Gain-loss-induced non-Abelian Bloch braids

TL;DR

This paper presents a theoretical framework for topological braiding in non-Hermitian energy bands induced by gain and loss, demonstrating non-Abelian braids in three-band systems with potential applications in topological quantum computing.

Contribution

It introduces a gain-loss-induced topological braiding principle and proposes effective Hamiltonians for realizing non-Abelian braid groups in multiband lattice models.

Findings

Braid phase transition occurs at exceptional point degeneracy.

Effective Hamiltonians for $ ext{B}_2$ and $ ext{B}_3$ braid groups are proposed.

Non-Abelian braiding arises from collective behavior of exceptional points.

Abstract

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.