Theory of magnetic phase transitions in helimagnetism

Magnetism is woven into everyday life, from the massive generators that power entire cities to the tiny memory chips that store information in our devices. Yet beneath this familiar phenomenon lies a world of remarkable complexity—one that originates from a purely quantum mechanical property of electrons, the spin. A rich variety of interactions emerge from this property that govern the macroscopic properties in magnetic materials. Beyond conventional magnetism, where individual magnetic moments have collinear alignment, helimagnetism arises from a non-collinear arrangement of magnetic moments across the crystal. The fascination deepens with two-dimensional materials, which are crystals just a few atoms thick, where magnetism exhibits fundamentally different and often exotic behaviors compared to bulk crystals. These complex states are not only of fundamental interest but also host novel magnetic textures that promise advanced spintronic devices and next-generation technologies.

Understanding the origins of these properties allows us to manipulate magnetism at the atomic level. The collective excitation modes arising from magnetic order, known as magnons form the fundamental excitations spectra of the material. Since the functionality of a magnetic material in devices depends on the temperature range over which magnetic order persists, the critical temperature and the associated phase transition are of central importance.

In this work, we develop a comprehensive theoretical framework to describe magnetic phase transitions in single-𝑄 helimagnetic systems, with a particular focus on two-dimensional materials. We present two distinct approaches, the Holstein–Primakoff bosonization and Green’s function method using linear response, to calculate magnon energies at higher temperatures and determine the critical temperature of phase transition. Within the Green’s function formalism, two decoupling schemes are explored, the Random Phase Approximation (RPA) and the Callen decoupling (CD) method, where the latter is often overlooked in the literature. We also present an implementation strategy that preserves the generality of interactions, making it applicable to a broad class of helimagnetic materials.

The framework was thoroughly benchmarked against experimental and first-principles data, demonstrating its accuracy and reliability. We identified the most suitable approach for different classes of magnetic materials. The methods accurately capture thermally renormalized magnon spectra and enable reliable predictions of critical temperatures. Overall, the framework provides a versatile and practical tool for predicting magnetic behavior, enabling the exploration and design of complex spin textures and low-dimensional magnetic devices for future technologies.