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Dynamically Reentrant Skyrmion Phase in Oscillating Magnetic Fields

Using Monte Carlo simulations, this study reveals a novel nonequilibrium skyrmion phase in a periodically driven two-dimensional magnet that emerges between conventional skyrmion and ferromagnetic states, characterized by a dynamically induced reversal of topological charge due to an imbalance in skyrmion creation and annihilation.

Original author: Z. Demir Vatansever, E. Vatansever, A. Vasilopoulos, N. G. Fytas, A. Berger

Published 2026-11-05

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CC0 1.0

Original author: Z. Demir Vatansever, E. Vatansever, A. Vasilopoulos, N. G. Fytas, A. Berger

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of magnetic materials, tiny whirlpools of spinning atoms can form stable, particle-like structures known as skyrmions. These are not made of water or air, but of the magnetic orientation of electrons, swirling in a way that gives them a unique topological identity, much like a knot that cannot be untied without cutting the rope. Because they are so small and robust, scientists have long been interested in using them to store information in future computers. In a calm, steady environment, these magnetic whirlpools exist only within a specific range of temperatures and magnetic fields; if the external field becomes too strong, the whirlpools are destroyed, and the material settles into a uniform, flat state where all spins point in the same direction. For decades, this behavior was understood as a fixed rule of nature, determined by the balance of forces inside the material.

However, nature often behaves differently when it is pushed and pulled rather than left alone. A team of researchers has now discovered that by shaking a magnetic material with a rapidly oscillating magnetic field, they can create a new state of matter that simply does not exist when the system is at rest. Using powerful computer simulations to model a two-dimensional magnetic sheet, the scientists applied a magnetic field that flipped back and forth like a square wave. They found that under the right conditions, the material does not just lose its skyrmions and stay lost. Instead, as the strength of the shaking field increases, the skyrmions disappear and then suddenly reappear, but with their magnetic swirl reversed. This "dynamically re-entrant" phase is a fleeting, nonequilibrium state where the material creates more skyrmions of one type than it destroys during a single cycle of the field, resulting in a net charge that is the opposite of what was there before.

The researchers, working with a model of a magnetic material on a triangular grid, observed that this phenomenon depends entirely on how fast the magnetic field changes. When the field flips slowly enough for the atomic spins to keep up, a window opens up where the reversed skyrmions thrive. In this window, the external field acts as a precise control knob. During the first half of the field's cycle, the forces combine to create a burst of skyrmions with a positive twist. During the second half, the forces align to destroy them, but not completely. Because the creation happens more effectively than the destruction within that specific timeframe, the material is left with a leftover population of skyrmions that have the opposite sign to the original ones. This is not a temporary glitch or a momentary fluctuation; it is a stable, repeating state that persists as long as the oscillating field continues.

This discovery challenges the traditional view that strong magnetic fields simply wipe out complex magnetic textures. The simulations showed that if the field oscillates too quickly, the atomic spins cannot follow the rapid changes, and the special reversed state vanishes, leaving the material in a standard, uniform state. But when the timing is right, the oscillating field becomes a tool to not only stabilize these magnetic whirlpools but to flip their fundamental nature. The study confirms that this behavior is a genuine feature of the system's dynamics, distinct from what happens when the field is simply turned on and left on. It suggests that by tuning the rhythm and strength of an external magnetic field, scientists could potentially manipulate magnetic information in ways that are impossible in a static environment, opening a new path for controlling the topology of magnetic materials.

Technical Summary: Dynamically Reentrant Skyrmion Phase in Oscillating Magnetic Fields

Problem Statement
While magnetic skyrmions are well-established topological spin textures with potential for spintronic applications, their behavior under time-dependent driving fields remains an open question. Existing literature has largely focused on the stability of individual skyrmions or collective dynamics induced by static fields and electric currents. However, the collective nonequilibrium dynamics of extended skyrmion assemblies subjected to periodically oscillating magnetic fields are far less explored. Specifically, it is unknown how periodic driving affects the topological charge and stability of skyrmion phases, and whether such driving can induce novel topological states that have no equilibrium counterpart.

Methodology
The authors investigate the nonequilibrium response of a two-dimensional classical ferromagnetic Heisenberg model on a triangular lattice, incorporating Dzyaloshinskii–Moriya interactions (DMIs), easy-axis anisotropy, and a time-dependent magnetic field. The system is described by a Hamiltonian including nearest-neighbor exchange ( $J$ ), DMI ( $D=1$ ), single-ion anisotropy ( $A=0.1$ ), and an external field $H(t) = h_b + h(t)$ . The time-dependent component $h(t)$ is a square wave with amplitude $h_0$ and half-period $t_{1/2}$ .

The study employs extensive Monte Carlo simulations using the single-spin Metropolis algorithm on lattices of size $L=120$ with periodic boundary conditions. The system is cooled from high temperatures, and after thermalization, simulations run over $10^3$ field periods to accumulate thermal averages. Key observables include:

Period-averaged magnetization ( $Q$ ): Defined as the time average of the instantaneous magnetization over one field period.
Period-averaged skyrmion number ( $\langle \chi_Q \rangle$ ): The time average of the topological charge $\chi_Q$ , calculated via lattice discretization on elementary triangles.

The study systematically varies the bias field ( $h_b$ ), oscillation amplitude ( $h_0$ ), and half-period ( $t_{1/2}$ ) to map the phase diagram in the nonequilibrium regime.

Key Contributions and Results
The primary finding is the discovery of a dynamically re-entrant skyrmion phase that emerges within a finite window of field amplitudes and bias fields, sandwiched between the conventional skyrmion phase and the polarized ferromagnetic phase.

Topological Charge Reversal: In equilibrium, increasing the bias field $h_b$ drives the system from a skyrmion phase (negative $\chi_Q$ ) to a polarized ferromagnetic phase ( $\chi_Q \approx 0$ ). Under periodic driving with intermediate amplitudes ( $h_0 \approx 1.0 - 1.5$ ), the period-averaged skyrmion number $\langle \chi_Q \rangle$ re-emerges with a positive sign (opposite to the equilibrium skyrmion charge) before vanishing at higher amplitudes. This creates a distinct "yellow" region in the $(h_b, h_0)$ phase diagram where the topological charge is reversed.
Mechanism of Re-entrance: The reversal arises from an imbalance between skyrmion creation and annihilation over a single field cycle. During the first half-period, the superposition of the negative oscillatory field and the positive bias field favors the formation of positive skyrmions. During the second half-period, the fields align to drive the system toward a ferromagnetic state, annihilating skyrmions. For specific parameters, the positive contribution generated in the first half-cycle exceeds the negative contribution in the second, resulting in a net positive $\langle \chi_Q \rangle$ .
Dynamic Response Regime: The re-entrant phase exists only when the system can effectively follow the driving field. It is observed for long half-periods ( $t_{1/2} = 100$ and $200$ MCSS) but is strongly suppressed at shorter periods ( $t_{1/2} = 20$ MCSS). In the fast-driving regime, spins cannot follow rapid field reversals, weakening the asymmetric creation and annihilation dynamics required for the re-entrant effect.
Phase Characteristics: The re-entrant region exhibits an inclined, ellipsoidal shape in the $(h_b, h_0)$ plane with a slope of approximately unity, indicating the response is strongest along lines of nearly constant $h_0 - h_b$ . The real-space configurations within this phase remain disordered (skyrmion liquid) rather than forming an ordered lattice, consistent with the energy transfer from periodic driving.

Significance
The paper establishes periodic magnetic driving as a powerful nonequilibrium control parameter for stabilizing and reversing skyrmion topology. The dynamically re-entrant phase represents an intrinsic nonequilibrium response, distinct from transient relaxation effects, which has no equilibrium counterpart. By demonstrating that oscillating fields can induce a topological charge reversal and stabilize a new phase between conventional states, the work highlights the richness of nonequilibrium topological phenomena. These findings provide a general framework for probing collective dynamical phenomena in topological magnetic systems using time-dependent fields, offering a method to manipulate skyrmion creation, annihilation, and topological sign without relying solely on electric currents or static field tuning.

Technical Summary: Dynamically Reentrant Skyrmion Phase in Oscillating Magnetic Fields

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