Spontaneous $\mathcal{PT}$-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models

The spontaneous parity-time $\left(\mathcal{PT}\right)$ symmetry breaking is discussed in non-Hermitian $\mathcal{PT}$-symmetric Kitaev and extended Kitaev models whose Hermiticity is broken by the presence of two conjugated imaginary potentials $\pm i\gamma$ at two end sites. In the case of the non-Hermitian Kitaev model, a spontaneous $\mathcal{PT}$-symmetry breaking transition $\left(S\mathcal{PT}BT\right)$ occurs at a certain ${\gamma }_c$ in the topologically trivial phase (TTP) region, similar to that of the Su-Schrieffer-Heeger (SSH) model. However, unlike the SSH model, the system also undergoes such a transition in the topologically nontrivial phase (TNP) region. We study an extended Kitaev model by combining the superconducting pairing in the Kitaev model and the staggered hopping in the SSH model. This model contains three different topological phases: the TTP, the Kitaev-like TNP, and the SSH-like TNP. For the non-Hermitian extended Kitaev model, a $S\mathcal{PT}BT$ occurs in the Kitaev-like TNP region, as well as in part of the TTP and SSH-like TNP regions, whereas the $\mathcal{PT}$ symmetry is broken for an arbitrary nonzero $\gamma$ in the rest of the TTP and SSH-like TNP regions. Therefore, we can conclude that there is no universal correlation between topological properties and the $S\mathcal{PT}BT$.

Figure 1

(a) Energy spectrum for the Kitaev model with parameters $\mathrm{\Delta }=0.8, t=1$, and $N=100$ under the OBC. (b) The phase diagram of the extended Kitaev model in the parameter space spanned by $\mathrm{\Delta }, \delta$, and $\mu$. (c) Energy spectrum for the extended Kitaev model with parameters $\mathrm{\Delta }=0.4, \mu =0.4, t=1, N=100, {\delta }_{c_1}=-0.447$, and ${\delta }_{c_2}=0.447$ under the OBC. In our calculations we set $t=1$ as the unit of energy.

Figure 2

The real and the imaginary parts of the eigenvalue spectra of the non-Hermitian Kitaev model as functions of $\mu$ with parameters $\mathrm{\Delta }=0.8$ and $N=100$ for different $\gamma$: (a) $\gamma =0.1$, (b) $\gamma =0.93$, (c) $\gamma =0.975$, (d) $\gamma =0.99$, and (e) $\gamma =3$ under the OBC. Left figures represent the real part of the spectra and right figures represent their imaginary part. The red points represent the real (left) part and imaginary (right) part of the complex eigenvalues.

Figure 3

The real (left) and imaginary (right) parts of the eigenvalue spectra versus $\gamma$ for the non-Hermitian Kitaev model in different topological regions. (a) System in the TNP region with $\mu =1$; (b) system in the TNP region with $\mu =0$; and (c) system in the TTP region with $\mu =3$. The red points represent the real (left) part and imaginary (right) part of the complex eigenvalues.

Figure 4

The sketch of the ${\gamma }_c\left(\mu \right)$ in the non-Hermitian Kitaev model as a function of $\mu$.

Figure 5

The real and the imaginary parts of the eigenvalue spectra of the non-Hermitian extended Kitaev model as functions of $\delta$ with parameters $\mathrm{\Delta }=0.4, \mu =0.4$, and $N=100$ for different $\gamma$: (a) $\gamma =0.01$, (b) $\gamma =0.2$, (c) $\gamma =0.6$, and (d) $\gamma =1.5$ under the OBC. Left figures represent the real part of the spectra and right figures represent their imaginary part. The red points represent the real (left) part and imaginary (right) part of the complex eigenvalues.

Figure 6

The real (left) and imaginary (right) parts of the eigenvalue spectra versus $\gamma$ for the non-Hermitian extended Kitaev model in different topological regions. (a) System in the TTP region with $\delta =-0.7$; (b) system in the TTP region with $\delta =-0.5$; (c) system in the Kitaev-like TNP region with $\delta =-0.3$; (d) system in the Kitaev-like TNP region with $\delta =0.3$; (e) system in the SSH-like TNP region with $\delta =0.5$; and (f) system in the SSH-like TNP region with $\delta =0.7$. The red points represent the real (left) part and imaginary (right) part of the complex eigenvalues.

Figure 7

The sketch of the ${\gamma }_c\left(\delta \right)$ in the non-Hermitian extended Kitaev model as a function of $\delta$.