Berry Phase Determination from Quantum Oscillations Challenged by Landé-factor Influence

The accurate determination of the Berry phase, a critical geometric phase for understanding the topological properties of materials, presents a significant challenge to condensed matter physics. Bogdan M. Fominykh, Valentin Yu. Irkhin, and Vyacheslav V. Marchenkov investigate the ambiguities inherent in extracting this phase solely from quantum oscillations, specifically Shubnikov-de Haas oscillations. Their work reveals that the conventional interpretation of oscillation phase, based on the Lifshitz-Kosevich theory, is often complicated by the influence of the spin factor and the unknown g-factor, potentially leading to misinterpretations of topological properties. By demonstrating how the interplay between the Berry phase, Zeeman effect, and Fermi level obscures unambiguous determination, this research highlights the necessity for combining quantum oscillation data with complementary experimental techniques to accurately characterise novel materials. This analysis offers crucial insight for researchers working to understand and harness the potential of topological systems.

De Haas-van Alphen Oscillations and Spin Factors

De Haas-van Alphen (SdH) oscillations are frequently employed to determine the phase of electronic states, yet unambiguous identification of this phase continues to present difficulties. This research addresses the inherent ambiguities in interpreting the oscillation phase using only SdH data, principally due to the influence of the spin factor RS, which is dependent on the Landé g-factor and effective mass. Although the Lifshitz-Kosevich (LK) theory offers a framework for analysing these oscillations, uncertainty arises from the typically unknown g-factor. A zero oscillation phase, for example, could result from either a nontrivial Berry phase or a negative RS value.

The study demonstrates that overlooking RS in contemporary investigations, particularly concerning topological materials exhibiting strong spin-orbit coupling, can lead to misinterpretations of the observed phase. Researchers employed a combination of theoretical modelling and analysis of experimental data to disentangle the contributions of RS and the Berry phase. This approach involved detailed calculations of the Fermi surface and band structure to accurately determine the g-factor and, consequently, the spin factor. A specific contribution of this work is the development of a methodology for reliably extracting the Berry phase even in the presence of a significant spin factor. The researchers present a clear distinction between the effects of RS and the Berry phase on the SdH oscillations, offering a more robust interpretation of experimental results. Furthermore, the findings highlight the importance of considering RS when analysing data from topological materials, ensuring a more accurate understanding of their electronic properties and topological characteristics.

Fermi Surface Mapping via Quantum Oscillations

This body of work constitutes a comprehensive collection of research papers focused on quantum oscillations, particularly in two-dimensional electron systems, semiconductor heterostructures, and topological materials. The studies collectively aim to deepen the understanding of electronic structure and charge carrier dynamics through phenomena such as the Shubnikov–de Haas (SdH) effect, providing both theoretical foundations and experimental insights.

The foundational papers establish the core physics of quantum oscillations, including SdH oscillations and related effects, and demonstrate how these oscillations can be used to map Fermi surfaces and extract key parameters such as effective mass and g-factor. These works examine the dependence of oscillation amplitude and phase on magnetic field strength and temperature, while also addressing corrections to classical quantization rules that arise from non-parabolic energy bands and other complexities. A significant subset of these studies explores the role of Berry phase and other geometric effects, showing how they modify Landau level quantization and lead to characteristic phase shifts in oscillation patterns. Several papers also investigate quantum oscillations beyond the conventional quantum limit, where extremely high magnetic fields produce large energy level spacings and unconventional behavior.

Building on this theoretical framework, another major group of papers applies quantum oscillation techniques to two-dimensional electron gases formed in semiconductor heterostructures. These studies use SdH oscillations to probe material properties such as effective mass, g-factor, band structure, and spin-related effects. Particular attention is given to Rashba spin–orbit coupling, which leads to spin splitting of energy bands and produces distinctive oscillation signatures that can be experimentally resolved. Research on heterostructures with inverted band structures further reveals how band inversion alters oscillation behavior and provides insight into the underlying electronic topology. Connections between quantum oscillations and the quantum Hall regime are also explored, linking oscillatory transport phenomena with quantized Hall conductance.

A further set of papers focuses on quantum oscillations in topological materials, including topological insulators and semimetals. In these systems, quantum oscillations serve as a powerful probe of topological surface and bulk states, such as Dirac fermions at interfaces and in three-dimensional semimetals. Several studies report anomalous phase shifts in oscillations that directly reflect the topological nature of the electronic bands and the associated Berry phase. Other works demonstrate how the topology of the Fermi surface wavefunction itself can be inferred from magnetic quantum oscillation measurements.

Taken together, this collection represents a substantial and coherent body of research that spans theoretical development, experimental investigation, and the discovery of novel quantum phenomena. It highlights the central role of quantum oscillations as a diagnostic tool for exploring electronic structure, spin–orbit interactions, and topological properties, with a strong emphasis on two-dimensional systems and emerging topological materials.

Berry Phase Determination Limited by Spin Effects

Scientists have demonstrated inherent ambiguities in determining the Berry phase, a crucial geometric phase in quantum systems, using only Shubnikov-de Haas (SdH) oscillations. The research highlights how neglecting the spin factor, influenced by the Landé g-factor and effective mass, can lead to inaccurate conclusions, particularly in topological materials exhibiting strong spin-coupling. Experiments reveal that the interplay between the Berry phase and the Zeeman effect complicates precise phase determination, demanding a more nuanced approach to data interpretation. Theoretical analysis and numerical examples underpin the findings, emphasizing the need for complementary experimental techniques.

The Lifshitz-Kosevich (LK) theory describes the oscillatory component of magnetoconductivity, expressed as ∆σxx ∝ r BF RSRT RD cos 2π F B⁻ + β + δ, where RT and RD define temperature and field-dependent damping. Crucially, the spin factor, RS = cos π ES ∆E, introduces complications, originating from spin degeneracy lifting in magnetic fields. When the spin splitting arises from the Zeeman effect, RS = cos π gm∗ 2m0, becoming independent of the magnetic field, yet still dependent on both effective mass and the g-factor. Scientists determined that while temperature and field-dependent factors do not affect phase, the spin factor introduces critical uncertainties, as the effective mass can be determined from temperature dependence, but the g-factor remains elusive from standard transport measurements.

Contemporary approaches often utilize Landau level fan diagrams constructed using the rule N = F B + γ to determine oscillation phases. The team showed that assigning integer Landau level indices to maxima of ∆σxx, with γ = −1/2 + β + δ, can yield a phase of zero when β = 0.5, but only if all prefactors in the LK equation remain positive. However, analysis of magnetoresistivity ∆ρxx, related to magnetoconductivity by σxx = ρxx / (ρxx + ρxy), reveals that phase shifts can occur if ρxx is significantly larger than ρxy. Measurements confirm that in materials where ρxx ≈ ρxy, the general relationship between σxx and ρxx must be considered for accurate phase determination, establishing a practical rule for correct oscillation phase analysis. The work demonstrates that for hole pockets, LK formula analysis indicates a non-trivial Berry phase for electron pockets, but this inconsistency is attributed to a potential Zeeman effect arising from a large g-factor, particularly relevant for topological semimetals with strong spin-orbit coupling. The research underscores the importance of accounting for the g-factor in modern studies and calls for further investigation to refine the interpretation of quantum oscillations in topological systems.

Berry Phase Extraction Hindered by Spin Effects

This work clarifies inherent difficulties in determining the Berry phase in quantum materials using only Shubnikov-de Haas oscillations. Researchers demonstrate that the oscillation phase is not a direct measure of the Berry phase due to the significant, often overlooked, influence of the spin factor, dependent on the Landé g-factor and effective mass. A zero oscillation phase, for example, could stem from a nontrivial Berry phase or simply a negative spin factor, creating ambiguity in interpretation. The study highlights the complex interplay between the Berry phase and the Zeeman effect, particularly in materials exhibiting strong spin-orbit coupling, and also points to the importance of considering magnetic field dependence of the Fermi level. Authors acknowledge limitations arising from the reliance on the Lifshitz-Kosevich theory, which requires knowledge of the g-factor, a value frequently unknown in standard transport measurements. Future research should focus on employing complementary experimental techniques to resolve these ambiguities and refine the analysis of quantum oscillations in topological systems, ultimately leading to more robust determination of Berry phase values.