A theoretical model built around a six-atom repeating cell has produced a surprisingly intricate map of topological phases. In the calculations, the Chern-number range widened as the band index rose from 1 to 3: band 1 reached ±2, while band 3 reached ±7. A zoomed region of the band-3 map showed −7.
The Chern number is the integer label used here to distinguish the model’s band topologies. The preprint asks how those labels and the boundaries between them change as the phases θ and ϕ of complex nearest- and next-nearest-neighbor hopping terms are varied, with the hopping-magnitude ratio held fixed.
A compact model with a crowded phase map
The repeating primitive cell contains three hexagons and six atoms. The total flux through the cell is zero, while two constituent hexagons carry opposite fluxes.
The main detailed case fixes t2/t1 at 4/10. That ratio was described as having many crossings, making it useful for examining how topological phases change. At each k-point, the researchers numerically diagonalized the Hamiltonian, evaluated Berry curvature, and integrated it across the Brillouin zone to obtain Chern numbers.
The main displayed phase diagrams used a 161 × 321 grid for the two phase directions. The calculation treated a band as well separated when its energy gap was at least 10−4 t1. Fewer than 2% of the phase-diagram points had smaller gaps, and most of those points lay on crossing lines.
The crossings behind the changes
Analytic crossing lines at γ, k− and k+ explained a substantial fraction of the topological phase transitions. They provided an analytic framework for reading part of the numerical phase map.
The crossing pattern also tracked the size of a topological change. Isolated crossings were almost always associated with a one-unit change in Chern number. Away from the high-symmetry locations, changes were often multiples of three and appeared as groups of three or six isolated crossings.
One representative merger produced an apparent charge-two crossing: four crossings together had a net charge of two before the process ended in a quadratic crossing at k−. The study describes quadratic crossings as extremely rare. In the specified merger scenario, a high-symmetry linear crossing joined an opposite-sign triple crossing and was associated with a two-unit Chern-number change.
Small changes in the ratio, large changes in the map
The reported comparison showed a marked difference between the two hopping-amplitude ratios: the phase diagrams at t2/t1 = 3/10 were much simpler than those at 4/10. The detailed analysis centered on 4/10 because of its many crossings, rather than presenting a full topological map for every possible ratio.
Symmetry relations reduced the amount of calculation needed. The stated transformations allow all-band phase diagrams to be generated from the first three bands over θ ∈ (0, π] and ϕ ∈ (0, π]. Under the transformation (θ, ϕ) → (−θ, −ϕ), the Chern-number sign changes.
A detailed case, not a complete map
The central phase diagram is therefore a detailed 4/10 case, supported by a comparison with 3/10. Ratios higher than 4/10 were not taken through full topological phase diagrams, leaving the behavior of the broader parameter space unresolved.
The analytic high-symmetry lines explained a substantial fraction of the transitions but did not account for every crossing or boundary. The numerical treatment also depended on a finite gap threshold, with points below 10−4 t1 handled separately; most of those points were on crossing lines.
Very small regions with high Chern numbers depended on increased resolution, and identifying the extrema was not the main purpose of the calculation. The reported ±7 should therefore be read as a feature of the displayed model calculations, not as a general limit on Chern numbers.
The document is identified as arXiv:2608.25875v1 and dated 26 August 2026. Its acknowledgments report computational support from the University of Manchester, Ministry of Education of Oman support for Ahmed Al-kharusi, and UK Science and Technology Funding Council support for Niels R. Walet under grant ST/Y000323/1.
Paper data and sources
Original title: Topological phases of a generalised tripartite Haldane model
Authors: Ahmed Al-kharusi, Alessandro Principi, Niels R. Walet
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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