🔬 materials science

Theory of phonon-magnon hybridization and angular momentum in CrI $_3$ and CrBr $_3$

This paper develops a constrained Hamiltonian framework to quantify phonon-magnon hybridization in CrI $_3$ and CrBr $_3$ , revealing significant mixing (up to 25% in CrBr $_3$ ) and demonstrating how total angular momentum is conserved while being shared between the hybridized phononic and magnonic subsystems.

Original authors: Maxime Mignolet, Miquel Royo, Massimiliano Stengel, Matthieu J. Verstreate

Published 2026-07-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maxime Mignolet, Miquel Royo, Massimiliano Stengel, Matthieu J. Verstreate

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the inside of a solid object, like a crystal, not as a static block of matter, but as a bustling dance floor. In this microscopic world, two very different types of dancers are constantly moving. First, there are the phonons. Think of these as the rhythm section: they are the collective vibrations of the atoms themselves, jiggling and shaking in perfect sync, creating the sound and heat within the material. Second, there are the magnons. These are the spin dancers; they represent the tiny magnetic compasses (spins) attached to every atom, wobbling and precessing in unison. Usually, these two groups dance to their own tunes, ignoring each other. But in certain magnetic materials, something magical happens: if the beat of the atomic vibrations matches the rhythm of the magnetic spins, and if the material has the right "shape" (symmetry), the two groups can start to interact. They might even grab hands and start dancing a new, hybrid routine together. This paper explores exactly how that happens, how they share their energy, and how they swap a very specific "move" called angular momentum.

Why should we care? Because this swapping of moves isn't just a cool party trick; it's the secret sauce behind some of the most exciting technologies of the future. Understanding how vibrations and magnetism talk to each other helps scientists design better ways to store data, create faster computers, and even detect invisible particles like dark matter. It's about learning the rules of the dance so we can choreograph new materials with superpowers.

The Great Dance-Off: When Atoms and Magnets Mix

In this paper, the authors, Maxime Mignolet and his team, set out to write the ultimate rulebook for this dance. They wanted to understand what happens when phonons (vibrating atoms) and magnons (wobbling magnetic spins) get close enough to interact. They focused on two specific materials, CrI₃ (Chromium Iodide) and CrBr₃ (Chromium Bromide), which are like perfect dance halls for this experiment because their atoms and spins are arranged in a way that allows them to mix.

The team built a sophisticated mathematical framework—a "constrained Hamiltonian"—to describe this interaction. You can think of this framework as a super-precise choreographer's notebook. It doesn't just track where the dancers are; it tracks their energy, their momentum, and how they influence each other. A key part of their theory involves something called "Berry curvature," which is a bit like a hidden magnetic field generated by the electrons moving around the atoms. This field acts like a subtle wind on the dance floor, pushing the dancers in specific directions and allowing them to gain "angular momentum," which is essentially a measure of how much they are spinning or rotating.

The Big Discovery: The Hybrid Dance
The most exciting finding is that when the phonons and magnons get close in energy, they don't just bump into each other; they actually merge to form new, hybrid dancers. The authors calculated exactly how much of each original dancer is in this new hybrid.

In CrI₃, they found a specific pair of dancers (a phonon and a magnon) that mixed so well that the phonon became about 8% "magnon-like."
In CrBr₃, the mixing was even more dramatic. One phonon grabbed onto a magnon so tightly that it became 25% magnon-like.

This is a big deal because, before this paper, scientists didn't have a clear, universal way to measure exactly how much of a phonon had turned into a magnon. The authors developed a new method to break down the total energy of the system, allowing them to say, "This mode is 75% vibration and 25% magnetism."

The Great Heist: Stealing Angular Momentum
Here is where the story gets really playful. In physics, there is a rule that the total amount of "spin" or angular momentum in a closed system must be conserved. It's like a bank account: money can move between accounts, but the total balance stays the same.

The authors discovered that when a phonon and a magnon hybridize, they don't just share energy; they steal angular momentum from each other.

Normally, a pure magnon carries a specific amount of angular momentum (like a spinning top).
A pure phonon usually carries zero angular momentum.
But when they hybridize, the magnon "steals" some of the phonon's mechanical rotation, and the phonon "steals" some of the magnon's magnetic spin.

In their simulations of CrBr₃, the optical magnon (the magnetic dancer) gained about 25% phononic character. This means it literally "stole" 25% of the mechanical angular momentum from the vibrating atoms. The paper shows that this transfer is incredibly precise; the total angular momentum is conserved to within a tiny fraction (about $10^{-4}$ to $10^{-3}$ of the unit $\hbar$ ), proving that the "theft" is a balanced transaction.

What They Didn't Find (and Why It Matters)
It's important to note what didn't happen. The authors found that this mixing only happens with specific types of dancers.

Non-degenerate modes (dancers that are unique and don't have a partner) stayed pure. They didn't mix at all.
Degenerate modes (dancers that come in pairs with the same energy) were the ones that split and mixed.
Furthermore, the mixing is chiral selective. This means a phonon and a magnon only dance together if they are spinning in the same direction (same "handedness"). If they spin in opposite directions, they barely interact. This explains why only one specific pair in the CrI₃ and CrBr₃ crystals decided to hybridize, while their neighbors stayed separate.

How Sure Are We?
The authors are very confident in their theoretical framework, which is built on rigorous math and established physics principles. However, the specific numbers they report (like the 8% and 25% hybridization) come from computer simulations (Density Functional Theory calculations), not direct laboratory measurements of the hybridization itself.

They note that their predicted frequencies for the magnetic spins (magnons) in these materials are a bit higher than what experiments have measured in the past. This is a known limitation of the specific computer model they used (the LDA functional), which tends to overestimate these values.
Despite this, the mechanism they describe—the mixing, the energy decomposition, and the angular momentum transfer—is presented as a solid theoretical result. They have provided the tools to measure these effects, and their simulations suggest these effects are real and significant.

The Takeaway
This paper gives us a new pair of glasses to look at magnetic materials. Instead of seeing vibrations and magnetism as separate things, we can now see them as a fluid mix that can swap energy and spin. The authors have shown that in materials like CrI₃ and CrBr₃, this mixing is strong enough to be measured (up to 25% in CrBr₃). They have also proven that angular momentum is conserved during this swap, even though it moves from the "magnetic" account to the "mechanical" account. This work lays the groundwork for future experiments where scientists might try to control these hybrid dancers to create new types of electronic devices or to understand how heat and magnetism interact in the quantum world.

Technical Summary: Theory of Phonon-Magnon Hybridization and Angular Momentum in CrI3 and CrBr3

Problem Statement
In magnetic materials, angular momentum is mediated by electrons, magnons, and phonons. While the interaction between phonon angular momentum (PAM) and spins is often modeled phenomenologically (e.g., via Raman spin-lattice coupling) or through explicit magneto-elastic terms, a rigorous unified quantum theory for coupled phonon-magnon dynamics has been lacking. Specifically, there is no established framework to quantify the degree of hybridization between phonons and magnons or to track the exchange of angular momentum between these subsystems upon hybridization. Existing literature often lacks explicit methods for decomposing hybrid modes or defining orthonormalization schemes for coupled eigenmodes.

Methodology
The authors develop a constrained Hamiltonian framework that treats ionic displacements and atomic spin deviations on equal footing. The theoretical development proceeds through the following steps:

Lagrangian and Hamiltonian Formulation: Starting from an effective adiabatic Lagrangian, the authors derive equations of motion for coupled spin-nuclei dynamics. This includes the molecular Berry curvature, which acts as a velocity-dependent force analogous to a magnetic field. A constrained Hamiltonian is formulated using Lagrange multipliers to handle the constraint arising from the spin Berry connection, allowing for a natural decomposition of the total energy.
Second Quantization: The equations of motion are second-quantized using the Holstein-Primakoff transformation for spins and a Bogoliubov transformation for the coupled system. This yields a generalized eigenvalue problem involving mass-weighted displacement, momentum, and spin variables.
Energy Decomposition and Hybridization Metric: A key theoretical innovation is the derivation of a novel orthonormalization scheme based on the total energy. By decomposing the total energy of a mode into phononic, magnonic, and coupled contributions, the authors define a metric ( $\eta$ ) to quantify the degree of hybridization. This generalizes the norm decomposition used for pure phonons in the literature.
Angular Momentum Analysis: The framework explicitly calculates the phonon angular momentum (PAM) and the spin angular momentum carried by magnons, allowing for the tracking of angular momentum transfer between subsystems.
First-Principles Application: The theory is applied to bulk ferromagnetic Van der Waals materials CrI $_3$ and CrBr $_3$ using Density Functional Theory (DFT) with the ABINIT software. Calculations are performed at the $\Gamma$ point using constrained magnetic moment DFPT to obtain derivatives with respect to atomic spins.

Key Results
The application of this theory to CrI $_3$ and CrBr $_3$ yields the following quantitative results:

Hybridization Degrees: The authors quantify the hybridization between phonons and magnons. In CrI $_3$ , the maximum hybridization is approximately 8% (specifically, an optical magnon and a specific $E_u$ phonon mode exchange character). In CrBr $_3$ , the hybridization is significantly stronger, reaching approximately 25% for the same interacting modes.
Frequency Splitting and Shifts: Non-degenerate phonon modes ( $A_g, A_u$ ) show no frequency shift or hybridization. Doubly-degenerate modes ( $E_g, E_u$ ) split into left and right circularly polarized modes due to the molecular Berry curvature and magnon coupling. The largest frequency splitting occurs in the $E_u$ mode pair closest in energy to the optical magnon.
Angular Momentum Exchange: The study demonstrates that upon hybridization, the optical magnon "steals" a portion of the phonon angular momentum. In CrI $_3$ , the hybridized magnon gains phononic character and corresponding PAM; similarly in CrBr $_3$ . The total PAM is conserved within numerical precision ( $\sim 10^{-4} \hbar$ ) when accounting for transfers between the optical magnon and all phonons, though local metric modifications by the mixed spin-phonon Berry curvature mean conservation is not an absolute rule in all contexts.
Chirality Selectivity: The results confirm that hybridization is chirality-selective; phonons and magnons with the same handedness interact strongly, while those with opposite handedness interact weakly.

Significance and Claims
The paper claims several significant theoretical and practical contributions:

Unified Framework: It provides a rigorous, unified quantum theory for coupled phonon-magnon dynamics that treats ionic and spin degrees of freedom symmetrically, avoiding the rigid spin approximation in the incorporation of adiabatic corrections.
Quantification of Hybridization: The proposed energy-decomposition method offers a clear, explicit way to quantify the phononic, magnonic, and coupled characters of hybrid modes, addressing a gap in current literature where such decomposition is often unclear or implicit.
Orthonormalization Scheme: The authors establish a necessary orthonormalization condition for hybrid phonon-magnon eigenvectors, which they note has been overlooked in previous works but is crucial for accurately computing quantities dependent on mode eigenvectors, such as PAM.
Mechanism of Angular Momentum Transfer: The paper identifies and quantifies a "stealing mechanism" where magnons acquire mechanical angular momentum from phonons during hybridization. This mechanism was not accounted for in literature using naive orthonormalization schemes.
Foundation for Future Work: The authors state that this framework enables the study of angular momentum transport in magnetic systems and the conversion of angular momentum between magnetic degrees of freedom and the ionic lattice (e.g., in ultrafast demagnetization). They note that extending these results to the full Brillouin zone would allow for the characterization of anti-crossing points and the computation of band topology and topological transport.

The authors maintain a modest stance regarding future applications, noting that the current theory deals with equilibrium states and that studying non-equilibrium processes (like ultrafast demagnetization) would require including scattering via higher-order force constants and coupling to external driving sources.

The Great Dance-Off: When Atoms and Magnets Mix

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