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arXiv:2606.06626

Non-Hermitian Crystalline Braid Topology from Hermitian Projection: A Zero-Mode Resonance Mechanism

Stefan Đorđević, Vladimir Juričić

cond-mat.mes-hallquant-phSubmitted Jun 4, 2026• Announced Jun 8, 2026
v1 · Jun 4, 2026v2 · Jun 15, 2026
arXivHTMLPDF

Abstract

Non-Hermitian topological phases are usually engineered through gain, loss, asymmetric couplings, or explicit environmental channels. Here we show that non-Hermitian crystalline braid topology can instead emerge from projection alone, starting from a fully Hermitian and topologically trivial parent lattice. The mechanism is zero-mode-resonant projection. When the eliminated complement is zero-mode free, projection has a smooth low-frequency limit and reduces to a static Schur complement, yielding conventional SSH-type descendants. When a complement zero mode couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency limit becomes singular, and topology is carried by the finite-frequency projected Green's function-where frequency is a tunable parameter, the drive frequency in a circuit realization, for instance. We demonstrate this mechanism in an exactly solvable model, a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. Odd-parity periodic sectors are resonant: a sublattice-imbalance zero mode generates the singular self-energy, and the complex spectrum forms an abelian two-band braid whose transitions occur only at isolated finite frequencies. Although the internal class is $\text{AI}^†$ featuring only trivial phases, embedding parity induces conjugated pseudo-Hermiticity (CPH), quantizes the complex Berry phase, and identifies it with the braid count. The model is free of the non-Hermitian skin effect, making the invariant a genuine Bloch bulk quantity. In topolectrical realizations, the same finite-frequency braid transitions appear as transmission zeros and admittance features at the predicted drive frequencies.

Comments
49 pages main text, 17 pages SM, 31 figures

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Non-Hermitian topological phases are usually engineered through gain, loss, asymmetric couplings, or explicit environmental channels. Here we show that non-Hermitian crystalline braid topology can instead emerge from projection alone, starting from a fully Hermitian and topologically trivial parent lattice. The mechanism is zero-mode-resonant projection. When the eliminated complement is zero-mode free, projection has a smooth low-frequency limit and reduces to a static Schur complement, yielding conventional SSH-type descendants. When a complement zero mode couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency limit becomes singular, and topology is carried by the finite-frequency projected Green's function-where frequency is a tunable parameter, the drive frequency in a circuit realization, for instance. We demonstrate this mechanism in an exactly solvable model, a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. Odd-parity periodic sectors are resonant: a sublattice-imbalance zero mode generates the singular self-energy, and the complex spectrum forms an abelian two-band braid whose transitions occur only at isolated finite frequencies. Although the internal class is AI†\text{AI}^†AI† featuring only trivial phases, embedding parity induces conjugated pseudo-Hermiticity (CPH), quantizes the complex Berry phase, and identifies it with the braid count. The model is free of the non-Hermitian skin effect, making the invariant a genuine Bloch bulk quantity. In topolectrical realizations, the same finite-frequency braid transitions appear as transmission zeros and admittance features at the predicted drive frequencies.
The synergy between non-Hermiticity and topology makes the bulk-boundary correspondence (BBC) highly elusive. Here we study a non-Hermitian Creutz ladder incorporating both gain-loss and nonreciprocity, and construct multiple BBCs involving scale-free, normal, and anomalous skin modes, as well as topological zero- energy modes. We characterize the skin effect induced by the parity-time (PT) phase transition through an average winding number, thereby formulating a PT-related BBC. Furthermore, a hidden chiral symmetry ensures that topological phase transitions can be reliably detected via a Z 2 invariant. Although the gain-loss breaks the standalone P symmetry, it preserves the combined PT symmetry, allowing a modified Z 2 invariant to sustain the topological BBC. Notably, sublattice symmetry facilitates the precise analytical determination of non-Bloch spectra. Leveraging this, we introduce a hybrid spectral winding that encodes the localization (or delocalization) characteristics of two counterintuitive bulk modes coexisting with normal skin modes, thus defining a non-Hermitian BBC. One of these modes exhibits exponential boundary accumulation in a direction opposite to the nonreciprocity, while the other manifests as an anomalous surge of Bloch-wave states within the nonreciprocal lattice. These results reveal a series of unexpected phenomena governed by underlying symmetry, significantly broadening our fundamental understanding of the BBC mechanism in non-Hermitian topological systems.
Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.
Topological transitions in non-Hermitian systems are generally boundary sensitive: a point-gap winding transition under periodic boundary condition (PBC) and a non-Bloch bulk real-line-gap transition under open boundary condition (OBC) at Re(E)=0\mathrm{Re}(E)=0Re(E)=0 are governed by different spectra and therefore need not coincide. Here we show, for a class of chiral non-Hermitian Su--Schrieffer--Heeger (SSH)-type lattices, that these two criticalities can be locked by an exceptional-point-constrained (EP-constrained) parameter evolution. The key requirement is not the occurrence of isolated exceptional points, but the persistence of a zero-energy Bloch degeneracy along the entire sweep, which is generically exceptional in the non-Hermitian regime. In an analytically tractable limit of an extended non-Hermitian SSH chain, the EP-constrained manifolds and both transition boundaries are obtained in closed form, making the locking explicit. Away from this limit, numerical generalized-Brillouin-zone (GBZ) calculations confirm the correspondence for representative constrained sweeps, whereas unconstrained paths show that isolated exceptional points or Hermitian degeneracies do not enforce locking. We further verify the mechanism in a spinful four-band extension with branch-resolved GBZs, including strongly branch-imbalanced regimes. These results establish a path-dependent diagnostic principle: along EP-constrained sweeps in this SSH-type class, changes in PBC point-gap winding can indicate OBC non-Bloch bulk real-line-gap transitions and the corresponding changes in zero-energy boundary modes.
Non-Hermitian physics endows the non-Abelian systems with exceptional topology characterized by non-commutative braid patterns. Interplay of distinct competing sources of non-Hermiticity may induce novel topological effects. Here, we provide a generalized Hatano-Nelson model with higher-order nonreciprocal hoppings, non-Abelian gauge fields, and staggered gain-loss processes showing the exceptional topological structure of the Hopf-link braids that undergoes a EP-mediated topological phase transition. We demonstrate that mixing multiple nonreciprocal channels drives the system into highly intricate, nested complex energy Hopf-link braids and expands the topological landscape up to higher-order braiding sectors. Furthermore, utilizing biorthogonal eigenvector tracking, we map the structural evolution of the exceptional phase boundaries via the maximum Petermann factor in the phase angle parameter planes. We show that the gain-loss non-Hermiticity drives a topological crossover where the extended exceptional contours constrict into isolated regimes. The demonstration of EP-mediated topological phase transition of Hopf-link braids and the associated rich exceptional phase portraits may offer new physical insights with promising applications in robust, fault-tolerant communication channels and quantum computing platforms.
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are populated by compact localized states (CLS) - a special class of localization with wavefunctions confined within a small region. In this work, we discover that the non-Hermitian skin effect (NHSE) can appear in a flat band. Unlike conventional NHSEs for dispersive bands that are protected by nontrivial point-gap topology, the flat band remains a point on the complex-energy plane and is therefore always topologically trivial. We found that, intriguingly, the flat-band skin effect (FBSE) is associated with the non-trivial spectral topology of the dispersive bands enclosing the flat band on the complex-energy plane, so it only emerges within a finite range of non-Hermitian parameters and can counterintuitively disappear at large non-Hermiticity. Moreover, the gaps between the flat and the dispersive bands can close at higher-order exceptional points under both periodic and open boundary conditions. The flat-band wavefunctions are discontinuous in quantum distance across these exceptional points, signifying that the gap-closing is singular. The FBSE was experimentally observed in a non-Hermitian mechanical lattice. Our work reveals flat-band phenomena unique to non-Hermitian systems and highlights new possibilities in quantum geometry and localization control.
Chirality is more than a geometric curiosity; it governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals. We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present. Built from the non-commutative fundamental group and its braid representation, topological chirality acts as a singularity selector: clockwise EP loops occupy a homotopy class that avoids EPs, whereas counter-clockwise mirrors are equivalent only if they cross the EPs themselves. We confirm this binary rule in an optical microcavity and a non-Hermitian topological band. The same two-sheeted topology governs EP pairs in spin systems, photonic crystals and hybrid light-matter structures, where EP encirclements have already been demonstrated, so the framework transfers without alteration and confirms its experimental viability. Our findings lay the cornerstone for interpreting loop-sensitive observables such as spectral vorticity, the complex Berry phase and the non-Abelian holonomy. Finally, a gluing-of-planes construction extends the invariant to an n-sheeted surface hosting 2m EPs, unifying higher-order EP pairs.