Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points

Capital to topological insulators, the bulk-boundary correspondence ties a topological invariant computed from the bulk (extended) states with those at the boundary, which are hence robust to disorder. Here we put forward a different ordering unique to non-Hermitian lattices whereby a pristine system becomes devoid of extended states, a property which turns out to be robust to disorder. This is enabled by a peculiar type of non-Hermitian degeneracy where a macroscopic fraction of the states coalesce at a single point with a geometrical multiplicity of 1.

Figure 1

(a) Scheme representing three cells of the lattice model introduced in the text. The imaginary on-site energies lead to the non-Hermiticity of the Hamiltonian. (b) Map of the $\mathcal{P}T$-broken and $\mathcal{P}T$-unbroken regions as a function of the Hamiltonian parameters $v$ and $r$ (both in units of $\gamma )$. (c) and (d) [(e) and (f)] show the real and imaginary parts of the spectrum obtained for a system of $N=30(N=31)$ unit cells and $r=0.5\gamma$.

Figure 2

(a) Real and (b) imaginary parts of the eigenenergies obtained as a function of $v$. This corresponds to a finite system with $N=30$. The color scale shows the inverse participation ratio. (c) Details of the evolution of the spectrum as $v$ increases from zero to some value above the singular point $v=0.5\gamma$ where EPs and higher-order EPs emerge. The red arrows indicate whether the points tend to coalesce or separate as $v$ increases. (d) Probability density associated with the eigenstates obtained for different values of $v$. Note that all states remain localized at one edge.

Figure 3

Effect of increasing amounts of disorder $d$ (from left to right) in $\gamma ,v,r$, and on-site Anderson disorder on the spectrum. The results for 300 realizations of disorder and $N=30$ are superimposed. The starting points for all plots are $r=v=0.5$ and $\gamma =1$ as in the third plot in Fig. 2. The color scale encodes the inverse participation ratio and shows the resilience of localization against disorder.

Figure 4

(a) As a measure of the biorthogonality of the Hamiltonian eigenstates, we plot the sum of the absolute values of the phase rigidities $r_{\alpha }$ defined in Eq. (3) normalized so that it gives the values of 1 when the Hamiltonian is Hermitian and 0 when all the states are biorthogonal. The main panel is for $N=30$, whereas the inset shows three values of $N$ over a larger range of $v$. (b) Rank of the Hamiltonian for a system containing $N$ unit cells. The inset shows a zoom over the $v$ region close to the exceptional points.