Topological parafermion corner states in $\mathbb{Z}_{3}$ clock-symmetric non-Hermitian second-order topological insulator.

Parafermions are generalization of Majorana fermions, whose braiding is known to generate all the Clifford gates in topological quantum computation. We consider a breathing Kagome lattice with complex hoppings by imposing the $\mathbb{Z}_{3}$ clock symmetry in the complex energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We demonstrate that the topological corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric model. It is also shown that the model is realized in electric circuits properly designed, where the parafermion corner states are observed by impedance resonance.