Winding number and bulk-boundary correspondence in a one-dimensional non-Hermitian photonic lattice

A topological phase transition and the related bulk-boundary correspondence in a class of one-dimensional non-Hermitian photonic lattice with chiral symmetry are studied. We find that the corresponding topological number should be defined by adding up winding numbers calculated under the chiral symmetry. The Wilson loop method is used to calculate this topological number. We reveal the topological phase transition in one-dimensional non-Hermitian systems with chiral symmetry and provide a convenient and general pathway to characterize the related topology.

Figure 1

Open chain described by Eq. (1). The dashed box denotes the unit cell.

Figure 2

Spectrum of the open-boundary condition, with $M=50$. There are two eigenstates with energy $E=\pm iA$, labeled by the diamonds, in the topological phase, and no such edge state in the trivial phase.

Figure 3

Spectra of the bulk Hamiltonian with different parameters. Circles, pluses, diamonds, and crosses denote $n=1$, 2, 3, and 4, respectively, and $n^{\prime }=4,3,2,1$ and $n_c=2,1,4,3$.

Figure 4

Topological numbers of systems with $w=2$ while $\gamma$ varies from 0 to 4. The change in topological numbers indicates the topological phase transition. There is a slight shift in the blue lines for avoiding overlap. The phase transition point is analytically $\gamma =\sqrt{3}$.

Figure 5

Numerical solution and theoretical approximate solution for $w=2$ and $\gamma =1$. After normalization, the approximate solution is $a_1=0.7557$, $b_1=-0.3528$, $a_2=-0.4602$, and $\lambda =0.3045$.