🔬 mesoscale physics

Double anticrossings induced by nonlinear magnon interactions

This paper demonstrates that nonlinear magnon interactions, specifically pump-induced nondegenerate three-magnon splitting, dynamically generate double anticrossings by creating two distinct magnon populations that independently couple to the Kittel mode.

Original authors: Geil Emdi, Tomosato Hioki, Koujiro Hoshi, Takahiko Makiuchi, Aoi Yamauchi, Eiji Saitoh

Published 2026-07-22

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CC0 1.0

Original authors: Geil Emdi, Tomosato Hioki, Koujiro Hoshi, Takahiko Makiuchi, Aoi Yamauchi, Eiji Saitoh

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 a world where tiny, invisible waves dance inside a piece of metal, carrying energy and information. These waves are called "magnons," and they are the quantum footsteps of a magnet's spin. Usually, when scientists want to make these waves talk to each other, they build a rigid track with a fixed number of lanes. If you want two waves to mix, you build two lanes; if you want three, you build three. It's like a train system where the tracks are laid in concrete before the train even arrives. But what if the tracks could appear out of thin air, created by the train's own speed? This is the question physicists are asking in the field of condensed matter physics. They are exploring how "nonlinear" effects—where the rules change depending on how hard you push—can create new pathways for these magnetic waves to interact. Understanding this is crucial because it could lead to faster, smarter ways to process information without using electricity, potentially revolutionizing how our future computers and communication devices work.

In a recent study, a team of researchers discovered a way to make these invisible magnetic waves create their own extra lanes, dynamically generating a complex dance that was previously thought to require a more complicated setup. They set up a simple experiment using a thin film of a magnetic metal called permalloy, sandwiched between layers of insulation and gold wires. They sent a steady stream of microwave energy (a "pump") into the material at a specific frequency, around 4 billion cycles per second (4 GHz). Under normal circumstances, you might expect the magnetic waves to simply respond to this push or perhaps split into two identical twins. However, the researchers observed something far more surprising: the system spontaneously created two distinct interaction channels where there should have been only one.

The evidence for this came in the form of "double anticrossings." In the world of wave physics, when two different modes of vibration get close to each other, they usually repel, creating a gap in their frequencies. This looks like an "X" shape on a graph where the lines avoid touching. The researchers saw not just one "X," but two of them appearing simultaneously. As they turned up the power of their microwave pump, these two gaps grew wider and shifted position. Crucially, they found that these double gaps vanished completely when they increased the magnetic field to a high level (98 mT). This disappearance was the smoking gun: it proved that the phenomenon relied on a specific process called "three-magnon splitting," where one high-energy wave breaks apart into two lower-energy waves. At high magnetic fields, the energy rules of the universe forbid this splitting, so the extra lanes simply disappeared.

The team explains that this happens because the microwave pump doesn't just create a uniform wave; it excites waves that are moving in specific directions with different speeds. These moving waves then split into two distinct groups of "children" waves, each with its own unique frequency. One group forms a standing wave that couples with the main magnetic mode, and the second group does the same thing independently. It's as if a single parent wave gave birth to two different families of children, and each family built its own separate bridge to the main highway. The researchers confirmed this by measuring how the size of the gaps and the shift in frequency changed with the power of the pump. They found that the gaps grew in a way that perfectly matched the math for this "nondegenerate" splitting (where the two new waves have different frequencies), rather than a simpler, symmetric split.

The study explicitly rules out other possibilities, such as simple "Kerr-type" effects (where the material just changes its properties uniformly) or the idea that the extra channels were pre-existing features of the device. The fact that the effect disappears when energy conservation forbids the splitting proves that the nonlinear dynamics of the waves themselves are the architects of these new channels. The researchers used computer simulations to model this behavior, and the results matched their real-world measurements perfectly, showing that as the magnetic field changed, the two gaps would naturally become more symmetric.

In short, this paper demonstrates that by using a specific type of microwave push, you can force a single magnetic system to dynamically generate multiple, independent pathways for energy to flow. This isn't just a neat trick with magnets; it suggests a new way to build systems where the connections aren't fixed in stone but can be created, changed, or erased on the fly by the energy flowing through them. It opens the door to exploring complex phenomena like "bright-dark" mode manifolds and other exotic states of matter, all driven by the chaotic, creative energy of nonlinear interactions rather than the rigid geometry of a device.

Technical Summary: Double Anticrossings Induced by Nonlinear Magnon Interactions

Problem and Motivation
Coupling between oscillators is fundamental to physical systems, typically resulting in hybridized eigenmodes separated by a frequency gap, observed as an anticrossing (avoided crossing) in spectra. While linear coupling channels are fixed by device geometry and the number of resonators, nonlinear dynamics offer a route for modes and coupling channels to emerge dynamically. In magnetic systems, magnons (quanta of collective spin excitations) exhibit strong intrinsic nonlinearities, enabling three-magnon splitting (a driven magnon decaying into two lower-frequency magnons) and confluence processes. Previous experiments utilizing uniform excitation ( $k=0$ ) to drive these processes were restricted by momentum conservation to counterpropagating modes with equal frequency, yielding only a single effective coupling channel. While multiple channels have been observed when multiple pre-existing magnetostatic standing-wave modes are present, these originate from independent eigenmodes defined by geometry rather than the nonlinear dynamics themselves. The central question addressed is whether nonlinear dynamics alone can generate multiple coupling channels within a single system without relying on multiple pre-engineered resonant modes.

Methodology
The authors investigated this phenomenon using a 50 nm-thick permalloy (NiFe) film patterned into a trapezoidal geometry on a silicon substrate. The sample was covered with an electrical insulating layer ( $Al_2O_3$ ) and two coplanar waveguides (CPWs) made of Ti/Au.

Experimental Setup: A static in-plane magnetic field ( $B$ ) was applied along the $y$ -axis. Microwaves were applied to one CPW using a signal generator to provide a continuous-wave pump at frequency $f_p$ and a vector network analyzer (VNA) to provide a weak swept probe at frequency $f_{VNA}$ to measure the transmission spectrum ( $S_{21}$ ).
Excitation Strategy: Unlike previous uniform excitation studies, the localized field profile of the CPW predominantly excited finite-wavevector magnons ( $k_p \neq 0$ ).
Analysis: The team analyzed the $S_{21}$ spectra as a function of pump frequency, pump power ( $P$ ), and magnetic field strength. They compared experimental data with micromagnetic simulations and a microscopic theoretical model based on a nondegenerate three-magnon splitting Hamiltonian.

Key Results

Observation of Double Anticrossings: Under microwave pumping ( $f_p = 4$ GHz), the single dip corresponding to the Kittel mode (standing-wave resonance) split into two distinct dips as pump power exceeded a threshold. As pump power increased, the frequency gap ( $f_{gap}$ ) between these dips widened, and their central frequency shifted.
Field Dependence and Three-Magnon Origin: At low magnetic fields ( $B \approx 10.5$ mT), a single anticrossing was observed. As the field increased to $17.2$ mT, a second gap emerged above the pump frequency, resulting in a clear "double anticrossing" structure. Crucially, at high magnetic fields ( $B = 98$ mT), both anticrossings and frequency shifts vanished. This suppression occurs because energy conservation forbids three-magnon splitting when the spin-wave band shifts above $f_p/2$ , confirming the three-magnon origin of the phenomenon and ruling out Kerr-type effects.
Power-Law Scaling: The frequency gap ( $f_{gap}$ $f_{g a p}$ ) and the central frequency shift ( $f_{shift}$ $f_{s hi f t}$ ) exhibited distinct power-law dependencies on the pump power above the threshold ( $P_{th}$ $P_{t h}$ ):
- $f_{gap} \propto (\sqrt{P} - \sqrt{P_{th}})^{0.49}$
- $f_{shift} \propto (\sqrt{P} - \sqrt{P_{th}})^{0.95}$
  These scalings align with the theoretical predictions for nondegenerate three-magnon splitting.
Asymmetry and Evolution: At intermediate fields, the double anticrossing gaps were asymmetric. As the magnetic field increased, the gaps became more symmetric. This evolution was attributed to the reduction of the frequency detuning ( $\Delta$ ) between the two generated magnon populations as the field increased, causing them to converge toward $f_p/2$ .

Mechanism and Theoretical Model
The authors attribute the double anticrossings to nondegenerate three-magnon splitting mediated by finite-wavevector pumped magnons.

Nondegenerate Splitting: Because the pump wavevector $k_p \neq 0$ , momentum conservation enforces nondegenerate solutions ( $k_1 \neq k_2$ ). The pump magnon decays into two secondary magnon populations with distinct frequencies ( $\omega_1, \omega_2$ ) and wavevectors, satisfying $\omega_p = \omega_1 + \omega_2$ and $k_p = k_1 - k_2$ .
Dual Coupling Channels: Within each population, counterpropagating modes form standing-wave states. Each population couples independently to the Kittel mode (via three-magnon confluence and splitting) while satisfying momentum conservation. This creates two effective coupling channels ( $g_1$ and $g_2$ ) between the Kittel mode and the two distinct magnon populations, resulting in the observed double anticrossing.
Model Validation: A three-mode coupled model, incorporating the steady-state amplitudes of the secondary magnons and their coupling to the Kittel mode, successfully reproduced the experimental $S_{21}$ spectra, including the emergence of the central branch at high fields and the power-law scaling of the gaps and shifts.

Significance
The paper demonstrates that nonlinear magnon interactions can dynamically generate multiple coupling channels within a single physical system, without the need for multiple pre-engineered resonant modes. By utilizing finite-wavevector pumping to induce nondegenerate three-magnon splitting, the authors created two distinct magnon populations that couple independently to a single Kittel mode. These results establish finite-wavevector nonlinear magnon dynamics as a viable route for creating emergent phenomena, such as bright–dark mode manifolds and higher-order exceptional points, driven by nonlinear interactions rather than fixed by device geometry.

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