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 ‫؍‬ 1͞2 in chiral superfluids, and N ‫؍‬ 1͞2 and N ‫؍‬ 1͞4 in chiral superconductors), which plays the part of the Alice string in relativistic theories and (ii) the hedgehog in theˆl 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 1͞4 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, ٌ⅐B ϭ 0, which implies that the magnetic flux through any closed surface is zero: ͛ S dS⅐B ϭ 0. If one tries to construct the monopole solution B ϭ gr͞r 3 , the condition that magnetic field is nondivergent requires that magnetic flux ⌽ ϭ 4g 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 fluxwhere 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 ϩ string can be constructed without violating of the condition ٌ⅐B ϭ 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 m...

Reentrant violation of special relativity in the low-energy corner

“…It is the same discrete symmetry which gives rise to the Alice string (the half-quantum vortex, for review see [5]). Such symmetry came from the fundamental level well above the first Planck scale E Planck = ∆ 2 0 /v F p F [3] at which the spectrum becomes nonlinear and thus the Lorentz invariance is violated.…”

Hyperbolic spin vortices and textures in exciton–polariton condensates

“…This shows the high-degree of universality and interest of such studies. Particularly fascinating topological excitations are predicted to occur in spinor quantum fluids 7 and condensates 8 , where the additional spin degree of freedom allows for the existence of mixed spin-phase topologies. The possible occurrence of spin vortices and hedgehogs (magnetic monopole-like excitations) in spinor fluids has been predicted by several theorists [9][10][11][12][13] Such topological entities consist of quantized spin currents with no phase winding, when circumventing the vortex core.…”

Warping and interactions of vortices in exciton-polariton condensates

“…This is the hedgehog vortex [see Fig. 4(a)], where the polarization points in the radial direction everywhere, similar to the field of the magnetic monopole [42,43]. For the other polarization vortices, Eq.…”