagneticfield
B. The thermal stability of system is obtained via minimization of energyof the system by using the wave function and the
total Hamiltonian. The best Hamiltonianof the system involvestwo parts; 1) BCS Hamiltonianand 2) LandauFermiliquid
Hamiltonian. Theeffectsofℓ = 2termsofLandauFermiliquidhavebeenconsideredinthispaper. ThethirdterminEq. (12)
containsLandauparametersZ2andF2,whichcomparetothesecondtermisorderof~2/2mR2εF. ThisquantitywithR=0.1
microntakesapproximately10−7. Therefore,onecan omitthis term in Eq. (12). Consequentlyforthe stability conditionof
HQVs,itissufficientthatthesecondtermofEq. (12)beconsideredand ρsp <1isobtainedforstabilitycondition.
ρs
∗ Electronicaddress:[email protected]
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