Topological phase transitions driven by non-Hermiticity in quantum spin Hall insulators

The interplay between non-Hermiticity and topology opens an exciting avenue for engineering novel topological matter with unprecedented properties. While previous studies have mainly focused on one-dimensional systems or Chern insulators, here we investigate topological phase transitions to and from quantum spin Hall (QSH) insulators driven by non-Hermiticity. We show that a trivial to QSH insulator phase transition can be induced by solely varying non-Hermitian terms, and there exists exceptional edge arcs in QSH phases. We establish two topological invariants for characterizing the non-Hermitian phase transitions: (i) with time-reversal symmetry, the biorthogonal ${\mathbb{Z}}_2$ invariant based on non-Hermitian Wilson loops, and (ii) without time-reversal symmetry, a biorthogonal spin Chern number through biorthogonal decompositions of the Bloch bundle of the occupied bands. These topological invariants can be applied to a wide class of non-Hermitian topological phases beyond Chern classes, and provides a powerful tool for exploring novel non-Hermitian topological matter and their device applications.

Figure 1

Topological phase transitions driven by non-Hermiticity. (a), (b) The non-Hermitian term $i{\lambda }_{23}{\mathrm{\Gamma }}_{23}$ drives the system from (a) trivial insulator (${\lambda }_{23}=0.1$) to (b) QSH insulator (${\lambda }_{23}=0.5$). The panels show open-boundary spectra on a zigzag ribbon. The parameters are $t=1$, ${\lambda }_{\text{SO}}=0.06$, ${\lambda }_{\nu }=0.4$, and ${\lambda }_{\text{Rb}}=0.05$. (c), (d) The non-Hermitian term $i{\lambda }_{14}{\mathrm{\Gamma }}_{14}$ drives the system from (c) QSH insulator (${\lambda }_{14}=0.05$) to (d) trivial insulator (${\lambda }_{14}=0.4$) with exceptional edge arcs. The parameters are the same as panels (a) and (b) except ${\lambda }_{\nu }=0.1$ and ${\lambda }_{\text{Rb}}=0$. The insets show the change of the real insulating gap $\mathrm{Re}({\mathrm{\Delta }}_I$) with respect to the non-Hermitian parameters. The dashed gray lines represent gap closing points, “IN” denotes the trivial insulator phase and the light purple triangles (disks) denote the non-Hermitian parameters for panels on the left (right).

Figure 2

Exceptional points and exceptional edge arcs. (a), (b) Real and imaginary parts of the edge states below the Fermi level. The parameters are the same as Figs. 1 and 1 except ${\lambda }_{14}=0.09$. The red arrows highlight the exceptional points (EPs) and edge arcs. (c) Evolution of (right) exceptional point along $k_x$ with respect to ${\lambda }_{14}$.

Figure 3

Bulk biorthogonal ${\mathbb{Z}}_2$ invariant from non-Hermitian Wilson loops. (a) The non-Hermitian Wilson loop in the Brillouin zone (dashed parallelogram) is defined along the purple arrow while the base point for each Wilson loop is given by the orange arrow. (b), (c) The biorthogonal Wannier centers plotted to varying base points, corresponding to the cases in Figs. 1, 1, 1, and 1 respectively. The blue (red) points represent weak (strong) non-Hermitian effects and the biorthogonal ${\mathbb{Z}}_2$ index $I$ is labeled in each panel.

Figure 4

Biorthogonal spin Chern number as the topological invariant for TR-broken non-Hermitian QSH insulator. The parameters are the same as in Figs. 1 and 1. The blue ($S=1/2$, spin “up”) and orange ($S=-1/2$, spin “down”) dots are biorthogonal spin Chern numbers $C_S$. The red curve represents the real insulating gap.

Figure 5

Imaginary bands in TR-symmetric non-Hermitian Kane-Mele model. (a)–(d) Imaginary parts of the spectra shown in Figs. 1 to 1, respectively. The edge states are plotted with the same color as those in Fig. 1 and we only plot the purely real edge states in panels (a) and (b).

Figure 6

The spectra under periodic boundary conditions at different momenta on complex plane. The parameters for each panel are the same as in Fig. 1 in the main text.

Figure 7

Non-Hermitian skin effect induced by ${\lambda }_{23}$. Panels (a) and (b) show the bulk modes with weak [Fig. 1] and strong [Fig. 1] non-Hermitian terms, respectively. We choose $k_x=0$ here.

Figure 8

(a), (b) Real and imaginary bands in momentum space across the phase transition. The parameters are ${\lambda }_{14}=0.2$, 0.295 (gap closing point), and 0.4. (c) Topological phase transition driven by $i{\lambda }_{14}{\mathrm{\Gamma }}_{14}$ with nonzero Rashba spin-orbit interaction. The parameters are the same as those in Fig. 2 in the main text except ${\lambda }_{\text{Rb}}=0.05$. The light-blue-coded areas correspond to the gapless phases.

Figure 9

Biorthogonal spin Chern number as topological invariant for TR-symmetric non-Hermitian QSH insulators. (a), (b) Biorthogonal spin Chern numbers computed for the TR-symmetric models in Figs. 1, 1, 1, and 1, respectively.

Figure 10

Biorthogonal spin Chern number as the topological invariant for TR-broken non-Hermitian QSH insulators. The topological phase transition is induced by a real exchange field ${\lambda }_{\mu }{\mathrm{\Gamma }}_{34}$ and a small non-Hermitian term $i0.05{\mathrm{\Gamma }}_{14}$ is included. Other parameters are the same as those in Figs. 1 and 1.

Figure 11

TR-broken QSH phase driven by the non-Hermitian term $i{\lambda }_4{\mathrm{\Gamma }}^4$. Open-boundary spectrum is plotted for (a) trivial insulator phase (${\lambda }_4=0.1$) and (b) TR-broken QSH phase (${\lambda }_4=0.5$). Other parameters are the same as those in Figs. 1 and 1.