Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks

Abstract

Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which describe the dynamics of field singularities using gauge fields. Homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups $\pi_n(S^n)$, $n\ge 1$. These homotopies govern monopole and hedgehog topological defects in $d=n+1$ dimensions, and enable the generalization of both weakly-interacting and fractional quantum Hall liquids of vortices to $d>2$. Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and $d=3$ topological materials. We predict novel topological orders in systems with U(1)$\times$Spin($d$) symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the RVB spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.