Monopoles and Fractional Vortices in Chiral Superconductors

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Monopoles and fractional vortices in chiral superconductors

G. E. Volovik*

Low Temperature Laboratory, Helsinki University of Technology, Box 2200, FIN-02015 HUT, Espoo, Finland; and Landau Institute for Theoretical Physics, 117334, Moscow, Russia

Communicated by Olli V. Lounasmaa, Helsinki University of Technology, Espoo, Finland, December 22, 1999 (received for review November 24, 1999) I discuss two exotic objects that must be experimentally identified in for the Standard Model (see ref. 5 for a review): the outgoing flux chiral superfluids and superconductors. These are (i) the vortex with of the hypermagnetic field is compensated by the incoming hyper- .(charge flux through the Z-string (Fig. 1 ؍ in chiral superfluids, and N 1͞2 ؍ a fractional quantum number (N in chiral superconductors), which plays the part of the In condensed matter, there are also topological objects that 1͞4 ؍ and N 1͞2 Alice string in relativistic theories and (ii) the hedgehog in the ˆl field, imitate magnetic monopoles. In chiral superconductors, their which is the counterpart of the Dirac magnetic monopole. These structure is very similar to the nexus: it is the magnetic monopole objects of different dimensions are topologically connected. They combined either with two Abrikosov vortices, each carrying the form the combined object that is called a nexus in relativistic theories. ͞ ⌽ flux (1 2) 0, or with four half-quantum vortices, each playing In chiral superconductors, the nexus has magnetic charge emanating the part of 1͞4 of the Dirac string. I also discuss the interaction radially from the hedgehog, whereas the half-quantum vortices play of such topological defects in superconductors with the ‘t the part of the Dirac string. Each half-quantum vortex supplies the Hooft–Polyakov monopole. If the latter exists, then the nexus fractional magnetic flux to the hedgehog, representing 1͞4ofthe provides a natural topological trap for the magnetic monopole. ‘‘conventional’’ Dirac string. I discuss the topological interaction of the superconductor’s nexus with the ‘t Hooft–Polyakov magnetic mono- Symmetry Groups. The similarity between the objects in Standard pole, which can exist in Grand Unified Theories. The monopole and Model and in chiral superconductors stems from the similar the hedgehog with the same magnetic charge are topologically symmetry breaking scheme. In the Standard Model, the local confined by a piece of the Abrikosov vortex. Such confinement makes ϫ the nexus a natural trap for the magnetic monopole. Other properties electroweak symmetry group SU(2)W U(1)Y at high energy is broken at low energy to the diagonal subgroup of the electro- of half-quantum vortices and monopoles are discussed as well, ϭ Ϫ including fermion zero modes. magnetism U(1)Q, where Q Y W3 is the electric charge. In amorphous chiral superconductors, the relevant symmetry above the superconducting transition temperature T is SO(3) ϫ agnetic monopoles do not exist in classical electromagne- c L U(1) , where SO(3) is a global group of the orbital rotations. tism. Maxwell equations show that the magnetic field is Q L M Below T , the symmetry is broken to the diagonal subgroup divergenceless, ٌ⅐B ϭ 0, which implies that the magnetic flux c Ϫ ͛ ⅐ ϭ U(1)Q L3. In high-energy physics, such symmetry breaking of through any closed surface is zero: SdS B 0. If one tries to construct the monopole solution B ϭ gr͞r3, the condition that the global and local groups to the diagonal global subgroup is magnetic field is nondivergent requires that magnetic flux ⌽ϭ called semilocal, and the corresponding topological defects are 4␲g from the monopole must be accompanied by an equal called semilocal strings (5). Thus, in chiral superconductors, the strings are semilocal; however, in electrically neutral chiral singular flux supplied to the monopole by an attached Dirac ϫ string. Quantum electrodynamics, however, can be successfully superfluids, they are global, because both groups in SO(3)L modified to include magnetic monopoles. In 1931, Dirac (1) U(1) are global there. showed that the string emanating from a magnetic monopole If one first neglects the difference between the global and local becomes invisible for electrons if the magnetic flux of the groups, the main difference between the symmetry breaking monopole is quantized in terms of the elementary magnetic flux schemes in high-energy physics and chiral superconductors is the discrete symmetry. It is the difference between SU(2) and hc SO(3) ϭ SU(2)͞Z and also one more discrete symmetry, Z , 4␲g ϭ n⌽ , ⌽ ϭ , [1] 2 2 0 0 e which comes from the coupling with the spin degrees of freedom. These discrete symmetries lead to the larger spectrum of the where e is the charge of the electron. strings and nexuses in superconductors, as compared with the In 1974, it was shown by ‘t Hooft (2) and Polyakov (3) that a Standard Model. magnetic monopole with quantization of the magnetic charge, according to Eq. 1, can really occur as a physical object if the Fractional Vortices in Chiral Superfluids͞Superconductors. Order pa- U(1) group of electromagnetism is a part of the higher symmetry rameter in chiral superfluids͞superconductors. The order param- group. The magnetic flux of a monopole in terms of the eter describing the vacuum manifold in a chiral p-wave super- elementary magnetic flux coincides with the topological charge fluid (3He-A) is the so-called gap function, which, in the of the monopole: this quantity remains constant under any representation S ϭ 1(S is the spin momentum of Cooper pairs) smooth deformation of the quantum fields. Such monopoles can and L ϭ 1(L is the orbital angular momentum of Cooper pairs), appear only in Grand Unified Theories, where all interactions depends linearly on spin ␴ and momentum k, namely: are united by, say, the SU(5) group.

In the Standard Model of electroweak interactions, such mono- ͑1͒ ͑2͒ ⌬͑k r͒ ϭ ␣ ͑r͒␴␣ ␣ ϭ ⌬ˆ␣͑ˆ ϩ ˆ ͒ [2] poles do not exist, but the combined objects monopole ϩ string can , A i ki, A i d ei iei . ,be constructed without violating of the condition ٌ⅐B ϭ 0. Further

following the terminology of ref. 4, I shall call such a combined *To whom reprint requests should be addressed. E-mail: [email protected].ﬁ. PHYSICS object a nexus. In a nexus, the magnetic monopole looks like a Dirac The publication costs of this article were defrayed in part by page charge payment. This monopole, but the Dirac string is physical and is represented by the article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. cosmic string. An example is the electroweak monopole discussed §1734 solely to indicate this fact.

PNAS ͉ March 14, 2000 ͉ vol. 97 ͉ no. 6 ͉ 2431–2436 Downloaded by guest on September 24, 2021 ␪ ⌬͑k, r͒ ϭ ͓sin2 k⅐a͑r͒ Ϫ sin2 k⅐b͑r͔͒ei . [4]

Because of the breaking of time reversal symmetry in chiral crystalline superconductors, persistent electric current arises not only because of the phase coherence but also because of deformations of the crystal:

e ប [j ϭ ␳ ͩv Ϫ Aͪ ϩ Ka ٌb , v ϭ ٌ␪. [5 s s mc i i s 2m

The parameter K ϭ 0 in nonchiral d-wave superconductors. ϫ ϫ 3 The symmetry breaking scheme SO(3)S SO(3)L U(1)N ϫ Ϫ ϫ U(1)S3 U(1)N L3 Z2, realized by the order parameter in Eq. 2, results in linear topological defects (vortices or strings) of group Z4 (8). Vortices are classified by the circulation quantum ϭ ͞ ͛ ⅐ number N (2m h) dr vs around the vortex core. The simplest realization of the vortex with integer N is eˆ(1) ϩ ieˆ(2) ϭ (xˆ ϩ iyˆ)eiN␾, where ␾ is the azimuthal angle around the string. Vortices with even N are topologically unstable and can be continuously transformed to a nonsingular configuration. N ϭ 1͞2 and N ϭ 1͞4 vortices. Vortices with a half-integer N result from the above identification of the points. They are combinations of the ␲-vortex and ␲-disclination in the dˆ field: ␾ ␾ ͑ ͒ ͑ ͒ ␾ dˆ ϭ ˆx cos ϩ ˆy sin , ˆe 1 ϩ ieˆ 2 ϭ ei /2͑ˆx ϩ iyˆ͒. [6] 2 2

The N ϭ 1͞2 vortex is the counterpart of Alice strings considered in particle physics (9): a particle that moves around an Alice string flips its charge. In 3He-A, the quasiparticle going ͞ around a 1 2 vortex flips its S3 charge, that is, its spin. The d-vector, which plays the role of the quantization axis for the spin of a quasiparticle, rotates by ␲ around the vortex, such that a quasiparticle adiabatically moving around the vortex insensibly finds its spin reversed with respect to the fixed environment. As Fig. 1. Dirac monopole, ‘t Hooft–Polyakov monopole, and electroweak a consequence, several phenomena (e.g., global Aharonov– monopole with physical Dirac string. Bohm effect) discussed in the particle physics literature have corresponding discussions in condensed matter literature (see refs. 10 and 11 for 3He-A and refs. 12 and 13 in particle physics). ˆ (1) Here, d is the unit vector of the spin-space anisotropy and eˆ In type II superconductors, vortices with N circulation quanta and eˆ(2) are mutually orthogonal unit vectors in the orbital space; ⌽ ϭ ͞ ⌽ ͞ carry a magnetic flux N (N 2) 0; the extra factor 1 2 comes they determine the superfluid velocity of the chiral condensate from the Cooper pairing nature of superconductors. According (ϭប (1)ٌ (2 vs /2 meˆi eˆi , where 2m is the mass of the Cooper pair and to the London equations, screening of the electric current far ˆ ϭ (1) ϫ (2) ϭ ͞ the orbital momentum vector is l eˆ eˆ . The important from the vortex leads to the vector potential A (mc e)vs and ˆ ͐ ⅐ ϭ ͛ ⅐ ϭ ͞ ͛ ⅐ ϭ ͞ ⌽ discrete symmetry comes from the identification of the points d, to the magnetic flux dS B dr A (mc e) dr vs (N 2) 0. eˆ(1) ϩ ieˆ(2) and Ϫdˆ, Ϫ(eˆ(1) ϩ ieˆ(2)): they correspond to the same Therefore, the conventional N ϭ 1 Abrikosov vortex in conven- ͞ ⌽ ϭ ͞ value of the order parameter in Eq. 2 and are thus physically tional superconductors carries (1 2) 0, whereas the N 1 2 ͞ ⌽ indistinguishable. vortex carries 1 4 of the elementary magnetic flux 0. The vortex The same order parameter describes the chiral superconduc- with N ϭ 1͞2 has been observed in high-temperature supercon- tor if the crystal lattice influence can be neglected, e.g., in an ductors (14): as predicted in ref. 15, this vortex is attached to the amorphous material. However, for crystals, the symmetry group tricrystal line, which is the junction of three grain boundaries must take into account the underlying crystal symmetry, and the (Fig. 2a). ⌽ ͞ classification of the topological defects becomes different. It is Objects with fractional flux below 0 2 are also possible (16). believed that chiral superconductivity occurs in the tetragonal They can arise if the time reversal symmetry is broken (17, 18). layered superconductor Sr2RuO4 (6, 7). The simplest represen- Such fractional flux can be trapped by the crystal loop, which tation of the order parameter, which reflects the underlying forms the topological object, a disclination: the orientation of the ␲͞ crystal structure, is crystal lattice continuously changes by 2 around the loop (Fig. 2b). The other topologically similar loop can be constructed by ␪ ⌬͑k, r͒ ϭ ͑d⅐␴͓͒sin k⅐a͑r͒ ϩ i sin k⅐b͑r͔͒ei , [3] twisting a thin wire by an angle ␲͞2 and then by gluing the ends (Fig. 3). ␪ where is the phase of the order parameter and a and b are the Figs. 2b and 3 illustrate fractional vortices in the cases of elementary vectors of the crystal lattice. The time reversal d-wave and chiral p-wave superconductivity in the tetragonal symmetry is broken in chiral superconductors. As a result, the crystal. Single valuedness of the order parameter requires that order parameter is intrinsically complex, i.e., its phase cannot be the ␲͞2 rotation of the crystal axis around the loop must be eliminated by a gauge transformation. On the contrary, in the compensated by a change of its phase ␪. As a result, the phase nonchiral d-wave superconductor in layered cuprate oxides, the winding around the loop is ␲ for a tetragonal d-wave supercon- order parameter is complex only because of its phase: ductor and ␲͞2 for a tetragonal p-wave superconductor. Con-

2432 ͉ www.pnas.org Volovik Downloaded by guest on September 24, 2021 Fig. 3. Fractional flux trapped by a loop of twisted wire with tetragonal cross section. The wire is twisted by 90°, and its surface behaves as 1͞2 of the Mo¨bius strip: the surface transforms into itself only after four circulations around the Fig. 2. (a) Experimentally observed fractional flux, topologically trapped by loop. After four circulations, the phase ␪ of the order parameter in chiral junctions of the grain boundaries in high-Tc superconductors. (b) Fractional p-wave superconductor changes by 2␲. Thus, the loop traps 1͞4 of the circu- flux topologically trapped by loops of monocrystals with tetragonal symme- lation quantum and correspondingly ⌽0͞8 of magnetic flux, if the parameter try. The tetragonal crystal with d-wave pairing traps N ϭ 1͞2 circulation K in the deformation current in Eq. 5 is neglected. quanta and thus 1͞4 of the quantum of magnetic flux. The same crystal with p-wave pairing traps N ϭ 1͞4 circulation quanta and thus 1͞8 of the quantum of magnetic flux (if the parameter K in the deformation current in Eq. 5 is nexus, appears in 3He-A when the topologically unstable vortex neglected). Note that, in my notation, the flux carried by a conventional N ϭ ϭ ͞ ⌽ with N 2 ends at the hedgehog in the orbital momentum field, 1 Abrikosov vortex in conventional superconductors is (1 2) 0. The empty ˆ ϭ space inside the loop represents the common core of the ␲͞2 crystal disclina- l rˆ (23, 24). In both cases, the distributions of the vector tion and an N ϭ 1͞2orN ϭ 1͞4 vortex. potential A of the hypermagnetic field and of the superfluid ϭ velocity vs field have the same structure, if one identifies vs (e͞mc)A. Assuming that the Z-string of the Standard Model or sequently, the loop of d-wave superconductor traps N ϭ 1͞2of its counterpart in the electrically neutral 3He-A, the N ϭ 2 ͞ ⌽ the circulation quantum and thus (1 4) 0 of the magnetic flux. vortex, occupy the lower half axis z Ͻ 0, one has The loop of the chiral p-wave superconductor traps 1͞4ofthe ប Ϫ ␪ circulation quantum. The magnetic flux trapped by the loop is c 1 cos [A ϭ ␾ˆ , B ϭ ٌ ϫ A ϭ B ϩ B , [7 obtained from the condition that the electric current in Eq. 5 is 2er sin ␪ mon string j ϭ 0 in superconductor. Therefore, the flux depends on the បc r បc parameter K in the deformation current in Eq. 5. In the limit ϭ ϭ Ϫ ⌰͑Ϫ͒␦ ͑␳͒ ϭ ⌽ ͞ Bmon 3 , Bstring z 2 . [8] case, when K 0, one obtains the fractional flux 0 8. In the 2e r e ⌽ ͞ same manner, 0 12 flux can be trapped, if the underlying crystal lattice has hexagonal symmetry. In amorphous chiral superconductors, Eq. 8 describes the dis- In 3He-B, the experimentally identified nonaxisymmetric N ϭ tribution of the real magnetic field. The regular part of the 1 vortex (19) can be considered as a pair of N ϭ 1͞2 vortices, magnetic field, radially propagating from the hedgehog, corre- ⌽ ϭប ͞ connected by a wall (20–22). sponds to a monopole with elementary magnetic flux 0 c e, whereas the singular part is concentrated in the core of the Nexus in Chiral Superfluids͞Superconductors. Nexus. The Z-string in doubly quantized Abrikosov vortex, which terminates on the the Standard Model, which has N ϭ 1, is topologically unstable, hedgehog and supplies the flux to the monopole (25). because N ϭ 0 (mod 1). Topological instability means that the Because of the discrete symmetry group, the nexus structures string may end at some point (Fig. 1c). The end point, a in 3He-A and in amorphous chiral superconductors are richer hedgehog in the orientation of the weak isospin vector, ˆl ϭ rˆ, than in the Standard Model. The N ϭ 2 vortex can split into two

⌽ ϭ ϭ ͞ PHYSICS looks like a Dirac monopole with the hypermagnetic flux 0 in N 1 Abrikosov vortices, into four N 1 2 vortices (Fig. 4), or Eq. 1, if the electric charge e is substituted by the hypercharge into their combination, provided that the total topological (5). The same combined object of a string and hedgehog, the charge N ϭ 0 (mod 2). Thus, in general, the superfluid velocity

Volovik PNAS ͉ March 14, 2000 ͉ vol. 97 ͉ no. 6 ͉ 2433 Downloaded by guest on September 24, 2021 If ˆl is fixed, the energy of the nexus in the spherical bubble of radius R is determined by the kinetic energy of mass and spin superflow:

1 e2 ͵ dV͑␳ v2 ϩ ␳ v2 ͒ ϭ ͵ dV͑␳ ϩ ␳ ͓͒͑A1 ϩ A2͒2 2 s s sp sp 2m2c2 s sp

e2 ϩ ͑A3 ϩ A4͒2] ϩ ͵ dV͑␳ Ϫ ␳ ͒͑A1 ϩ A2͒͑A3 ϩ A4͒. m2c2 s sp [11]

In the simplest case, which occurs in the ideal Fermi gas, one ␳ ϭ ␳ ͞ has s sp (28). In this case, the 1 2 vortices with positive spin-current circulation ␯ do not interact with 1͞2 vortices of negative ␯. The energy minimum occurs when the orientations of two positive-␯ vortices are opposite, such that these two 1͞4 fractions of the Dirac strings form one line along the diameter (see Fig. 4). The same happens for the other fractions with negative ␯. The mutual orientations of the two diameters is 3 ␳ Ͻ ␳ arbitrary in this limit. However, in real He-A, one has sp s ␳ ␳ ␯ ␯ (28). If sp is slightly smaller than s, the positive- and negative- Fig. 4. Nexus in a small droplet of superfluid 3He-A: the hedgehog is ϭ ͞ strings repel each other, such that the equilibrium angle between connecting four vortices with N 1 2 each. Blue arrows outward show the them is ␲͞2. In the extreme case ␳ ϽϽ ␳ , the ends of four radial distribution of the orbital momentum lˆ field; red arrows outward sp s half-quantum vortices form the vertices of a regular tetrahedron. illustrate the radial distribution of superfluid vorticity ٌϫvs or of magnetic field B in the superconducting counterpart, a chiral superconductor. The To fix the position of the nexus in the center of the droplet, one magnetic flux of the nexus ⌽0 is supplied by four half-quantum vortices, each must introduce a spherical body inside, which will attach the carrying the flux ⌽0͞4 to the hedgehog. The charge ␯ ϭϮ1͞2 is the number nexus because of the normal boundary conditions for the ˆl 4 3 of circulation quanta of the spin supercurrent velocity vsp around the half- vector. The body can be a droplet of He immersed in the He quantum vortex. The stability of the monopole in the center of the droplet is liquid. For the mixed 4He͞3He droplets, obtained via the nozzle 4 supported by the foreign body in the center, for example by a cluster of He beam expansion of He gas, it is known that the 4He component liquid, which provides the radial boundary condition for the lˆ-vector. of the mixture does form a cluster in the central region of the 3He droplet (29). field in the 3He-A nexus and the vector potential in its super- In an amorphous p-wave superconductor, but with preserved conducting counterpart obey layered structure, such a nexus will be formed in a spherical shell. In the crystalline Sr2RuO4 superconductor, the spin-orbit cou- e pling between the spin vector dˆ and the crystal lattice seems to v ϭ A, A ϭ ͸Aa, [9] ˆ ˆ s mc align the d vector along l (30). In this case, the half-quantum a vortices are energetically unfavorable, and instead of four half- quantum vortices, one would have two singly quantized vortices Aa where is the vector potential of the electromagnetic field in the spherical shell. a produced by the th string, i.e., the Abrikosov vortex with the Nexuses of this kind can be formed also in the so-called circulation number N terminating on the monopole, provided a ferromagnetic Bose condensate in optical traps. Such a conden- that ⌺ N ϭ 0 (mod 2). a a sate is described by a chiral order parameter, which is either Such topological coupling of monopoles and strings is realized vector or spinor (31). also in relativistic SU(n) quantum field theories, for example in Nexus with fractional magnetic flux. A nexus with fractional quantum chromodynamics, where n vortices of the group Z n magnetic charge can be constructed by using geometry with meet at a center (nexus) provided the total flux of vortices adds several condensates. Fig. 5 shows the nexus pinned by the to zero (mod n) (4, 26, 27). 3 3 interface between superfluid He-A and the nonchiral super- Nexus in a He droplet. A nexus can be the ground state of 3 3 ␮ fluid He-B. Because of the tangential boundary condition for He-A in a droplet, if its radius is less than 10 m. In this case, ˆ ˆ the lowest energy of the nexus occurs when all vortices termi- the l vector at the interface, the l field of the nexus covers only nating on the monopole have the lowest circulation number: half of the unit sphere. For the superconducting analogs, such a ϭ ϭ ϭ ϭ ͞ nexus represents the monopole with 1͞2 of the elementary flux there must be four vortices with N1 N2 N3 N4 1 2. ⌽ ϭ According to Eq. 6, each half-quantum vortex is accompanied 0. Thus, on the B-phase side, there is only one vortex with N by a spin disclination. Assuming that the dˆ-field is confined into 1, which terminates on the nexus. a plane, the disclinations can be characterized by the winding A monopole, which is topologically pinned by a surface or ␯ Ϯ ͞ interface is called a boojum (32). The topological classification numbers a of the dˆ vector, which have values 1 2 in half- quantum vortices. The corresponding spin-superfluid velocity of boojums is discussed in refs. 33–35). In high-energy physics, vsp is linear defects terminating on walls are called Dirichlet defects (36). 4 4 Gravimagnetic monopole. In addition to the symmetry break- 2e v ϭ ͸␯ Aa, ͸␯ ϭ 0, [10] ing scheme, there is another level of analogies between super- sp mc a a ͞ aϭ1 aϭ1 fluids superconductors and quantum vacuum. They are related to the behavior of quasiparticles in both systems. In chiral where the last condition means the absence of the monopole in superfluids, quasiparticles behave as chiral fermions living in the ␯ ϭ ␯ ϭ the spin sector of the order parameter. Thus, we have 1 2 effective gauge and gravity fields, produced by the bosonic Ϫ␯ ϭϪ␯ ϭ ͞ 3 4 1 2. collective modes of the superfluid vacuum (see ref. 37 for a

2434 ͉ www.pnas.org Volovik Downloaded by guest on September 24, 2021 Fig. 5. An N ϭ 1 vortex in 3He-B terminating on a boojum, a point defect at the interface between 3He-B and 3He-A. Blue arrows show the distribution of lˆ on the 3He-A side of the nexus, which resembles the gravimagnetic monopole whose Dirac string is the B-phase vortex. In superconductors, such a nexus accounts for 1͞2 of the magnetic charge of the Dirac monopole, whose ﬂux is supported by a single N ϭ 1 vortex on the B-phase side.

review). In particular, the superfluid velocity acts on quasipar- 0i ϭϪ i ticles in the same way as the metric element g vs acts on a relativistic particle in Einstein’s theory. This element g ϭϪg0i plays the part of the vector potential of the gravimagnetic field ϭٌϫ Bg g. For the nexus in Fig. 5, the ˆl vector, the superfluid velocity vs, and its ‘‘gravimagnetic field,’’ i.e., vorticity Bg, on the A-phase side are Fig. 6. Hedgehogs and magnetic monopoles in superconductors. (a)‘t ប 1 Ϫ cos ␪ ប r Hooft–Polyakov magnetic monopole in a conventional superconductor. Mag- ϭ ˆ ϭ ␾ˆ ϭ ٌជ ϫ ϭ netic flux of the monopole is concentrated in Abrikosov vortices because of ˆ l r, vs ␪ , Bg vs 3 . 2m3 r sin 2m3 r the Meissner effect. (b) Magnetic monopole in a chiral superconductor with [12] uniform lˆ vector. As distinct from the monopole inside conventional superconductors, the magnetic flux ⌽0 of the monopole can be carried away by four On the B-phase side, one has half-quantum vortices. (c) Nexus: hedgehog with two Abrikosov vortices emanating from the core. Magnetic flux ⌽0 enters the core of the hedgehog ␲ប ϭ ϭ ͞ ϭ ٌជ ϫ ϭ ␦ ͑␳͒ along two Abrikosov vortices with N 1 (or four vortices with N 1 2) and Bg vs 2 . [13] then flows out radially along the lines of the lˆ vector field. (d) ‘t Hooft– m3 Polyakov magnetic monopole ϩ hedgehog in a chiral superconductor. The The gravimagnetic flux propagates along the vortex in the B Abrikosov vortices attached to the ‘t Hooft–Polyakov magnetic monopole phase toward the nexus (boojum) and then radially and diver- annihilate the Abrikosov vortices attached to the hedgehog. Magnetic field of the monopole penetrates radially into the bulk chiral superconductor along gencelessly from the boojum into the A phase. In general the lines of the lˆ vector field. (e) Topological confinement of the ‘t Hooft– relativity, the gravimagnetic monopole has been discussed in Polyakov magnetic monopole and hedgehog by Abrikosov strings in a chiral ref. 38. superconductor. For simplicity, a single Abrikosov string with N ϭ 2 is depicted.

Topological Interaction of Magnetic Monopoles with Chiral Supercon- ductors. Because the ‘t Hooft–Polyakov magnetic monopole, magnetic flux is not necessarily concentrated in the tubes but can which can exist in Grand Unified Theories, and the monopole propagate radially from the hedgehog (Fig. 6c). If now the ‘t part of the nexus in chiral superconductors have the same magnetic and topological charges, there is a topological inter- Hooft–Polyakov magnetic monopole enters the core of the action between them. First, let us recall what happens when the hedgehog in Fig. 6c, which has the same magnetic charge, their magnetic monopole enters a conventional superconductor: be- strings, i.e., Abrikosov vortices carried by the monopole and cause of the Meissner effect—expulsion of the magnetic field Abrikosov vortices attached to the nexus, will annihilate each ϩ from the superconductor—the magnetic field from the mono- other. What is left is the combined point defect: hedgehog pole will be concentrated in two flux tubes of Abrikosov vortices magnetic monopole without any attached strings (Fig. 6d). Thus, with the total winding number N ϭ 2 (Fig. 6a). In a chiral the monopole destroys the topological connection of the hedge- amorphous superconductor, these can form four flux tubes, hog and Abrikosov vortices; instead, one obtains topological represented by half-quantum Abrikosov vortices (Fig. 6b). confinement between the monopole and hedgehog. The core of

However, the most interesting situation occurs if one takes the hedgehog represents the natural trap for one magnetic PHYSICS into account that, in a chiral superconductor, the Meissner effect monopole: if one tries to separate the monopole from the is not complete because of the ˆl texture. As discussed above, the hedgehog, one must create the piece or pieces of the Abrikosov

Volovik PNAS ͉ March 14, 2000 ͉ vol. 97 ͉ no. 6 ͉ 2435 Downloaded by guest on September 24, 2021 vortex or vortices) that connect the hedgehog and the monopole empty, there is a fractional entropy (1͞2)ln2 per layer related (Fig. 6e). to the vortex. The factor (1͞2) appears, because in superconductors, the particle excitation coincides with its antiparticle Discussion: Fermions in the Presence of Topological Defects. Fermi- (hole), i.e., the quasiparticle is a Majorana fermion (a nice ons in topologically nontrivial environments behave in a discussion of Majorana fermions in chiral superconductors can curious way, especially in the presence of such exotic objects be found in ref. 41). Also the spin of the vortex in a chiral as fractional vortices and monopoles discussed in this paper. In superconductor can be fractional. According to ref. 42, the the presence of a monopole, the quantum statistics can change; N ϭ 1 vortex in a chiral superconductor must have a spin S ϭ for example, the isospin degrees of freedom are transformed 1͞4 (per layer); consequently, N ϭ 1͞2 vortex must have the to spin degrees (39). There are also the so-called fermion zero spin S ϭ 1͞8 per layer. Similarly, there can be an anomalous modes: the bound states of fermions at monopole or vortex, fractional electric charge of the N ϭ 1͞2 vortex, which is 1͞2 that have exactly zero energy. For example the N ϭ 1͞2 vortex of the fractional charge e͞4 discussed for the N ϭ 1 vortex (43). in a two-dimensional chiral superconductor, which contains There is still some work to be done to elucidate the problem only one layer, has one fermionic state with exactly zero energy with the fractional charge, spin, and statistics related to the (40). Because the zero-energy level can be either filled or topological defects in chiral superconductors.

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