Gain-loss-induced non-Abelian Bloch braids

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups ${\mathbb{B}}_2$ and ${\mathbb{B}}_3$ of two and three bands, respectively. While ${\mathbb{B}}_2$ is trivially Abelian, the group ${\mathbb{B}}_3$ features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing.