Chiral vortical effect generated by chiral anomaly in vortex-skyrmions
CChiral vortical effect generated by chiral anomaly in vortex-skyrmions
G.E. Volovik
1, 2 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics RAS, Kosygina 2, 119334 Moscow, Russia (Dated: October 22, 2018)We discuss the type of the general macroscopic parity-violating effects, when there is the currentalong the vortex, which is concentrated in the vortex core. We consider vortices in chiral superfluidswith Weyl points. In the vortex core the positions of the Weyl points form the skyrmion structure.We show that the mass current concentrated in such a core is provided by the spectral flow throughthe Weyl points according to the Adler-Bell-Jackiw equation for chiral anomaly.
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I. INTRODUCTION
The problems with Weyl materials – Weyl semimetals and Weyl superconductors (see latest reviews in Ref. and Ref. correspondingly) – is that at first glance they should possess any type of a bulk response that exists inconventional non-Weyl materials with the same symmetry . That is why the task is to resolve the contribution ofthe anomalies, which accompany the physics of Weyl fermions, from the conventional contributions, which follow fromthe symmetry consideration. Here we consider the situation when the effect is fully determined by chiral anomaly.This is the chiral vortical effect (CVE) produced by the vortex-skyrmions in Weyl superfluids/superconductors. II. CVE BY SKYRMION IN CHIRAL SUPERFLUID WITH TWO WEYL NODES
For simplicity we consider the Galilean invariant system, where the mass current coincides with the momentumdensity, and in case of charged particles the electric current coincides with the mass current with the factor e/m .We discuss a particular type of the parity violating effects , when there appears the current along the vortexaxis, which is concentrated in the vortex core. As an example of vortices, which experience such a chiral vorticaleffect (CVE), we consider vortex-skyrmions in Weyl superfluids. The vortex-skyrmion is the continuous (non-singular)texture in Fig. 1, which has the skyrmion structure of the orbital magnetization, and the superflow with two quanta ofcirculation around the skyrmion, see review . For such inhomogeneous configuration the traditional response theory,which results depend on the order of limits q → ω → , is not applicable. Instead we can use the trick, whichhas already been used in chiral Weyl superfluids for the calculation of the angular momentum of the texture (see alsoRefs. ), and for the calculation of different manifestations of the chiral magnetic effect (CME) . We calculate thecurrent generated by deformation of the order parameter within the skyrmion, when the texture is deformed from thestate obeying the space inversion P to the state with the violated space inversion symmetry. In the initial state thereis no current along the axis of the skyrmion, since it is forbidden by the P -symmetry. The task is to find the currentin the final sate using the spectral flow in the process of deformation. We shall show that in this process the CVEcurrent emerges and it fully originates from the chiral anomaly.If the texture has no dependence on the coordinate z along the vortex-skyrmion, then in the process of deformationof the texture there is no force applied in z direction. In this situation the change of the total linear momentum orequivalently of the mass current J z may come only from the spectral flow from the occupied negative energy states ofthe inhomogeneous vacuum of skyrmion to the positive energy world, where the momentum of the created particlesis accumulated. Such spectral flow may take place either through the nodes in the bulk spectrum , or through theboundaries . Since the skyrmion is the localized object, which is not connected to the side walls of the cylindricalcontainer, the boundary effects can be ignored. Then, if the bulk state is fully gapped, there will be no current alongthe skyrmion axis, even if the space inversion (or the combined space inversion) is broken. But in the materialswith Weyl point nodes in bulk, the spectral flow through the Weyl nodes results in the total current, which is fullyregulated by the Adler-Bell-Jackiw equation for the chiral anomaly experienced by the Weyl fermions.We consider the system, which has two Weyl points with opposite chiralities, but the extension to the superfluidswith several Weyl points is straightforward . Such Weyl nodes cannot occur in the Galilean invariant normal state,but may occur in the pair-correlated gases and liquids, such as the A-phase of superfluid He. The latter is the spin-triplet chiral p -wave superfluid, where the Weyl points are at K ± = ± k F ˆ l , and ˆ l is the unit vector in the direction ofthe angular momentum of Cooper pairs. a r X i v : . [ c ond - m a t . o t h e r] J a n FIG. 1: Fig. 1
Right : Neel-type vortex skyrmion, which is symmetric under the combined symmetry
P T O x,π , which forbidsthe current along the vortex axis, since J z = P T O x,π J z = − J z . Left : Bloch-type skyrmion, which is symmetric under thecombined symmetry
T O x,π . The broken P -symmetry allows the current along the vortex axis – the chiral vortical effect. Themain difference from the Neel and Bloch skyrmions discussed in the nonsuperconducting magnetic materials in Ref. is that inthe chiral superfluids there is the circulation of superfluid velocity around the skyrmion. In the p -wave superfluids the skyrmionrepresents the vortex with two quanta of circulation . We start with the axi-symmetric texture of the vortex-skyrmion. In the texture, the positions of Weyl nodes inmomentum space are the following functions of the coordinates: K ± ( r ) = ± k F (cid:16) ˆ z cos η ( r ) + sin η ( r )(ˆ r cos α + ˆ φ sin α ) (cid:17) . (1)Here ( z, r, φ ) are cylindrical coordinates; η ( r ) changes from π to 0 ( η (0) = π and η ( ∞ ) = 0); the parameter α isconstant. For α = 0, the texture is shown in Fig. 1 ( right ). It is the Neel-type skyrmion, which is symmetricunder the combined symmetry P T O x,π , where P is space inversion, T is time inversion, and O x,π is π rotation abouthorizontal axis. This symmetry forbids the current along the vortex axis, since J z = P T O x,π J z = − J z . For α (cid:54) = 0 thissymmetry is violated and the current along the vortex axis may exist. For α = π/ T O x,π . Neel and Bloch skyrmions in nonsuperconductingmagnetic materials see e.g. in Ref. .Here we show that for α (cid:54) = 0 the current along the vortex comes from the chiral anomaly experienced by the Weylfermions, and it vanishes when the Weyl nodes annihilate each other. Let us consider deformation, at which α ( t )changes from zero to the finite value. The positions of the Weyl points play the role of the effective vector potentials A ± ( r , t ) = K ± ( r , t ) acting on fermions in the vicinity of these two Weyl points (in superconductors such syntheticgauge fields are induced by strain ). This results in the effective electric and magnetic fields: E ± = − ∂ t K ± , B ± = ∇ × K ± , (2)with E + · B + = E − · B − = − k F d cos ηdr sin η d sin αdt . (3)The production of the chiral charge at two Weyl points due to chiral anomaly compensate each other:˙ n = 14 π ( E + · B + − E − · B − ) = 0 , (4)But the created chiral charge carries with it the linear momentum K ± , which is not cancelled. As a result there isthe momentum production per unit time per unit volume:˙ j z = 14 π ( K z + ( E + · B + ) − K z − ( E − · B − )) = − k F π cos η d cos ηdr sin η d sin αdt . (5)The linear momentum accumulated by the vortex in the process of deformation, and thus the mass current along the FIG. 2: The cross section of the w -vortex skyrmion in superfluid He-A (from Ref. ). The texture is symmetric under thecombined symmetry T O x,π , while the parity P is broken. The broken P -symmetry in the skyrmion allows the current alongthe vortex axis in Eq.(8) – the chiral vortical effect. As distinct from the Bloch-type skyrmion in Fig. 1 ( left ), this vortex isnot axisymmetric and consists of two merons. Nevertheless its chiral vortical effect is also fully determined by chiral anomaly. skyrmion, is obtained by integration over time and space: J z (vortex) = (cid:90) d r (cid:90) dt ˙ j z = − k F π sin α (cid:90) d r cos η d cos ηdr sin η = − k F π sin α (cid:90) ∞ dr sin η . (6) III. CVE IN HE-A
Let us compare the value of the mass current in Eq.(6) with the current in the core of the vortex skyrmion insuperfluid He-A in Eq.(5.35) of review . The vortex-skyrmion with non-zero current is the so-called w -vortex with T O x,π symmetry shown in Fig. 2. Its structure is given by Eq.(5.29) in Ref. : K ± ( r ) = ± k F (ˆ y cos η ( r ) + sin η ( r )(ˆ x sin φ − ˆ z cos φ )) . (7)This w -vortex is analogous to the skyrmion with α = π/ l -vector is not axisymmetric: this vector changes from ˆ l ( r = 0) = − ˆ y to ˆ l ( r = ∞ ) = ˆ y . The reasonfor such asymmetry is that is that the NMR experiments are made in applied magnetic field, and far from the vortexcore the ˆ l -vector is oriented perpendicular to the field direction due to spin-orbit interaction.The current along the w -vortex axis is expressed in terms of the hydrodynamic parameters calculated by Cross from the microscopic BCS theory: J z (vortex) = − π m (cid:90) ∞ dr (cid:16) C sin η + ( ρ ⊥ s − ρ (cid:107) s )(1 + cos η ) sin η cos η (cid:17) . (8)Here m is the mass of the He atom, ρ (cid:107) s and ρ ⊥ s are superfluid densities for the superfluid motion along and perpen-dicular to the ˆ l -vector.At T = 0 the superfluid density tensor becomes isotropic, ρ (cid:107) s ( T = 0) = ρ ⊥ s ( T = 0) = ρ , and the hydrodynamicparameter C approaches the value C ( T = 0) = mk F / π . As a result the Eq.(8) is reduced to Eq.(6) for thecurrent in the axisymmetric skyrmion with α = π/
2. Note that in the traditional approach the hydrodynamicparameters have been calculated using fermionic spectrum very far from the nodes. Nevertheless, the CVE currentis fully determined by the spectral flow through the nodes. This demonstrates the universality of the contribution ofthe Weyl fermions to the CVE.It is interesting that the CVE current in Eq.(6) is determined solely by the positions of the nodes and do not dependon any other parameter of the system, including the mass m of the atom. For the considered Ansatz (1) one has: J z (vortex) = 16 π (cid:126) (cid:90) ∞ dr K ⊥ ( r ) K φ ( r ) , (9)where K ⊥ ( r ) = | K | sin ( r ) , K φ ( r ) = | K | sin α sin η ( r ) . (10)The current disappears when the Weyl nodes merge at | K | → one can obtain the CVE current not only in vortex-skyrmions, butalso in singular vortices with broken parity. The spectrum of fermions in the axisymmetric vortex E ( p z , Q ) dependson two quantum numbers, the linear momentum p z and angular momentum Q . Typically the minigap is much smallerthan the gap in bulk, and the quantization of Q can be ignored. In this case the current concentrated in the vortexcore (the total linear momentum of the vortex) is J z (vortex) = (cid:88) p z ,a p z Q a ( p z ) . (11)Here Q a ( p z ) are the values of Q , at which the spectrum crosses zero energy level as a function of Q at fixed p z , i.e.these are the solutions of equation E ( p z , Q ) = 0.Note that in the ground state of the system or in the equilibrium state in general, the total current is absent dueto the Bloch theorem, see e.g. Ref. . In our case the current J z (vortex) within the skyrmion in Eqs. (6) and (8) orwithin the vortex core in Eq.(11) in general, is compensated by the bulk current J z (bulk) = (cid:82) d r ρv sz . This can beseen from the following consideration. The equilibrium or the ground state of the system corresponds to the energyminimum of the superfluid with skyrmion. The relevant part of the free energy contains two terms – the energy ofthe superflow along z in bulk liquid and the interaction of the bulk superflow with the current inside the vortex: F = 12 (cid:90) d rρv sz + v sz J z (vortex) . (12)The ground state is determined by minimization of F with respect to the superfluid velocity v sz of the bulk flow, dF/dv sz = 0 . On the other hand the variation of the energy over v sz is nothing but the total current along z , i.e. dF/dv sz = J z (total). That is why the total current is zero in equilibrium, J z (total) = J z (bulk) + J z (vortex) = 0. Fora single vortex in a large volume of the bulk liquid, the density of the compensating bulk current is much smaller thanthe current density in the vortex core, j z (bulk) = ( R /R ) j z (vortex) (cid:28) j z (vortex). Here R core is the core size ofvortex or skyrmion, and R is the radius of the container.The Bloch theorem is also the reason, why the equilibrium CVE is not possible in the normal (non-superfluid) stateof the liquid: there is no non-dissipative supercurrent in bulk which could compensate the CVE current, and thus inthe ground state one has J z (vortex) = 0. The same refers to the chiral magnetic effect (CME), where it was shownthat there is no current in response to a strictly static magnetic field . In superconductors the non-uniform currentalong the magnetic field is possible if parity is properly violated, but again the total current is zero in equilibrium .In superconductors with broken parity it is difficult to resolve between the conventional CME and the CME originatingfrom the Weyl nodes, but in some cases the Weyl contribution can be dominating . IV. CONCLUSION
Superfluids, which contain the Weyl points, experience the type of chiral vortical effect, when there is the currentalong the vortex, which is concentrated in the vortex core. We considered the vortices, which cores have the skyrmionstructure. The current concentrated in such a core is provided by the spectral flow through the Weyl points accordingto the Adler-Bell-Jackiw equation for chiral anomaly. In a full equilibrium the current along the vortex is compensatedby the superfluid counter-current in the bulk superfluid outside the vortex core.I thank M. Zubkov for numerous discussions. This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement Shuang Jia, Su-Yang Xu and M.Z. Hasan, Weyl semimetals, Fermi arcs and chiral anomalies (a short review),arXiv:1612.00416. M. Sato and Y. Ando, Topological superconductors, arXiv:1608.03395. D.A. Pesin, Nonlocal electrodynamics of helical metals, in: 2016 IEEE INTERNATIONAL CONFERENCE ON MATHE-MATICAL METHODS IN ELECTROMAGNETIC THEORY (MMET), International Conference on Mathematical Methodsin Electromagnetic Theory, 115–118 (2016). J. Ma and D. A. Pesin, Chiral magnetic effect and natural optical activity in metals with or without Weyl points, Phys.Rev. B, , 235205 (2015). A. Vilenkin, Macroscopic parity-violating effects: Neutrino fluxes from rotating black holes and in rotating thermal radiation,Phys. Rev. D , 1807 (1979). A. Vilenkin, Equilibrium parity-violating current in a magnetic field, Phys. Rev. D , 3080 (1980). D.E. Kharzeev, J. Liao, S.A. Voloshin, G. Wang, Chiral magnetic effect in high-energy nuclear collisions – a status report,Progress in Particle and Nuclear Physics , 1–28 (2016), arXiv:1511.04050. M.M. Salomaa, G.E. Volovik, Quantized vortices in superfluid He, Rev. Mod. Phys. , 533–613 (1987). G.E. Volovik, Orbital momentum of vortices and textures due to spectral flow through the gap nodes: Example of the He-Acontinuous vortex, Pisma ZhETF , 935–941 (1995), JETP Lett. , 958–964 (1995). Y. Tada, Wenxing Nie, M. Oshikawa, Orbital angular momentum and spectral flow in two dimensional chiral superfluids,Phys. Rev. Lett. , 195301 (2015). G.E. Volovik, Orbital momentum of chiral superfluids and spectral asymmetry of edge states, Pis’ma ZhETF , 843–846(2014); JETP Lett. , 742–745 (2014). G.E. Volovik, On chiral magnetic effect in Weyl superfluid He-A, Pis’ma ZhETF , 31–32 (2017), arXiv:1611.06803. S. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev. , 2426–2438 (1969). S.L. Adler, Anomalies to all orders, in:
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