Anatomy of skin modes and topology in non-Hermitian systems

A non-Hermitian system can exhibit extensive sensitivity of its complex energy spectrum to the imposed boundary conditions, which is beyond any known phenomenon from Hermitian systems. In addition to topologically protected boundary modes, macroscopically many “skin” boundary modes may appear under open boundary conditions. We rigorously derive universal results for characterizing all avenues of boundary modes in non-Hermitian systems for arbitrary hopping ranges. For skin modes, we introduce how exact energies and decay lengths can be obtained by threading an imaginary flux. Furthermore, for one-dimensional topological boundary modes, we derive a generic criterion for their existence in non-Hermitian systems which, in contrast to previous formulations, does not require specific tailoring to the system at hand. Our approach is intimately based on the complex analytical properties of in-gap exceptional points, and gives a lower bound for the winding number related to the vorticity of the energy Riemann surface. It also reveals that the topologically nontrivial phase is partitioned into subregimes where the boundary mode's decay length depends differently on complex momenta roots.

Figure 1

(a) Illustration of Eq. (2), with unbalanced intra-unit-cell couplings $T_0$ and balanced inter-unit-cell couplings $T_{\pm 1}$. Spectra for (b)–(d) $H_{\text{SSH}}^{{\gamma }_x,0}$ and (e)–(g) $H_{\text{SSH}}^{0,{\gamma }_y}$ from Eq. (2). (d), (g) Both systems exhibit topological zero modes (black lines at $\text{Re}\left[E\right]=0$) at small ${\gamma }_x$ or ${\gamma }_y$, but only $H_{\text{SSH}}^{{\gamma }_x,0}$ exhibits BBC. For $H_{\text{SSH}}^{0,{\gamma }_y}$, all the OBC modes (black) differ from the PBC modes (red), not just for topological modes. (b), (c), (e), (f) Anatomy of PBC and OBC spectra in a topologically nontrivial (${\gamma }_{x,y}=0.4$) and trivial (${\gamma }_{x,y}=1.2$) regime. Light blue-magenta tapering curves illustrate the evolution of PBC modes into OBC modes as $\text{Im}\varphi$ increases from 0 to $\infty$. Pale background solid curves are contours of constant $\kappa =\text{Im}k$ with intervals of 0.1. For $H_{\text{SSH}}^{{\gamma }_x,0}$, the PBC and OBC spectra coincide along open arcs and no evolution occurs, except towards the isolated topological zero mode in (b). For $H_{\text{SSH}}^{0,{\gamma }_y}$, the OBC skin modes (black) morph as a whole from the PBC modes (red). Bulk modes become localized skin modes as soon as they lie along $\kappa \ne 0$ contours. From Eq. (7), skin modes (black) of (f) satisfy $\text{Re}\left[E^2\right]=\frac{5}{4}-{\gamma }_y^2$ and $4{\gamma }_y^2>1+{\left(\text{Im}\left[E^2\right]\right)}^2$.

Figure 2

PBC-OBC spectral flow of model (8). In (a), PBC bulk eigenvalues (red) flow along the blue-magenta curves, accumulating as OBC skin modes along the $\mathsf{\text{Y}}$-shaped black lines. The latter are the largest magnitude solutions to $|{\beta }_{\mu }|=|{\beta }_{\nu }|$ [Eq. (7)], as evident from the pale background contours at intervals $\left|\mathrm{\Delta }\kappa \right|=\mathrm{\Delta }log\left|\beta \right|=0.07$. In (b), this flow is visualized as eigenmodes “sliding down” the brown surface (solution of $\left|\beta \right|=e^{-\kappa }$) upon imaginary flux threading, until they stop along the black valley where two $\left|\beta \right|$ solutions (brown and yellow) intersect.

Figure 3

(a) Phase diagram of Eq. (13) with $\gamma =1.2$. Different colors represent regimes with topological mode decay rate $-{(log|\beta \left|\right)}^{-1}$ determined by $\beta =a_1$, $a_2$, $b_1$, or $b_2$, respectively. (b) Illustration of how the ordering of $-log\left|\beta \right|$ solutions determines the phase along the dashed line ($t_2=0.05$) of (a), with $\beta =a_1$, $a_2$, $b_1$, and $b_2$ solutions colored red, light red, dark green, and light green. From criterion (11), topological modes occur when no greenish (reddish) curve falls between two reddish (greenish) curves, with corresponding regimes colored as in (a).

Figure 4

Application of criterion (11) to a more complicated instance of (9) with $p_a=p_b=4$ and $r_a=3$, $r_b=2$, which is completely topologically characterized by the roots of their $a\left(z\right)$ and $b\left(z\right)$ (purple and orange dots). Noncontractible contours in the purple region ($W_a>0$) enclose at least $q_a+1=p_a-r_a+1=2$ purple roots, while contours in the orange region ($W_b<0$) enclose fewer than $q_b+1=p_b-r_b+1=2$ orange roots. In (a)/(b), the presence/absence of a zero mode corresponds to the presence/absence of an overlap region where $W_a>0$ (purple) and $W_b<0$ (orange) simultaneously (i.e., $W_a{W}_b<0$).