The research was led by scientists from the University of São Paulo (USP) in Brazil, Los Alamos National Laboratory, the University of Washington, and other U.S. institutions. The team combined electrical transport experiments performed in magnetic fields as strong as 60 tesla and at temperatures near 0.7 kelvin (-272.45 °C) with detailed theoretical calculations.

"This work expands our understanding of electron transport in exotic phases of matter and suggests that topological insulators support the transport of not only electric charge, but also another fundamental degree of freedom: electron spin," says Julio Larrea Jiménez, a professor at USP's Physics Institute (IF) and co-founder and director of the Laboratory for Quantum Matter under Extreme Conditions (LQMEC).

Larrea served as the doctoral advisor of Cauê Kaufmann Ribeiro, the study's first author. Ribeiro carried out a large portion of the experimental work during an internship at the National High Magnetic Field Laboratory in Los Alamos, United States, supported by a FAPESP Research Internship Abroad. While there, he was co-advised by Johanna Palmstrom and Sean Thomas.

A Material With Two Electronic Personalities

Topological insulators have an unusual combination of properties. Their interiors behave as electrical insulators, while their surfaces can conduct electricity. This behavior arises from the topology of their electronic bands, which describes broad features of the quantum structure of electronic states that are protected by crystal symmetries.

ZrTe₅ is especially valuable for studying this type of physics because it sits close to the boundary separating different topological phases. Small changes in temperature, mechanical deformation, chemical composition, or magnetic field can significantly alter its electronic behavior. That sensitivity has made ZrTe₅ an important material for investigating topological phase transitions and relativistic quasiparticles in solids.

When electrons move through a magnetic field, their possible energies no longer form a continuous range. Quantum mechanics restricts them to specific energy values known as Landau levels, named after the Soviet physicist and mathematician Lev Landau (1908-1968).

In very pure metals, these Landau levels can repeatedly pass through the Fermi level, the energy boundary between occupied and unoccupied electronic states. Each crossing can produce an oscillation in electrical resistance. These effects, known as Shubnikov-de Haas oscillations, normally follow a predictable periodic pattern in 1/B, where B represents the magnetic field.

Quantum Oscillations That Refused To Disappear

The researchers found that ZrTe₅ did not follow this familiar pattern. Its magnetoresistance oscillations were not conventionally periodic in 1/B, and they continued well beyond the quantum limit.

That result was surprising because, beyond the quantum limit, electrons should be restricted to the lowest Landau level. Under conventional expectations, the familiar oscillations should therefore disappear.

"In materials near topological phase transitions, electrons may cease to behave like ordinary particles within a metal. Their electronic excitations begin to behave like quasiparticles similar to Dirac fermions -- that is, relativistic particles. In our work, we show that the spin of these quasiparticles plays a central role: when we apply strong magnetic fields, the interaction between spin and the magnetic field profoundly alters the energy levels of the electrons. As a result, Landau levels that would normally move away from the system's relevant energy can 'return' and cross it again. This unusual behavior is what we call reentrant Landau levels," says Kaufmann.

To explain the effect, the researchers propose a process known as the "back-bending" of Landau levels. Rather than changing their energy in a simple, straight progression as the magnetic field increases, some Landau levels can bend back toward the Fermi level and cross it again. Those renewed crossings generate additional oscillations in a regime where conventional theory predicts that oscillations should already have vanished.

Electron Spin Changes the Picture

The unusual behavior emerges from the interaction between two different physical effects. One is cyclotron energy, which comes from the orbital motion of electrons through a magnetic field. The other is the Zeeman effect, which describes the coupling between the magnetic field and electron spin.

In a material such as ZrTe₅, where spin-orbit interaction is strong, these two contributions cannot be considered independently. Electron spin and orbital motion become coupled, causing the energies of the Landau levels to evolve in a nonlinear way as the magnetic field changes.

A major goal of the study was to determine which of two possible mechanisms was responsible for the anomalous oscillations. One possibility involved many-body effects produced by collective interactions among numerous electrons. The other involved intrinsic topological properties associated with the material's electronic structure.

The researchers found that interactions among many electrons were not required to explain the behavior observed in their sample. Instead, a single-particle model based on a three-dimensional Dirac Hamiltonian and including strong spin-orbit coupling was able to reproduce the experimental regimes.

"What we saw is that the effect doesn't stem from many-body interactions, but rather from a nontrivial topology of the electronic bands," Larrea summarizes.

Resolving Conflicting Results in ZrTe₅

The findings may also help clarify a long-running debate surrounding experiments with ZrTe₅. Different samples of the same material have produced apparently different kinds of quantum oscillations.

Some samples display conventional oscillations that are periodic in 1/B. Others show oscillations that are not periodic in 1/B. Still others produce signals that seem to have logarithmic periodicity in B.

According to the new results, those differences may not require separate physical explanations. Instead, all of these behaviors could arise from the same underlying Dirac electronic structure, with the outcome largely determined by the carrier density and the size of the Fermi surface in each sample.

"In samples with low carrier density, such as the one investigated here, the Zeeman and cyclotronic effects become comparable in experimentally accessible magnetic fields. That favors the re-entry of Landau levels and makes the anomalous oscillations visible. In samples with higher carrier density, the conventional term dominates, and the oscillations retain their usual periodicity of 1/B," Larrea comments.

Two Spin Channels Interfere

The researchers also identified two separate contributions to the quantum oscillations that are linked to spin-separated states. These two contributions have different effective masses and can interfere with each other.

That interference helps explain another unexpected feature in the measurements. According to the conventional Lifshitz-Kosevich model, the amplitude of quantum oscillations should steadily decrease as temperature rises. Instead, the researchers observed a local minimum in the oscillation amplitude across certain temperature ranges.

The result indicates that two electronic channels are interfering with one another rather than simply fading in the standard way.

Measurements of angular magnetoresistance provided additional information about the material. Under low magnetic fields, the Fermi surface appeared to be three-dimensional and roughly ellipsoidal.

The researchers calculated a very low carrier density of approximately 10¹⁶ per cubic centimeter. That value is consistent with ZrTe₅ being positioned extremely close to a topological phase transition.

Testing Matter Under Extreme Conditions

The experiments were performed at the National High Magnetic Field Laboratory in Los Alamos, one of only a small number of facilities worldwide capable of combining pulsed magnetic fields reaching 60 tesla with temperatures below 1 kelvin.

"This type of experiment can only be performed in a few places around the world. Access to those facilities is highly competitive," Larrea notes.

Beyond explaining the unusual quantum oscillations themselves, the findings strengthen the case for using ZrTe₅ as a platform for exploring additional topological phases of matter.

The researchers suggest that carefully adjusting factors such as symmetry, carrier density, mechanical stress, temperature, and magnetic field could produce even more exotic electronic states. These could include phases involving Weyl quasiparticles.

"Our experiment provided the first empirical demonstration of a process that had previously been shrouded in controversy," Larrea summarizes.

The research was also supported by FAPESP through a Young Investigator Grant awarded to Larrea. Additional funding came from U.S. institutions including Los Alamos National Laboratory, the National High Magnetic Field Laboratory, the National Science Foundation, and the U.S. Department of Energy.