Su-Schrieffer-Heeger chain with one pair of $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric defects

Abstract

Abstract The topologically nontrivial edge states induce $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 transition in Su-Schrieffer-Heeger (SSH) chain with one pair of gain and loss at boundaries. In this study, we investigated a pair of $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric defects located inside the SSH chain, in particular, the defects locations are at the chain centre. The $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 symmetry breaking of the bound states leads to the $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 transition, the $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric phases and the localized states were studied. In the broken $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric phase, all energy levels break simultaneously in topologically trivial phase; however, two edge states in topologically nontrivial phase are free from the influence of the $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric defects. We discovered $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric bound states induced by the $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric local defects at the SSH chain centre. The $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 -symmetric bound states significantly increase the $$\mathcal{P}\mathcal{T}$$ 𝒫 𝒯 transition threshold and coalesce to the topologically protected zero mode with vanishing probabilities on every other site of the left-half chain and the right-half chain, respectively.