Spin polarization of half-quantum vortex in systems with equal spin pairing
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Spin polarization of half-quantum vortex in systems with equal spin pairing.
Victor Vakaryuk ∗ and Anthony J. Leggett Department of Physics, University of Illinois, Urbana, Illinois 61801, USA (Dated: October 29, 2018)We present a variational analysis for a half-quantum vortex (HQV) in the equal-spin-pairingsuperfluid state which, under suitable conditions, is believed to be realized in Sr RuO and He-A.Our approach is based on a description of the HQV in terms of a BCS-like wave function with aspin-dependent boost. We predict a novel feature: the HQV, if stable, should be accompanied bya non-zero spin polarization. Such a spin polarization would exist in addition to the one inducedby the Zeeman coupling to the external field and hence may serve as an indicator in experimentalsearch for HQV.
When He is liquified and cooled into the millidegreeregime it enters a new phase which has been proved topossess a spin triplet paired condensate. There also ex-ists by now a growing body of experimental evidence thatSr RuO below 1.5 K is a spin triplet superconductor [1].This implies the possibility of many interesting phenom-ena not expected in systems with spin singlet pairing.One of them is the existence of half-quantum vortices(HQV’s) in the equal-spin-pairing (ESP) state of the spintriplet condensate [2, 3].The pairing symmetry for He is well established andthe so-called A phase is confidently believed to realize anESP spin triplet state [4, 5]. While there is, to the bestof our knowledge, no unambiguous observation of HQVin He-A [6], there is a strong theoretical argument infavor of their existence at least under some assumptionson the geometry of the experiment. On the contrary, thepairing state of Sr RuO is currently poorly understoodand the observation (or not) of HQV, along with otherexperimental information, would facilitate identificationof the underlying pairing symmetry. In addition, thereis a significant interest in HQV for topological quantumcomputing [13].It should be noted right away that the identificationof the superconducting phase of Sr RuO with an ESPstate does not by itself guarantee thermodynamic stabil-ity of HQV in this compound. Even under the assump-tion of negligible spin-orbit coupling, the kinetic energyof unscreened spin currents which accompany HQV dis-favors its formation vis-`a-vis the formation of a regularvortex where electromagnetic currents are screened overthe length of the penetration depth. Such an unfavorableenergy balance can be avoided by limiting the sample sizeto a few microns [7]. This however further complicatesthe experimental detection of HQV in Sr RuO .One of the most direct ways for detection of HQV isto look for spin currents which circulate around it. Theusual techniques for spin current detection are based onthe accumulation of spin and their straightforward ap-plication to this situation seems to be difficult. One canhowever use a fact that spin currents generate electricfield. A very rough conservative estimate shows that fora ring of size 1 µ the quadrupole electric field generated by the spin currents of HQV will create a potential differ-ence of 1nV across the ring – quite small, but not beyondthe capabilities of current experimental techniques.There is also a possibility of detecting HQV by lookingfor specific features in the magnetization curves of smallrings made of Sr RuO . Experiments in this directionare currently underway in the Budakian group at UIUC.In this paper we suggest an apparently new effectwhich may be utilized for the detection of HQV both inSr RuO and in He-A. The effect consists in the pres-ence of an effective Zeeman field in the HQV state of theESP condensate. In thermodynamic equilibrium such aneffective Zeeman field will produce a non-zero spin po-larization in addition to that created by external fields.In particular, such a spin polarization would exist evenin the absence of external Zeeman coupling provided thecondensate is in the HQV state. At the same time thisfield would not exist in a normal vortex state thus allow-ing one to distinguish between the two. For a 1 µ ringof Sr RuO the magnitude of the effective Zeeman fieldis about 10 Gauss and, taking the spin susceptibility tobe of order 10 − emu / mol [1], the spin polarization pro-duced by such a field can be seen in T or even Knightshift measurements.We start by noticing that in the ESP state the Cooperpair is always in a linear superposition of states in whichboth spins in the pair are either aligned (“up”) or an-tialigned (“down”) with a given direction in space. Thecorresponding many-body wave function for a systemwith N/ ϕ ↑ and ϕ ↓ canbe written asΨ ESP = A n(cid:2) ϕ ↑ ( r , r ) |↑↑i + ϕ ↓ ( r , r ) |↓↓i (cid:3) . . . (cid:2) ϕ ↑ ( r N − , r N ) |↑↑i + ϕ ↓ ( r N − , r N ) |↓↓i (cid:3)o , (1)where A is the antisymmetrization operator with respectto particles’ coordinates r i and spins. In the weak cou-pling limit, provided the pairing interaction conservesspin, the up and the down spin particles can be consid-ered as independent subsystems. In this case the HQVstate of the ESP condensate has a simple physical inter-pretation: It is a state in which the two spin systemshave different winding numbers, i.e. accommodate differ-ent number of vortices.To avoid complications related to the presence of thevortex core we specialize to an annular geometry. Let R be the radius of the annulus and d be the wall thickness;it will be assumed that d/R ≪ d/R or higher can be ignored. Then specializing to thezero-temperature case and choosing the spin axis alongthe symmetry axis of the annulus, a conceptually simpleansatz for the HQV state of the condensate isΨ HQV = exp n iℓ ↑ X i = ↑ θ i + iℓ ↓ X i = ↓ θ i o Ψ ESP , (2)where θ i denotes the azimuthal coordinate of the i -thparticle on the annulus. The integer ℓ σ is a projectionof the angular momentum of the σ -th component of thewave function of the pair on the symmetry axis; the coef-ficient 1/2 in the exponent reflects the fact that ℓ σ is themomentum of the pair, and, the case ℓ ↑ = ℓ ↓ correspondsto a regular full vortex. As can be seen from the above,state Ψ HQV is obtained from the initial state Ψ
ESP bya uniform spin-dependent boost. While there might bedoubts that the actual HQV is described by such a sim-ple form, it nevertheless should be considered as a goodstarting point for a variational analysis.In the d -vector formalism Ψ HQV as written above pro-duces a d vector which lies in the plane perpendicularto the spin axis i.e. to the symmetry axis of the annu-lus. For an annulus made of single crystal Sr RuO withthe c -axis along the symmetry axis, this corresponds to d being in the ab -plane of the crystal. Although this config-uration is not favored by the spin-orbit interaction, thereare theoretical indications that even a very small exter-nal magnetic field along c -axis can stabilize it [8]. Similarconsiderations also apply to He-A, and in what followsthe in-plane position of the d vector will be assumed. Itshould be emphasized however that our qualitative con-clusion about non-zero spin polarization in HQV doesnot depend on this assumption.Knowledge of the state (2) allows one to obtain a vari-ational energy of HQV which can then be minimized toyield its detailed structure. For definiteness from now onwe will consider only charged systems. In this case theappropriate for minimization thermodynamic potential isGibbs energy: G = hHi + Z d r n π B − π H · B o , (3)where hHi is the expectation value of the Hamiltonianof the system and B and H are the magnetic field andinduction respectively. For an annulus in a shape of aninfinite cylinder with the fields along the symmetry axis H is the external magnetic field and B is the field insidethe cylinder.As it will be seen below the actual form of H is crucialfor the stability of the HQV. In the simplest case H can be taken to contain only the reduced BCS Hamiltonian H BCS with the spin triplet pairing term. However such achoice of H combined with the ansatz (2) never makesHQV thermodynamically stable; at best the HQV, inwhich ℓ ↑ = ℓ ↓ , is degenerate with a full vortex ℓ ↑ = ℓ ↓ at the transition point between states with differentvorticities. To lower the energy of HQV below that of afull vortex one needs to account for strong interparticleforces. This can be done in the framework of Fermi liquidtheory which is also applicable in the superconductingstate [9]. With that purpose we write the Hamiltonianof the system as H = H BCS + H FL . (4)Here H FL describes energy corrections due to Fermi liq-uid effects and H BCS is a reduced BCS Hamiltonian withspin triplet pairing term representing the weak couplingpart of the theory. We will first evaluate the expectationvalue of the weak coupling Hamiltonian on the state (2).It can be written as a sum of three terms which havedifferent physical origins: E BCS = E + E S + T. (5)The first term in the equation above is the energy con-tribution coming from the internal degrees of freedomof Cooper pairs. For the radius of the annulus R muchlarger than the BCS coherence length ξ this contribu-tion will depend on neither the center of mass motion ofthe Cooper pairs, i.e. on quantum numbers ℓ ↑ and ℓ ↓ , northe magnitude of the magnetic field [10]. Assuming thatwe are dealing with a big enough annulus this term willnot be included in the subsequent considerations.The second term is the spin polarization energy of thesystem. Let N σ be the number of particles with spinprojection σ . Defining S as a projection of the totalnumber spin polarization on the symmetry axis S ≡ ( N ↑ − N ↓ ) / , (6)and g S as the gyromagnetic ratio for the particles in ques-tion, the spin polarization energy takes the following form E S = ( g S µ B S ) χ ESP − g S µ B B · S , (7)where χ ESP is the spin susceptibility of the ESP statecalculated in the weak-coupling limit. It should be notedthat at this point the total spin polarization S is a vari-ational parameter with the actual value of S to be foundby the minimization of energy.The third term on the r.h.s of eqn. (5) is the kineticenergy of the currents circulating in the system. Let Φbe the total flux through the annulus and Φ ≡ hc/ e bethe flux quantum. Introducing the notation ℓ sΦ ≡ ℓ ↑ + ℓ ↓ − Φ / Φ , ℓ sp ≡ ℓ ↑ − ℓ ↓ , (8)one obtains for T the following expression: T = ~ m ∗ R n ( ℓ + ℓ ) N + 4 ℓ sp ℓ sΦ S o , (9)where m ∗ is the effective mass of the particles due toFermi liquid corrections [14]. In the expression abovethe first term in the brackets is proportional to the to-tal number of particles N ≡ N ↑ + N ↓ and is thus fixedfor given values of ℓ s , ℓ sp . The second term is propor-tional to the spin polarization S and creates an effectiveZeeman field in the HQV state due to a mismatch be-tween velocities of the up and down spin components.The value of this field and hence the magnitude of thethermal equilibrium spin polarization should be found byenergy minimization. However, as have already been em-phasized, minimization of E BCS (or corresponding Gibbspotential G when self inductance is important) does notproduce a stable HQV; at best the HQV is degeneratewith a full vortex at the transition point at which theeffective Zeeman field vanishes due to vanishing of ℓ sΦ .To make an HQV stable one needs to go beyond theweak coupling Hamiltonian and introduce strong cou-pling effects. This can be done in the framework of Fermiliquid theory, in the way indicated by eqn. (4). For thatwe need to calculate the change of the Fermi liquid en-ergy E FL caused by the presence of spin and momentumcurrents in the HQV state. These currents are generatedby the spin-dependent boost (2) and can be expressedin terms of spin-up and spin-down quasiparticle distri-butions. Using the standard formalism of Fermi liquidtheory one obtains: E FL = 12 (cid:18) dndǫ (cid:19) − Z S + N − ~ m ∗ R h ( ℓ F + ℓ Z N + 4( ℓ F + ℓ Z S + 4 ℓ sΦ ℓ sp ( F + Z SN i . (10)Here ( dn/dǫ ) is the density of states at the Fermi leveland Z , Z and F are Landau parameters [15]. The firstterm, proportional to Z , is the energy cost produced bya spin polarization and the rest describes Fermi liquidcorrections due to the presence of the currents.It is worth pointing out that expressions (9) and (10),which describe energy transformation under the spin de-pendent boost (2), can also be written down in terms ofmomentum and spin currents and are limiting forms ofmore general transformation rules given in, e.g. [11].Now we are in a position to find the equilibrium spinpolarization in the HQV state. Minimizing the energy E BCS + E FL with respect to S we obtain that in equilib-rium S = ( g S µ B ) − χ B , (11)where χ is the spin susceptibility of the system which,up to terms of order ǫ − ~ / m ∗ R , is the spin suscepti-bility of the ESP state with Fermi liquid corrections; for He-A at low temperatures the value of χ is about 0.37of the normal state susceptibility [9]. The other quan-tity of interest in eqn. (11), the Zeeman field B , has twocontributions: B = B + B eff , (12)which are the external Zeeman field B and the effectiveZeeman field B eff caused by the presence of spin currents: B eff = − ~ ( g S µ B ) − m ∗ R ℓ sp ℓ sΦ n F / Z / o . (13)In thermal equilibrium the effective field is a periodicfunction of the total flux Φ with period Φ and changesits sign at flux values equal to half-integer number of fluxquanta. Since at least some of the constants enteringeqn. (13) are currently not known for Sr RuO it is notpossible to give an accurate prediction of the field’s mag-nitude. It is, however, of order µ − ~ / mR = Φ /πR ;since the first HQV, if stable, exists at about the samevalue of the external field, this means that the spin po-larization produced by the effective field is comparable tothat induced by the external field. It is this phenomenonwhich may provide additional ways for the experimen-tal detection of HQV. Its signature would be a sawtoothcontribution given by eqn. (13), to the otherwise linearfield dependence of the Zeeman spin polarization.Taking into account both types of spin polarizationand omitting the internal energy contribution (cf. thediscussion after eqn. (5)), the energy of the system E ≡hHi can be written as E = − χ B + ~ N mR (cid:26) ℓ + ℓ Z /
121 + F / (cid:27) , (14)where m is the bare particle mass related to m ∗ by theusual relation of Fermi liquid theory. For reasonable val-ues of the external field the contribution of the spin po-larization energy given by the first term on the r.h.s.of eqn. (14) relative to the total energy E is of order ~ ǫ − / mR and thus can be safely ignored for the anal-ysis of the stability of HQV.The region of stability of the HQV depends on (1 + Z / / (1 + F /
3) which is a zero temperature value forthe ratio of spin superfluid and superfluid densities ρ sp /ρ s [4]. The stability criterion found by direct minimizationof (14) yields the condition ρ sp /ρ s < et al. [7] the self inductance effect, whose treatment ne-cessitates the use of the Gibbs potential (3) constructedout of the energy (14), replaces this condition with amuch more stringent one. In particular, for a cylindricalannulus the stability of HQV requires that ρ sp /ρ s < (cid:0) Rd/ λ (cid:1) − , (15)with λ L denoting the London penetration depth. Thevalue of ρ sp /ρ s in Sr RuO is currently unknown, how-ever condition (15) makes the existence of HQV in largerings practically impossible.The physical interpretation of the stability condition(15) is quite transparent: While the electromagneticcurrents which accompany both the full and the half-quantum vortex are well screened in the bulk, the spincurrents which are present only in the HQV are not, pro-ducing an additional energy cost over the full vortex. Tomitigate such a cost and make HQV stable one needs toreduce the spin current energy by reducing either spinsuperfluid density ρ sp or the “effective volume” Rd/ λ over which the spin currents flow such that the condition(15) is satisfied.It is further to be remarked that in an annular geome-try where one does not have to deal with the vortex core,the stability condition ρ sp /ρ s < He-A. In He-A the ratio ρ sp /ρ s is known to be well below 1 for all temperaturesbelow critical, hence making the existence of HQV pos-sible in a large part of the phase diagram. By contrast,recent numerical results [12] in a solid cylindrical geom-etry claim that HQV exists only in a high field regionand at temperatures sufficiently close to the transition.We believe, however, that the narrowness of the regionof HQV stability obtained in [12] is due to the omissionof the Fermi liquid effects from the consideration.The real-life question about the thermodynamic sta-bility of HQV is complicated and may depend on manyfactors not included in the preceding discussion. Amongthese are deviations from the annular geometry, inclusionof spin-orbit interaction and the possibility of the d vec-tor lying in the plane other than ab . It should however beemphasized that once the HQV has been stabilized we donot expect our qualitative conclusion about the presenceof the spin polarization to be altered by the aforemen-tioned factors since this conclusion originates from oneof the defining properties of the HQV, namely velocitymismatch between different spin components. This ve-locity mismatch shifts the chemical potentials of up- anddown-spin components by an amount of order ~ /mR which, in thermal equilibrium, produces an effective spinpolarization.In conclusion, we have shown that the thermal equi-librium state of the half-quantum vortex in the annulargeometry should be spin polarized. This effective spinpolarization is a periodic function of flux and contributesadditively to the spin polarization induced by the exter- nal Zeeman coupling. The magnitudes of the two con-tributions are comparable, thus making the effect po-tentially observable through, e.g., NMR measurements.This suggests a new way for the experimental detectionof half-quantum vortices.One of us (V.V.) would like to thank D. Fergusonfor a number of valuable discussions and C.P. Slichterfor a discussion on the possibility of NMR measure-ments. This work was supported by the U.S. Depart-ment of Energy, Division of Materials Sciences underAward No. DE-FG02-07ER46453 through the FrederickSeitz Material Research Laboratory at the University ofIllinois at Urbana-Champaign. ∗ [email protected][1] A. Mackenzie and Y. Maeno, Rev. Mod. Phys. , 657(2003).[2] G. E. Volovik and V. P. Mineev, JETP Lett. , 561(1976).[3] M. C. Cross and W. F. Brinkman, J. Low Temp. Phys. , 683 (1977).[4] A. J. Leggett, Rev. Mod. Phys. , 331 (1975).[5] D. Vollhardt and P. W¨olfle, The Superfluid Phases ofHelium 3 (Taylor and Francis, 1990).[6] M. Yamashita, K. Izumina, A. Matsubara, Y. Sasaki,O. Ishikawa, T. Takagi, M. Kubota, and T. Mizusaki,Phys. Rev. Lett. , 025302 (2008).[7] S. B. Chung, H. Bluhm, and E. A. Kim, Phys. Rev. Lett. , 197002 (2007).[8] J. Annett, G. Litak, B. Gyorffy, and K. Wysokinski,Physica C , 995 (2007).[9] A. J. Leggett, Phys. Rev. Lett. , 536 (1965).[10] V. Vakaryuk, Phys. Rev. Lett. , 167002 (2008).[11] M. C. Cross, J. Low Temp. Phys. , 525 (1975).[12] T. Kawakami, Y. Tsutsumi, and K. Machida, Phys. Rev.B , 092506 (2009).[13] see e.g. D. A. Ivanov, Phys. Rev. Lett. , 268 (2001).[14] It is implicitly assumed that the system under consid-eration is Galilean invariant, e.g. can be adequately de-scribed by the jellium model. We ignore complicationsrelated to the presence of the periodic lattice potentialwhich are discussed in A. J. Leggett, Ann. Phys. , 76(1968).[15] The commonly used set of parameters F al , F sl is relatedto these by F sl = F l and F al = Z l /70