Gain-loss induced non-Abelian Bloch braids

Onsite gain-loss induced topological braiding principles 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 degeneracies. 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. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesize exceptional materials for applications in topological quantum photonic computation and information processing.