Defects Superdiffusion and Unbinding in a 2D XY Model of Self-Driven Rotors

Abstract: We consider a non-equilibrium extension of the two-dimensional (2D) XY model, equivalent to the noisy Kuramoto model of synchronization with short-range coupling, where rotors sitting on a square lattice are self-driven by random intrinsic frequencies. We study the static and dynamic properties of topological defects (vortices) and establish how self-spinning affects the Berezenskii-Kosterlitz-Thouless phase transition scenario. The non-equilibrium drive breaks the quasi-long-range ordered phase of the 2D XY model into a mosaic of ordered domains of controllable size and results in self-propelled vortices that generically unbind at any temperature, featuring superdiffusion $\langle r^2(t)\rangle\sim t^{3/2}$ with a Gaussian distribution of displacements. Our work provides a simple framework to investigate topological defects in active matter and sheds new light on the problem of synchronization of locally coupled oscillators.

• The paper demonstrates that introducing self-driving in a 2D XY model disrupts the quasi-long-range order through intrinsic frequency dispersion.
• The study shows that vortices unbind and exhibit superdiffusive motion with a scaling law of ⟨r²(t)⟩ ∼ t^(3/2), deviating from the classic BKT behavior.
• The paper reveals that increased noise reduces the correlation length and ordered domain size, fundamentally altering the phase transition dynamics.

Defects Superdiffusion and Unbinding in a 2D XY Model of Self-Driven Rotors

The paper "Defects Superdiffusion and Unbinding in a 2D XY Model of Self-Driven Rotors" examines a non-equilibrium extension of the two-dimensional (2D) XY model. This extension is conceptually linked to the noisy Kuramoto model of synchronization, with self-driven rotors located on a lattice. The primary focus is on the behavior of topological defects, specifically vortices, and how the introduction of self-driving mechanisms influences their dynamics and the Berezinsky-Kosterlitz-Thouless (BKT) phase transition.

Key Findings and Contributions

The paper presents a comprehensive study of the self-driven 2D XY model, highlighting the following:

1. Self-Spinning Dynamics: The incorporation of random intrinsic frequencies causes the breaking of the quasi-long-range ordered phase observed in the standard 2D XY model, resulting in domains of ordered patches of variable sizes.
2. Vortex Behavior and Phase Transition: Unlike the equilibrium BKT scenario where vortices remain bound and display diffusive behavior below a critical temperature, self-spinning vortices unbind and exhibit superdiffusion characterized by a $\langle r^2(t)\rangle\sim t^{3/2}$ scaling with Gaussian distributed displacements. Notably, this behavior leads to the absence of a BKT transition in the presence of self-driving mechanisms.
3. Correlation Length and Domain Structure: The size of ordered domains is inversely proportional to the dispersion of intrinsic frequencies, $\sigma^{-1}$. With increasing $\sigma$, the correlation length systematically decreases, altering the critical properties of the model.
4. Implications on Interacting Defects: Vortex pairs demonstrate a significant deviation from the expected logarithmic interaction of a BKT system, showing instead an unbinding and independent evolution due to the self-spinning dynamics.

Implications and Future Directions

From a theoretical perspective, this investigation expands the understanding of how active matter systems with local energy input behave in contrast to their passive counterparts. The implications are twofold:

• Synchronization and Topological Defects: The study bridges the active matter framework with classical synchronization phenomena, offering insights into how topological defects behave in coupled oscillator systems under non-equilibrium conditions.
• Universal Superdiffusion Behavior: The characterization of superdiffusion in vortices adds to the growing evidence of universal scaling behavior in active 2D systems, previously observed in dense colloidal and active nematic experiments.

Looking forward, the results open avenues for exploring synchronization in other physical systems characterized by intrinsic noise and local energy input, such as colloidal monolayers and spin torque oscillators. Furthermore, the exploration of models with different interaction ranges and noise types may yield insights into broader classes of phase transitions in non-equilibrium systems. Overall, this study establishes a foundational framework for understanding complex dynamics in self-driven lattice systems, contributing significantly to the field of active matter and statistical mechanics.