Topological phase transition independent of system non-Hermiticity

Non-Hermiticity can vary the topology of system, induce topological phase transitions, and even invalidate the conventional bulk-boundary correspondence. Here, we show the introduction of non-Hermiticity without affecting the topological properties of the original chiral symmetric Hermitian systems. Conventional bulk-boundary correspondence holds; the topological phase transition and the (non)existence of edge states are unchanged even though the energy bands are inseparable due to non-Hermitian phase transitions. The Chern number for energy bands of the generalized non-Hermitian system in two dimensions is proved to be unchanged and favorably coincides with the simulated topological charge pumping. Our findings provide insights into the interplay between non-Hermiticity and topology. A topological phase transition independent of the non-Hermitian phase transition is a unique feature that is beneficial for future applications of non-Hermitian topological materials.

Figure 1

(a) Schematic of the comb lattice. Numerical simulations of $Q_N\left(t\right)$ for the band $(\rho ,\lambda )=(-,-)$ in two quasiadiabatic processes (insets) of (b) ${\delta }_0=0$ and (c) ${\delta }_0=1$ at $\gamma =0.5, \omega ={10}^{-3}$, and $T=2\pi /\omega$. Real (imaginary) part of $Q_N\left(t\right)$ is in black (red); the corresponding $j_N\left(t\right)$ is shown in Appendix pp5.

Figure 2

Energy bands of the edge Hamiltonian for ${\delta }_0=0$ at (a) $\gamma =1.0$ and (b) $\gamma =2.8$, the real (imaginary) part is in black (red), the blue dots are EPs. Other parameters are $J=1, {\kappa }_0=2, R=0.6$, and $N=10$.

Figure 3

(a) [(c)] Schematic of a bipartite lattice in 1D (2D), the non-Hermitian gain and loss are introduced in the sublattices $A$ and $B$, respectively. (b) [(d)] Schematic of the equivalent lattice of (a) [(c)], the double arrows indicate the asymmetric couplings. For the sake of clarity, the long-range couplings in the 2D lattices are not shown in the schematics.

Figure 4

Energy bands of the edge Hamiltonian for ${\delta }_0=0$ at (a) $\gamma =0$, (b) $\gamma =1.0$, (c) $\gamma =1.5$, (d) $\gamma =1.8$, (e) $\gamma =2.8$, (f) $\gamma =3.3$. The real (imaginary) part is in black (red), the blue dots are EPs. Other parameters are $J=1, {\kappa }_0=2, R=0.6$, and $N=10$.

Figure 5

Particle current $j_N\left(t\right)$ for the numerical simulations in Figs. 1 and 1 of the main text for $\mathcal{H}\left(t\right)$. Numerical simulations are performed for (a) topological nontrivial phase $R>\left|{\delta }_0\right|$ at ${\delta }_0=0$ and (b) topological trivial phase $R<\left|{\delta }_0\right|$ at ${\delta }_0=1$. The speed of time evolution is $\omega =0.001$ and the period is $T=2\pi {\omega }^{-1}$. Other parameters are $\gamma =0.5, J=1, R=0.6$, and $N=10$. Two quasiadiabatic processes are illustrated in the insets.

Figure 6

Particle current $j_N\left(t\right)$ and topological charge pumping $Q_N\left(t\right)$ for the edge states of ${\mathcal{H}}_{\mathrm{edge}}\left(t\right)$. Numerical simulations are performed for (a) topological nontrivial phase $R>\left|{\delta }_0\right|$ at ${\delta }_0=0$ and (b) topological trivial phase $R<\left|{\delta }_0\right|$ at ${\delta }_0=1$. The speed of time evolution is $\omega =0.001$ and the period is $T=2\pi {\omega }^{-1}$. Other parameters are $\gamma =0.5, J=1, {\kappa }_0=2, R=0.6$, and $N=10$. Two quasiadiabatic processes are illustrated in the insets.