Topological enhancement of nonnormality in non-Hermitian skin effects

The non-Hermitian skin effects are representative phenomena intrinsic to non-Hermitian systems: the energy spectra and eigenstates under the open boundary condition (OBC) drastically differ from those under the periodic boundary condition (PBC). Whereas a nontrivial topology under the PBC characterizes the non-Hermitian skin effects, their proper measure under the OBC has not been clarified yet. This paper reveals that topological enhancement of nonnormality under the OBC accurately quantifies the non-Hermitian skin effects. Corresponding to spectrum and state changes of the skin effects, we introduce two scalar measures of nonnormality and argue that the non-Hermitian skin effects enhance both macroscopically under the OBC. We also show that the enhanced nonnormality correctly describes phase transitions causing the non-Hermitian skin effects and reveals the absence of non-Hermitian skin effects protected by average symmetry. The topological enhancement of nonnormality governs the perturbation sensitivity of the OBC spectra and the anomalous time-evolution dynamics through the Bauer-Fike theorem.

Figure 1

Skin effect. (a) The blue and yellow dots represent the PBC and the OBC spectra of Eq. (2) with $t=1.5$, $g=0.5$, $L=100$. (b) Site dependence of the weight function for a right eigenstate of Eq. (2) under the OBC with $E=-2.664$. (c) Site dependence of the weight function for the left eigenstate corresponding to the right eigenstate in (b). The left and right eigenstates are localized at opposite boundaries.

Figure 2

Symmetry-protected skin effect. (a) The blue and yellow dots represent the PBC and the OBC spectra of Eq. (18) with $t=1$, $g=0.3$, $\mathrm{\Delta }=0.2$, $L=100$. (b) Site dependence of the weight function for a right eigenstate of Eq. (18) under the OBC with $E=-9.091\times {10}^{-2}$. (c) Site dependence of the weight function for the left eigenstate corresponding to the right eigenstate in (b). The left and right eigenstates are localized at different ends. (d) A right eigenstate corresponding to the Kramers partner of the right eigenstate in (b).

Figure 3

(a) The blue and yellow dots represent the PBC and the OBC spectra of Eq. (I1) with $t=1.5$, $g=0.5$, $L=100$, $\gamma =0.5$. Eigenstates with the OBC energy inside the PBC spectrum are skin modes, and eigenstates with the OBC energy outside the PBC spectrum are Anderson modes. (b) Site dependence of weight function for a right eigenstate of Eq. (I1) under the OBC with eigenenergy at the black cross in (a). (c) Site dependence of weight function for the left eigenstate corresponding to the right eigenstate in (c). The right eigenstate is localized on one side while the left counterpart is localized on the other side. (d) Site dependence of weight function for a right eigenstate of Eq. (I1) under the OBC with eigenenergy at the red cross in (a). (e) Site dependence of weight function for the left eigenstate corresponding to the right eigenstate in (d). The right and left eigenstates are localized at the same position.

Figure 4

(a) The blue and yellow dots represent the PBC and the OBC spectra of Eq. (24) with $t=1$, $g=0.3$, $\mathrm{\Delta }=0.2$, $N=100$, $\gamma =0.2$. Eigenstates with the OBC energy inside the PBC spectrum are skin modes, and eigenstates with the OBC energy outside the PBC spectrum are Anderson modes. (b) Site dependence of weight function for a right eigenstate of Eq. (24) under the OBC with eigenenergy at the black cross in (a). (c) Site dependence of weight function for the left eigenstate corresponding to the right eigenstate shown in (b). (d) Site dependence of weight function for the Kramers partner corresponding to the right eigenstate shown in (b). The right eigenstate is localized at one end while the left counterpart and the Kramers partner are localized at the other end. (e) Site dependence of weight function for a right eigenstate of Eq. (24) under the OBC with eigenenergy at the red cross in (a). (f) Site dependence of weight function for the left eigenstate corresponding to the right eigenstate shown in (e). (g) Site dependence of weight function for the Kramers partner corresponding to the right eigenstate shown in (e). The right, left eigenstates and the Kramers partner are localized at the same position.

Figure 5

The blue dots represent the numerical PBC spectrum of Eq. (I1) with $t=1.5,g=0.5,L=1000$, and $\gamma =0.5$. The black curve represents the analytical form of the loop structure.

Figure 6

The phase diagram of the OBC Hatano-Nelson model. The blue and yellow regions correspond to the phase where all eigenstates are Anderson modes ($W=0$) and the phase where skin modes exist ($W=1$), respectively. In addition, the red point represents only the origin. At this point, both Anderson localization and skin effect do not occur.

Figure 7

[(a),(c)] Disorder dependence of ${\kappa }_{\mathrm{R}}={\kappa }_{\mathrm{OBC}}\left(V\right)/{\kappa }_{\mathrm{PBC}}\left(V\right)$. We numerically calculate ${\kappa }_{\mathrm{OBC}}\left(V\right),{\kappa }_{\mathrm{PBC}}\left(V\right)$, and their ratio ${\kappa }_{\mathrm{OBC}}\left(V\right)/{\kappa }_{\mathrm{PBC}}\left(V\right)$. We perform these calculations using one hundred different configurations of random potential and calculate the geometric mean of these one hundred obtained ratios. We plot these geometric means in these figures. [(b),(d)] Disorder dependence of ${({\mathrm{dep}}_{\mathrm{D}}/\sqrt{{\sum }_j{s}_j^2})}^{-1}$. We numerically calculate ${({\mathrm{dep}}_{\mathrm{D}}/\sqrt{{\sum }_j{s}_j^2})}^{-1}$. We perform this calculation using five different configurations of random potential and calculate the arithmetic mean of these five obtained differences. We plot these arithmetic means in these figures. (a), (b) Hatano-Nelson model (I1) with $t=1.75g=0.25$. (a) $L=250$. (b) $L=1000$. (c), (d) Time-reversal invariant Hatano-Nelson model (24) with $t=1,g=0.3,\mathrm{\Delta }=0.2$. (c) $L=140$. (d) $L=1000$.

Figure 8

The blue and yellow dots represent the PBC and the OBC spectra of the model (I4) with time-reversal symmetry broken disorder, respectively ($t=1,g=0.3,\mathrm{\Delta }=0.2,\gamma =0.1$, and $L=100$). The disorder follows the Cauchy distribution.

Figure 9

System size dependence of nonnormality. [(a),(d)] Nonnormality of the Hatano-Nelson model (I1) with $t=1.75$ and $g=0.25$. [(b),(e)] Nonnormality of the time-reversal invariant Hatano-Nelson model (24) with $t=1,g=0.3$, and $\mathrm{\Delta }=0.2$. [(c),(f)] Nonnormality of the time-reversal symmetry broken Hatano-Nelson model (I4) with $t=1,g=0.3$, and $\mathrm{\Delta }=0.2$.

Figure 10

Three eigenstates ($|E^1\rangle$, $|E^1\rangle \rangle$, $|E^2\rangle$) under the normalization conditions ($\langle E^1|E^1\rangle =1$, $\langle E^2|E^2\rangle =1$) and the biorthogonal condition ($\langle \langle E^1|E^2\rangle =0$, $\langle E^1|E^1\rangle \rangle =1$). For almost parallel $|E^1\rangle$ and $|E^2\rangle$, the angle $\alpha$ is close to $\pi /2$, resulting in significantly large $\langle \langle E^1|E^1\rangle \rangle$.

Figure 11

The energy spectrum of Eq. (B1) with $t=1.5$, $g=0.5$. The PBC spectrum (blue line) encloses the origin of the complex energy plane twice, $W\left(E\right)=2$ around $E=0$. The yellow dots represent skin modes under OBC with the system size $L=21$. Since the number of skin modes coincides with the system size, skin modes do not have degeneracy even in the region of $W\left(E\right)=2$.

Figure 12

(a) The blue and yellow dots represent the energy spectra of Eq. (C1) and Eq. (C3), respectively with $t=1.5$, $g=0.5$, and $L=101$. (b) System size $L$ dependence of the condition number in Eq. (C1) and Eq. (C3) with $t=1.5$, and $g=0.5$. Here $V_{\mathrm{I}/\mathrm{F}}$ and $V_{\mathrm{SD}}$ are matrices diagonalizing the matrix representation of ${\widehat{H}}_{\mathrm{I}/\mathrm{F}}$ and ${\widehat{H}}_{\mathrm{DS}}$, respectively. (c) System size $L$ dependence of the departure from normality in Eq. (C1) and Eq. (C3) with $t=1.5$, and $g=0.5$.

Figure 13

The blue line represents the spectrum of $H=H_0+W$, where $W$ follows the Cauchy distribution. The black line represents the energy spectrum that shifts the energy spectrum of $H_0$ parallel to the imaginary axis by $\pm \gamma$.

Figure 14

The flow of $\arg (detH\left(\theta \right))$ with respect to $\theta$ for the Hatano-Nelson model (I1) with $t=1.5$, $g=0.5$, $L=100$, $\gamma =0.5$. (a) The reference energy is inside the PBC spectrum ($E=0$). (b) The reference energy is outside the PBC spectrum ($E=10+10i$).

Figure 15

The flow of $arg(det[(H\left(\theta \right)-E\left])T\right)$ with respect to $\theta$ for the model (24) with $t=1$, $g=0.3$, $\mathrm{\Delta }=0.2$, $L=100$, and $\gamma =0.2$. (a) The reference energy is inside the PBC spectrum ($E=0$). (b) The reference energy is outside the PBC spectrum ($E=10+10i$).

Figure 16

Schematic illustration of energy spectrum for $\tilde{H}$ under the PBC and the OBC. $s_{min}(H-z)$ and $s_{max}(H-z)$ represent the smallest and largest singular value of $H-z$, respectively. (a) Trivial case of the Hamiltonian (K4). (b) Nontrivial case of the Hamiltonian (K4).

Figure 17

The blue (yellow) dot represents the disorder dependence of $\kappa \left(H\right)$ under the PBC (OBC). We numerically calculate $\kappa \left(H\right)$ under the PBC and the OBC under one hundred different configurations of random potential, and calculate the geometric mean of these one hundred obtained ratios. We plot these geometric means in these figures. (a) The condition number of the Hatano-Nelson model (I1) with $t=1.5$, $g=0.5$, and $L=3000$. The black line denotes $\gamma =1$. (b) The condition number of the time-reversal invariant Hatano-Nelson model (23) with $t=1,g=0.3,\mathrm{\Delta }=0.2$, and $L=2500$. The black line denotes $\gamma =0.44$. (c) The condition number of the time-reversal symmetry broken Hatano-Nelson model (I4) with $t=1,g=0.3$, $\mathrm{\Delta }=0.2$, and $L=1500$.