Relation between $\mathcal{PT}$-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models

Non-Hermitian systems with $\mathcal{PT}$ symmetry can possess purely real eigenvalue spectra. In this work two one-dimensional systems with two different topological phases, the topological nontrivial phase (TNP) and the topological trivial phase (TTP), combined with $\mathcal{PT}$-symmetric non-Hermitian potentials are investigated. The models of choice are the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain. The interplay of a spontaneous $\mathcal{PT}$-symmetry breaking due to gain and loss with the topological phase is different for the two models. The SSH model undergoes a $\mathcal{PT}$-symmetry breaking transition in the TNP immediately with the presence of a nonvanishing gain and loss strength $\gamma$, whereas the TTP exhibits a parameter regime in which a purely real eigenvalue spectrum exists. For the Kitaev chain the $\mathcal{PT}$-symmetry breaking is independent of the topological phase. We show that the topologically interesting states—the edge states—are the reason for the different behaviors of the two models and that the intrinsic particle-hole symmetry of the edge states in the Kitaev chain is responsible for a conservation of $\mathcal{PT}$ symmetry in the TNP.

Figure 1

First row: isolated ($\gamma =0$) energy spectrum for (a) the Kitaev chain with the parameters $t=1.0, \delta =1.0$ and (b) the SSH chain with $t=1.0, \mathrm{\Delta }=0.3$. In both cases the spectrum is computed for a chain with $N=200$ sites. Second row: One of the two edge states for (c) the Kitaev model with the parameters $\mu =1.0, t=1.0$, and $\delta =1.0$ and (d) for the SSH model with $\mathrm{\Theta }=0.1\pi , t=1.0$, and $\mathrm{\Delta }=0.3$. The edge states shown feature a vanishing energy eigenvalue.

Figure 2

First row: Imaginary parts of all energies of the SSH model with potentials $U_1$ (left) and $U_2$ (right). In both calculations the parameters $t=1.0, \mathrm{\Delta }=0.3, N=200$, and $\gamma ={10}^{-5}$ were used. Second row: Imaginary parts of all energies of the Kitaev chain with the parameters $t=1.0, \delta =1.0, N=200$, and $\gamma ={10}^{-5}$.

Figure 3

Imaginary parts of all energies in dependence on the gain and loss strength of all eigenstates of the SSH model. For each plot the parameters $t=1.0, \mathrm{\Delta }=0.3$, and $N=200$ are used. For (a) and (b) the potential $U_1$ is used, whereas for (c) and (d) the potential $U_2$ is applied. The black crosses in (a) and (c) represent the purely imaginary energies of the two edge states appearing in the TNP.

Figure 4

Imaginary parts of all energy eigenvalues for the Kitaev chain with length $N=200$ and the parameters $t=1.0$ and $\delta =t$. The data in (a) and (b) are calculated with the potential $U_1$. In (c) and (d) $U_2$ is applied.

Figure 5

The number of states fulfilling the property of Eq. (9) for the Kitaev chain with $N=200$ lattice sites and the parameters $t=1.0$ and $\delta =t$ in combination with the potential $U_1$ in (a) and $U_2$ in (b). The color represents the number of states, where yellow (bright) stands for the value 2 and black (dark) for the value 0.