Field-induced multipolar character in the dipolar ground state of the honeycomb rare-earth chalcohalide NdOF

This study establishes the honeycomb rare-earth chalcohalide NdOF as a model system where field-tunable reconstruction of crystalline electric field doublets drives a continuous evolution of the ground state from dipolar to dipolar-multipolar character, as confirmed by Raman spectroscopy and magnetization measurements.

The Gist

Imagine a tiny, magical building block inside a crystal. This block is a Neodymium ion (Nd), and it has a very specific job: it acts like a tiny magnet. In most materials, these little magnets are simple; they just point "up" or "down" like a standard compass needle. Scientists call this a "dipolar" state.

However, in a special honeycomb-shaped crystal called NdOF, these tiny magnets are more complex. They can behave like compass needles and like more exotic, multi-faceted shapes (like an octopus with eight arms) at the same time. This complex behavior is called "multipolar."

The big question this paper answers is: Can we force these simple magnets to become complex ones just by using a magnetic field?

Here is the story of how they found out, explained simply:

1. The Stage: A Honeycomb Crystal

Think of the NdOF crystal as a flat, two-dimensional honeycomb net (like a beehive). Inside each hexagon sits a Neodymium ion. These ions are surrounded by oxygen and fluorine atoms, creating a specific "room" for them. This room has a triangular symmetry, which is like a three-sided mirror.

The researchers first checked the crystal's structure using X-rays (like taking a high-res photo) to make sure it was pure and didn't change shape when it got cold. They also used a laser (Raman spectroscopy) to listen to the "vibrations" of the atoms. It's like tapping a glass to hear its ring; this helped them identify the specific "notes" (energy levels) the Neodymium ions could play.

2. The Discovery: Four Special Notes

When they looked at the energy levels, they found four distinct "notes" the ions could jump between. These are called Crystal Field Excitations.

• One note was very low energy (1.7 meV), meaning the gap between the "ground floor" and the "first floor" of the ion's energy building was very small.
• Because this gap was so small, the ion was very "jittery" and sensitive to outside influences.

3. The Experiment: Pushing with a Magnet

The researchers applied a strong magnetic field (up to 9 Tesla, which is incredibly strong) to the crystal. They wanted to see what would happen to those four "notes."

• The Result: Instead of just shifting slightly, the notes split and twisted in a very complicated, non-linear way. One note split into two, another into three, and so on, eventually creating seven distinct branches.
• The Analogy: Imagine a spinning top. If you push it gently, it wobbles a bit. But if you push it from a specific angle, it might suddenly start spinning in a completely different, complex pattern. The magnetic field acted like that specific push, forcing the ions to change how they spin.

4. The Big Reveal: From Simple to Complex

The most important finding is what happened to the "ground state" (the lowest energy level where the ion usually sits).

• At Zero Field: The ion behaves like a simple compass needle (dipolar). It's straightforward.
• With a Magnetic Field: As they turned up the magnetic field, the researchers found that the ion's behavior started to change. It didn't just stay a simple needle; it began to mix in the "exotic" behavior (multipolar).
• The Transformation: By the time they reached 9 Tesla, the ground state had evolved. It was no longer just a simple magnet; it had acquired a "multipolar" character. The magnetic field acted like a dial or a knob that the scientists could turn to continuously transform the magnet from simple to complex.

5. Why This Matters (According to the Paper)

The paper claims that NdOF is a perfect "test kitchen" for this phenomenon. Because the energy gap is so small, it is incredibly easy to tune the magnet's personality using:

1. Magnetic Fields: Turning the "knob" of the external magnet.
2. Pressure: Squeezing the crystal (which the paper mentions as a complementary way to tune it).

The researchers successfully built a mathematical model that predicted exactly how the energy levels would split and how the magnetism would change. Their model matched the experimental data perfectly, proving that they understood exactly how the magnetic field was rewriting the rules of the ion's behavior.

Summary

In short, the paper shows that in the honeycomb crystal NdOF, you can take a simple magnetic atom and, by applying a magnetic field, continuously reshape its quantum nature from a simple "compass needle" into a complex "multipolar" object. They didn't just guess this; they measured the energy "notes" the atoms sang, watched them split under pressure, and proved that the magnetic field is the tool that drives this transformation.

Technical Summary

Technical Summary: Field-induced multipolar character in the dipolar ground state of the honeycomb rare-earth chalcohalide NdOF

Problem Statement
In rare-earth ions with half-integer total angular momentum $J$, the interplay between strong spin-orbit coupling and the crystalline electric field (CEF) generates a local ground-state Kramers doublet. While these doublets are often purely dipolar (giving rise to conventional Ising or XY anisotropy), specific wave functions can exhibit "dipolar-octupolar duality," where different pseudospin components transform as magnetic dipoles or higher-order multipoles (e.g., octupoles). This duality is theoretically predicted to host rich phases, such as multipolar quantum spin liquids and octupolar quantum spin ice. However, experimentally observing and, crucially, controlling the transition from a purely dipolar ground state to one with significant multipolar character in real materials remains a significant challenge. Layered rare-earth chalcohalides (REOX) offer a promising structural platform due to their clean honeycomb networks and local $C_{3v}$ symmetry, but the specific response of their CEF ground states to external perturbations requires detailed investigation.

Methodology
The authors investigate NdOF, a honeycomb rare-earth chalcohalide with the SmSI-type trigonal structure ($R3m$ space group), where Nd$^{3+}$ ions ($J=9/2$) form a 2D honeycomb lattice with local $C_{3v}$ symmetry. The study employs a multi-faceted experimental and theoretical approach:

1. Structural and Vibrational Characterization: Powder X-ray diffraction (XRD) was used to confirm phase purity and structural stability down to 3 K. Raman spectroscopy was utilized to distinguish between lattice phonons and electronic excitations, using non-magnetic LaOF as a reference to assign vibrational modes.
2. CEF Excitation Mapping: Temperature-dependent Raman spectroscopy (1.7 K to 150 K) was performed to identify intrinsic CEF excitations within the $^4I_{9/2}$ manifold.
3. Field-Dependent Analysis: Raman measurements were conducted under magnetic fields up to 9 T at 1.8 K. To analyze the powder-averaged response, the authors developed a computational scheme calculating Zeeman splitting for random crystallographic orientations and averaging the results.
4. Magnetometry: Magnetization ($M$) and susceptibility ($M/H$) measurements were performed over 0.1–9 T and various temperatures.
5. Theoretical Modeling: A CEF Hamiltonian constrained by $C_{3v}$ symmetry was constructed. The model parameters were refined by simultaneously fitting zero-field Raman spectra and field-dependent splitting patterns. The resulting wave functions were analyzed to quantify the dipolar versus multipolar character under varying field strengths.

Key Results

• CEF Level Scheme: Raman spectroscopy identified four intrinsic CEF excitations at 1.7, 15.6, 19.2, and 80.9 meV. These correspond to transitions from the ground-state doublet to four excited Kramers doublets, consistent with the splitting of the $J=9/2$ multiplet under $C_{3v}$ symmetry.
• Nonlinear Field Splitting: Under applied magnetic fields, the four CEF modes split into seven distinct branches. The experimental powder-averaged spectra were quantitatively reproduced by the CEF model including Zeeman splitting. The analysis reveals that the nonlinear splitting is primarily driven by in-plane magnetic fields ($H \perp c$), which induce strong mixing between closely spaced doublets, whereas $c$-axis fields produce nearly linear shifts.
• Ground State Character: At zero field, the ground state is identified as a purely dipolar doublet ($\Gamma_4$), separated from the first excited doublet ($\Gamma_{5,6}$) by a small gap of 1.7 meV.
• Field-Induced Evolution: Magnetization data ($M-H$ and $M/H-T$) show a crossover from Curie-Weiss behavior at low fields to a broad, nearly temperature-independent plateau at high fields. This behavior is well-described by the CEF model without invoking complex exchange interactions.
• Multipolar Admixture: Calculations of the ground-state wave functions under magnetic fields (3 T to 9 T) demonstrate a continuous reconstruction. The weight of the first excited state ($\Gamma_{5,6}$), which carries octupolar character, increases systematically with field strength (from ~2% at 3 T to ~14% at 9 T). This indicates a continuous evolution from a purely dipolar ground state to a mixed dipolar-multipolar state.

Significance and Claims
The paper establishes NdOF as a model system for exploring the controlled induction of multipolar components in rare-earth magnets. The primary significance lies in demonstrating that the CEF ground state, while predominantly dipolar at zero field, can be continuously reconstructed under external magnetic fields to acquire substantial multipolar character.

The authors claim that:

1. Tunability: The small CEF gap (1.7 meV) renders the NdOF ground state exceptionally sensitive to external perturbations, allowing for the continuous tuning of the dipolar-multipolar admixture via magnetic fields (and potentially pressure, via lattice distortion).
2. Methodological Advance: The developed powder-averaged CEF framework successfully reproduces complex nonlinear field splitting, providing a robust method for extracting CEF parameters in polycrystalline samples where single-crystal data is unavailable.
3. Physical Mechanism: The emergence of a field-induced ferromagnetic state with mixed dipolar-multipolar nature explains the observed susceptibility plateau. This mechanism highlights a general route for manipulating the symmetry of ground-state wave functions in rare-earth honeycomb magnets.

The study concludes that while precise extraction of exchange parameters requires future single-crystal work, the CEF physics alone confirms that external fields act as a continuous control knob to transform the ground state nature in NdOF.