Point-Gap Topology in Non-Hermitian Physics

• Point-gap topology is a spectral phase in non-Hermitian systems defined by the absence of specific eigenvalues and characterized by integer or Z2 winding invariants.
• Its nontrivial winding leads to unique phenomena such as the non-Hermitian skin effect, exceptional boundary modes, and a generalized bulk–boundary correspondence.
• Practical realizations in photonic crystals, cold atoms, and quantum-optical setups demonstrate robust topological features that persist even in the presence of disorder and interactions.

A point-gap is a spectral property unique to non-Hermitian systems, wherein the spectrum of a complex-energy Hamiltonian or time-evolution operator avoids a reference point in the complex plane, but may wind nontrivially around it. Unlike the line-gap familiar from Hermitian band theory—which separates the spectrum by a line in the plane—point-gap topology is inaccessible in Hermitian systems and leads to phenomena such as the non-Hermitian skin effect, exceptional boundary and hinge modes, and the breakdown or novel generalization of bulk–boundary correspondence. Point-gap phases are classified by integer or $\mathbb{Z}_2$ topological invariants associated with the winding of the spectrum about the reference point, and are deeply intertwined with non-Hermitian symmetry classes, boundary conditions, real-space disorder, and many-body interactions.

1. Definition of Point-Gap and Topological Invariants

In a non-Hermitian system, the spectrum $\mathrm{Sp}(H)$ of an operator $H$ (Hamiltonian or Floquet operator) is generally a subset of $\mathbb{C}$. A point gap at a reference $E_0 \in \mathbb{C}$ satisfies

$\det[H(k)-E_0] \neq 0 \ \forall\,k.$

This condition implies the reference point is not an eigenvalue for any $k$, and $(H-E_0 I)$ is invertible everywhere. The topological invariant for point-gap topology is normally a winding number—quantifying how many times the PBC spectrum winds around $E_0$: $$W(E_0) = \frac{1}{2\pi i} \int_0^{2\pi} dk\, \partial_k \ln \det[H(k) - E_0] \in \mathbb{Z}.$$ Generalizations in higher dimensions use Chern numbers of associated “flattened” Hermitian matrices or more elaborate invariants, often depending on symmetry class (Pan et al., 2020, Nakamura et al., 2023, Nakamura et al., 2022, Denner et al., 2023).

2. Physical Consequences: Skin Effect and Bulk–Boundary Correspondence

A hallmark of non-Hermitian point-gap topology is the non-Hermitian skin effect (NHSE): under open boundary conditions (OBC) or semi-infinite boundary conditions (sIBC), extensive numbers of bulk modes localize at the system edge, rather than remaining extended. For models where the PBC spectrum forms a loop winding $\mathrm{Sp}(H)$0, OBC spectra collapse onto lines or arcs within the loop, with the physical edge-localized states precisely corresponding to the winding number $\mathrm{Sp}(H)$1 (Pan et al., 2020).

The bulk–boundary correspondence in point-gap topology can be “complete” in systems such as the dissipative quantum harmonic oscillator Liouvillian on Fock space—where the natural sIBC enables the entire interior of the PBC spectral loop to support skin modes, in one-to-one correspondence with the bulk invariant (Pan et al., 2020). In lattice models, the point-gap skin effect is quantified by the appearance and number of exponentially localized modes, classified by the integer winding (Nakamura et al., 2023, Schindler et al., 2023).

In higher-order topological systems, point-gap topology also predicts mid-gap corner or hinge modes, with the correspondence extending to real-space winding numbers derived from singular-value decomposition of $\mathrm{Sp}(H)$2 or Floquet operators (Yang et al., 4 Jan 2026).

3. Boundary Sensitivity, Symmetry, and Classification

Point-gap topology is distinguished from Hermitian insulator topology by boundary sensitivity—the dependence of spectral topology on the choice of boundary conditions. For example, the PBC and OBC spectra of the non-Hermitian SSH chain differ drastically in the topological regime; only under sIBC (as in Fock-space realizations) do the full contents of the point-gap invariant manifest at the boundary (Pan et al., 2020).

Classification of point-gap phases relies on internal symmetry (Altland–Zirnbauer and their “daggered” extensions in non-Hermitian physics), boundary geometry, and potentially spatial symmetries like pseudo-inversion which can protect nontrivial topological invariants at the boundary even when the Hermitian bulk is trivial (Schindler et al., 2023). The full periodic table of non-Hermitian topological phases details which symmetry classes in $\mathrm{Sp}(H)$3 or $\mathrm{Sp}(H)$4 dimensions admit nontrivial point-gap invariants, and whether these encode skin effects, boundary states, or higher-order phenomena (Nakamura et al., 2023, Nakamura et al., 2022, Denner et al., 2023).

Point-gap invariants also possess dual meaning in the context of Floquet operators—allowing classification of non-unitary quantum walks or driven evolutions under both Hamiltonian- and evolution-operator-type symmetry constraints (Jiang et al., 2024).

4. Beyond One Dimension: Higher-Order and 2D/3D Bulk–Boundary Physics

In one dimension, a nonzero point-gap winding implies the NHSE, as all bulk states become boundary-localized. In two dimensions, point-gap topology yields richer boundary phenomena. Edge physics can realize two distinct families: “infernal points,” where skin effects occur only at isolated edge momenta, and “exceptional-point dispersions,” with finite sets of edge states and nontrivial spectral flow along high-symmetry directions (Denner et al., 2023). The general bulk–boundary correspondence is established via the extended Hermitian doubling construction, mapping boundary states in the non-Hermitian model to zero modes in an associated Hermitian system of one higher dimension (Nakamura et al., 2022).

In three dimensions, non-Hermitian boundary or hinge modes arise with 2D point-gap invariants (Chern or $\mathrm{Sp}(H)$5), producing topological chiral or helical hinge modes unique to the non-Hermitian context (Nakamura et al., 2023, Schindler et al., 2023).

5. Experimental and Numerical Realizations

Point-gap topology extends to photonic, cold atom, and quantum–optical platforms. In photonic crystals, complex-frequency band structures of lossy or non-reciprocal dielectric arrays manifest topologically nontrivial winding and the skin effect, observable as wholesale accumulation of electromagnetic eigenmodes at sample boundaries when truncated (Zhong et al., 2021). Dissipative boundary engineering on a Hermitian bulk enables robust realization of point-gap topological phases in quantum materials, as proposed for topological insulators and superconductors where only the boundary is rendered non-Hermitian (Nakamura et al., 2023, Schindler et al., 2023). Non-Hermitian skin and point-gap effects have been numerically and experimentally probed in various 1D and 2D lattice models, non-unitary quantum walks and quadratic-bosonic squeezed systems (Wan et al., 2023, Yang et al., 4 Jan 2026, Jiang et al., 2024).

Notably, point-gap invariants retain robustness against significant bulk disorder—winding numbers measured in real space remain quantized up to substantial disorder, and the associated anomalous chiral currents or localized states are robust until a disorder-driven transition point is crossed (Sarkar et al., 2022). In many-body contexts, point-gap topology persists in collective doublon–holon excitations even where Pauli exclusion suppresses the skin effect in the ground state (Kim et al., 2023).

6. Interactions, Correlated Systems, and Classification Reductions

Correlation effects can nontrivially reduce or collapse the free-fermion point-gap classification. For example, in zero dimension, interactions reduce the $\mathrm{Sp}(H)$6 classification of chiral-symmetric point-gap topology to $\mathrm{Sp}(H)$7, enabling adiabatic connection of free-fermion topological phases that appeared distinct in the non-interacting limit (Yoshida et al., 2021). In 1D, U(1) and spin-parity symmetric systems reduce from $\mathrm{Sp}(H)$8 to $\mathrm{Sp}(H)$9 classification under two-body interactions, concomitantly destroying the skin effect (Yoshida et al., 2022). However, in specific excitonic or collective-quasi-particle sectors, correlated systems can sustain nontrivial point-gap topology and bulk–boundary correspondence (Kim et al., 2023). The role of symmetry and system dimension is thus crucial in determining the stability of point-gap phases under interactions.

7. Bath Duality and Emergent Multi-Sheet Topology

In open quantum systems and quantum optics, point-gap spectral topology has further consequences, such as the emergence of a “mirage bath” on a second Riemann sheet. This virtual bath mediates long-range interactions between emitters—even when conventional bulk modes are forbidden by the point gap. Interactions inherit the topology of the mirage bath (typically line-gap) rather than the physical system (point-gap), demonstrating a fundamental duality and the need for multi-sheet analytic continuation to fully understand dynamical and topological properties in the presence of dissipation (Sun et al., 13 Feb 2025).

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References:

• (Pan et al., 2020) Pan, Li, Gong, "Point-gap topology with complete bulk-boundary correspondence in dissipative quantum systems"
• (Nakamura et al., 2023) Nakamura et al., "Universal platform of point-gap topological phases from topological materials"
• (Yang et al., 4 Jan 2026) "Non-Hermitian second-order topological insulator with point gap"
• (Denner et al., 2023) "Infernal and Exceptional Edge Modes: Non-Hermitian Topology Beyond the Skin Effect"
• (Jiang et al., 2024) "Dual Symmetry Classification of Non-Hermitian Systems and $H$0 Point-Gap Topology of a Non-Unitary Quantum Walk"
• (Zhong et al., 2021) "Non-trivial point-gap topology and non-Hermitian skin effect in photonic crystals"
• (Sun et al., 13 Feb 2025) "Anomalous Point-Gap Interactions Unveil the Mirage Bath"
• (Yoshida et al., 2022) "Reduction of one-dimensional non-Hermitian point-gap topology by correlations"
• (Yoshida et al., 2021) "Correlation effects on non-Hermitian point-gap topology in zero dimension: reduction of topological classification"
• (Nakamura et al., 2022) "Bulk-boundary correspondence in point-gap topological phases"
• (Wan et al., 2023) "Quantum-Squeezing-Induced Point-Gap Topology and Skin Effect"
• (Sarkar et al., 2022) "Interplay of Disorder and Point-Gap Topology: Chiral Modes, Localization and Non-Hermitian Anderson Skin Effect in One Dimension"
• (Kim et al., 2023) "Collective non-Hermitian skin effect: Point-gap topology and the doublon-holon excitations in non-reciprocal many-body systems"
• (Wang et al., 26 Mar 2026) "Exceptional-point-constrained locking of boundary-sensitive topological transitions in non-Hermitian lattices"
• (Schindler et al., 2023) "Hermitian Bulk -- Non-Hermitian Boundary Correspondence"