Spin vortex lattice in the Landau vortex-free state of rotating superfluids
aa r X i v : . [ c ond - m a t . o t h e r] A p r Spin vortex lattice in the Landau vortex-free state of rotating superfluids
G.E. Volovik
1, 2 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia (Dated: April 14, 2020)We show that the Landau vortex-free state in rotating container may give rise to the lattice of spinvortices. We consider this effect on example of spin vortices in magnon Bose-Einstein condensate(the phase coherent spin precession) in the B-phase of superfluid He, and on example of spinvortices in the polar phase of He.
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I. INTRODUCTION
In the rotating vessel the lattice of mass vortices repre-sents the ground state or the thermal equilibrium state ofthe rotating superfluid. For spin vortices the situation isdifferent. Orbital rotations does not act on the spin vor-tices, and the other external or internal fields are neededfor the formation of the lattice, such as Dzyaloshinskii-Moriya interaction, which leads to the formation ofskyrmion lattice, or some deformations of the crystal.Here we show that the lattice of spin vortices can becreated in rotating vessel, if the formation of the massvortices is suppressed,. The Landau vortex-free state inthe rotating vessel (the analog of Meissner state in su-perconductors) acts on spin vortices in the same way asrotation acts on mass vortices.
II. MIXTURE OF SPIN AND MASSSUPERFLUIDS
The hydrodynamics of mixture of two superfluids con-tains the term in the hydrodynamic energy, which mixestwo superfluid velocities . Here we consider magnonBEC in superfluid He-B – the homogeneously precess-ing domain (HPD) . When the spontaneous coherentspin precession takes place in the moving superfluid He-B, we have the mixture of spin superfluidity of magnonBEC and mass superfluidity of He-B.The magnon BEC or the phase coherent precession ofmagnetization is characterized by the density of magnonsin the condensate n M = S (1 − cos β ) = S − S z , where S = χH is spin density in magnetic field; S z is the projectionof the precessing spin on magnetic field; and β is thetilting angle of the precessing magnetization, see review .The magnon BEC has the superfluid velocity v M = ~ m M ∇ α , (1)where α is the angle of the coherent precession, whichplays the role of the phase of magnon BEC; and m M ismagnon mass (for simplicity we neglected the anisotropyof this mass). In the spin dynamics of magnon BEC,the magnon density n M and the phase α are canonically conjugate variables, and thus P = n M ∇ α = m M n M v M , (2)represents the momentum density of the magnon field.Since magnon BEC is the excited state of the back-ground superfluid, in the moving superfluid the magnonBEC acquires the Doppler shift energy term: F mix = P · ( v s − v n ) = m M n M v M · ( v s − v n ) . (3)It describes the interaction of magnon BEC with the massflow in the He-B, where v s and v n are correspondinglysuperfluid and normal velocities of the B-phase. Thisterm, which mixes superfluid velocity of the backgroundmass superfluid and superfluid velocity of magnon BEC,represents another realization of the Andreev-Bashkin ef-fect in superfluid mixtures, when the superfluid currentof one component depends on the superfluid velocity ofanother component. On the other hand the counter-flow v s and v n together with spin density plays the simi-lar role as Dzyaloshinskii-Moriya interaction in magnets,which violates the space inversion symmetry. In magnetsthis leads to formation of skyrmion lattices, see e.g. .The mixed term modifies the kinetic energy of magnonBEC: F grad = 12 m M n M v + m M n M v M · ( v s − v n ) . (4)This equation is valid in the zero temperaturelimit, where however the HPD experiences the Suhlinstability. For finite temperatures, at which the HPDis stable, the mixed term is smaller, but it has the sameorder of magnitude.
III. FORMATION OF SPIN VORTEX LATTICEIN MAGNON BEC
Now let us consider the Landau state in the containerrotating with angular velocity Ω . This is the state inwhich the normal component of the liquid has the solidbody rotation with velocity v n = Ω × r , while the super-fluid component of the B-phase is vortex-free and thusis not rotating, v s = 0. In He-B the Landau state istypical, because the critical velocity of the creation ofquantized vortices is large due to the large energy barrier,which is necessary to overcome for the creation of massvortices. In this case the gradient energy of magnonBEC (omitting the constant term) becomes F grad = 12 m M n M ( v M − Ω × r ) . (5)This means that the Landau state in the rotating vesselacts on spin superfluid (magnon BEC) in the same wayas rotation acts on mass superfluid, i.e. it should leadto formation of spin vortices, in which the phase α has2 π winding (single spin vortex in magnon BEC has beenobserved in Ref. ). So, if the creation of mass vortices issuppressed, but the creation of spin vortices is allowed,one obtains the state with the lattice of spin vortices.The number of these spin vortices in the Landau statein rotating vessel is determined by the circulation quan-tum of spin vortex κ M = 2 π ~ /m M and by angular ve-locity. That is why the number of spin vortices in thelattice in the Landau state can be expressed in terms ofthe equilibrium number of quantized mass vortices in thefully equilibrium rotating state: N spin N mass = κ κ M = m M m , (6)where κ = 2 π ~ / m is the quantum of circulation insuperfluid He-B and m is the mass of He atom. Inthis equation we compared two rotating states: the fullyequilibrium rotating state where the mass vortices formthe vortex lattice, while the spin vortices are absent; andthe metastable Landau state in rotating vessel, wheremass vortices are absent, while spin vortices form thelattice.In the isotropic approximation, which we used, themagnon mass is m M = ω L / c , where ω L is Larmor fre-quency, and c s is the speed of spin waves in He-B. Intypical experimental situations one has N M <
1, and thusspin vortices were not created in the Landau state. Thespin vortex lattice could become possible in the highermagnetic fields. The condition for N M > v c atwhich the energy barrier for formation of the mass vortexvanishes: ~ m M R < v c . (7)Here R is the radius of the container. IV. FROM HELICAL SPIN TEXTURE TO SPINVORTEX LATTICE
Similar effect of the formation of the vortex latticetakes place can be applied to the conventional spin vor-tices in superfluid He. The main problem there is that the spin vortices are influenced by the spin-orbit interac-tion, due to which they become the termination lines ofthe topological solitons. Spin vortices with soliton tailhave been observed in He-B. In the magnon BEC discussed in the previous sectionsthe spin-orbit problem is absent: in the precessing statethe spin-orbit interaction is averaged over the fast pre-cession, and as a result the spin vortices have no solitonictails, and thus may form the lattice in the Landau state.The solitonic tails appear in the applied radio-frequencyfield, where the energy of the condensate explicitly de-pends on α . But in the free precession the spin vorticesare free.For the conventional spin vortices, the spin-orbit prob-lem is resolved in the polar phase of He, which exists inthe nano-scale confinement (in the so called nafen).
If the magnetic field H is parallel to the strands of nafen,the solitonic tails are absent. As a result both the spinvortices and Alice strings (the objects, which combinethe half-quantum mass vortex and the half-quantum spinvortex) have been observed. The Landau states havebeen also observed in the polar phase in spite of zerovalue of the Landau critical velocity in this superfluidwith Dirac nodal line. So, let us consider the Landaustate in the polar phase.According to Brauner and Moroz, in the presence ofboth the counterflow v s − v n and magnetic field H , thespin texture is formed, which they call the helical spintexture. The spin texture originates form the similarmixed term in Eq.(3): F mix = S ∇ α · ( v s − v n ) , (8)where α is the angle of the unit ˆ d -vector, which describesthe spin part of the order parameter, and S = χH is spindensity in magnetic field. Let us mention that Braunerand Moroz considered the formation of the ˆ d -texture ina different phase – in the chiral A-phase. Both the A-phase and the polar phase belong to the class of spin-triplet p -wave superfluids with equal spin pairing. Inboth phases the spin part of the order parameter is de-scribed by the ˆ d -vector, but in the A-phase the spin-orbit problem exists. To avoid the effect of spin-orbit in-teraction, Brauner and Moroz considered the quasi-two-dimensional system – thin film of He-A.Extension of the constant counterflow discussed byBrauner and Moroz to the Landau state of the polarphase in the rotating cryostat, with v n = Ω × r and v s = 0, is straightforward. The gradient energy for spintextures becomes (omitting the constant term): F grad = 12 ρ spin (cid:18) ∇ α − Sρ spin Ω × r (cid:19) . (9)This corresponds to Eq.(5) with magnon mass m M = S/ρ spin , and thus instead of the helical texture suggestedin Ref. one obtains the lattice of spin vortices. Thenumber of equilibrium spin vortices in the Landau statein the vessel of radius R is: N spin = χHρ spin Ω R . (10)It has the same order of magnitude as in the case ofmagnon BEC, and also requires large magnetic field forthe experimental realization of spin-vortex lattice. V. CONCLUSION
For the spin-triplet superfluids, such as superfluid He, the Landau state of the superfluid in the ro-tating container can be the source of the formation ofthe vortex lattice of spin vortices. In the polar phase of He, the lattice of the conventional spin vortices is pos- sible. In the B-phase of superfluid He, the lattice ofvortices can be formed in the magnon Bose condensate.The spin-vortex lattice is formed, if the Feynman criticalvelocity for the creation of a spin vortex is lower than thereal critical velocity for creation of the mass vortex. Insuperfluid He the latter critical velocity is rather large,because of the large energy barrier.Similar phenomenon may occur in rotating neutronstars (review on superfluidity and superconductivity inneutron stars see in Ref. ). Acknowledgements . This work has been supportedby the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovationprogramme (Grant Agreement No. 694248). I thankVladimir Eltsov for discussions. I. Dzyaloshinskii, A thermodynamic theory of ”weak” fer-romagnetism of antiferromagnetics, J. Phys. Chem. Solids , 241 (1958). T. Moriya, Anisotropic superexchange interaction andweak ferromagnetism, Phys. Rev. , 91 (1960). A.F. Andreev and E.P. Bashkin, Three-velocity hydrody-namics of superfluid solutions, JETP , 164–167 (1975). G.E. Volovik, V.P. Mineev, I.M. Khalatnikov, Theory ofthe solution of the superfluid fermi liquid in superfluid boseliquid, JETP , 342–348 (1975). A.S. Borovik-Romanov, Yu.M. Bunkov, V.V. Dmitriev,Yu.M. Mukharskiy and K. Flahbart. Stratification of Hespin precession in two magnetic domains, JETP , 1199(1985). I.A. Fomin, Long-lived induction signal and spatiallynonuniform spin precession in He-B, JETP Lett. ,1037–1040 (1984). I.A. Fomin, Separation of magnetization precession in He-B into two magnetic domains. Theory, JETP , 1207–1213 (1985). Yu.M. Bunkov and G.E. Volovik, Spin superfluidity andmagnon BEC, in
Novel Superfluids , eds. K.H. Bennemannand J.B. Ketterson, International Series of Monographs onPhysics , 253 (2013). T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa,Yu. Yamasaki, H. Sagayama, H. Nakao, Ya. Taguchi, Taka-hisa Arima, Yo.Tokura, Skyrmion lattice with a giant topo-logical Hall effect in a frustrated triangular-lattice magnet,Science , 914–918 (2019). Yu.M. Bunkov, V.S. L’vov, and G.E. Volovik, Solutionof the problem of catastrophic relaxation of homogeneousspin precession in superfluid He-B, Pis’ma ZhETF ,624–629 (2006), JETP Lett. , 530–535 (2006). ¨U. Parts, V.M.H. Ruutu, J.H. Koivuniemi, Yu.N. Bunkov,V.V. Dmitriev, M. Fogelstr¨om, M. Huenber, Y. Kondo,N.B. Kopnin, J.S. Korhonen, M. Krusius, O.V. Lounas-maa, P.I. Soininen, G.E. Volovik, Single-vortex nucleationin rotating superfluid He-B, Europhys. Lett. , 449-454(1995). A.S. Borovik-Romanov, Yu.M. Bunkov, V.V. Dmitriev,Yu.M. Mukharskiy, D.A. Sergatskov, Observation of vortex-like spin supercurrent in He-B, Physica
B 165 ,649 (1990) V.P. Mineyev, G.E. Volovik, Planar and linear solitons insuperfluid He, Phys. Rev. B , 3197–3203 (1978). Y. Kondo, J.S. Korhonen, M. Krusius, V.V. Dmitriev, E.V.Thuneberg and G.E. Volovik, Combined spin - mass vor-tices with soliton tail in superfluid He-B, Phys. Rev. Lett. , 3331 (1992). K. Aoyama and R. Ikeda, Pairing states of superfluid He in uniaxially anisotropic aerogel, Phys. Rev. B ,060504(R) (2006). R.Sh. Askhadullin, V.V. Dmitriev, D.A. Krasnikhin, P.N.Martynov, A.A. Osipov, A.A. Senin, A.N. Yudin, Phase di-agram of superfluid 3He in ”nematically ordered” aerogel,JETP Lett. , 326 (2012). V.V. Dmitriev, A.A. Senin, A.A. Soldatov, and A.N.Yudin, Polar phase of superfluid He in anisotropic aerogel,Phys. Rev. Lett. , 165304 (2015). V.V. Dmitriev, A.A. Soldatov, A.N. Yudin, Influence ofmagnetic scattering on superfluidity of He in nematicaerogel, Phys. Rev. Lett. , 075301 (2018). V.V. Dmitriev, M.S. Kutuzov, A.A. Soldatov, and A.N.Yudin, Superfluid He in squeezed nematic aerogel, JETPLett. , 734–738. (2019). W.P. Halperin, J.M. Parpia, J.A. Sauls, New phases ofsuperfluid He confined in aerogels, Physics Today , 30(2018). G.E. Volovik, Polar phase of superfluid He: Dirac linesin the parameter and momentum spaces, JETP Lett. ,324–326 (2018). J. Nissinen and G.E. Volovik, Effective Minkowski-to-Euclidean signature change of the magnon BEC pseudo-Goldstone mode in polar He, JETP Lett. , 234–241(2017). T. Hisamitsu, M. Tange, R. Ikeda, Impact of stronganisotropy on phase diagram of superfluid He in aerogels,Phys. Rev. B , 100502 (2020). S. Autti, V.V. Dmitriev, J.T. M¨akinen, A.A. Soldatov,G.E. Volovik, A.N. Yudin, V.V. Zavjalov, and V.B. Eltsov,Observation of half-quantum vortices in superfluid He,Phys. Rev. Lett. , 255301 (2016) J.T. M¨akinen, V.V. Dmitriev, J. Nissinen, J. Rysti, G.E.Volovik, A.N. Yudin, K. Zhang, V.B. Eltsov, Half-quantumvortices and walls bounded by strings in the polar-distortedphases of topological superfluid He, Nat. Comm. , 237(2019). G. E. Volovik, K. Zhang, String monopoles, string walls,vortex-skyrmions and nexus objects in polar distorted B-phase of He, arXiv:2002.07578. S. Autti, J.T. Mkinen, J. Rysti, G.E. Volovik, V.V.Zavjalov, V.B. Eltsov, Exceeding the Landau superflowspeed limit with topological Bogoliubov Fermi surfaces,arXiv:2002.11492. T. Brauner and S. Moroz, Helical spin texture in a thin film of superfluid He, Phys. Rev. B , 214506 (2019). D. Vollhardt and P. W¨olfle,
The superfluid phases of helium3 , Taylor and Francis, London (1990). T. Mizushima, Ya. Tsutsumi, T. Kawakami, M. Sato, M.Ichioka, K. Machida, Symmetry protected topological su-perfluids and superconductors, From the basics to He, J.Phys. Soc. Jpn. , 022001 (2016).31