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Spin Diffusion Coefficient of A Phase of Liquid 3He at Low Temperatures and Stability of Half Quantum Vortex PDF

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Preview Spin Diffusion Coefficient of A Phase of Liquid 3He at Low Temperatures and Stability of Half Quantum Vortex

SpinDiffusionCoefficient ofAPhaseofLiquid 3HeatLow Temperatures and Stability ofHalf Quantum Vortex Shokouh Haghdani1,∗ and Mohammad Ali Shahzamanian1 1Department of Physics, Faculty of Sciences, University of Isfahan, 81744 Isfahan, Iran We theoretically investigate the spin diffusion coefficient tensor in the A phase of liquid 3He in term of quasiparticle life-time by using the Kubo formula approach at low temperatures. In general, the coefficient isafourth rank tensor for theanisotropic states and canbe defined as afunction of both normal component of spin-current and magnetization. The quasiparticle life-timeis obtained by using the Boltzmann equation. 2 1 Wefind that components of the spin diffusion coefficient are proportional to T−2 at low temperatures. The 0 normalcomponentsofspincurrent,hence, arestronglydiffusiveandonecanignorethecontributionofthese 2 componentstothestabilityofhalfquantumvortices(HQVs)intheequal-spin-pairingof3He−Astate.Hence tomakeastableHQV,itisenoughforonetoconsiderweakinteractionplustheeffectsofLandauFermiliquid. n a PACSnumbers: J 4 1. INTRODUCTION ] n o After the discovery of new phases of 3He , a large amount of theoretical effort has been done on the spin collective c excitations1,2. Oneviewtotheliquidphaseof3Hecontainstwonormalandsuperfluidpartswhichat3×10−3 Kelvinthesu- - r perfluidpartstartstobeoccurredmostlyintripletpairing1,2. ThespintripletpairinginthesuperconductorcompoundSr RuO p 2 4 bellow 1.5 Kelvin is observed experimentally3 like the A phase of superfluid 3He . The main similarity between 3He−A u s and Sr2RuO4 phases is that only Cooper’s pairs correspond to pair spin projections Sz = 1 and −1 or equal-spin-pairing . (ESP)inthesephases.ThenormalstatepropertiesofSr RuO areobtainedintermsofaquasitwodimensionalFermiliquid t 2 4 a quantitatively3whereas3He−AisathreedimensionalFermiliquid. Thisdiscrepantingmanifestitselfonthestructureofthe m gapenergiesofthem. Thetripletpairingcontainsparticleswiththesamespindirectionsthatphenomenonleadstospincurrent. - Intwofluidmodel,thisspincurrentincludesnormalandcondensate(Cooper’spairs)components.ThespincurrentofCooper’s d pairsleadstointerestingandimportantphenomenasuchasHQVsintheESP state.Unlikecommonvortices,thehalf-quantum n vorticescontainhalf-integermultiplicationsofthefluxquantumΦ =hc/2e. o 0 c VakaryukandLeggett4representedadifferentanalysisintheequal-spin-pairingsuperfluidstateforaHQV. TheyusedaBCS [ likewavefunctionwithaspin-dependentboostandpredictedthatinthestabilityofHQV aneffectiveZeemanfieldexists. In thethermodynamicstabilitystate,theeffectiveZeemanfieldproducesanon-zerospinpolarizationinadditiontothepolarization 1 ofexternalmagneticfield. Intheirtheorytheeffectofdiffusiveflowofthenormalcomponentdidnottakeintoaccount.Inthis v 3 paperweshowthatonlyatverylowtemperaturethisisthecaseandinsomelimitsoftemperaturethenormalcomponentmight 2 benotdiffusiveandtakepartinstabilityofHQV. 8 0 . 1 2. THEORETICALAPPROACH 0 2 OurformulationonspindiffusioncoefficientismainlyconcentratedontheKuboformulaapproach5. FortheAphaseof3He 1 superfluiditisbrieflywritteninthefollowing.Ingeneral,thespindiffusionD isafourthranktensorwhichmaybeexplained : v bytherelation: i X ∂ ar J(iαn) =XDiαjβ(∂xβ)(Sj −χjlHl), (1) βjl whereJ ,S,Handχarethespincurrentdyadicofthenormalcomponent,themagnetization,thestaticmagneticfield,andthe n staticmagneticsusceptibilitytensor,respectively.Forthemomentthedipoleforcesareignoredandaccordingtotheconservation lawonecanwrite: ∂S ∂ ( i)+( )J =0, (2) iα ∂t ∂x α inwhichthecomponentsofthetotalspincurrentJare 1 J =J(n)+J(s) and J(s) =( )ραβΩ , (3) iα iα iα iα 2m ij jβ 2 hereΩ isthecomponentofthespinsuperfluidvelocitydyadic5. Thedynamicsusceptibilitytensorisdefinedby: jβ S(k,ω)= χ (k,ω)H (k,ω). (4) ij j X j BysubstitutingEq. (3)andEq. (1)inEq.(2),andtocomparewithEq.(4)isobtained: lim lim ω χ′′(k,ω)= Dαβk k χ , (5) ω→0k→0k2 il X ij α β jl αβj b b ′′ hereχ (k,ω)is the imaginarypartof the spindynamicsusceptibilityχ (k,ω). FollowingKadanoffandMartin’sprocedure il il ontheimaginarypartofthedynamicsusceptibilitywiththeanti-commutatorofthemagnetizationinthenormalphaseonemay generalizetheirresulttotheanisotropicsuperfluidstates5. Intheabsentofthedipoleinteractiononemaywrite: ′ ′ dω dk βω ′′ ′ ′ h{S (r,t),S (r ,t )}i= coth( )χ (k,ω)×exp[ik.(r−r )−iω(t−t )]. (6) i j Z π Z (2π)3 2 ij Withsupposethatonlythespincurrentassociatedwiththenormalcomponenttakespartinthediffusiveflow,theeffectsof thesuperfluidcomponentbeingnegligible.ByusingEq.(6)andEq.(2)inEq. (5),isobtained ∞ 1 Dαβχ = β lim dr dteiωth{J(n)(t),J(n)(0)}i, (7) ij jl 4 ω→0Z0 Z iα lβ hi representsthe expectationvalue in the equilibriumensemble. By introducingthe correlationfunctionK(τ) as Kαβ(τ) = ij hT {J(n)(τ),J(n)(0)}iinwhichthetimeτ variesbetween−β to+β. ThespindiffusioncoefficientDmaybewritteninterms τ iα jβ ofthecorrelationfunction5; 1 ImKαβ(τ)(k =0,iω →ω+iη) Dαβχ = lim il l , (8) ij jl 4ω→0 ω where µ2 d3p 1 Kαβ(ω ,k)= eξmη(k +2P )+(k +2P )× ij n 8m2 Z (2π)3β α α β β X m Tr[αiG(k+P,ω +ξ )αjG(P,ξ )]. (9) n m m inwhichαi arethePaulimatricesinthefour-dimensionalrepresentation.AfterabitofalgebraEq.(8)canbewrittenas: Dαβχ = βχnVF2(1+ 14Z0) dξdΩdωp p sech2(βω)× ij jl 32 Z 4π 2π α β 2 Tr{αi[G(P,ω+)−G(P,ω−)]αl[G(P,bω+b)−G(P,ω−)]}. (10) Forthe3He−Aphase,theGreen’sfunctionmaybewrittenas ωZ (ω)+ξρ ×1−Z (ω)∆(α.d)σ ×ρ G(p,ω)= p 3 p 2 2, (11) ω2Z2(ω)−ξ2−Z2(ω)∆2(Ω) p p p whereZ (ω)isthere-normalizationfunction.Wedefined≡−1i (σ σ) ∆ anduseG(p,ω)forobtainingDαβχ ; p 2 α,β 2 αβ αβ ij jl P Z dΩ ∞ sech2(βω) Dαβχ =βχ V2(1+ 0) p p dω 2 ×[(2ω2−∆2)δ +∆2d d ], (12) ij jl n F 4 Z 4π α βZ∆ 8Z2(ω2−∆2)32 il i l b b hereZ istheimaginarypartofZ (ω)andissimplyhandledbyusingthepolesofthesingle-particleGreen’sfunctiontodefine 2 p thequasiparticleenergyandlifetime. Hencetolowestorderintheimaginarypartswemaywrite EZ (E) Z = 1 . (13) 2 2τ(E,T)(E2−∆2) 3 Therefor,thespindiffusioncoefficientsforthe3He−Aphasearewrittenas Z dΩ ∞ βE βτ(E,T) ∆2 Dαβχ =χ V2(1+ 0) p p dξsech2( )× [2δ − (δ −d d)]. (14) ij jl n F 4 Z 4π α βZ 2 4Z (E) il E2 il i l 0 1 b b The aboveformulahas been obtainedwith the conditionω τ < 1 where ω is the Larmor frequencyand τ is the spin L D L D diffusionlifetime. Astotheexperimentalvaluesofω 6 thisconditionisfulfillevenatverylowtemperatures. Hence,wemay L computeτ(E,T)atlowtemperatures. Inthislimitoftemperatureonlythequasiparticleswhicharelocatedatthenodesofthe gapenergyinthe3He−Astatetakeparttoscatteringprocessesandwemaywrite7 πθ2 [(πk T)2+E2] τ−1(E,T)=( m )A2 B (15) 256εF s[1+exp(−kBET)] where8θ ≃ πkBT ,A = S ,S =As−3Aa andAs,a = Fls,a ,whereFs,a aretheLandua’sparameters7. m ∆(0) s Pl l l l l l [1+(F2lls+,a1)] l Finally, the spin diffusioncoefficientin 3He−A phase at low temperaturesare Dzz = Dzz ≃ 3Dzz = C/T2 in which zz xx yy C ≃ 16VF2(1+F0a)εF~. A2π2k2 As itcansbeBunderstoodatlow temperatures,the spindiffusioncoefficientsincrease asT−2 andthe normalcomponentsof spincurrentarestronglydiffusiveandonecanignorethecontributionofthesecomponentstothestabilityofHQV. 3. CONCLUSIONS Inconclusion,we have investigatedthe temperaturedependanceofspin diffusioncoefficientof superfluid3He−Aat low temperatures. We haveemployedthe Kuboapproachandderivedthe spin-diffusivecoefficientof A phase of the Liquid3He in term of quasiparticle life-time. The quasiparticle life-time is obtained by using the Boltzmann equation. We have found thatthespin-diffusivecoefficientsatlowtemperaturesregimeareproportionalto1/T2. Thefindingsuggestsstronglydiffusive normalcomponentsof spin-currentand then weak contributionof them to the stability of the half-quantumvortexstate. Our workof agreementwith the assumptionis in the recentworkby Vakaryukand Leggett4. The authorsstudy the half-quantum vortexphenomenonintheabsenceofnormalcomponentsofspin-currentatT =0andshowthepossibilityofformationofthe half-quantumvorticesin theequal-spinparingofLandauFermiliquid. Thenormalcomponentsofspin-currentareimportant quantitieswhich mightgenerateimportantinfluencesand our findingsshow goodagreementwith their assumption regarding thespin-current. ∗ Electronicaddress:[email protected] 1 LeggettAJ1975Rev.Mod.Phys.47331-414. 2 VollhardtDandWolfleP1990TheSuperfluidPhasesofHelium3(London:TaylorandFrancis). 3 MackenzieAPandMaenoY2003Rev.Mod.Phys.75657-712. 4 VakaryukVandLeggettAJ2009Phys.Rev.Lett.103057003. 5 ShahzamanianMA1976J.LowTemp.Phys.2227. 6 WheatleyJ1970Rev.Mod.Phys.47415. 7 ShahzamanianMA1989Condens.Matter. 11965-1970. 8 GreavesNAandLeggettAJ1983J.Phys.C:SolidStatePhys164383.

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