Pseudospectral phenomena and the origin of the non-Hermitian skin effect

Abstract

The non-Hermitian skin effect (NHSE), characterized by a macroscopic accumulation of eigenstates at the edge of a system with open boundaries, is often ascribed to a non-trivial point-gap topology of the Bloch Hamiltonian. We revisit this connection and show that the eigenspectrum of non-normal operators is highly sensitive to boundary conditions and generic perturbations, and therefore does not constitute a stable object encoding topological information. Instead, topological properties are reflected in the singular-value spectrum of finite systems and, in the semi-infinite limit, correspond to boundary-localized eigenmodes implied by the index of the corresponding Toeplitz operator. For a Hatano-Nelson ladder, where point-gap winding and non-normality can be varied independently, we demonstrate that the NHSE can occur without point-gap winding and, conversely, that point-gap winding can persist without the NHSE. These results establish that the NHSE originates from spectral instability and non-reciprocity rather than topology, and that the commonly assumed relation between spectral winding and boundary localization relies on translational invariance and is therefore not generic.

Figures (7)

• Figure 1: Spectrum of the Hatano-Nelson model with $N=50$ and $t_L=1$, $t_R=0.4$, $\mu=0$ for PBC and OBC. Also shown are the pseudospectral contour lines $\|(H-\lambda\mathbf{1})^{-1}\|_2=1/\varepsilon$ for the open chain, which will all converge to $h(k)$ for $N\to\infty$ and which show the extreme instability of the eigenspectrum.
• Figure 2: Perturbed spectrum of the open Hatano-Nelson model for $N=50$ and $t_L=1$, $t_R=0.4$, $\mu=0$. The open blue squares on the real axis are obtained by perturbing each individual onsite potential and hopping by an additive random perturbation drawn from the interval $[-0.4,0.4]$. The other symbols denote 5 realizations of the Hamiltonian perturbed by a random complex matrix $A$ with $|a_{ij}|\leq 0.1/\sqrt{N}$.
• Figure 3: Hatano-Nelson model with OBC and parameters as in Fig. \ref{['Fig2']}. The main panels show the center $X/N$ of each wave function versus its IPR, providing a simultaneous measure of localization strength and spatial bias. The large red dots represent the unperturbed system and the black circles in (a) $100$ realizations of the local perturbation with strength $\varepsilon=0.2$, and in (b) $100$ realizations of the global perturbation with strength $0.2/\sqrt{N}$. The insets show the mean IPR as a function of disorder strength $\varepsilon$ with the blue shaded bands showing the $5-95$ percentile range.
• Figure 4: Singular value spectrum for the open Hatano-Nelson model with $N=50$, $t_L=1$, $t_R=0.4$, $\mu=0$. Circles denote the unperturbed system, open squares the system with a global perturbation of strength $0.1/\sqrt{N}$ with an average taken over $1000$ realizations. The topologically protected value near zero remains separated from the bulk spectrum by a gap. Inset: The corresponding right singular vector is exponentially localized at the left edge.
• Figure 5: Two Hatano-Nelson chains coupled by non-reciprocal couplings $t_U$, $t_D$, and $t_D\gamma$; see Eq. \ref{['HN_doubled']}.