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Numerical analysis of the Landau--Lifshitz--Bloch equation with spin-torques

This paper establishes the existence and uniqueness of global strong solutions for the Landau–Lifshitz–Bloch equation with spin-torques above the Curie temperature and proposes fully discrete, linearly implicit finite element schemes that achieve optimal-order convergence and unconditional energy stability, supported by numerical simulations.

Original authors: Agus L. Soenjaya

Published 2026-07-28

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CC BY 4.0

Original authors: Agus L. Soenjaya

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 the tiny magnets inside your hard drive or your smartphone's memory chip are not just static little arrows, but a bustling crowd of dancers. In the cold, they move in perfect, rigid unison. But as things heat up—like when your phone gets warm from heavy use—these dancers start to jitter, wobble, and even change their height, not just their direction. This chaotic dance is the heart of micromagnetism, the science of how magnetic materials behave on a microscopic scale.

For decades, scientists used a classic rulebook called the Landau–Lifshitz equation to predict how these magnetic dancers move. It worked great when things were cool. But when temperatures rise high enough to melt the "order" of the material (a point known as the Curie temperature), the old rulebook breaks down. It forgets that the dancers can shrink or grow in size, not just spin. To fix this, physicists invented a new, more complex set of rules called the Landau–Lifshitz–Bloch (LLB) equation. This new model accounts for the "longitudinal" wiggles—the changes in the strength of the magnet itself. Now, imagine adding a twist: what if you could push these dancers around using an electric current? This is the world of spintronics, where electricity doesn't just carry charge but also carries "spin," allowing us to manipulate magnets with currents. The challenge is that when you combine high heat, changing magnet strength, and electric pushes, the math becomes incredibly messy, almost impossible to solve with a pencil and paper.

This paper is like a master architect stepping in to build a reliable bridge across that mathematical chaos. The author, Agus L. Soenjaya, tackles the problem of how to simulate these hot, current-driven magnetic dances on a computer. The paper does two main things. First, it proves that the new, complex rules actually make sense: it shows that if you start with a specific magnetic state, there is exactly one way the system will evolve over time, and it won't suddenly explode or behave unpredictably. Crucially, this guarantee holds for any starting condition in one or two dimensions, but for three dimensions, the starting magnetic state must be "small" enough to prevent the system from running away. This is a crucial "sanity check" that the underlying physics is sound.

Second, and perhaps more importantly for engineers, the paper designs two different computer algorithms (numerical schemes) to solve these equations. The first is a linear scheme, which is like a fast, efficient assembly line. It's quick to run and gives good results, and the paper proves it converges at the best possible rate in standard distance measures (the $L^2$ and $H^1$ norms). However, the paper admits it cannot mathematically guarantee that this fast scheme will perfectly preserve the system's energy in every single scenario, nor can it prove it converges in the specific "energy norm" that tracks the total energy of the system. The second is a nonlinear scheme, which is more like a careful, slow-motion simulation. It is harder to compute, but the paper proves it is rock-solid: it guarantees stability and, crucially, it preserves the rule that energy should naturally dissipate (fade away) when no current is pushing the system. However, this rigorous guarantee for the nonlinear scheme only holds when the "non-adiabatic torque" (a specific type of spin-torque effect) is negligible. The authors ran simulations to test these methods, showing that both work well in practice, with the fast one being great for general use and the careful one being the champion for scenarios where energy conservation is critical (provided that specific torque is small). The paper doesn't claim to have solved every mystery of magnetism, but it provides the first rigorous, unconditionally stable tools to simulate these high-temperature, current-driven magnetic systems without needing to cheat with approximations.

Problem Statement
This paper addresses the numerical analysis of the Landau–Lifshitz–Bloch (LLB) equation augmented with spin-torque terms, specifically in the regime above the Curie temperature ( $T_c$ ). While the standard Landau–Lifshitz equation fails to capture longitudinal magnetization fluctuations at elevated temperatures, the LLB equation provides a continuum model valid both below and above $T_c$ . The inclusion of spin-torque terms is critical for modeling current-induced magnetization dynamics in spintronics applications, such as heat-assisted magnetic recording (HAMR). The governing equation is a quasilinear vector-valued partial differential equation (PDE) where the effective field includes exchange, internal exchange (Ginzburg–Landau type), and anisotropy contributions. A significant challenge in the analysis of this system is the lack of magnetization length conservation and the complexity introduced by the spin-torque terms, which prevents the system from being written as a variational derivative of an energy functional.

Methodology
The authors employ a two-pronged approach combining rigorous mathematical analysis of the continuous problem with the development and analysis of finite element schemes.

Analytical Framework: The existence and uniqueness of global strong solutions are established for spatial dimensions $d=1, 2, 3$ . For $d=3$ , a smallness assumption on the initial data is required. The proof utilizes Faedo–Galerkin approximations and derives uniform a priori estimates in stronger norms (specifically $L^\infty$ and $H^2$ ). A key difficulty addressed is obtaining a uniform $L^\infty$ bound, which is non-trivial due to the absence of length conservation. The authors also derive physical smoothing and decay estimates, showing that in the absence of current, the magnetization decays to zero and energy dissipates.
Numerical Schemes: Two fully discrete finite element schemes are proposed:
- Linearly Implicit Scheme (Section 4): A linear scheme is developed that applies directly to the LLB equation with spin-torques without requiring regularization. This scheme uses an elliptic projection adapted to the problem.
- Nonlinear Energy-Stable Scheme (Section 5): A fully implicit, nonlinear scheme is proposed for the specific case where the non-adiabatic torque constant $\beta_2 = 0$ . This restriction is physically justified for transition-metal ferromagnets dominated by exchange fields. This scheme is designed to preserve energy dissipation at the discrete level when no current is present.

Key Contributions and Results

Well-Posedness: The paper proves the existence and uniqueness of a global strong solution to the LLB equation with spin-torques. For $d=1, 2$ , this holds unconditionally; for $d=3$ , it holds under a smallness condition on the initial data. Physically relevant smoothing and decay estimates are also derived.
Linear Scheme Convergence: The linearly implicit scheme is shown to achieve optimal-order convergence in the norm $\ell^\infty(0, T; L^2) \cap \ell^2(0, T; H^1)$ . Specifically, for a finite element space of degree $r$ , the spatial convergence rate is $r+1$ and the temporal rate is $1$. This result removes the need for regularization parameters used in previous literature. However, the authors note that proving convergence or stability in the stronger $\ell^\infty(0, T; H^1)$ norm for this linear scheme remains an open analytical challenge due to nonlinear cross-product terms.
Nonlinear Scheme Convergence and Stability: For the case $\beta_2 = 0$ , the nonlinear scheme is proven to be unconditionally stable and convergent in the energy norm $\ell^\infty(0, T; H^1)$ with an optimal rate. Crucially, in the absence of current ( $\nu=0$ ), this scheme preserves the energy dissipation property of the continuous model at the discrete level.
Numerical Experiments: Simulations using the FEniCS package validate the theoretical convergence rates for both schemes. The experiments demonstrate that while the linear scheme is computationally more efficient and produces qualitatively similar results to the nonlinear scheme in the presence of current, the nonlinear scheme is superior for capturing energy dissipation in current-free scenarios.

Significance and Claims
The authors claim this work represents the first time a linear scheme has been proposed to directly approximate the LLB equation (with or without spin-torques) above the Curie temperature, with a rigorous proof of unconditional convergence at an optimal rate. Furthermore, they assert that their second scheme is the first to maintain energy dissipativity at the discrete level for the LLB equation without spin-torques while still achieving optimal convergence rates.

The paper positions these results as improvements over existing literature, which either relied on regularization (introducing parameter-dependent convergence rates) or failed to guarantee energy stability and unconditional convergence in the energy norm. The work bridges the gap between theoretical well-posedness and practical, stable numerical methods for spintronic applications involving high-temperature magnetization dynamics. The authors conclude by noting that while the linear scheme is efficient, the nonlinear scheme is preferable for problems requiring precise energy relaxation modeling, and future work aims to develop a scheme that is both linear and energy-dissipative.

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