A mathematical model of a ferromagnet carrying a spin-polarized current links temperature to the way the material’s magnetization evolves. The study proposes a temperature-dependent thermal gauge potential, a mathematical quantity used to track how a system changes, and a temperature-dependent form of the Landau-Lifshitz-Gilbert equation, an equation for magnetization dynamics.
The calculations compare scenarios at 100 K, 200 K and 300 K under a temperature gradient of 1 K/nm. In that selected setup, the proposed gauge potential changes only slightly with temperature, while the position-dependent magnetization solution varies across the tested temperatures.
A broader gauge description for spin
The manuscript asks whether the operator that carries a quantum state forward in time can be represented as a gauge transformation. In the basic case, the model describes this structure with U(1). When spin freedom is included, it uses the non-Abelian groups SU(2) for spin-1/2 particles and SU(3) for spin-1 particles. Here, “non-Abelian” means the order of mathematical transformations can matter.
The paper then applies that framework to a ferromagnet with a spin-transfer-torque term and a spin-polarized current. It introduces temperature into the conduction-electron wavefunction under an adiabatic approximation, in which magnetization changes more slowly than the conduction-electron motion. The resulting construction is the proposed thermal gauge potential.
What the equations predict
In the model, temperature enters the magnetization equation through the spin average. The authors analytically derive this temperature-dependent LLG equation and obtain a linear approximation in the semi-classical limit, where the system is treated partly with classical methods while retaining its spin-based quantum description.
To make that approximation, the calculation assumes small deviations from the reference state and drops a higher-order nonlinear term. Solving the resulting characteristic equation gives one zero characteristic value and paired positive and negative square-root values based on the three field components. The magnetization plot shown in the paper uses the solution associated with the zero characteristic value.
The reported pattern becomes more strongly oscillatory at higher temperature. At the same time, the z-component decreases with position relative to the x- and y-components. These are features of the selected numerical solution.
A modest change in the proposed field
The temperature dependence is not equally pronounced in every part of the calculation. Across the 100 K, 200 K and 300 K scenarios, the three plotted components of the thermal gauge potential show only a slight change in the chosen model.
The authors also compare two energy scales. They report a thermal-fluctuation energy of about 0.0172 eV, smaller than an s-d interaction energy of about 0.1 eV. The comparison is part of the paper’s interpretation of the modeled system.
A calculation still awaiting a test
This is a model calculation built from analytic equations and numerical scenarios, so its predictions depend on the assumptions used to derive them. The temperature treatment uses an adiabatic approximation, and the linear solution relies on small deviations after a higher-order nonlinear term has been omitted.
The paper shows only the zero-characteristic-value magnetization solution; the other solutions are described as similar but are not displayed. The reported temperature response therefore applies to the selected solution and numerical setup.
What comes next
The paper presents its construction as a way to describe time evolution with gauge transformations and to represent temperature-dependent magnetization dynamics in a spin-polarized ferromagnet. The supplied evidence supports those conclusions within the stated model equations and assumptions.
The study reports support from the National Key R&D Program of China under grant 2022YFA1402703. It says generated data sets are available from the corresponding author on reasonable request. The supplied competing-interest statement is incomplete.
Paper data and sources
Original title: The temperature-dependent non-Abelian gauge potential
Authors: Zheng-Chuan Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text