Non-Hermitian Topological Domain Walls

• Non-Hermitian topological domain walls are interfaces between regions with distinct non-Hermitian phases, supporting robust, localized modes via generalized invariants.
• They employ modified non-Bloch band theory and transfer-matrix methods to restore bulk-boundary correspondence, even amid skin effects and exceptional points.
• Experimental realizations in photonic, quantum, and mechanical systems validate the control of interface modes, enabling applications like topological lasing and robust state localization.

A non-Hermitian topological domain wall is an interface or boundary between regions of distinct non-Hermitian topological phases in systems described by non-Hermitian Hamiltonians—those for which $H \neq H^\dagger$—that support robust, often localized, modes with spectral and spatial characteristics dictated by a generalized topological invariant. Non-Hermiticity (reflecting gain/loss, nonreciprocity, or dissipative effects) fundamentally modifies bulk-boundary correspondence, the protection and localization of edge states, and the analytic structure of associated invariants. Central features include the breakdown of conventional bulk-boundary correspondence, the emergence of the non-Hermitian skin effect, generalized (“non-Bloch”) topological invariants, real-space exceptional points, nontrivial spectral topology (including fractional winding numbers, point and line gaps), and, in certain regimes, restoration or modification of topological protection at domain walls compared to open boundaries.

1. Breakdown and Reconstruction of Bulk-Boundary Correspondence

Non-Hermitian systems frequently exhibit a breakdown of the conventional (Hermitian) bulk-boundary correspondence (BBC), whereby bulk invariants such as integer winding numbers or Chern numbers cease to predict localized edge (or domain wall) states under realistic boundary conditions. This breakdown is rooted in the fact that imposing open boundary conditions (OBC), in contrast to periodic ones (PBC), can drive the system through exceptional points (EPs)—degeneracies at which both eigenvalues and eigenvectors coalesce. As a result, the structure of the energy spectrum and the topology of the bulk bands are heavily altered (Xiong, 2017, Kunst et al., 2018).

For example, in non-Hermitian systems with “fractional winding numbers”—a hallmark of models exhibiting square-root branch point structures in their Bloch Hamiltonians—the process of reducing the boundary coupling from PBC to OBC necessitates crossing EPs, changing the topological structure of the spectrum and causing all states to localize at one edge. The zero-energy boundary states (ZEBS) appearing under these conditions are not protected by a bulk topological invariant but result from the defective nature of the Hamiltonian at the EP.

However, for domain walls—interfaces within the bulk where parameters are smoothly or abruptly changed—such passage through EPs is generally not required. Here, the topological indices remain robust: the generalized index theorem and bulk-boundary correspondence can be restored, with the number and nature of interface states (domain-wall modes) still tied to differences in bulk invariants (Xiong, 2017, Deng et al., 2019, Kunst et al., 2018).

2. Generalized Topological Invariants and Non-Bloch Band Theory

To capture the topological nature of domain walls in non-Hermitian systems, invariants must be constructed beyond the Hermitian Bloch theory. The non-Hermitian skin effect drives bulk eigenstates to accumulate (skin) at boundaries; thus, the relevant “Brillouin zone” for topological classification is no longer the unit circle for $e^{ik}\in U(1)$ but a generalized Brillouin zone (GBZ) defined by decay parameters $\lambda$ ($|\lambda|\neq 1$) determined by real-space recurrence relations (Deng et al., 2019, Kunst et al., 2018).

In the non-Hermitian Su–Schrieffer–Heeger (SSH) model, for example, bulk wavefunctions take the form $\psi_j\sim\lambda^j$, with $\lambda$ constrained by a characteristic equation that depends on boundary conditions. The corresponding non-Bloch winding number is defined as

$$\tilde{\nu}_\alpha = \frac{1}{2\pi} \int dp \frac{-\tilde{h}_x \partial_p \tilde{h}_y + \tilde{h}_y \partial_p \tilde{h}_x}{\tilde{h}_x^2 + \tilde{h}_y^2}$$

where $(\tilde{h}_x, \tilde{h}_y)$ are the Hamiltonian components written in terms of $\lambda$ rather than $e^{ik}$.

Under domain wall configurations—joining two bulks L and R with different parameters—the non-Bloch winding numbers $\tilde{\nu}_L$ and $\tilde{\nu}_R$ generally depend nonlocally on both sides of the interface. The bulk-boundary correspondence is recovered in the form that the number of protected zero-energy domain-wall modes is determined by the difference $\tilde{\nu}_R - \tilde{\nu}_L$ (Deng et al., 2019).

Analytically, the transfer matrix formalism offers a powerful vantage, where localization properties, real-space EPs, and robust domain-wall modes are encoded in the eigenvalues ($\rho_\pm$) and determinant ($\Gamma$) of a 2x2 transfer matrix $T$. The condition $|\rho_+|=|\rho_-|=\sqrt{\Gamma}$ for OBC generalizes the Bloch theory and ensures the correct topological counting (Kunst et al., 2018): $k_x \rightarrow \phi - \frac{i}{2} \log(\Gamma)$ for computing modified topological invariants (e.g., Chern number) on the complexified Brillouin zone.

3. Role of Exceptional Points, Skin Effect, and Real-Space Anomalies

Non-Hermitian systems can exhibit non-Hermitian skin effects, where an extensive number of eigenstates are exponentially localized at a particular boundary or domain wall (Kunst et al., 2018, Deng et al., 2019). This is closely connected with the breakdown of the conventional BBC and the emergence of real-space EPs—parametric values where the entire spectrum collapses to a single boundary.

For domain walls, the transfer matrix determinant $\Gamma$ serves as a “tuning knob”:

• $\Gamma < 1$: bulk and domain-wall states localize at the left boundary.
• $\Gamma > 1$: localization at the right boundary.
• $\Gamma \to 0$ or $\Gamma \to \infty$: high-order EP, complete coalescence of all states.

At domain walls (or “internal boundaries”), however, the bulk spectrum and associated topological invariants remain well-defined, so long as the local transfer matrix parameters avoid critical EPs. This “protection” is lost at free edges, where the system is forced to be at (or very near) an EP (Xiong, 2017).

The field theory of non-Hermitian topological systems establishes the skin effect as an anomalous response: a macroscopic number of boundary modes emerge as a spatially extensive anomaly, encoded via the effective action

$S[A] = W_1 \int A(x)\,dx$

and a nonvanishing winding current—distinct from Hermitian TQFT, which features isolated edge anomalies (Kawabata et al., 2020).

4. Interface Physics: Spontaneous Symmetry Breaking, Complex Spectra, and Point Gaps

Interfacing two non-Hermitian bulks with distinct topological invariants can drive spontaneous breaking of underlying symmetries (e.g., $\mathcal{PT}$) at the domain wall (Yuce, 2018). For instance, in $\mathcal{PT}$-symmetric SSH chains with gain/loss parameter $\gamma$, edge or interface states transition from real-valued energies (unbroken phase) to complex energies (broken phase) as parameters cross exceptional points. This is generically accompanied by amplification or decay localized at the domain wall, providing new mechanisms for interface-selective mode control (such as edge-mode topological lasing).

In non-Hermitian systems, topology is frequently characterized not by energy line gaps but by point gaps in the complex spectrum. The nontrivial winding of the energy bands around a reference point in the complex plane underlies the existence of robust interface states. The associated point-gap winding number,

$$w = \int_{-\pi}^{\pi} \frac{dk}{2\pi i} \partial_k \ln (E_k - E_B)$$

predicts the existence and chiral nature of domain-wall modes in non-Hermitian settings (and, via doubling constructions, is in one-to-one correspondence with anomalous Hermitian boundary invariants) (Lee et al., 2019, Hamanaka et al., 16 May 2024).

5. Smooth and Sharp Domain Walls: Analytical Solutions and Universal Relations

Exact analytical approaches—such as the solution of generalized non-Hermitian Jackiw–Rebbi equations—demonstrate how the spatial profile and decay of domain-wall zero modes in line-gapped non-Hermitian systems are determined by the complex asymptotic values of mass $m(x)$ and velocity $v(x)$ fields (Marra et al., 9 Apr 2025): $$\varphi''(x) + 2 s v(x)\varphi'(x) - m(x) \varphi(x) = 0$$ yielding universal relations between decay rate $\mu$ and bulk parameters: $$\mu_{L,R} + i \kappa_{L,R} = \pm v_{L,R} \pm \sqrt{v_{L,R}^2 + m_{L,R}}$$ The number and localization of domain-wall zero modes depends solely on the difference in generalized topological invariants of the left and right bulks, irrespective of the microscopic details of the smooth interface. This framework extends the bulk-boundary correspondence to non-Hermitian line-gapped systems and unifies point-gap and line-gap physics for experimental observables.

6. Symmetry, Higher-Order Topology, and Robustness

Symmetries, particularly chiral and particle–hole, further enrich non-Hermitian domain wall physics. In systems with conserved symmetries, the non-Hermitian skin effect and the associated generalized bulk-boundary correspondence are subject to symmetry constraints, such as the $Z_2$ skin effect in particle–hole symmetric second-order topological superconductors (Ji et al., 2023). Here, both bulk and edge modes accumulate at system corners in a symmetry-enforced manner, with nontrivial domain-wall (Majorana) corner modes exhibiting degeneracies and localization determined by non-Bloch band theory. At the same time, chiral symmetry breaking in non-Hermitian quantum walks requires more general, non-Bloch topological invariants (pairs), yet still enables experimentally robust interface-localized modes (Zhang et al., 7 Apr 2025).

Tables below summarize key features for domain wall states compared to open boundaries:

Boundary Type	Bulk-Boundary Correspondence	Invariant Type
Open boundary	Typically fails (EPs traversed)	Point-gap/skin effect; no direct link to bulk invariant
Smooth domain wall	Holds (no EP crossing)	Non-Bloch/transfer-matrix/topological index
Disordered wall	Modified; disorder-averaged invariant	Real-space winding or Chern number

7. Experimental Realizations and Applications

Experimental platforms for observing non-Hermitian topological domain-wall phenomena include photonic lattices (topological microlasers (Li et al., 2023)), frequency-modulated lasers in non-Hermitian quasicrystals (Longhi, 2019), and quantum walk interferometry with single photons (Zhang et al., 7 Apr 2025). In topologically nontrivial photonic devices, domain walls between regions of different effective winding number serve as robust locations for state localization and signal amplification, provided the system avoids exceptional-point-induced breakdowns. The transfer-matrix and similarity transformation approaches enable analytic prediction of phase boundaries and topological invariants, which aids in the design of reconfigurable or disorder-robust devices.

In non-Hermitian higher-order topological insulators, domain walls manifest as interface corners that support robust zero-dimensional (e.g., Majorana) modes, even in the presence of particle–hole symmetric skin effects (Ji et al., 2023). In fluid/mechanical systems, non-Hermitian wall modes (such as those in rotating Rayleigh–Bénard convection) are characterized by quantized Chern numbers defined over complexified wavevector space (Zhang et al., 2023), offering new routes for controlling chiral transport in open, dissipative environments.

References to Core Results and Methodologies

• Fractional winding number and domain wall protection: (Xiong, 2017)
• Transfer-matrix and biorthogonal polarization: (Kunst et al., 2018)
• Non-Bloch invariants and skin effect at domain walls: (Deng et al., 2019)
• Point-gap topology and boundary physics: (Lee et al., 2019, Hamanaka et al., 16 May 2024)
• Non-Hermitian Jackiw–Rebbi equation—smooth domain wall analytics: (Marra et al., 9 Apr 2025)
• Experimental observation in single-photon quantum walks: (Zhang et al., 7 Apr 2025)
• Higher-order topology with domain-wall accumulation: (Ji et al., 2023)
• Topological microlasers and photonic platforms: (Li et al., 2023)
• Non-Hermitian topological wall modes in fluids: (Zhang et al., 2023)
• Generalized field theory for non-Hermitian topological phenomena: (Kawabata et al., 2020)

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In summary, the non-Hermitian topological domain wall is a locus where the interplay between complex spectral topology, non-Bloch band theory, and generalized bulk-boundary correspondences give rise to robust (potentially skin-affected) localized modes that cannot be inferred from Hermitian paradigms. The analytic and experimental understanding of such domain walls provides a basis for engineering highly controllable, symmetry-tunable boundary phenomena in open quantum, photonic, mechanical, and classical systems.