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 .