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arXiv:2601.09560

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

Bogdan M. Fominykh, Valentin Yu. Irkhin, Vyacheslav V. Marchenkov

cond-mat.mtrl-sciphysics.app-phquant-phSubmitted Jan 14, 2026• Announced Jan 15, 2026
arXivHTMLPDF

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.

Journal reference
Physics Letters A, 569 (2026) 131238
DOI
10.1016/j.physleta.2025.131238

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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 RSR_SRS​, which depends on the Landé ggg-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 RSR_SRS​. We demonstrate that neglecting RSR_SRS​ 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.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
Abstract Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometrical forms of the interaction, or due to interactions which do not result from conservative forces acting on the system (see Section 1.2). A conscientious reader may point out that all these effects could be entirely explained using the Schrödinger equation only. But the geometrical phase formalism is useful because it sometimes plays a role in fundamental quantum phenomena and it shows that additional phase factors exist which are easily overlooked, making the interpretation of measured effects more mysterious than necessary. Thus, our understand-ing of quantum mechanics has deepened since the existence of geometrical phases was explicitly revealed (Shapere and Wilczek 1989). This happened due to the seminal works of Pancharatnam (1956), Berry (1984), Anandan (1992), and Bhandari (1997). It also shows that a wavefunction often carries much more information than is usually extracted in a standard experiment. A typical example is the assumption of adiabaticity, where the magnetic moment of the neutron produces (inside a slowly varying magnetic field) a dynamical phase shift proportional to the path integral ∫ B ds, but when the direction of the field varies slowly an additional, i.e. geometrical, phase also appears. The back reaction of a quantum measurement can also be described as a kind of an induced topological phase (Aharonov et al. 1998).
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.
Abstract Preservation of quantum coherence can be achieved by harnessing a system’s topology—encoding information in global geometric properties of quantum states insensitive to local perturbations. The robustness arises as certain geometric properties depend only on the trajectory of a quantum operation and the curvature in the controlled-parameter space, rather than on details of the system’s dynamics, thereby offering a topology-based framework for fault-tolerant quantum computation. To probe a system’s topology, geometric phase interference (Berry phase) has been studied in various optical systems and condensed matter physics, including molecular magnets (MMs). However, its direct detection in 4f-based MMs—prospective qudit (multi-level qubits) platforms—has remained elusive due to large tunnel gaps and environmental coupling. Here, we report a magneto-spectroscopic approach: the μSQUID-EPR technique, to directly resolve tunnel splittings in the 160 Gd-based MM [ 160 GdPc₂]⁻ (Pc = phthalocyanine). By irradiating single crystals with microwaves under transverse magnetic fields, we map the spin manifold and report the first direct observation of pronounced oscillations in the tunnel splitting—a signature of quantum phase interference. These oscillations reveal topological quenching of tunnelling and highlight the role of fourth-order transverse anisotropy in shaping the spin orientation space. Our findings uncover a nontrivial topological structure in 4f-MMs, opening possibilities towards holonomic quantum computation.
Berry's phase often appears in quantum two-level systems with a degeneracy. An example of such a system is a spin-1/2 particle in a magnetic field. As the magnetic field is slowly evolved through a closed path, the particle has been shown to acquire an additional phase called Berry's phase, in addition to the usual dynamical phase. This phase has been found in two-level quantum systems intrinsic to many superconducting qubits and has particularly been calculated for the charge, flux and Josephson phase qubit. Here, we present an alternative derivation of the Berry's phase in a current-biased Josephson junction qubit. We also calculate the complete Berry's phase evaluated over a closed loop in a frequency parameter space and compare this with the geometric phase. From this comparison, we examine the possibility of using a single phase qubit for topological quantum computing.