🧮 Mathematical Foundations of HELIOS-3D

To rigorously understand the behavior of hopfions as stable information carriers, it is essential to ground their existence in field theory. This document details the defining topological properties that afford hopfions their stability and the mechanics of their eventual decay.

🌀 The Hopf Invariant ($H$)

A magnetic hopfion is a three-dimensional topological soliton characterized by a non-zero Hopf index, $H$. While skyrmions are defined by a 2D winding number, hopfions are defined by the linking of pre-images in 3D space.

The Hopf number is defined as the integral of the magnetic helicity over all space:

$H = \frac{1}{(4\pi)^2} \int_{\mathbb{R}^3} \mathbf{A} \cdot \mathbf{B} \, d^3r$

Where:

• $\mathbf{B}$ is the emergent magnetic field, defined as $\mathbf{B}_i = \frac{1}{2} \epsilon_{ijk} \mathbf{n} \cdot (\partial_j \mathbf{n} \times \partial_k \mathbf{n})$, with $\mathbf{n}$ being the normalized local magnetization vector.
• $\mathbf{A}$ is the vector potential corresponding to the emergent magnetic field such that $\mathbf{B} = \nabla \times \mathbf{A}$.

The Hopf index $H$ is an integer that represents the linking number of the pre-images of any two points on the target sphere $S^2$. Physically, this means that the “tubes” of constant magnetization are linked or knotted. A standard hopfion has $H=1$, where every pair of pre-image curves links exactly once.

🛡️ Topological Protection vs. Decay

The mathematical nature of the Hopf index—being an integer—provides topological protection. A hopfion cannot be continuously deformed into a trivial ferromagnetic state (where all spins align uniformly) without the magnetization locally dropping to zero (a Bloch point singularity). This “unzipping” process requires significant energy.

The stability of the hopfion is governed by the energy barrier ($\Delta E$) separating the hopfion state from the trivial state. The probability of decay due to thermal fluctuations follows an Arrhenius-like law:

$\Gamma = f_0 e^{-\frac{\Delta E}{k_B T}}$

In the HELIOS-3D architecture, the material parameters (exchange stiffness, DMI, and particularly the massive perpendicular magnetic anisotropy of the $Fe_3GaTe_2$ / $RuO_2$ heterostructure) are engineered to create a deep energy minimum for the hopfion state.

• Energy Barriers: Simulations indicate energy barriers typically in the range of $10–50 \, k_B T$ at room temperature.
• Non-Volatile Storage: This robust energy barrier prevents the hopfion from collapsing into a trivial state, providing the physical logic for long-term, non-volatile data storage. The hopfion acts as a discrete, stable bit of information purely by virtue of its knotted geometry.