A theoretical preprint reports distinct topological phases in three modeled non-Hermitian lattice systems, with integer labels from a generalized Euler class and matching counts from Wilson-loop winding. In plain language, the calculations use two mathematical checks that track the same modeled structure in the non-Hermitian topological phases. The result comes from constructed models, not an experiment.
The work asks whether two-dimensional, three-band non-Hermitian lattices with Hamiltonians unchanged by transposition can support nontrivial Euler topology. It examines the models' Euler bands, entanglement spectra and bulk-boundary correspondence, which compares properties calculated in the bulk with signatures appearing at an edge.
The supplied front matter identifies the work as arXiv:2608.25411v1, dated 26 Aug 2026. The study uses analytical derivations and numerical calculations across three representative models, with no empirical participants or biological samples involved.
Why the spectral gap matters
The analysis first separates gapped phases from critical regions by measuring the smallest distance between the third energy band, E3(k), and zero. That definition is used because the first two bands, E1 and E2, are pinned at zero energy. The resulting gap measure is part of the calculation of phase boundaries and spectral gaps.
At the modeled non-Hermitian critical phases, however, neither the Euler class nor the Wilson-loop winding is a reliable topological index. That finding limits the calculation's reach: the reported integer labels concern the modeled gapped phases, not the gapless regions.
The first model has three gapped phases
Model I has three topologically distinct gapped phases. Two are nontrivial, with e12 = ±2, while the third is trivial, with e12 = 0. Distinct gapped phases in this model are separated by gapless phases, so the phase changes pass through regions where the topological indices are not reliable.
The nontrivial phases also show a specific boundary pattern. Their energy edge bands have quadratic touchings, meaning the bands meet with a curved, quadratic dependence near the contact rather than a simple linear crossing. Matching quadratic touchings appear in the entanglement spectrum, another spectrum calculated in the study alongside the edge bands. The study treats these as corresponding edge and entanglement signatures of the modeled bulk topology.
A larger Euler label appears in Model II
Model II extends the range of labels in the construction. Its three gapped phases have e12 = 4, e12 = 2 and e12 = 0. In the phase with e12 = 4, the entanglement-spectrum edge bands contain two quadratic touchings, located at kx = 0 and kx = π. The calculation therefore includes two separate edge contacts within that entanglement spectrum.
Model III produces an overlap instead of a single contact
Model III has three gapped phases, including nontrivial phases with e12 = ±2 and a trivial phase with e12 = 0. In its nontrivial phases, the edge-band branches overlap across a range of momentum. In the Hermitian limit, that overlap shrinks to a single quadratic touching at kx = π.
The imaginary part of the Euler class is reported as zero in all the gapped regions considered. This is a statement about the regions analyzed in the model, alongside the separate result that the Euler class and Wilson-loop winding are not reliable in the modeled critical phases.
A result tied to a specific construction
The paper also gives the relation e12 = 2c−. In the paper's notation, the equation says that the generalized Euler label is twice the quantity written as c−. Together with the integer-quantized Wilson-loop winding, which matches the Euler class in magnitude, the relation provides the study's central internal comparison between its topological measures.
The evidence is limited to analytical and numerical work on three constructed two-dimensional, three-band non-Hermitian lattice models with transposition-invariant Hamiltonians. The study does not experimentally validate the proposed phases or signatures, and it does not establish nontrivial Euler topology in fully gapless non-Hermitian systems. Its bulk-boundary interpretation is likewise tied to the stated model construction and symmetry structure.
Paper data and sources
Original title: Non-Hermitian topological Euler insulators
Authors: Longwen Zhou
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text