Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values

TL;DR

The paper addresses the challenge that eigenvalue-based bulk–boundary descriptions fail for two-dimensional non-Hermitian systems due to spectral instability. It develops a rigorous framework based on Toeplitz operator theory and singular values to establish bulk–boundary correspondences for half- and quarter-plane truncations, linking topological indices to the presence of finite-size singular values that decouple from the bulk. Distinct results are provided for scalar and matrix-valued symbols, including clear criteria for edge and corner modes, as well as higher-order topological phases, demonstrated through Hatano–Nelson-type models and a non-Hermitian BBH generalization. The work yields a practical, operator-theoretic paradigm for identifying and characterizing topological phases in non-Hermitian systems, independent of crystalline symmetries, with robust finite-size diagnostics via singular values.

Abstract

In contrast to eigenvalue-based approaches, we formulate the bulk-boundary correspondence for two-dimensional non-Hermitian quadratic lattice Hamiltonians in terms of Toeplitz operators and singular values, which correctly capture the stability, localization, and scaling of edge and corner modes. We show that singular values, rather than eigenvalues, provide the only stable foundation for topological protection in non-Hermitian systems because they remain robust under translational-symmetry-breaking perturbations that destabilize the eigenvalue spectrum, rendering it unsuitable for topological classification. Building on Toeplitz operator theory, we establish general results for non-Hermitian Hamiltonians defined on half and quarter planes, relating the topological indices of the associated Toeplitz operators to the number of finite-size singular values that are separated from the bulk singular-value spectrum and vanish in the thermodynamic limit. This yields a precise bulk-boundary correspondence for edge and corner modes, including higher-order topological phases, without requiring crystalline symmetries. We illustrate our general results with detailed examples exhibiting topologically protected families of edge states, coexisting edge and corner modes, and phases with both gapped bulk and edges supporting only stable corner modes. The latter is exemplified by a non-Hermitian generalization of the Benalcazar-Bernevig-Hughes model.

Figures (5)

• Figure 1: Hatano-Nelson model \ref{['HN']} with parameters $t_R=0.5$, $t_L=0.9$, $t_U=1.2$, $t_D=0.5$ corresponding to a point gap $\Delta_0\approx 0.26$ and windings $(I_x,I_y)=(0,-1)$: (a) $F(k_x,k_y)=E_x+E_y$. (b) First $100$ singular values for a $20\times 20$ lattice. The filled symbols represent the unperturbed system, the open symbols a perturbed system with $\varepsilon=0.1$, see Eq. \ref{['pert']}. The perturbation destroys degeneracies but the spectrum is stable. (c) Main: Scaling of five smallest singular values for a clean $N_x=N_y$ system. Upper inset: Same for a perturbed system with $\varepsilon=0.1$, averaged over $20$ realizations. Lower inset: Number of topologically protected singular values as a function of $N_x$ for $N_y=20$. (d) $|\Psi(x,y)|^2$ for the singular vector belonging to the smallest singular value of a perturbed $40\times 40$ system with $\varepsilon=0.1$ averaged over $100$ samples showing stable edge localization.
• Figure 2: (a) The eigenspectrum of the unperturbed $20\times 20$ Hatano-Nelson model \ref{['HN']} with parameters as in Fig. \ref{['Fig_HN']}, and (b) the spectrum for 5 realizations of the perturbed system with $\varepsilon=0.1$. (c) For $N_x,N_y\to\infty$ the eigenspectrum of the perturbed system will fill out the entire image of $F(k_x,k_y)$ for any arbitrarily small perturbation $\varepsilon$TrefethenEmbree2005.
• Figure 3: Edge state belonging to the smallest singular value of a $40\times 40$ extended Hatano-Nelson model \ref{['HN_ext']} with a perturbation $\varepsilon=0.1$ averaged over $100$ samples showing localization along two edges but no corner mode.
• Figure 4: (a) The five smallest singular values for the Hamiltonian \ref{['product']} as function of unit cells in the phase with $(I_L^x,I_L^y,I_R^x,I_R^y)=(0,0,-1,-1)$. The singular value belonging to the corner mode scales much faster to zero than those of the edge modes (almost on top of each other on this scale). (b) Spectrally protected corner mode of the Hamiltonian \ref{['product']} in the phase with $(I_L^x,I_L^y,I_R^x,I_R^y)=(0,0,-1,-1)$ obtained after averaging over $10$ samples with perturbation strength $\varepsilon=0.1$.
• Figure 5: Smallest singular values---averaged over $20$ samples with a disorder $\varepsilon=0.1$ that respects the sublattice symmetry---for the non-Hermitian BBH model \ref{['BBH3']} on a lattice with $30\times 30$ unit cells in the topological $(1,-1,1,-1)$ phase where both the bulk and the edges are gapped. The inset shows one of the four corner modes. The parameters are $\gamma_{x1}=0.2,\gamma_{x2}=0.1,\gamma_{y1}=0.5,\gamma_{y2}=0.05,\lambda_{x1}^R=\lambda_{x2}^R=\lambda_{y1}^R=\lambda_{y2}^R=2.7,\lambda_{x1}^L=2.5,\lambda_{x2}^L=2.6,\lambda_{y1}^L=2.4,\lambda_{y2}^L=2.8$ and all symmetries except for the sublattice symmetry $\Pi$ are broken.