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Over the past 40 years, the quantum Hall effect (QHE) has inspired new theories and led to experimental discoveries in a range of fields going beyond solid-state electronics to photonics and quantum entanglement. In this Viewpoint, physicists reflect on how the QHE has influenced their research.

A positive magneto-thermoelectric conductance is observed in the Weyl semimetal niobium phosphide, suggesting the presence of the elusive mixed axial–gravitational anomaly.

The ultra-quantum limit refers to the high magnetic-field regime where electrons are confined to the lowest Landau level and is most easily reached in topological semimetals due to their low carrier density. Here, the authors study this regime in the Dirac semimetal ZrTe₅ and find evidence for a Lifshitz transition at moderate field, leading to the emergence of a 1D-Weyl band structure at high field.

Although novel topological quasiparticles have recently been evidenced, their electrical transport properties remain elusive. Here, the authors report ultra-low resistivity down to 6 nΩcm at 2 K in MoP with a large mean free path, which hints on the exotic properties of triple point fermions.

A 3D quantum Hall effect has been reported in Dirac semimetal ZrTe₅ due to a magnetic-field-driven Fermi surface instability. Here, the authors show evidence of quasi-quantized Hall response without Fermi surface instability, but they argue that it is due to the interplay of the intrinsic properties of ZrTe₅ electronic structure and Dirac semi-metallic character.

There is a long-standing experimental effort to observe field-induced correlated states in three-dimensional materials. Here, the authors observe an unconventional Hall response in the quantum limit of the bulk semimetal HfTe₅ with a plateau-like feature in the Hall conductivity.

Advances in the fabrication of low-disorder metallic materials have made it possible to reach the hydrodynamic regime of electronic transport. Here the authors investigate a hydrodynamic electron fluid in tungsten diphosphide and find that both electrical and thermal transport are limited by the quantum indeterminacy.

In the charge-density-wave Weyl semimetal (TaSe₄)₂I, an axion is observed and identified as a sliding mode in the charge-density-wave phase characterized by anomalous magnetoelectric transport effects.

The anomalous Nernst effect (ANE) in topological materials with large Berry curvature shows great potential for transverse thermoelectrics, but antiferromagnets typically show small ANEs. The antiferromagnet YbMnBi₂ has an ANE thermopower of 3 μV K⁻¹, similar to ferromagnets, and a larger ANE conductivity.

We report a high anomalous Nernst thermopower (\(S_{yx}^A\)) -value of ~6.0 µV K⁻¹ at room temperature in the ferromagnetic topological Heusler compound Co₂MnGa. The measured value is seven-times larger than any anomalous Nernst thermopower value ever reported for a conventional ferromagnet. The high anomalous Nernst effect originates from a large net Berry curvature near the Fermi level associated with nodal lines and Weyl points.

When interactions between electrons in a material are strong, they can start to behave hydrodynamically. Spatially resolved imaging of current flow in a three-dimensional material suggests that electron–electron interactions are mediated by phonons.

Axion fields provide a unique way to understand large quantized electromagnetic responses in topological insulators and dynamics in Weyl semimetals. This Review discusses the theory of axion fields in condensed matter, their experimental realization and their application in next-generation devices.