Anomalous phonon magnetic moments

This paper identifies three anomalous cases—rotationless axial phonons, divergent gyromagnetic ratios, and anisotropic gyromagnetic ratios—that demonstrate phonon magnetic moments cannot be fully explained by conventional frameworks, thereby revealing new aspects of phonon magnetism and suggesting the existence of phononomagnetic hidden order.

The Gist

Imagine a crystal lattice as a giant, microscopic dance floor. Usually, when we think of "phonons" (the particles that represent sound or heat vibrations in a solid), we picture atoms spinning in circles, like ice skaters twirling. Because they are spinning, they carry two things: angular momentum (the "spin" itself) and a magnetic moment (a tiny magnetic field, like a miniature bar magnet).

In the old textbook view, these two things were always locked together. If an atom spun, it had momentum and magnetism. If it stopped spinning, both disappeared.

This paper says: "Not so fast." The authors have found three weird, "anomalous" cases where this rule breaks down. They discovered that atoms don't need to spin in circles to create magnetism, and that magnetism and momentum don't always point in the same direction.

Here are the three strange cases they found, explained with everyday analogies:

1. The "Ghost Dancers" (Rotationless Axial Phonons)

The Old View: To get a magnetic effect, atoms must physically rotate in a circle.
The New Discovery: Atoms can move in a straight line (up and down) and still create a magnetic effect, as long as they move in a specific, coordinated rhythm.

The Analogy: Imagine a line of people standing in a circle.

• Normal Phonon: Everyone spins around in a circle. They have "spin" and "magnetism."
• Rotationless Phonon: Everyone stands still but jumps up and down. However, they jump in a specific pattern: Person A jumps, then Person B jumps a split-second later, then Person C. Even though no one is spinning, the timing of their jumps creates a "phase difference."
• The Result: The authors found that this coordinated "jumping" creates a "pseudo" spin (a mathematical property) that acts just like real spin. In a material called Cerium Trichloride, they showed that these non-spinning atoms can still react to magnetic fields and generate a magnetic moment, purely because of their synchronized timing. It's like a wave moving through a stadium crowd; the people aren't running around the stadium, but the "wave" has momentum.

2. The "Tug-of-War" (Divergent Gyromagnetic Ratios)

The Old View: If the total spin of a group is zero, the total magnetism must also be zero.
The New Discovery: You can have zero total spin but a huge amount of magnetism.

The Analogy: Imagine two people on a seesaw.

• Person A is heavy and spins clockwise.
• Person B is light but spins counter-clockwise.
• If they spin at just the right speeds, their "spin" cancels out perfectly. The total spin is zero.
• However: Imagine Person A is holding a positive charge and Person B is holding a negative charge. When they spin, they create electric currents. Because their charges are opposite, their magnetic fields actually add up instead of canceling out.
• The Result: The authors found this in a material called Boron Nitride. The atoms are spinning in opposite directions so perfectly that their total "spin" is zero, but their magnetic fields are strong. It's like a tug-of-war where the rope doesn't move (zero momentum), but the tension is immense (high magnetism).

3. The "Twisted Arrow" (Anisotropic Gyromagnetic Ratios)

The Old View: If an object has spin pointing "North," its magnetism must also point "North." They are always parallel.
The New Discovery: The spin can point one way, while the magnetism points a completely different way.

The Analogy: Imagine a spinning top.

• Normal Case: The top spins on its axis (pointing up), and its magnetic field points up too.
• The New Case: Imagine a group of dancers. Some are spinning on the floor (creating a magnetic field pointing sideways), while others are spinning on the ceiling (creating a magnetic field pointing up). When you look at the whole group, the "spin" of the group might point North, but the combined "magnetic field" points East.
• The Result: In Gallium Arsenide (a common semiconductor), the authors showed that the atoms' circular motions are misaligned. The "spin" vector and the "magnetic" vector are not lined up; they are twisted relative to each other. This means you could theoretically push the magnetism in one direction while the spin goes in another.

Why This Matters (According to the Paper)

The authors suggest these findings change how we understand the "hidden order" inside materials.

• Hidden Magnetism: We might have been missing magnetic effects in materials because we were only looking for spinning atoms. Now we know that coordinated, non-spinning atoms can also be magnetic.
• New Tools: This suggests that sound waves (phonons) could be used to detect or manipulate hidden magnetic orders that we couldn't see before.
• Fundamental Physics: It forces us to ask: Is the "spin" or the "magnetism" the more important thing when sound interacts with magnetism? The paper shows they can be separated, which opens up new questions about how energy moves through solids.

In short, the paper reveals that the "dance" of atoms in a crystal is more complex than we thought. They don't just need to twirl to create magnetism; they can jump in rhythm, pull in opposite directions, or spin in different directions to create strange and powerful magnetic effects.

Technical Summary

Technical Summary: Anomalous Phonon Magnetic Moments

Problem Statement
Conventional understanding posits that phonon magnetic moments ($m_{ph}$) arise strictly from the circular motion of atoms, which generates a real angular momentum ($l_{ph}$). In this framework, $m_{ph}$ and $l_{ph}$ are assumed to be collinear and proportional via a scalar phonon gyromagnetic ratio ($\gamma_{ph}$), such that $m_{ph} = \gamma_{ph} l_{ph}$. Furthermore, it is generally assumed that non-zero magnetic responses require non-zero real angular momentum. This paper challenges these assumptions by investigating phonon eigenmodes where the relationship between angular momentum and magnetic moment deviates from the conventional picture. Specifically, the authors address cases where magnetic moments arise without circular atomic motion, where magnetic moments exist despite vanishing angular momentum, and where the two vectors are non-collinear.

Methodology
The authors employ a combination of group theoretical analysis and first-principles calculations to derive and verify anomalous phonon behaviors.

1. Formalism: The study distinguishes between real phonon angular momentum (derived from the cross product of displacement and velocity vectors, $u \times \dot{u}$) and phonon pseudo angular momentum (PAM), which arises from phase differences in atomic motions under $n$-fold rotational operations.
2. Material Systems: The analysis focuses on specific lattice symmetries and materials:
   • Hexagonal Boron Nitride (h-BN): Used to model honeycomb lattices and demonstrate rotationless axial phonons and near-divergent gyromagnetic ratios.
   • Kagome Lattices (e.g., Co$_3$Sn$_2$S$_2$, FeGe): Analyzed to show how out-of-plane linear motions can carry PAM via intracell and intercell phase differences.
   • Cerium Trichloride (CeCl$_3$): A rare-earth paramagnet with strong orbit-lattice coupling, used to calculate the magnetic response of rotationless $E_{2u}$ modes.
   • Gallium Arsenide (GaAs): A noncentrosymmetric crystal used to demonstrate anisotropic gyromagnetic ratios.
3. Calculations: First-principles calculations (cited from references [44–47] for h-BN and [56, 57] for GaAs) were used to determine phonon dispersion, eigenvectors, and effective charges. For CeCl$_3$, the authors utilized a formalism involving orbit-lattice coupling to compute phonon frequency splitting (Zeeman effect) in external magnetic fields.

Key Contributions and Results
The paper identifies and characterizes three distinct anomalous cases of phonon magnetism:

1. Rotationless Axial Phonons:

   • The authors demonstrate that phonons can possess effective magnetic moments despite having zero real angular momentum ($l_{ph} = 0$) and involving only linear atomic motion.
   • In hexagonal boron nitride (h-BN) and kagome lattices, out-of-plane linear motions at $K/K'$ valleys or zone centers can carry non-zero PAM due to phase differences between sublattices.
   • In CeCl$_3$, the $E_{2u}$ mode involves linear out-of-plane motion of Cl$^-$ ions with a relative phase difference of $\pm 2\pi/3$. While $l_{ph}$ vanishes, the mode carries spin PAM ($l^p_s = \pm 1$).
   • Result: These modes exhibit a Zeeman splitting in an external magnetic field. The authors define an effective phonon magnetic moment ($m_{ph}^{eff}$) based on this splitting, finding values on the order of $1 \mu_B$ at low temperatures, which generates a real magnetic moment via spin polarization in the paramagnetic host.

2. Divergent Gyromagnetic Ratios:

   • The study identifies cases where a finite magnetic moment exists despite vanishing net angular momentum, leading to a divergent gyromagnetic ratio ($\gamma_{ph} \to \infty$).
   • In monolayer h-BN, the fast transverse acoustic (TA) branch at the $K$ point features B and N sublattices revolving in opposite directions. This results in a near-zero net angular momentum ($l_{ph}^+ \approx -l_{ph}^-$) but a substantial net magnetic moment because the effective charges ($Z^*$) are opposite, causing the magnetic contributions to align rather than cancel.

3. Anisotropic Gyromagnetic Ratios:

   • The authors show that in noncentrosymmetric crystals like GaAs, the phonon angular momentum vector ($l_{ph}$) and the magnetic moment vector ($m_{ph}$) can be non-collinear.
   • This occurs when atomic angular momentum vectors on different sublattices are misaligned (e.g., Ga ions aligned along $z$, As ions in the $xy$ plane).
   • Result: The relationship between $m_{ph}$ and $l_{ph}$ becomes tensorial ($m_{ph} = \gamma_{ph} l_{ph}$), with the vectors being nearly orthogonal along certain trajectories in the Brillouin zone.

Significance and Claims
The paper claims that these findings fundamentally alter the understanding of phonon magnetism by decoupling magnetic response from the necessity of circular atomic motion.

• Hidden Orders: The existence of rotationless axial phonons suggests the possibility of "phononomagnetic hidden order," where phonons couple to intrinsic orders with atomic-scale phase structures (e.g., chiral charge-density waves) that are otherwise difficult to probe.
• Spin-Phonon Coupling: The results challenge the assumption that real angular momentum is the primary driver for spin-phonon coupling, suggesting that PAM and effective magnetic moments are sufficient.
• New Physical Effects: The anisotropic and divergent ratios imply the existence of novel phenomena, such as a planar Zeeman effect (where the field is perpendicular to angular momentum) and an orthogonal phonon Barnett effect.
• Multipolar Interactions: The non-collinear nature of these moments indicates that phonons can generate higher-order magnetic multipoles, potentially allowing coupling to magnetic field gradients and electric fields, which could be detected via X-ray or neutron scattering.

The authors conclude that these anomalous behaviors are not limited to the specific materials studied but likely represent a broader class of phenomena in noncentrosymmetric crystals, offering new avenues for manipulating ultrafast magnetism and spin transport.