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

Technical Summary: Ambiguities in Determining the Berry Phase from Quantum Oscillations

Problem Statement
The Berry phase ($\phi_B$) is a fundamental geometric phase used to probe the topological properties of materials, particularly in identifying Dirac and Weyl fermions where $\phi_B = \pi$. In contemporary research, Shubnikov-de Haas (SdH) quantum oscillations are the primary experimental tool for extracting this phase. However, the unambiguous determination of the Berry phase solely from oscillation data remains a significant challenge. The core problem lies in the interpretation of the oscillation phase within the Lifshitz-Kosevich (LK) framework. The observed phase is not solely a function of the Berry phase but is also critically dependent on the spin factor $R_S$, which is governed by the Landé $g$-factor and the effective mass ($m^*$). Since the $g$-factor is typically unknown in standard transport measurements, a zero oscillation phase (often interpreted as a signature of non-trivial topology, $\beta = 0.5$) can equally arise from a trivial Berry phase ($\beta = 0$) combined with a negative $R_S$ value. Furthermore, other mechanisms, such as the magnetic field dependence of the Fermi level and the orbital magnetic moment, can mimic or mask the Berry phase, leading to potentially erroneous conclusions regarding the topological nature of a material.

Methodology
The authors employ a combination of theoretical analysis and numerical modeling to deconstruct the factors influencing quantum oscillation phases.

1. Theoretical Framework: The analysis is grounded in the Lifshitz-Kosevich theory of magnetoconductivity oscillations. The authors rigorously examine the phase term $\gamma = -1/2 + \beta + \delta$ in the Landau level fan diagram, where $\beta$ relates to the Berry phase and $\delta$ is a geometry-dependent factor.
2. Spin Factor Analysis: The study explicitly models the spin factor $R_S = \cos(\pi g m^* / 2m_0)$. The authors demonstrate how variations in the $g$-factor can cause $R_S$ to switch signs, effectively introducing a $\pi$ phase shift in the oscillations.
3. Numerical Simulations: Using parameters derived from the Weyl semimetal WTe$_2$ (oscillation frequency $F=129.5$ T, $E_F=60$ meV, $m^*=0.25m_0$), the authors simulate Landau level spectra for different $g$-factors ($g=48, 52, 56$). They construct Landau level (LL) fan diagrams to visualize how the Zeeman effect shifts the intercept $\gamma$.
4. Dispersion Relations: The paper compares systems with quadratic dispersion (conventional electrons) and linear dispersion (Dirac/Weyl fermions) to show how the Zeeman effect and Fermi level dependence affect the phase differently in each case.
5. Extended Mechanisms: The authors incorporate the magnetic field dependence of the Fermi level ($E_F(B)$) and the orbital magnetic moment contribution ($\phi_R$) into the quantization rule, analyzing their impact on the total phase $\lambda$.

Key Contributions and Results

• The $R_S$ Ambiguity: The paper demonstrates that neglecting the spin factor $R_S$ leads to fundamental ambiguities. A system with a trivial Berry phase ($\beta=0$) and a specific $g$-factor (yielding $R_S < 0$) produces an oscillation phase indistinguishable from a system with a non-trivial Berry phase ($\beta=0.5$) and $R_S > 0$. For instance, in WTe$_2$, a large $g$-factor can shift split Landau levels to positions midway between unsplit levels, creating a $\pi$ phase shift that mimics a non-trivial Berry phase.
• Revised Fan Diagrams: The authors propose an extended Landau level fan diagram equation that includes a term dependent on the $g$-factor: $N = F/B + \gamma + \frac{1}{2}(\frac{g m^*}{2m_0} \mod 2)$. This highlights that the intercept is not a constant solely determined by topology but is modulated by the $g$-factor.
• Fermi Level Dependence: The study identifies the magnetic field dependence of the Fermi level ($E_F(B)$) as a previously underappreciated mechanism. Even in the absence of Zeeman splitting, a linear dependence $E_F = E_0^F + \alpha B$ can shift the LL fan diagram intercept, potentially mimicking a non-trivial Berry phase in trivial systems.
• Total Phase Decomposition: Adhering to modern semiclassical theory, the paper emphasizes that the experimentally observed phase is the sum $\lambda = \phi_B + \phi_R + \phi_Z$ (Berry phase + orbital moment + Zeeman). Without independent knowledge of $\phi_R$ and $\phi_Z$, isolating $\phi_B$ is impossible.
• Case Studies: The authors cite specific examples where these ambiguities manifest, including HgTe quantum wells (where a giant $g$-factor created a spurious zero phase) and Te nanoribbons (where the spin-zero effect masked the true Berry phase).
• Alternative Method: The paper notes that analyzing the temperature dependence of the oscillation frequency, $F(T)$, offers a more robust method for identifying topological carriers. This approach relies on the distinct temperature coefficients ($\Theta$) for linear vs. quadratic dispersion and does not require knowledge of the $g$-factor, thereby avoiding the $R_S$ ambiguity.

Significance and Claims
The paper asserts that determining the Berry phase exclusively from quantum oscillation data is fundamentally unreliable in the general case. A zero phase offset in a Landau fan diagram is not a definitive "smoking gun" for Dirac or Weyl physics, as it can be equally explained by a combination of a trivial Berry phase and a negative spin factor, or by other mechanisms like Fermi level shifts and orbital moments.

The authors conclude that unambiguous identification of topological states requires a combined experimental approach. They advocate for:

1. Independent determination of the $g$-factor (e.g., via magneto-infrared spectroscopy or observing the spin-zero effect).
2. Using this fixed $g$-factor to constrain the analysis of quantum oscillations.
3. Utilizing symmetry considerations or first-principles calculations to estimate the orbital moment contribution.
4. Employing the temperature dependence of the oscillation frequency as a complementary, $g$-factor-independent diagnostic tool.

The work serves as a cautionary analysis, urging researchers to refine their interpretation protocols for quantum oscillations in topological systems to avoid misidentifying trivial electronic structures as topological, or vice versa.