Model non-Hermitian topological operators without skin effect: A general principle of construction

Abstract

We propose a general principle of constructing non-Hermitian (NH) operators for insulating and gapless topological phases in any dimension ($d$) that over an extended NH parameter regime feature real eigenvalues and zero-energy topological boundary modes, when in particular their Hermitian counterparts are also topological. However, the topological zero modes disappear when the NH operators simultaneously accommodate real and imaginary (in periodic systems) or display complex (in systems with open boundary conditions) eigenvalues. These systems are always devoid of NH skin effects, as has also been confirmed from the scaling of the inverse participation ratio, thereby extending the realm of the bulk-boundary correspondence to NH systems in terms of solely the left or right zero-energy boundary localized eigenmodes. We showcase these general and robust outcomes for NH topological insulators in $d=1,2$ and $3$, encompassing their higher-order incarnations, as well as for NH topological Dirac, Weyl, and nodal-loop semimetals. Possible realizations of proposed NH topological phases in designer materials, optical lattices, and classical metamaterials are highlighted.

Figures (9)

• Figure 1: Non-Hermitian Su-Schrieffer-Heeger model in $d=1$. (a) Eigenvalue spectrum for $\alpha=0.5$ with a periodic boundary condition (PBC) and an open boundary condition (OBC), showing their guaranteed reality condition and the existence of two near zero-energy topological modes (inset) for $|\alpha|<1$. (b) Amplitude square of the right ($\beta=R$) or left ($\beta=L$) eigenvectors of two zero-energy modes, showing their sharp localization near the ends of the chain. (c) The same as (b), but for all the right or left eigenvectors, showing no left-right asymmetry and confirming the absence of any NH skin effect (inset). (d) Eigenvalues for $\alpha=10$, showing its generic purely real or imaginary nature (with PBC) and complex nature (with OBC), the absence of any zero-energy topological modes and skin effect for $|\alpha|>1$. Here, we set $t=B=1$ and $\Delta_1=1$. See Eqs. \ref{['eq:Dirac']}, \ref{['eq:firstorderWD']} and \ref{['eq:NHGeneral']}, and Sec. \ref{['Sec:skinfreeNH']}.
• Figure 2: Non-Hermitian Qi-Wu-Zhang model in $d=2$. (a) Eigenvalues for $\alpha=0.5$ with PBCs and OBCs, confirming their reality condition and the existence of near zero energy topological edge modes (inset) for $|\alpha|<1$. (b) Amplitude square of the right ($\beta=R$) or left ($\beta=L$) eigenvectors of two closest to zero energy modes, showing their sharp edge localization. (c) The same as (b), but for all the right or left eigenvectors, showing no left-right or top-bottom asymmetry about the center of the system, thus no NH skin effect. (d) Purely real or imaginary (with PBCs) and complex (with OBCs) eigenvalues for $\alpha=10$, showing absence of any zero energy topological mode and NH skin effect for $|\alpha|>1$. Here, we set $t=B=1$ and $\Delta_1=6$. See Eqs. \ref{['eq:Dirac']}, \ref{['eq:firstorderWD']} and \ref{['eq:NHGeneral']}, and Sec. \ref{['Sec:skinfreeNH']}.
• Figure 3: Non-Hermitian second-order topological insulator in $d=2$. (a) Real eigenvalue spectrum with PBCs and OBCs, accommodating four near zero-energy topological modes (inset) for $\alpha=0.5$. (b) Amplitude square of the right ($\beta=R$) or left ($\beta=L$) eigenvectors of four closest to zero-energy modes, showing their sharp corner localization. (c) Same as (b), but for all the right or left eigenvectors, showing no left-right or top-bottom asymmetry about the center of the system, and thus no NH skin effect. (d) Generic real or imaginary (with PBCs) and complex (with OBCs) eigenvalues, and the absence of zero-energy topological modes and NH skin effect for $\alpha=10$. Here, we set $t=B=1$, $\Delta_1=6$ and $\Delta_2=1$. See Eqs. \ref{['eq:Dirac']}-\ref{['eq:NHGeneral']}, and Sec. \ref{['Sec:skinfreeNH']}.
• Figure 4: Hierarchy of non-Hermitian topological insulators (TIs) in $d=3$. Real eigenvalue spectrum for $\alpha=0.5$ with PBCs and OBCs, showing the existence of near zero-energy topological modes (insets) for (a) first-order, (d) second-order and (g) third-order TIs. Amplitude square of the right or left eigenvectors of (b) four, (e) four and (h) eight near zero-energy modes, showing sharp localization (b) on six surfaces, (e) on four $z$-directional hinges and two $xy$ surfaces, and (h) at eight corners. Here, we set $t=B=1$, $\Delta_1=10$, $\Delta_2=1$ and $\Delta_3=1$ throughout. See Eqs. \ref{['eq:Dirac']}-\ref{['eq:NHGeneral']}, and Sec. \ref{['Sec:skinfreeNH']}. Panels (c), (f), and (i) show purely real or imaginary (with PBCs) and complex (with OBCs) eigenvalue spectra and the absence of near-zero-energy modes in the spectrum of NH operator as in panels (a), (d), and (g), respectively, but for $\alpha=10$.
• Figure 5: Non-Hermitian topological semimetals. Boundary modes of (a) a two-dimensional NH Dirac semimetal, featuring Fermi arcs between two Dirac points at $k_y=\pm \pi/(2a)$ for $\Delta_1=0$ and $t_2=B=1$, (b) a three-dimensional NH nodal-loop semimetal, showing drumhead surface states for $\Delta_1=0$ and $t_2=t_3=B=1$, with images of the bulk nodal ring determined by $\cos(k_y a)+\cos(k_z a)=0$ on the top and bottom surfaces, (c) a three-dimensional NH Weyl semimetal, displaying Fermi arcs in between the Weyl nodes at $k_z=\pm \pi/(2a)$, for $\Delta_1=0$ and $t_3=B=1$, and (d) a three-dimensional NH second-order Dirac semimetal with hinge Fermi arcs between two Dirac points at $k_z=\pm \pi/(2a)$ for $\Delta_1=0$ and $\Delta_2=t_3=B=1$. Here, we set $\alpha=0.5$. See Eqs. \ref{['eq:Dirac']}-\ref{['eq:NHGeneral']}, and Sec. \ref{['Sec:skinfreeNH']}. Results are obtained from the local density of states (probability amplitude) of the right topological eigenstates. We arrive at the same conclusion from the left topological eigenstates (not shown explicitly). These findings show that the conventional bulk-boundary correspondence is operative for the NH gapless topological phases in terms of the left or right topological modes in our general principle of constructing NH skin effect-free operators, see Sec. \ref{['sec:NHgapless']}.