I will discuss the Dirac quasiparticles in the lattice systems of electrons, exemplified by graphene, in the presence of topological defects of the allowed order parameters. These orders appear as possible mass terms in the Dirac equation, and their topological defects have been known to carry non-trivial quantum numbers such as fractionalized charge since the work of Jackiw and Rebby in 1976. In the talk I will discuss their additional internal degree of freedom: irrespectively of the nature of orders that support the defect, an extra mass-order-parameter spontaneously emerges in the defect's core. The determination of the quantum state of the topological defect in Dirac systems turns out to be an interesting problem in the (real) representation theory of Clifford algebras; with the Clifford algebra C(2,5) playing a fundamental role in graphene, for example. Ultimately, the particle-hole symmetry restricts the defects to always carry the quantum numbers of a single effective "isospin" - 1/2, quite independently of the values of their electric charge or true spin. Examples of this new degree of freedom in graphene and other Dirac-like systems, such as graphene bilayers or d-wave superconductors will be given.
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