A theoretical study predicts two possible signatures of a linking change in a Hopf-link semimetal: a ring-shaped pattern in a quantum-metric quantity and sharp peaks in optical conductivity. The calculations compare an unlinked phase, assigned m = 0, with a linked Hopf-link phase at m > 0. The model also has a calculated nonlinear Hall response in the presence of a specified symmetry-breaking perturbation.
The work involves no participants or physical experimental sample. Its unit of analysis is a low-energy continuum model, with k = (kx, ky, kz) representing crystal momentum and C set to 0.5. Analytical and numerical calculations use that model to evaluate the bands, quantum metric, optical conductivity and nonlinear Hall response.
A geometric pattern that changes with the phase
One central quantity is gzz, a component of the quantum metric, a mathematical way of describing the model's quantum geometry in momentum space. In the calculations, gzz formed a pronounced ring for finite values of m. The ring expanded as m increased, shrank as m approached zero and diverged at points where the bands touched.
The study expresses the real part of the interband optical conductivity, σzz, in terms of the quantum metric. In the calculated response, m = 0, 0.5 and 1.0 produced pronounced peaks for finite m, while no peak was observed in the m = 0 unlinked case. The peak positions moved to higher photon frequencies as m increased.
The joint density of states, or JDOS, provided a second way to compare the calculated transition energies. JDOS maxima occurred at exactly the same frequencies as the optical-conductivity peaks. Those frequencies also matched the maximum band gap between the nodal line and the loop in the model. Larger m was accompanied by a larger gap and a higher peak.
A separate route to a nonlinear Hall signal
The model's symmetry analysis reported preserved PT symmetry, vanishing Berry curvature for each nondegenerate band and a vanishing conventional Berry-curvature-dipole contribution to the nonlinear Hall response. The bare model's intrinsic nonlinear Hall response was also reported to vanish.
The researchers then calculated the model with a symmetry-breaking perturbation written as H' = βσx. At β = 0, positive and negative contributions to the quantum-metric-dipole, or QMD, kernel were symmetric about kx = 0, producing zero net QMD. At finite β, the distribution became increasingly asymmetric, and a contribution appeared along kx = 0.
For the linked cases examined at m = 0.5 and m = 1.0, finite-β calculations produced a nonlinear Hall conductivity labelled σyxx. The unlinked phase showed no finite nonlinear Hall response in the reported analysis. Across the finite-β calculations, σyxx had a pronounced peak near chemical potential µ = 0, identified as charge neutrality in the model. Its peak magnitude was not monotonic with β and could change sign as β varied.
What the calculation can establish
These results are predictions within one low-energy continuum model, not measurements from a material. Their scope is also tied to the specified perturbation H' = βσx and the parameter settings examined in the nonlinear Hall calculation. The findings therefore do not establish that every Hopf-link semimetal or every symmetry-breaking perturbation will produce the same response.
Whether the predicted optical signatures and intrinsic nonlinear Hall response can be measured in a material realization, or whether they generalize to material-specific lattice models, remains open. The paper is an arXiv version 1 preprint dated 26 Aug 2026. The authors state that the datasets generated and analyzed during the study are available from the corresponding author upon reasonable request.
Paper data and sources
Original title: Quantum geometric signatures of Link-Unlink transitions and nonlinear Hall response in Hopf-link semimetals
Authors: Kamalesh Bera, Arijit Saha, Debashree Chowdhury
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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