🔬 condensed matter

Linear response across interaction regimes in two-dimensional ferromagnets

This paper introduces an efficient Gaussian-based method to solve the quantum Boltzmann equation for two-dimensional ferromagnets, successfully modeling the temperature-driven crossover from ballistic to hydrodynamic magnon transport and demonstrating strong agreement with experimental data on monolayer CrCl $_3$ .

Original authors: Aaron Müller, Pavel E. Dolgirev, Oleksii Malyshev, Eugene Demler

Published 2026-08-17

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CC BY 4.0

Original authors: Aaron Müller, Pavel E. Dolgirev, Oleksii Malyshev, Eugene Demler

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 a world where tiny particles don't just bounce off each other like billiard balls, but instead dance in a complex, coordinated rhythm. This is the realm of quantum physics, specifically the study of how energy and information move through materials. In this corner of science, researchers look at "ferromagnets"—materials that act like tiny, permanent magnets. Inside these materials, the magnetic properties aren't carried by individual atoms alone, but by collective waves of spin called "magnons." Think of magnons like ripples on a pond; when you throw a stone (add energy), the water moves in a wave.

Usually, scientists have had to choose between two ways of describing how these ripples move. One view sees them as independent travelers, zooming through the material without much trouble (called "ballistic" transport). The other view sees them as a thick, sticky fluid where everyone bumps into everyone else, creating a collective flow (called "hydrodynamic" transport). The tricky part is that in real materials, both things happen at the same time, depending on how fast the ripples are moving and how hot the material is. Understanding exactly how these two behaviors mix is crucial for building faster, more efficient computers that use spin instead of electricity, a field known as spintronics. But solving the math to see this mix has been like trying to count every single grain of sand on a beach while the tide is coming in—too complicated for standard calculators.

This paper introduces a clever new way to solve that math problem and uses it to watch how a specific type of magnetic material changes its behavior as it gets warmer. The researchers developed a method to represent the messy, complex distribution of these magnetic waves as a simple sum of smooth, bell-shaped curves (Gaussians). This trick turns an impossible calculation into a manageable one. When they applied this method to a thin, atom-sized sheet of a material called Chromium Trichloride (CrCl₃), they discovered a fascinating temperature-driven switch.

At very low temperatures, the magnetic waves are sparse and move freely, like ghosts gliding through a hallway. In this "ballistic" regime, the magnetic noise detected by a sensor increases as the material gets slightly warmer. However, as the temperature rises, the material heats up enough to excite high-energy waves that crash into each other violently. This forces the system into a "hydrodynamic" regime, where the waves behave like a thick, viscous fluid. Surprisingly, in this fluid state, the magnetic noise actually starts to decrease as the temperature goes up. This reversal is a direct fingerprint of the specific way these magnetic waves interact, a phenomenon that had been hinted at in recent experiments but never fully explained until now.

The authors didn't just guess this; they ran detailed computer simulations based on the fundamental laws of quantum mechanics to map out exactly how the material responds to different temperatures and magnetic fields. Their results show a clear crossover point, around 6 to 7 Kelvin, where the material shifts from the ghost-like ballistic behavior to the fluid-like hydrodynamic behavior. They also compared this to a hypothetical material where the waves interact in a simpler, constant way (like sticky balls). In that simpler case, the noise would just keep going up as it gets hotter. The fact that the real material does the opposite confirms that the unique, momentum-dependent way these waves interact is the key to the anomaly.

This work suggests that the strange "anomalous" magnetic noise observed in recent experiments on CrCl₃ is indeed caused by this transition into a collective, fluid-like state. The researchers are confident in their simulations, which match the experimental data well, though they note that small uncertainties in the distance between the sensor and the material could affect the exact numbers. Ultimately, this paper provides a new, powerful tool for scientists to predict how quantum materials will behave, opening the door to designing better materials for future technologies that rely on controlling the flow of spin.

Technical Summary: Linear Response Across Interaction Regimes in Two-Dimensional Ferromagnets

Problem Statement
Recent discoveries of atomically thin two-dimensional (2D) ferromagnets have highlighted a unique transport phenomenon driven by the momentum dependence of exchange interactions. In these systems, magnon-magnon scattering vertices are proportional to the scalar product of incoming momenta ( $T_{ex} \propto -\mathbf{k} \cdot \mathbf{p}$ ). Consequently, low-momentum magnons interact weakly, while high-momentum magnons scatter strongly, potentially exhibiting collective hydrodynamic behavior. Understanding the transport properties of such systems requires a microscopic framework capable of treating ballistic and hydrodynamic regimes on equal footing. The natural framework is the Quantum Boltzmann Equation (QBE); however, its solution is notoriously difficult due to the multidimensional collision integrals constrained by energy and momentum conservation. Existing approaches often resort to uncontrolled approximations (e.g., single relaxation time) or computationally prohibitive brute-force discretization, making it impossible to accurately compute the full frequency- and momentum-resolved linear response $\chi(\mathbf{q}, \omega)$ across varying temperatures and magnetic fields.

Methodology
The authors develop a numerical method to solve the linearized QBE by representing the equilibrium Bose-Einstein distribution functions as a sum of Gaussians (Gaussian Mixture Model, GMM).

Gaussian Representation: The equilibrium distribution $n^{(0)}_p$ is approximated as a sum of a small number of Gaussians ( $n^{(0)}_p \approx \sum_s a_s e^{-p^2/(2\gamma_s^2)}$ ). This representation, accurate to order $10^{-10}$ with roughly ten Gaussians, renders the multidimensional collision integrals analytically tractable.
Linearized Kinetic Theory: The authors formulate the linear response to a weak external potential perturbation. By expanding the deviation from equilibrium in angular harmonics, the 2D integro-differential equation is reduced to a hierarchy of coupled 1D integral equations.
Numerical Implementation: The integral equations are discretized on a radial momentum grid and truncated in angular harmonics, converting the problem into a sparse linear algebraic system. This approach allows for the efficient computation of the collision kernel matrix elements across a broad range of temperatures and magnetic fields.
Application to CrCl $_3$ : The framework is applied to monolayer CrCl $_3$ , modeled as a 2D bosonic system with a parabolic dispersion and an exchange-driven interaction vertex. The model accounts for the specific lattice structure, exchange coupling ( $J \approx 10.9$ K), and anisotropy.

Key Results

Ballistic-to-Hydrodynamic Crossover: The study resolves a temperature-driven crossover in magnon transport. At low temperatures, the magnon gas is dilute, dominated by weakly interacting low-momentum modes, resulting in ballistic transport. As temperature increases, high-momentum modes become thermally populated; these modes interact strongly, driving the system into a collective hydrodynamic regime characterized by a propagating sound mode.
Full Linear Response: The authors compute the full density-density response function $\chi(\mathbf{q}, \omega)$ . In the hydrodynamic limit, the response exhibits a sharp peak corresponding to a sound mode with linear dispersion. The analysis reveals that the sound mode propagates in the isentropic regime rather than the isothermal regime.
Qubit Dephasing Signatures: The paper calculates the dephasing rate ( $1/T_2$ $1/ T_{2}$ ) of a nitrogen-vacancy (NV) center spin qubit probing the magnetic noise of the magnon gas. The results predict a non-monotonic temperature dependence:
- Low Temperature: Noise increases with temperature (ballistic regime).
- High Temperature: Noise decreases with temperature (hydrodynamic regime).
  This reversal is attributed to the specific scaling of the exchange-driven magnon viscosity ( $\mu_{ex} \sim 1/T$ ), which compensates for the explicit temperature factor in the noise formula.
Comparison with Contact Interactions: The authors contrast exchange-driven fluids with contact-interacting Bose fluids. While contact fluids also exhibit hydrodynamic sound modes, their high-momentum susceptibility decays rapidly rather than saturating. Consequently, qubit sensors (which filter for high momenta) probe the non-interacting ballistic tail in contact fluids, leading to a monotonic increase in noise with temperature, unlike the anomalous decrease seen in exchange fluids.

Significance and Claims
The paper claims to establish a general framework for computing momentum- and frequency-resolved linear response in interacting 2D quantum systems describable within quantum Boltzmann kinetics.

Resolution of Anomalous Noise: The method provides a microscopic interpretation for the anomalous magnetic noise (decreasing with temperature) recently observed in NV-center experiments on monolayer CrCl $_3$ . The authors argue this is a direct fingerprint of exchange-driven magnon hydrodynamics.
Computational Tractability: By overcoming the collision-integral bottleneck via the GMM representation, the work enables accurate solutions of the linearized QBE without resorting to uncontrolled approximations or intractable brute-force methods.
Generality: The framework is not limited to ferromagnets but applies to any 2D system with parabolic dispersion and soft scattering phase spaces, such as flexural phonons in suspended graphene or Bose-Fermi mixtures in semiconductors. The authors note that the method can be extended to multi-mode systems and potentially generalized to non-parabolic bands in the future.

The authors maintain a modest stance regarding quantitative comparisons with specific experiments, noting that the predicted dephasing rates are sensitive to the sample-probe distance ( $d_{NV}^{-4}$ ) and that their simplified theory neglects potential temperature dependencies of the pseudogap and magnon mass. However, the qualitative agreement and the explanation of the non-monotonic trend are presented as a robust validation of the exchange-hydrodynamic picture.

Technical Summary: Linear Response Across Interaction Regimes in Two-Dimensional Ferromagnets

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