Nonsymmorphic Topological Phases of Non-Hermitian Systems

TL;DR

This work extends the non-Hermitian topological classification by incorporating nonsymmorphic symmetry, introducing pseudo-nonsymmorphic variants, and mapping to Hermitian problems via Hermitization. It identifies new non-Hermitian topological crystalline phases, including $\mathbb{Z}_2$ and $\mathbb{Z}_4$ classifications, with distinctive boundary states such as loop-like edge spectra, hourglass modes, and skin effects protected by NSG symmetry. The authors provide concrete 2D and 3D models across symmetry classes A, AII$^{\dag}$, and AIII, deriving invariant formulas (Berry-phase, time-reversal polarization) and presenting continuum and lattice realizations. The results expand the landscape of open-system topological phases and suggest experimental routes in synthetic platforms to realize NSG-protected non-Hermitian boundary phenomena.

Abstract

Non-Hermiticity appears ubiquitously in various open classical and quantum systems and enriches classification of topological phases. However, the role of nonsymmorphic symmetry, crystalline symmetry accompanying fractional lattice translations, has remained largely unexplored. Here, we systematically classify non-Hermitian topological crystalline phases protected by nonsymmorphic symmetry and reveal unique phases that have no counterparts in either Hermitian topological crystalline phases or non-Hermitian topological phases protected solely by internal symmetry. Specifically, we elucidate the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ non-Hermitian topological phases and their associated anomalous boundary states characterized by distinctive complex-valued energy dispersions.

Figures (4)

• Figure 1: Complex energy spectra of Eqs. (\ref{['model:A+Udag:UneqV']}) and (\ref{['model:A+Udag:U=V']}) with $f(k_y)=-i(m+\cos k_y) + \sin k_y$ and $g(k_x)=\delta\sin (3k_x/2)$. $xy{\rm PBC}$ and $x{\rm PBC}y{\rm OBC}$ denote the periodic boundary conditions in both $x$ and $y$ directions and the open boundary conditions in the $y$ direction, respectively. We set the parameters to $m=0.3$, $\delta=0.7$, $L_x=120$, and $L_y=50$, where $L_x$ and $L_y$ represent the system lengths in the $x$ and $y$ directions, respectively. (a) $\mathcal{U}\neq\mathcal{V}$. Edge modes (orange) described by $g(k_x)e^{ik_x/2}\sigma_z$ appear for $x{\rm PBC}y{\rm OBC}$. (b) $\mathcal{U}=\mathcal{V}$. Skin modes (orange) from $f(k_y)$ appear for $x{\rm PBC}y{\rm OBC}$.
• Figure 2: Complex energy spectra of Eq. (\ref{['eq:nh_aIIdagger']}) [$g(k_x)=c \sin(k_x/2)$; $t_x=0.7$, $t_y=1$, $\Delta=1$, and $c=0.2$] with (a,c,d) $m=0$ and (b) $m=2$. The system size is set to $L_x = L_y = 30$. (a, b) Periodic boundary conditions (PBC) in the $x$ direction and open boundary conditions (OBC) in the $y$ direction for the bulk (blue dots) and edge (orange dots). The gray region represents the spectrum under PBC in both $x$ and $y$ directions. For (a), the orange curve is $\pm ig(k_x)e^{\pm ik_x/2}$. (c) OBC in both $x$ and $y$ directions for the bulk (blue dots) and corner (orange dots). The gray region and curve represent the spectrum for (a). (d) Real-space distribution of the right eigenvector with the energy $E=0.0554+0.00458i$ (the star symbol in the inset). The inset shows the energy spectrum of (c) around $E=0$.
• Figure 3: Complex energy spectra of Eq. (\ref{['model:AIII+U+']}) under the (a) periodic boundary conditions in all directions ($xyz{\rm PBC}$) and (b) open boundary conditions in the $y$ direction ($xz{\rm PBC}y{\rm OBC}$) [$a(k_x,k_z)=-\delta\sin (k_x/2)\sin (k_x/2)$ and $b(k_x,k_z)=\delta\cos (k_x/2)\sin (k_x/2)$; $m=0.5, \delta=0.2$, and $L_x=L_y=L_z=60$].
• Figure S1: Energy spectra of the Hamiltonian in Eq. (\ref{['eq:DIII']}) under the (a) periodic boundary conditions (PBC) in both $x$ and $y$ directions, (b) PBC in the $x$ direction and open boundary conditions (OBC) in the $y$ direction, and (c) OBC in the $x$ direction and PBC in the $y$ direction [$g(k_x)=c \sin(k_x/2)$, $m=0.3$, $t=\Delta=c=1$, $v=0.4$, $t'=0.3$, $L_i=30$ ($i=x,y$)].