Fig. 1: Spin-charge-separated Kramers pairs of defect or flux states.
From: Topological zero-dimensional defect and flux states in three-dimensional insulators
a–d An inversion- (\({{{{{{{\mathcal{I}}}}}}}}\)-) related pair of Kramers pairs of 0D defect or flux states in a spinful, time-reversal- (\({{{{{{{\mathcal{T}}}}}}}}\)-) symmetric insulator (where the \({{{{{{{\mathcal{I}}}}}}}}\) center is represented with a red × symbol in a–d). b, c When the Fermi level lies at charge neutrality, each Kramers pair is filled by only a single electron and therefore carries an excess chargeless spin-1/2 moment (Q = 0, S = 1/2). Hence at half filling, and taking the spins of the electrons occupying each pair of states to point in opposite directions, \({{{{{{{\mathcal{I}}}}}}}}\) (which relates the positions of the Kramers pairs) and \({{{{{{{\mathcal{T}}}}}}}}\) symmetries are "softly'' broken17,34,39,59, and each half-filled Kramers pair of states forms an effective spinon quasiparticle with a free-angle spin-1/2 moment (depicted in b, c in configurations that preserve \({{{{{{{\mathcal{I}}}}}}}}\times {{{{{{{\mathcal{T}}}}}}}}\) symmetry). By a removing or d adding two electrons to the system (one electron per Kramers pair), we may realize a system configuration in which each Kramers pair respectively carries a net charge of ± e (taking electrons to carry a charge −e), but carries a net-zero spin (Q = ± e, S = 0). Hence, each Kramers pair of states either carries chargeless spin or spinless charge and therefore exhibits the same reversed spin-charge relations as the solitons in polyacetylene60.
