Stable boundary modes for fragile topology from spontaneous PT-symmetry breaking

TL;DR

This work shows that in PT-symmetric non-Hermitian systems, fragile Euler topology can be stabilized into robust in-gap edge modes via spontaneous PT breaking, by creating an imaginary spectral gap between two bands. It develops an operator-based Chern-Euler duality where the two complex bands carry $C_\pm=\pm|\chi|$, and boundary spectral flow reproduces these bulk invariants, even in the presence of skin effects. The results prove that the net number of in-gap boundary modes, counted with their spectral-flow direction, equals $|\chi|$, and that this boundary signature is robust to boundary perturbations and non-Hermitian skin phenomena. Practically, loss and gain can thus drive fragile topological phenomena into stable topological behavior, enabling diagnostic and control schemes for fragile topology in photonic and acoustic platforms via boundary pumping and spectral-flow measurements.

Abstract

Two-dimensional topological insulators protected by nonlocal symmetries or with fragile topology usually do not admit robust in-gap edge modes due to the incompatibility between the symmetry and the boundary. Here, we show that in a parity-time (PT) symmetric system robust in-gap topological edge modes can be stably induced by non-Hermitian couplings that spontaneously break the PT symmetry of the eigenstates. The topological edge modes traverse the imaginary spectral gap between a pair of fragile topological bands, which is opened by the presence of the non-Hermitian perturbation. We demonstrate that the net number of resulting in-gap modes is protected by an operator version of anomaly cancellation that extends beyond the Hermitian limit. The results imply that loss and gain can in principle drive fragile topological phenomena to stable topological phenomena.

Figures (5)

• Figure 1: Top: In the spontaneous PT-breaking transition, a pair of real bands transitions into a pair of complex conjugate bands without hybridizing with any other bands. The topological invariant of the real band pair with PT symmetry is the Euler number $\chi$, featuring $|\chi|$ unremovable Dirac points between the pair PhysRevX.9.021013. The complex bands arising from the spontaneous breaking of PT symmetry have Chern numbers $C_{\pm} = \pm \chi$yang2025spontaneous. Bottom: With open boundary conditions, the complex bands are linked via in-gap boundary modes (magenta). In a cylinder geometry (periodic boundary conditions in $x$ direction, open boundary conditions in $y$ direction), the boundary modes exhibit spectral flow, such that the total number of boundary modes, weighed with their spectral flow direction, equals $|C_{\pm}| = |\chi|$.
• Figure 2: The in-gap edge modes (in red color) during the spontaneous symmetry-breaking transition, computed at $L=40$. (a) In the Euler-band model, the system exhibits trivial edge modes at each boundary. (b)-(c) Adding anti-Hermitian terms lifts the in-gap modes to the complex plane. (d) After the bulk modes are lifted away from the axis, in-gap modes connecting the two bulk spectra appear. The spectral flow of the top-boundary ($y=L$) in-gap modes indicated in magenta arrow. The bottom-boundary in-gap modes ($y=0$) are degenerate with the right-boundary modes and carry opposite spectral flow. (e) Top: the momentum-dependence of the in-gap modes at the top boundary, which gives the spectral flow in (d). Bottom: the biorthogonal localization $|\langle y |\psi_\textrm{ingap}(k_x)\rangle\langle\bar{\psi}_\textrm{ingap}(k_x)|y\rangle|=|\textrm{tr }P_\textrm{ingap}(k_x)P(y)|$ of the top-boundary in-gap modes (solid) as well as the localization of their PT partners---the bottom boundary modes (dashed).
• Figure 3: Spectra of the systems under boundary perturbations. (a) The in-gap modes for the Euler bands can be pushed into the bulk by the boundary perturbation. (b) After the symmetry breaking, the in-gap modes connecting the complex bands persist and the net flow remains unchanged.
• Figure S1: The spectrum of the model of Eqs. (\ref{['eq:Hlambda']})--(\ref{['eq:H1']}) with $\gamma=0$, which is separately $P$- and $T$-symmetric. The spectrum for open and periodic boundary conditions is shown in blue and grey, respectively. The open boundary bulk spectrum coincides with the periodic boundary bulk spectrum during the entire real-complex transition.
• Figure S2: A Hermitian inversion-symmetry breaking coupling (which still respects PT symmetry) can lift the degeneracy between the bottom boundary modes and the top boundary modes. Results are computed at different $\lambda$ for $\gamma=0,\gamma'=0.2$.