Monopoles and fractional vortices in chiral superconductors

Abstract

I discuss two exotic objects that must be experimentally identified in chiral superfluids and superconductors. These are (i) the vortex with a fractional quantum number (N = 1/2 in chiral superfluids, and N = 1/2 and N = 1/4 in chiral superconductors), which plays the part of the Alice string in relativistic theories and (ii) the hedgehog in the ^l field, which is the counterpart of the Dirac magnetic monopole. These objects of different dimensions are topologically connected. They form the combined object that is called a nexus in relativistic theories. In chiral superconductors, the nexus has magnetic charge emanating radially from the hedgehog, whereas the half-quantum vortices play the part of the Dirac string. Each half-quantum vortex supplies the fractional magnetic flux to the hedgehog, representing 1/4 of the “conventional” Dirac string. I discuss the topological interaction of the superconductor's nexus with the ‘t Hooft–Polyakov magnetic monopole, which can exist in Grand Unified Theories. The monopole and the hedgehog with the same magnetic charge are topologically confined by a piece of the Abrikosov vortex. Such confinement makes the nexus a natural trap for the magnetic monopole. Other properties of half-quantum vortices and monopoles are discussed as well, including fermion zero modes.

Magnetic monopoles do not exist in classical electromagnetism. Maxwell equations show that the magnetic field is divergenceless, ∇⋅B = 0, which implies that the magnetic flux through any closed surface is zero: ∮_SdS⋅B = 0. If one tries to construct the monopole solution B = gr/r³, the condition that magnetic field is nondivergent requires that magnetic flux Φ = 4πg from the monopole must be accompanied by an equal singular flux supplied to the monopole by an attached Dirac string. Quantum electrodynamics, however, can be successfully modified to include magnetic monopoles. In 1931, Dirac (1) showed that the string emanating from a magnetic monopole becomes invisible for electrons if the magnetic flux of the monopole is quantized in terms of the elementary magnetic flux

1

where e is the charge of the electron.

In 1974, it was shown by ‘t Hooft (2) and Polyakov (3) that a magnetic monopole with quantization of the magnetic charge, according to Eq. 1, can really occur as a physical object if the U(1) group of electromagnetism is a part of the higher symmetry group. The magnetic flux of a monopole in terms of the elementary magnetic flux coincides with the topological charge of the monopole: this quantity remains constant under any smooth deformation of the quantum fields. Such monopoles can appear only in Grand Unified Theories, where all interactions are united by, say, the SU(5) group.

In the Standard Model of electroweak interactions, such monopoles do not exist, but the combined objects monopole + string can be constructed without violating of the condition ∇⋅B = 0. Further, following the terminology of ref. 4, I shall call such a combined object a nexus. In a nexus, the magnetic monopole looks like a Dirac monopole, but the Dirac string is physical and is represented by the cosmic string. An example is the electroweak monopole discussed for the Standard Model (see ref. 5 for a review): the outgoing flux of the hypermagnetic field is compensated by the incoming hypercharge flux through the Z-string (Fig. 1).

Figure 1.

Dirac monopole, ‘t Hooft–Polyakov monopole, and electroweak monopole with physical Dirac string.

In condensed matter, there are also topological objects that imitate magnetic monopoles. In chiral superconductors, their structure is very similar to the nexus: it is the magnetic monopole combined either with two Abrikosov vortices, each carrying the flux (1/2)Φ₀, or with four half-quantum vortices, each playing the part of 1/4 of the Dirac string. I also discuss the interaction of such topological defects in superconductors with the ‘t Hooft–Polyakov monopole. If the latter exists, then the nexus provides a natural topological trap for the magnetic monopole.

Symmetry Groups.

The similarity between the objects in Standard Model and in chiral superconductors stems from the similar symmetry breaking scheme. In the Standard Model, the local electroweak symmetry group SU(2)_W × U(1)_Y at high energy is broken at low energy to the diagonal subgroup of the electromagnetism U(1)_Q, where Q = Y − W₃ is the electric charge. In amorphous chiral superconductors, the relevant symmetry above the superconducting transition temperature T_c is SO(3)_L × U(1)_Q, where SO(3)_L is a global group of the orbital rotations. Below T_c, the symmetry is broken to the diagonal subgroup U(1)_{Q − L3}. In high-energy physics, such symmetry breaking of the global and local groups to the diagonal global subgroup is called semilocal, and the corresponding topological defects are called semilocal strings (5). Thus, in chiral superconductors, the strings are semilocal; however, in electrically neutral chiral superfluids, they are global, because both groups in SO(3)_L × U(1) are global there.

If one first neglects the difference between the global and local groups, the main difference between the symmetry breaking schemes in high-energy physics and chiral superconductors is the discrete symmetry. It is the difference between SU(2) and SO(3) = SU(2)/Z₂ and also one more discrete symmetry, Z₂, which comes from the coupling with the spin degrees of freedom. These discrete symmetries lead to the larger spectrum of the strings and nexuses in superconductors, as compared with the Standard Model.

Fractional Vortices in Chiral Superfluids/Superconductors.

Order parameter in chiral superfluids/superconductors.

The order parameter describing the vacuum manifold in a chiral p-wave superfluid (³He-A) is the so-called gap function, which, in the representation S = 1 (S is the spin momentum of Cooper pairs) and L = 1 (L is the orbital angular momentum of Cooper pairs), depends linearly on spin σ and momentum k, namely:

2

Here, d̂ is the unit vector of the spin-space anisotropy and ê⁽¹⁾ and ê⁽²⁾ are mutually orthogonal unit vectors in the orbital space; they determine the superfluid velocity of the chiral condensate v_s = ℏ/2 mê_i⁽¹⁾∇ê_i⁽²⁾, where 2m is the mass of the Cooper pair and the orbital momentum vector is l̂ = ê⁽¹⁾ × ê⁽²⁾. The important discrete symmetry comes from the identification of the points d̂, ê⁽¹⁾ + iê⁽²⁾ and −d̂, −(ê⁽¹⁾ + iê⁽²⁾): they correspond to the same value of the order parameter in Eq. 2 and are thus physically indistinguishable.

The same order parameter describes the chiral superconductor if the crystal lattice influence can be neglected, e.g., in an amorphous material. However, for crystals, the symmetry group must take into account the underlying crystal symmetry, and the classification of the topological defects becomes different. It is believed that chiral superconductivity occurs in the tetragonal layered superconductor Sr₂RuO₄ (6, 7). The simplest representation of the order parameter, which reflects the underlying crystal structure, is

3

where θ is the phase of the order parameter and a and b are the elementary vectors of the crystal lattice. The time reversal symmetry is broken in chiral superconductors. As a result, the order parameter is intrinsically complex, i.e., its phase cannot be eliminated by a gauge transformation. On the contrary, in the nonchiral d-wave superconductor in layered cuprate oxides, the order parameter is complex only because of its phase:

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:

5

The parameter K = 0 in nonchiral d-wave superconductors.

The symmetry breaking scheme SO(3)_S × SO(3)_L × U(1)_N → 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 number N = (2m/h)∮dr⋅v_s around the vortex core. The simplest realization of the vortex with integer N is ê⁽¹⁾ + iê⁽²⁾ = (x̂ + iŷ)e^iNφ, where φ is the azimuthal angle around the string. Vortices with even N are topologically unstable and can be continuously transformed to a nonsingular configuration.

N = 1/2 and N = 1/4 vortices.

Vortices with a half-integer N result from the above identification of the points. They are combinations of the π-vortex and π-disclination in the d̂ field:

6

The N = 1/2 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 π around the vortex, such that a quasiparticle adiabatically moving around the vortex insensibly finds its spin reversed with respect to the fixed environment. As a consequence, several phenomena (e.g., global Aharonov–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).

In type II superconductors, vortices with N circulation quanta carry a magnetic flux Φ_N = (N/2)Φ₀; the extra factor 1/2 comes from the Cooper pairing nature of superconductors. According to the London equations, screening of the electric current far from the vortex leads to the vector potential A = (mc/e)v_s and to the magnetic flux ∫dS⋅B = ∮dr⋅A = (mc/e)∮dr⋅v_s = (N/2)Φ₀. Therefore, the conventional N = 1 Abrikosov vortex in conventional superconductors carries (1/2)Φ₀, whereas the N = 1/2 vortex carries 1/4 of the elementary magnetic flux Φ₀. The vortex with N = 1/2 has been observed in high-temperature superconductors (14): as predicted in ref. 15, this vortex is attached to the tricrystal line, which is the junction of three grain boundaries (Fig. 2a).

Figure 2.

(a) Experimentally observed fractional flux, topologically trapped by junctions of the grain boundaries in high-T_c superconductors. (b) Fractional flux topologically trapped by loops of monocrystals with tetragonal symmetry. The tetragonal crystal with d-wave pairing traps N = 1/2 circulation quanta and thus 1/4 of the quantum of magnetic flux. The same crystal with p-wave pairing traps N = 1/4 circulation quanta and thus 1/8 of the quantum of magnetic flux (if the parameter K in the deformation current in Eq. 5 is neglected). Note that, in my notation, the flux carried by a conventional N = 1 Abrikosov vortex in conventional superconductors is (1/2)Φ₀. The empty space inside the loop represents the common core of the π/2 crystal disclination and an N = 1/2 or N = 1/4 vortex.

Objects with fractional flux below Φ₀/2 are also possible (16). They can arise if the time reversal symmetry is broken (17, 18). Such fractional flux can be trapped by the crystal loop, which forms the topological object, a disclination: the orientation of the crystal lattice continuously changes by π/2 around the loop (Fig. 2b). The other topologically similar loop can be constructed by twisting a thin wire by an angle π/2 and then by gluing the ends (Fig. 3).

Figure 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 1/2 of the Möbius strip: the surface transforms into itself only after four circulations around the loop. After four circulations, the phase θ of the order parameter in chiral p-wave superconductor changes by 2π. Thus, the loop traps 1/4 of the circulation quantum and correspondingly Φ₀/8 of magnetic flux, if the parameter K in the deformation current in Eq. 5 is neglected.

Figs. 2b and 3 illustrate fractional vortices in the cases of d-wave and chiral p-wave superconductivity in the tetragonal crystal. Single valuedness of the order parameter requires that the π/2 rotation of the crystal axis around the loop must be compensated by a change of its phase θ. As a result, the phase winding around the loop is π for a tetragonal d-wave superconductor and π/2 for a tetragonal p-wave superconductor. Consequently, the loop of d-wave superconductor traps N = 1/2 of the circulation quantum and thus (1/4)Φ₀ of the magnetic flux.

The loop of the chiral p-wave superconductor traps 1/4 of the circulation quantum. The magnetic flux trapped by the loop is obtained from the condition that the electric current in Eq. 5 is j = 0 in superconductor. Therefore, the flux depends on the parameter K in the deformation current in Eq. 5. In the limit case, when K = 0, one obtains the fractional flux Φ₀/8. In the same manner, Φ₀/12 flux can be trapped, if the underlying crystal lattice has hexagonal symmetry.

In ³He-B, the experimentally identified nonaxisymmetric N = 1 vortex (19) can be considered as a pair of N = 1/2 vortices, connected by a wall (20–22).

Nexus in Chiral Superfluids/Superconductors.

Nexus.

The Z-string in the Standard Model, which has N = 1, is topologically unstable, because N = 0 (mod 1). Topological instability means that the string may end at some point (Fig. 1c). The end point, a hedgehog in the orientation of the weak isospin vector, l̂ = r̂, looks like a Dirac monopole with the hypermagnetic flux Φ₀ in Eq. 1, if the electric charge e is substituted by the hypercharge (5). The same combined object of a string and hedgehog, the nexus, appears in ³He-A when the topologically unstable vortex with N = 2 ends at the hedgehog in the orbital momentum field, l̂ = r̂ (23, 24). In both cases, the distributions of the vector potential A of the hypermagnetic field and of the superfluid velocity v_s field have the same structure, if one identifies v_s = (e/mc)A. Assuming that the Z-string of the Standard Model or its counterpart in the electrically neutral ³He-A, the N = 2 vortex, occupy the lower half axis z < 0, one has

7

8

In amorphous chiral superconductors, Eq. 8 describes the distribution of the real magnetic field. The regular part of the magnetic field, radially propagating from the hedgehog, corresponds to a monopole with elementary magnetic flux Φ₀ = ℏc/e, whereas the singular part is concentrated in the core of the doubly quantized Abrikosov vortex, which terminates on the hedgehog and supplies the flux to the monopole (25).

Because of the discrete symmetry group, the nexus structures in ³He-A and in amorphous chiral superconductors are richer than in the Standard Model. The N = 2 vortex can split into two N = 1 Abrikosov vortices, into four N = 1/2 vortices (Fig. 4), or into their combination, provided that the total topological charge N = 0 (mod 2). Thus, in general, the superfluid velocity field in the ³He-A nexus and the vector potential in its superconducting counterpart obey

9

where A^a is the vector potential of the electromagnetic field produced by the ath string, i.e., the Abrikosov vortex with the circulation number N_a terminating on the monopole, provided that Σ_aN_a = 0 (mod 2).

Figure 4.

Nexus in a small droplet of superfluid ³He-A: the hedgehog is connecting four vortices with N = 1/2 each. Blue arrows outward show the radial distribution of the orbital momentum l̂ field; red arrows outward illustrate the radial distribution of superfluid vorticity ∇ × v_s or of magnetic field B in the superconducting counterpart, a chiral superconductor. The magnetic flux of the nexus Φ₀ is supplied by four half-quantum vortices, each carrying the flux Φ₀/4 to the hedgehog. The charge ν = ±1/2 is the number of circulation quanta of the spin supercurrent velocity v_sp around the half-quantum vortex. The stability of the monopole in the center of the droplet is supported by the foreign body in the center, for example by a cluster of ⁴He liquid, which provides the radial boundary condition for the l̂-vector.

Such topological coupling of monopoles and strings is realized also in relativistic SU(n) quantum field theories, for example in quantum chromodynamics, where n vortices of the group Z_n meet at a center (nexus) provided the total flux of vortices adds to zero (mod n) (4, 26, 27).

Nexus in a ³He droplet.

A nexus can be the ground state of ³He-A in a droplet, if its radius is less than 10 μm. In this case, the lowest energy of the nexus occurs when all vortices terminating on the monopole have the lowest circulation number: there must be four vortices with N₁ = N₂ = N₃ = N₄ = 1/2.

According to Eq. 6, each half-quantum vortex is accompanied by a spin disclination. Assuming that the d̂-field is confined into a plane, the disclinations can be characterized by the winding numbers ν_a of the d̂ vector, which have values ±1/2 in half-quantum vortices. The corresponding spin-superfluid velocity v_sp is

10

where the last condition means the absence of the monopole in the spin sector of the order parameter. Thus, we have ν₁ = ν₂ = −ν₃ = −ν₄ = 1/2.

If l̂is fixed, the energy of the nexus in the spherical bubble of radius R is determined by the kinetic energy of mass and spin superflow:

11

In the simplest case, which occurs in the ideal Fermi gas, one has ρ_s = ρ_sp (28). In this case, the 1/2 vortices with positive spin-current circulation ν do not interact with 1/2 vortices of negative ν. The energy minimum occurs when the orientations of two positive-ν vortices are opposite, such that these two 1/4 fractions of the Dirac strings form one line along the diameter (see Fig. 4). The same happens for the other fractions with negative ν. The mutual orientations of the two diameters is arbitrary in this limit. However, in real ³He-A, one has ρ_sp < ρ_s (28). If ρ_sp is slightly smaller than ρ_s, the positive-ν and negative-ν strings repel each other, such that the equilibrium angle between them is π/2. In the extreme case ρ_sp ≪ ρ_s, the ends of four half-quantum vortices form the vertices of a regular tetrahedron.

To fix the position of the nexus in the center of the droplet, one must introduce a spherical body inside, which will attach the nexus because of the normal boundary conditions for the l̂ vector. The body can be a droplet of ⁴He immersed in the ³He liquid. For the mixed ⁴He/³He droplets, obtained via the nozzle beam expansion of He gas, it is known that the ⁴He component of the mixture does form a cluster in the central region of the ³He droplet (29).

In an amorphous p-wave superconductor, but with preserved layered structure, such a nexus will be formed in a spherical shell. In the crystalline Sr₂RuO₄ superconductor, the spin-orbit coupling between the spin vector d̂ and the crystal lattice seems to align the d̂ vector along l̂ (30). In this case, the half-quantum vortices are energetically unfavorable, and instead of four half-quantum vortices, one would have two singly quantized vortices in the spherical shell.

Nexuses of this kind can be formed also in the so-called ferromagnetic Bose condensate in optical traps. Such a condensate is described by a chiral order parameter, which is either vector or spinor (31).

Nexus with fractional magnetic flux.

A nexus with fractional magnetic charge can be constructed by using geometry with several condensates. Fig. 5 shows the nexus pinned by the interface between superfluid ³He-A and the nonchiral superfluid ³He-B. Because of the tangential boundary condition for the l̂ vector at the interface, the l̂field of the nexus covers only half of the unit sphere. For the superconducting analogs, such a nexus represents the monopole with 1/2 of the elementary flux Φ₀. Thus, on the B-phase side, there is only one vortex with N = 1, which terminates on the nexus.

Figure 5.

An N = 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 l̂ 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 1/2 of the magnetic charge of the Dirac monopole, whose flux is supported by a single N = 1 vortex on the B-phase side.

A monopole, which is topologically pinned by a surface or interface is called a boojum (32). The topological classification of boojums is discussed in refs. 33–35). In high-energy physics, linear defects terminating on walls are called Dirichlet de- fects (36).

Gravimagnetic monopole.

In addition to the symmetry breaking scheme, there is another level of analogies between superfluids/superconductors and quantum vacuum. They are related to the behavior of quasiparticles in both systems. In chiral superfluids, quasiparticles behave as chiral fermions living in the effective gauge and gravity fields, produced by the bosonic collective modes of the superfluid vacuum (see ref. 37 for a review). In particular, the superfluid velocity acts on quasiparticles in the same way as the metric element g⁰ⁱ = −v_sⁱ acts on a relativistic particle in Einstein’s theory. This element g = −g⁰ⁱ plays the part of the vector potential of the gravimagnetic field B_g = ∇ × g.

For the nexus in Fig. 5, the l̂ vector, the superfluid velocity v_s, and its “gravimagnetic field,” i.e., vorticity B_g, on the A-phase side are

12

On the B-phase side, one has

13

The gravimagnetic flux propagates along the vortex in the B phase toward the nexus (boojum) and then radially and divergencelessly from the boojum into the A phase. In general relativity, the gravimagnetic monopole has been discussed in ref. 38.

Topological Interaction of Magnetic Monopoles with Chiral Superconductors.

Because the ‘t Hooft–Polyakov magnetic monopole, which can exist in Grand Unified Theories, and the monopole part of the nexus in chiral superconductors have the same magnetic and topological charges, there is a topological interaction between them. First, let us recall what happens when the magnetic monopole enters a conventional superconductor: because of the Meissner effect—expulsion of the magnetic field from the superconductor—the magnetic field from the monopole will be concentrated in two flux tubes of Abrikosov vortices with the total winding number N = 2 (Fig. 6a). In a chiral amorphous superconductor, these can form four flux tubes, represented by half-quantum Abrikosov vortices (Fig. 6b).

Figure 6.

Hedgehogs and magnetic monopoles in superconductors. (a) ‘t Hooft–Polyakov magnetic monopole in a conventional superconductor. Magnetic flux of the monopole is concentrated in Abrikosov vortices because of the Meissner effect. (b) Magnetic monopole in a chiral superconductor with uniform l̂ vector. As distinct from the monopole inside conventional superconductors, the magnetic flux Φ₀ of the monopole can be carried away by four half-quantum vortices. (c) Nexus: hedgehog with two Abrikosov vortices emanating from the core. Magnetic flux Φ₀ enters the core of the hedgehog along two Abrikosov vortices with N = 1 (or four vortices with N = 1/2) and then flows out radially along the lines of the l̂ vector field. (d) ‘t Hooft–Polyakov magnetic monopole + hedgehog in a chiral superconductor. The Abrikosov vortices attached to the ‘t Hooft–Polyakov magnetic monopole annihilate the Abrikosov vortices attached to the hedgehog. Magnetic field of the monopole penetrates radially into the bulk chiral superconductor along the lines of the l̂ vector field. (e) Topological confinement of the ‘t Hooft–Polyakov magnetic monopole and hedgehog by Abrikosov strings in a chiral superconductor. For simplicity, a single Abrikosov string with N = 2 is depicted.

However, the most interesting situation occurs if one takes into account that, in a chiral superconductor, the Meissner effect is not complete because of the l̂ texture. As discussed above, the magnetic flux is not necessarily concentrated in the tubes but can propagate radially from the hedgehog (Fig. 6c). If now the ‘t Hooft–Polyakov magnetic monopole enters the core of the hedgehog in Fig. 6c, which has the same magnetic charge, their strings, i.e., Abrikosov vortices carried by the monopole and Abrikosov vortices attached to the nexus, will annihilate each other. What is left is the combined point defect: hedgehog + magnetic monopole without any attached strings (Fig. 6d). Thus, the monopole destroys the topological connection of the hedgehog and Abrikosov vortices; instead, one obtains topological confinement between the monopole and hedgehog. The core of the hedgehog represents the natural trap for one magnetic monopole: if one tries to separate the monopole from the hedgehog, one must create the piece or pieces of the Abrikosov vortex or vortices) that connect the hedgehog and the monopole (Fig. 6e).

Discussion: Fermions in the Presence of Topological Defects.

Fermions in topologically nontrivial environments behave in a curious way, especially in the presence of such exotic objects as fractional vortices and monopoles discussed in this paper. In the presence of a monopole, the quantum statistics can change; for example, the isospin degrees of freedom are transformed to spin degrees (39). There are also the so-called fermion zero modes: the bound states of fermions at monopole or vortex, that have exactly zero energy. For example the N = 1/2 vortex in a two-dimensional chiral superconductor, which contains only one layer, has one fermionic state with exactly zero energy (40). Because the zero-energy level can be either filled or empty, there is a fractional entropy (1/2)ln2 per layer related to the vortex. The factor (1/2) appears, because in superconductors, the particle excitation coincides with its antiparticle (hole), i.e., the quasiparticle is a Majorana fermion (a nice discussion of Majorana fermions in chiral superconductors can be found in ref. 41). Also the spin of the vortex in a chiral superconductor can be fractional. According to ref. 42, the N = 1 vortex in a chiral superconductor must have a spin S = 1/4 (per layer); consequently, N = 1/2 vortex must have the spin S = 1/8 per layer. Similarly, there can be an anomalous fractional electric charge of the N = 1/2 vortex, which is 1/2 of the fractional charge e/4 discussed for the N = 1 vortex (43). There is still some work to be done to elucidate the problem with the fractional charge, spin, and statistics related to the topological defects in chiral superconductors.