aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Half - Quantum Vortices
V.P.Mineev
Service de Physique Statistique, Magn´etisme et Supraconductivit´e,Institut Nanosciences et Cryog´enie, UMR-E CEA/UJF-Grenoble1, F-38054 Grenoble, France (Dated: August 8, 2018)Unlike to superfluid He the superfluid He-A support the existence of vortices with half quantumof circulation as well as single quantum vortices. The singular single quanta vortices as well asnonsingular vortices with 2 quanta of circulation have been revealed in rotating He-A. However,the half quantum vortices in open geometry always possess an extra energy due to spin-orbit couplingleading to formation of domain wall at distances larger than dipole length ∼ − cm from the vortexaxis. Fortunately the same magnetic dipole-dipole interaction does not prevent the existence of half-quantum vortices in the polar phase of superfluid He recently discovered in peculiar porous media”nematically ordered” aerogel. Here we discuss this exotic possibility.The discoveries of half-quantum vortices in triplet pairing superconductor Sr RuO as well inthe exciton-polariton condensates are the other parts of the story about half-quantum vortices alsodescribed in the paper. I. INTRODUCTION
The quantization of circulation around vortex lines in superfluid He has been pointed out first by Lars Onsagerin his famous remark at the conference on statistical mechanics in Florence in 1949. The states of superfluid aredescribed by the order parameter which is complex function | ψ | e iϕ , hence the phase ϕ ”may be multiple-valued, butits increment over any closed path must be a multiple of 2 π , so that the wave-function will be single-valued. Thus thewell known invariant called hydrodynamic circulation is quantized; the quantum of circulation is h/m , ... ” Indeed,the superfluid velocity is given by the gradient of phase v s = ~ ∇ ϕ/m , hence, velocity circulation over a closed path γ is Γ = I γ v s d r = N hm . (1)The quantized vortices differ each other by the integer number of circulation quanta N. In superfluid He the vorticeswith one quantum of circulation h/m are usually created by the vessel rotation such that in equilibrium the totalcirculation around all vortices corresponds to the circulation of classic liquid rotating with given angular velocity. The energy of vortex per unit length E v = Z ρ s v s d r = ρ s Γ π ln Ra (2)is proportional to square of circulation, hence, the vortices with circulation quanta higher than 1 are unstable to decayfor the vortices with one circulation quanta. Here, R is vessel size and a is the coherence length which is of the orderof the interatomic distance in liquid He.Magnetic field acts as rotation in case of charged superfluids, so in superconductors the quantized vortex linesalso carry one quantum of magnetic flux φ = hc/ e and fixed value of magnetic moment φ / π . The vortices withmultiple flux quantum are energetically unstable in respect to decay to the single quantized vortices. This processcan be written as sort of conservation law, for instance 2=1+1, of quanta of circulation.The superfluid phases of He discovered in 1972 were proved to be much reacher in respect of types of stabledefects in the order parameter distribution. For instance, in superfluid He-A there were predicted 4 type of stablevortices with N = 0 , ± / , He-A till now were not registered. On the contrary, half-integer flux quantization were observedin cuprate superconductors where it was proved to be a powerful tool for probing the d-wave symmetry of thesuperconducting gap. The discovery of vortices with half-quantum flux has been reported recently in mesoscopicsamples of spin-triplet superconductor Sr RuO similar to superfluid He-A. Even earlier half-quantum vortices havebeen revealed in quite different ordered media - an exciton-polariton condensate. Here we discuss the new possibility of half-quantum vortices realization that appeared with stabilization of polarphase of superfluid He in so called ”nematically ordered” aerogel. . With this purpose in chapter 2 we introducethe notion of half-quantum vortices in superfluid He-A. The advantages of realization of such type vortices in thesuperfluid polar state will be described in the following chapter. The story about discovery of half-quantum fluxstates in Sr RuO is the subject of the chapter 4. We conclude by mention of other even more exotic possibilities ofhalf-vortices realization in a supersolid and in particular in Fulde-Ferrel-Larkin-Ovchinnikov superconducting state. II. VORTICES IN SUPERFLUID HE-A
The matrix of order parameter of superfluid He-A A Aαi = ∆ V α (∆ ′ i + i ∆ ′′ i ) / √ by the product of its spin and orbital parts. The unit spin vector V is situated in the plane perpendicular tothe direction of spin up-up | ↑↑i and down-down | ↓↓i spins of the Cooper pairs. The vectorial product ∆ ′ × ∆ ′′ = l of orthogonal unit vectors ∆ ′ , ∆ ′′ determines the direction of the Cooper pairs orbital momentum l . The superfluidvelocity in such a liquid is determined by v s = ~ m ∆ ′ i ∇ ∆ ′′ i . (4)The velocity circulation is given by Γ = N h m . (5)The half-quantum vortices are admissible because a change of sign of orbital part of the order parameter acquiredover any closed path in the liquid corresponding to half quantum vortex can be compensated by the change of signof the spin part of the order parameter, so that the whole order parameter will be single-valued. These vortices inthe superflow field are simultaneously disclinations in the magnetic anisotropy field V with half -integer Frank index,analogous to the disclinations in nematic liquid crystals.More visual picture of half-quantum vortices can be given assuming that all vectors change their directions leavingin ( x, y ) plane: V α = ˆ x α cos φ − ˆ y α sin φ , ∆ ′ i + i ∆ ′′ i = (ˆ x i + i ˆ y i ) e iϕ . Then the A-phase order parameter is written as A Aαi = Ψ Aα (ˆ x i + i ˆ y i ) / √
2, where Ψ Aα = ∆ (cid:0) e iϕ | ↑↑i α + e iϕ | ↓↓i α (cid:1) / √ , (6) | ↑↑i α = (ˆ x α + i ˆ y α ) / √ | ↓↓i α = (ˆ x α − i ˆ y α ) / √ ϕ = ϕ + φ, ϕ = ϕ − φ . Thus, the order parameter of superfluidA-phase is presented as the sum of the order parameters of spin up-up and spin down-down superfluids. The singlequantum vortex corresponds to the order parameter distribution such that the phase increment of the orbital part ofthe order parameter over a closed path is ∆ ϕ = 2 π . Here, the both condensates with up-up and down-down spinsacquires the same phase increment ∆ ϕ = ∆ ϕ = 2 π , whereas the spin part of the order parameter is homogeneous V = const . On the opposite, the half-quantum vortex is characterized by the increments ∆ ϕ = ± π, ∆ φ = ± π . In two-condensates language this corresponds to the single quantum vortex either only in spin up-up ∆ ϕ = ± π, ∆ ϕ = 0,or only in spin down-down condensate that is ∆ ϕ = ± π, ∆ ϕ = 0.The gradient energy in superfluid He is F ∇ = Z d r (cid:18) K ∂A αi ∂x j ∂A ⋆αi ∂x j + K ∂A αi ∂x j ∂A ⋆αj ∂x i + K ∂A αi ∂x i ∂A ⋆αj ∂x j (cid:19) (7)It is easy to check that the energy corresponding to the combined defect consisting of half-quantum vortex in theorbital part of the order parameter and the disclination in the vector V field is twicely smaller than the gradientenergy of single quantum vortex. More generally, for superfliud phases with order parameter consisting of productorbital and spin vectors (3) the energy of a defect is proportional to sum of squares of winding numbers of orbital andspin vector fields along a closed path around defect axis. For an half-quantum vortex it is F ∇ = [(1 / + (1 / ] π | ∆ | (2 K + K + K ) ln Rξ = π | ∆ | (2 K + K + K ) ln Rξ , whereas for a single quantum vortex it is F ∇ = π | ∆ | (2 K + K + K ) ln Rξ .
Thus, the half-quantum vortices looks as energetically more profitable.The singular single quanta vortices as well as nonsingular vortices with 2 quanta of circulation have been revealedin rotating He − A (for review see ) but half - quantum vortices were not. The reason for this is the spin - orbitalinteraction caused by magnetic dipole interaction of Helium nucleus. In a superfluid phase with triplet pairing thedensity of SO coupling energy is F so = g D | ∆ | (cid:18) A αα A ⋆ββ + A αi A ⋆iα − A αi A ⋆αi (cid:19) , (8)that in case of A-phase with order parameter (3) is F Aso = g D (cid:18) − ( Vl ) (cid:19) . (9)Hence, at distances larger than dipole length ∼ − cm from the vortex axis the spin-orbital coupling suppress theinhomogeneity in the spin part of the order parameter distribution: vector V tends to be parallel or antiparallelto the direction of the Cooper pairs orbital momentum. At these distances a disclination transforms in the domainwall (a planar soliton) possessing energy proportional to its surface. The neutralization of the dipole energycan be reached in the parallel plate geometry where Helium fills the space between the parallel plates with distancesmaller then dipole length under magnetic field
H >>
25 G applied parallel to the normal to the plates. This casethe half quantum vortices can energetically compete with N=1 vortices. However, even in this case the rotation ofa ”parallel plate” vessel with He-A will create lattice of half quantum vortices which at the same time presentstwo-dimensional plasma of ± / The half quantum vortices in superfluid He − A till now have not been revealed. III. VORTICES IN SUPERFLUID POLAR PHASE OF HE IN ”NEMATICALLY ORDERED”AEROGEL
Filling by liquid He an aerogel porous media allows to study influence of impurities on superfluid states withnontrivial pairing.
There was found that both known in bulk liquid A and B superfluid phases of He also existin aerogel. The new chapter in the investigations was opened when there was recognized that anisotropy of aerogelcan influence superfluid He NMR properties. This way several states of He-A with orbital and spin disorderinghave been discovered (see and references therein). The following experimental investigations has been performed on He confined in a new type of aerogel consisting of Al O · H O strands with a characteristic diameter ∼
50 nm anda chatacteristic separation of ∼
200 nm. The strands are oriented along nearly the same direction (say along ˆ z axis)at macroscopic distance ∼ He inthis type aerogel there were obtained indications that at low pressures the pure polar phase may exist in some rangeof temperatures just below critical temperature. The pairing states of superfluid He in a random medium with global uniaxial anisotropy have been investigatedby Aoyama and Ikeda . The corresponding second order in the order parameter GL free energy density consists ofisotropic part common for all the superfluid phases with p-pairing and the anisotropic part F (2) = F (2) i + F (2) a = α ( T − T c ( x )) A αi A ⋆αi + η ij A αi A ⋆αj , (10)where the media uniaxial anisotropy with anisotropy axis parallel to ˆ z direction is given by the traceless tensor η ij = η − (11)The B-phase state A Bαi = ∆ R αi e iϕ is indifferent to the presence of uniaxial anisotropy F (2) a ( A Bαi ) = 0, whereas theequal spin pairing states with the order parameter of the form A αi = V α A i creates the various possibilities.(i) A-phase A i = ∆ √ (ˆ x i + i ˆ y i ) F a = η | ∆ | (12)(ii) A-phase A i = ∆ √ (ˆ z i + i (ˆ x i cos α + ˆ y i sin α )) F a = − η | ∆ | / A i = ∆ˆ z i e iϕ F a = − η | ∆ | (14)Which phase has the highest transition temperature from the normal state depends on the sign of η . At negative η < T c belongs to A-state (i) with the Cooper angular momentum direction ~l parallel to the anisotropyaxis. At positive η > the energy of a thin disk shape body immersed in He-A depends of orientationof disk surface in respect to ~l vector and the minimum of this energy corresponds to the parallel orientation of thenormal to the disk surface to the ~l . Hence if there are multiple disks homogeneously distributed in space with somehowfixed orientation parallel each other this should stimulate the phase transition to the A-phase state with ~l parallel tothe disks normal direction that corresponds to the η < He-A is that the cigar axisperpendicular to ~l . The multiple cigars homogeneously distributed in space with axis parallel each other shouldstimulate the A-phase state with ~l vectors randomly directed in the plane perpendicular to the cigars axis thatcorresponds to the η > ~l perpendicular tothe cigars axis. Hence, the cigars type objects with parallel axis will stimulate phase transition to the polar state. Itmeans that first there will be phase transition to the polar state (iii) and then at lower temperature, when the fourthorder terms in free energy are important, one must expect the second order type phase transition to the distorted A-phase A i ∝ (ˆ z i + ia (ˆ x i cos α + ˆ y i sin α )) transforming at low temperatures to the A-phase (ii) with vectors l randomlydistributed in (x,y) plane as it was predicted by Aoyama and Ikeda. It is interesting that the similar phenomenonwith two subsequent phase transitions has been revealed in multi-sublattice antiferromagnet CsNiCl . As forsuperfluid He there were already obtained indications on existence of the polar state in ”nematically ordered”aerogel.Substituting the order parameter of polar state A polαi = ∆ V α ˆ z i e iϕ (15)in the expression (8) for the spin-orbital energy density we get F polso = 2 g D (cid:18) ( V ˆ z ) − (cid:19) . (16)We see that the spin-orbit coupling settles vector V in the plane perpendicular to the directions of aerogel strands.From this observation trivially follows that along with the singular vortices with phase ϕ increment over any closedpath equal to a multiple of 2 π there are half-quantum vortices with increment ∆ ϕ = ± π accompanied by disclinationin the field V with Frank index 1/2. According the argumentation applied in previous chapter to A-phase the half-quantum vortices in polar state are more energetically profitable than single quantum vortices. However, unlikeA-phase where the spin-orbital coupling prevent existence of half-quantum vortices in rotating vessel this is not thecase in the polar state.Thus, rotation of vessel filled by the superfluid polar phase of He in ”nematically ordered” aerogel with angularvelocity larger than the lower critical one must be accompanied by creation of half-quantum vortices as the mostenergetically profitable objects imitating rotation of the polar state superfluid component.
IV. VORTICES IN SUPERCONDUCTING STRONTIUM RUTHENATE Sr RuO is nonconventional superconductor possessing many unusual properties (for review see ). Commonbelieve based on the absence of the Knight shift changes below the critical temperature is that here we deal withsuperconductivity with triplet pairing. The material crystal structure is tetragonal with the point group symmetry D h . This case the order parameter for superconducting states with triplet pairing are related either to one-dimensionalrepresentation or two dimensional representation of the point group. For example, the order parameter for A u representation is A αi ˆ k i = | ∆ | ˆ z α ˆ k z e iϕ and for E u representation is A αi ˆ k i = | ∆ | ˆ z α (ˆ k x + i ˆ k y ) e iϕ . In both cases thedirection of spin vector V = ˆ z fixed by spin-orbital coupling is pinned to the tetragonal axis. If the spin part of theorder parameter is fixed the only stable order parameter defects are the single flux quantum Abrikosov vortices. Atthe same time if one creates condition allowing vector V free rotation like in superfluid phases of He one can expectthe existence of half-quantum flux vortices. As we remember the energy of half-quantum vortices accompanied bydisclination in the V field is smaller than the energy of single quantum vortex, but it is true only at the scale ofdistances from the vortex axis not exceeding the spin-orbital length. At larger scales the increment of spin-orbitalenergy due to vector V inhomogeneity will be larger than the gain in gradient energy of half-quantum vortex incomparison with gradient energy of single quantum vortex with V || ˆ z . The spin-orbital length can be estimated infollowing manner.The configuration V || ˆ z means that the Cooper pair spins lie in the basal plane. Hence, below T c the magneticsusceptibility for the magnetic field oriented in basal plane should coincide with the susceptibility in the normal stateand must decrease for the field direction along the c axis. In practice it keeps the normal state value independentlyof field direction. There was found that the Knight shift is not changed for H k ˆ c for fields larger than 200G. Itmeans this field is already enough to rotate the Cooper pair spin system to be parallel or antiparallel to the fielddirection. In other words the 200 G field is enough to overcome the the spin-orbital coupling. The comparison ofcorresponding paramagnetic energy with gradient energy of inhomogenious vector V distribution allows estimate thespin-orbital coherence length ∼ µm . So, to register the flux changes corresponding to half quantum vortices onemust work with mesoscopic size samples.The authors of Science Report have used cantilever magnetometry to measure the magnetic moment of micrometer-sized annual sample of strontium ruthenate prepared such that ab crystal plane is parallel to the plane of ring ( xy plane). The usual expression for the superfluid current density4 πλ c j = φ π ∇ ϕ − A , where λ = p mc / πn s e is the London penetration depth and φ = hc/ e is the flux quantum, leads to the fluxoidquantization which is the phase increment over a closed pass around the ringΦ ′ = 4 πλ c I j d s + Φ = φ N. (17)Here Φ = H A d s is the magnetic flux. Then making use the expression for the ring magnetic moment µ = Z d r ( r × j ) / c one can write the ring magnetic moment for the magnetic field directed in ˆ z direction that is perpendicular to thering plane µ z = ∆ µ z N + χ M H z . (18)Below the lower critical field H c = 8 G the magnetic moment is the linear function of the external field with thenegative slope corresponding to the Meissner susceptibility χ M . At each fields 8 G, 16 G, 24 G . . . as well atcorresponding negative values of external field which are the multiples of the lower critical field there were revealedthe magnetic moment jumps equal to ∆ µ z = 4 . · − emu demonstrating penetration of single quantum vorticesinside the ring.The crucial observation was obtained by application of field both in ˆ z and ˆ x directions. This case each jump inz-component of magnetic moment starts to split at increasing ˆ x direction field component in two jumps of twicelysmaller heights ∆ µ z = · . · − emu. So, the experiment demonstrates the appearance of half flux quantumvortices.It is natural to ask why these vortices do not appear in the absence of H x field component. The plausible reason isthat the applied field in ˆ z direction does not exceed 50 G, which is probably smaller than necessary to overcome thespin-orbit coupling and settle vector V in the basal plane of crystal. On the contrary the measurements with H x fieldcomponent were performed up to lower critical field of ring in ˆ x direction which is of the order of ∼
250 G. The jumpssplitting were distinguishable starting the fields H x ≈
80 G that was enough to create an inhomogeneous distribution V = ˆ z cos α + ˆ y sin α with angle α increment equal to ± π along a closed path around the ring. The confirmation ofthe vector V nonhomogeneity follows from calculation of magnetic moment µ x = e mc Z d r ( r × j s ) x / c (19)created by the spin current j s = ~ n s ∇ α. (20)The estimation yields µ x ≈ − emu that corresponds to the measured value and points out that vector V is indeednonhomogeneously distributed around the ring. V. CONCLUSION
Each ordered media is characterized by particular type of coherence that can be probed through the propertiesdirectly reflecting the symmetry and topology of ordering such as the Josephson effect and quantized vortices. Afterdiscussion of several instructive examples one can say that the situation when the order parameter of some orderedmedia consists of product of two parameters opens the possibility of existence of combined defects. Each part ofsuch defect corresponds to the nonhomogeneous stable distribution of its part of the total order parameter. In someparticular cases like in polar state of superfluid He in ”nematically ordered” aerogel or in mesoscopic superconductingrings of strontium rhuthenate these combined defects consist of half quantum vortex and a disclination with the Frankindex 1/2 in the spin part of order parameter.The half quantum vortices have also been observed in exciton-polariton condensate . The order parameter of thisordered media Ψ = e λ | ψ | e iϕ is given by the product of condensate wave function and the vector of light polarization. Along with the ordinaryvortices with phase increment along a closed path a multiple of 2 π the ordering like this obviously allows the combineddefects consisting of half quantum vortex and disclination with the Frank index 1/2 in the field of polarization vector e λ ( r ).Finally, it is worth to mention not yet discovered half-quantum vortices in such ordered media as charge densitywaves - CDW, spin density waves - SDW, super solids, and Fulde-Ferrel-Larkin-Ovchinnikov - FFLO superconductingstate. For instance, in case of 2D periodic ordering in ( x, y ) plane all of these ”quantum crystal” orderings can becharacterized by the order parameter of the formΨ = A cos( k ρ + φ ) e iϕ . Then it is clear that the space increment of each phase φ or ϕ along a closed path can be multiple of 2 π as well themultiple of ± π . In the latter case the half quantum vortex in the field ϕ ( r ) should be accompanied by a half-quantumvortex in the field φ ( r ). More interesting possibilities one can find in the paper by O. Dimitrova and M.V.Feigel’man . L. Onsager, Discussion at an
International Conference on Statistical Mechanics , Florence, 17-19 May 1949, Nuovo CimentoSupplemento 2 (Series 9), 249 (1949). R. P. Feynman, Application of quantum mechanics to liquid Helium, in book
Progress in Low Temperature Physics , Vol.1,ed. C.J.Gorter, New York, 1955. A. A. Abrikosov,
Fundamentals of the Theory of Metals , North Holland, New York, 1988. G. E. Volovik and V. P. Mineev, Pis’ma Zh. Exp. Teor. Fiz. , 605 (1976) [JETP Lett. , 561 (1976)]. J. R. Kirtley, C. C. Tsuei, Martin Rupp, J. Z. Sun, Lock See Yu-Jahnes, A. Gupta, M. B. Ketchen, K. A. Moler, M. Bhushan,Phys. Rev. Lett. , 1339 (1996). J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, P. M. Goldbart, Y.Maeno, Science, , 186 (2011). K. G. Lagourdakis, T. Ostatnicky, A. V. Kavokin, Y. G. Rubo, R. Andre, B. Deveaud-Pledran, Science , 974 (2009). R. Sh. Askhadullin, V. V. Dmitriev, D. A. Krasnikhin, P. N. Martynov, A. A. Osipov, A. A. Senin, A. N. Yudin, Pisma vZhETF, , 355 (2012). V. P. Mineev, Usp. Fiz. Nauk. , 303 (1983) [Sov. Phys. Usp. , 160 (1983)]. V. P. Mineev, M. M. Salomaa and O. V. Lounasmaa, Nature , 333 (1986). G. E.Volovik and V. P. Mineev, Zh. Exp. Teor. Fiz. , 2256 (1977) [Sov. Phys. JETP , 1186 (1977)]. V. P. Mineev,
Topologically Stable Dedects and Solitons in Ordered Media” , Harwood Academic Publishers, 1998. J. V. Porto and J. M. Parpia, Phys. Rev. Lett. , 4667 (1995). D. T. Sprague, T. M. Haard, J. B. Kycia, M. Rand, Y. Lee, P. Hamot, and W. Halperin, Phys. Rev. Lett. , 661 (1995). B. I. Barker, Y. Lee, L. Polukhina, and D. D. Osheroff, Phys. Rev. Lett. , 2148 (2000). V. V. Dmitriev, D. A. Krasnikhin, N. Mulders, A. A. Senin, G. E. Volovik, A. N. Yudin, Pisma v ZhETF, , 669 (2010)[JETP Lett. , 599 (2010)]. K. Aoyama and R. Ikeda, Phys. Rev. B , 060504 (2006). D. Rainer and M. Vuorio, J. Phys C: Solid State Phys. , 3093 (1977). X. Zhu and M. B. Walker, Phys. Rev. B , 3830 (1987). M. L. Plumer, Kevin Hood, and A. Caille, Phys. Rev. Lett. , 45 (1988). A. P. Mackenzie, Y. Maeno, Rev. Mod. Phys. , 657 (2003). Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, Journ. Phys. Soc. Jpn. , 011009 (2012). H. Murakawa, K. Ishida, K. Kitagawa, H. Ikeda, Z. Q. Mao, and Y. Maeno, Journ. Phys. Soc. Jpn. , 024716 (2007). V. P. Mineev and K. V. Samokhin,
Introduction to Nonconventional Superconductivity , Gordon and Breach, New York, 1999. O. Dimitrova, M. V. Feigel’man, Phys. Rev. B76