Is it possible to determine unambiguously the Berry phase solely from quantum oscillations?

Abstract

The Berry phase, a fundamental geometric phase in quantum systems, has become a crucial tool for probing the topological properties of materials. Quantum oscillations, such as Shubnikov-de Haas (SdH) oscillations, are widely used to extract this phase, but its unambiguous determination remains challenging. This work highlights the inherent ambiguities in interpreting the oscillation phase solely from SdH data, primarily due to the influence of the spin factor $R_S$, which depends on the Landé $g$-factor and effective mass. While the Lifshitz-Kosevich (LK) theory provides a framework for analyzing oscillations, the unknown g-factor introduces significant uncertainty. For instance, a zero oscillation phase could arise either from a nontrivial Berry phase or a negative $R_S$. We demonstrate that neglecting $R_S$ in modern studies, especially for topological materials with strong spin-orbit coupling, can lead to doubtful conclusions. Through theoretical analysis and numerical examples, we show how the interplay between the Berry phase and Zeeman effect complicates phase determination. Additionally, we also discuss another underappreciated mechanism - the magnetic field dependence of the Fermi level. Our discussion underscores the need for complementary experimental techniques to resolve these ambiguities and calls for further research to refine the interpretation of quantum oscillations in topological systems.

Figures (6)

• Figure 1: a) Schematic illustration of a possible closed trajectory (dashed line) in the surface Brillouin zone around a Dirac point. Electrons moving along such a path acquire a nontrivial $\pi$ Berry phase. b) Parallel transport of a vector along a closed loop on a sphere.
• Figure 2: Oscillating part $\Delta\sigma_{xx}$ and corresponding Landau Level fan diagram for model (9).
• Figure 3: Dependence of $R_{S}=\cos\left(\pi\frac{gm^{*}}{2m_{0}}\right)$ on the $g$-factor at $m^{*}⁄m_{0} = 0.25$.
• Figure 4: The influence of the Zeeman effect on Landau levels, LL fan diagrams, and oscillations for carriers with a quadratic dispersion relation for Landau levels $N = 15$--$21$ and specific physical parameters: oscillation frequency $F = 129.5$ T, Fermi energy $E_{F} = 60$ meV, and effective mass $m^{*} = 0.25m_{0}$. (a-c) Energies of the unsplit and spin-split Landau levels as a function of $1/B$ for three cases: (a) $R_s = 1$ ($g = 48$): the split levels coincide in position with the unsplit ones; (b) $R_s = -1$ ($g = 56$): the split levels are shifted exactly midway between the unsplit ones; (c) $R_s = 0$ ($g = 52$): the levels do not overlap, leading to the suppression of oscillations. (d-f) The corresponding LL fan diagrams and the oscillating component $\Delta\sigma_{xx}$. Case (b) demonstrates how a strong Zeeman effect ($R_s = -1$) can create a $\pi$ phase shift, precisely mimicking the signal of a non-trivial Berry phase ($\beta = 0.5$) in case (a).
• Figure 5: (a) Dependencies of energies for unsplit and split Landau levels with indices $N = 15$--$21$ on inverse magnetic field for fermions with quadratic dispersion at $R_S = -1$, as well as for unsplit Landau levels of fermions with linear dispersion. (b, c) LL fan diagram and dependence of phase factor $\gamma$ on LL index for fermions with $E \sim k$, $g = 56$ and different $v_F$ values. (d) Corresponding oscillating components $\Delta\sigma_{xx}$, where black curves represent $\Delta\sigma_{xx}$ for unsplit Landau levels, while colored curves show split cases with different $v_F$.