Enhancing the Quantum Oscillation Toolbox
The transport properties of quantum materials often vary periodically with the strength of an applied magnetic field. These quantum oscillations have long provided physicists with an indispensable tool for extracting subtle, otherwise-inaccessible information on electronic phases of matter [1]. Now an experiment by Jinrui Zhong of the Beijing Institute of Technology and his colleagues has revealed a novel kind of quantum oscillation in moiré systems [2]. These are materials made from stacked monolayers that are twisted with respect to each other to create, in effect, atomic lattices with much wider unit cells. The experiment pointed to a special mechanism for facilitating the novel periodic fluctuations: the emergence of so-called Brown-Zak fermions. First observed in 2020 [3], these quasiparticles travel ballistically in 2D materials: Despite needing a strong magnetic field to be summoned into existence, Brown-Zak fermions move as if the applied field completely vanishes. This pioneering experiment showcases the power of nonlinear transport as a sensitive tool to characterize electronic properties of interest.
In the classical hall effect, an applied magnetic field engenders a voltage perpendicular to a longitudinal current flowing through the system. The transverse voltage is readily explained by electrons being diverted by the Lorentz force. But there’s another way to look at the Hall effect. If time were reversed, the transverse electric field would point in the same direction, but the transverse current would flow in the opposite direction. The only way the Hall voltage could preserve its value under time reversal would be if it were somehow zero. A magnetic field is needed to break time-reversal symmetry and engender a nonzero transverse voltage.
However, a second-order nonlinear Hall effect (NLHE) can arise even in the presence of time-reversal symmetry. If a 2D crystal breaks inversion symmetry, its charge carriers can occupy different energy states such that a Hall voltage proportional to the current squared starts flowing. The NLHE magnitude is typically related to the Berry curvature, a geometric property of quantum wave functions that describes how they respond to changes in a system’s parameters—in this case a longitudinal current [4]. Consequently, the observation and characterization of the NLHE could reveal valuable information on the quantum geometry of the electrons’ wave functions.
Current and other transport features can vary considerably—and revealingly—when an external magnetic field is applied to a system. For example, the oscillatory behavior of resistivity as a function of magnetic field, known as the Shubnikov–de Haas effect, can be used to investigate the shape of the Fermi surface (Fig. 1). This is because the period of the oscillations can be directly related to the area in momentum space enclosed by the Fermi surface and thus to the electronic density. Semiclassically, the quantization of electron orbits encircling the Fermi surface singles out a periodic series of magnetic-field values at which the electronic density of states and, with it, the resistivity change dramatically.
A strong enough magnetic field completely transforms this semiclassical picture. When the magnetic flux penetrating a unit cell is comparable to the flux quantum, a major reconstruction of the electronic band structure takes place. Specifically, at rational fractions (expressed as p/q, where p and q are integers) of the flux quantum, the original electronic bands are folded into q minibands, whose properties are partially inherited from the original bands. The particles occupying these minibands are Brown-Zak fermions, which are effectively oblivious to the external magnetic field [5, 6]. When p, q or both are not quite integers, the system is still well described by having Brown-Zak fermions, but now they experience a reduced effective magnetic field [7]. In conventional materials, the magnetic field needed to observe Brown-Zak fermions is unattainably strong. But in moiré materials, the enlargement of the unit cell by up to 4 orders of magnitude reduces the required field strength to values accessible in labs.
These are the systems that Zhong and his collaborators examined in their experiments. They extracted the NLHE signal and recorded its oscillations as a function of the perpendicular applied magnetic field. Remarkably, the oscillations first emerged at a magnetic field of just 0.5 tesla, which is too weak to induce any detectable oscillations of conventional linear-transport features in these systems.
Certain quantitative features of the NLHE oscillations, notably their magnitude, are expected to vary with the strength of an external electric field applied perpendicularly to the system and with the electron density. That’s because the electric field modulates the electrons’ wave functions and their spread across the stacked layers, while the electron density modulates the position of the Fermi surface. The oscillatory period remains unaltered. By contrast, the period of the Shubnikov–de Haas oscillations is inexorably tied to the electron density.
Zhong and his collaborators found that the NLHE oscillations peak at values of the magnetic-field strength that are commensurate with a rational fraction of flux quantum per moiré unit cell. Moreover, the researchers showed that at these specific fractions, the NLHE scaling behavior is consistent with the presence of Brown-Zak fermions. Far away from these commensurate values, however, the NLHE scales as one would expect for electrons under a weak field. This work has uncovered quantum oscillations of an unusual origin. They do not arise from the quantization of semiclassical orbits in momentum space but from the quantization of the size of the real-space magnetic unit cell. The fundamental nature of the low-energy quasiparticles—the Brown-Zak fermions—is modulated by the field.
Zhong and his collaborators have shown how nonlinear transport can reveal electronic properties of great interest, notably the quantum geometry of electronic wave functions [8]. The technique becomes exceptionally potent in the presence of an enlarged unit cell. A promising avenue for such experiments is to investigate scenarios in which a widely spaced lattice spontaneously forms through strong electron interactions. This is the case when the electrons’ density and temperature are low enough for the formation of a so-called Wigner crystal. Recent experiments in rhombohedral graphene systems have found that the formation of a Wigner crystal is intertwined with novel many-body topological phases and with metallic states whose Berry curvature is usually concentrated near the Fermi surface [9, 10]. Probing the NLHE oscillations in these systems could potentially isolate crystalline components from itinerant ones, unlocking improved understanding of these exotic phenomena.
References
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- J. Zhong et al., “Nonlinear Hall quantum oscillations to probe topological Brown-Zak fermions in graphene moiré systems,” Phys. Rev. Lett. 136, 246301 (2026).
- J. Barrier et al., “Long-range ballistic transport of Brown-Zak fermions in graphene superlattices,” Nat. Commun. 11, 5756 (2020).
- I. Sodemann and L. Fu, “Quantum nonlinear hall effect induced by Berry curvature dipole in time-reversal invariant materials,” Phys. Rev. Lett. 115, 216806 (2015).
- E. Brown, “Bloch electrons in a uniform magnetic field,” Phys. Rev. 133, A1038 (1964).
- J. Zak, “Magnetic translation group,” Phys. Rev. 134, A1602 (1964).
- R. K. Kumar et al., “High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices,” Science 357, 181 (2017).
- N. Verma et al., “Quantum geometry and the hidden scales in materials,” Nat. Rev. Phys. 8, 226 (2026).
- Z. Lu et al., “Extended quantum anomalous Hall states in graphene/hBN moiré superlattices,” Nature 637, 1090 (2025).
- T. Han et al., “Evidence of metallic Wigner crystal in rhombohedral graphene,” arXiv:2604.00113.