Topological defects in out-of-equilibrium systems

TL;DR

This thesis investigates how activity, mobility, and non-reciprocity influence topological defects and order in two-dimensional non-equilibrium systems, revealing the critical role of motility and non-reciprocal interactions.

Contribution

It introduces new insights into defect dynamics and phase transitions in active and non-reciprocal XY models, including a continuum theory for non-reciprocal interactions.

Findings

Mobility restores quasi-long-range order via BKT transition.

Defects unbind at all temperatures in noisy Kuramoto lattices.

Non-reciprocity affects defect shapes and pattern formation.

Abstract

In this PhD thesis, we study topological defects in two-dimensional non-equilibrium systems, focusing on active extensions of the XY model, including activity, mobility and non-reciprocity. In a noisy Kuramoto lattice with short-range coupling, intrinsic frequency heterogeneity destroys quasi-long-range order and fragments the system into finite domains. Defects unbind at all temperatures and exhibit superdiffusive random walks, advected by evolving domain boundaries. By contrast, when oscillators are allowed to move in space, the system undergoes a Berezinskii-Kosterlitz-Thouless transition and regains quasi-long-range order, revealing the fundamental role of motility in sustaining coherence. We also analyse a non-reciprocal O(2) model with vision-cone couplings and derive a continuum theory that captures the same large-scale physics. Non-reciprocity selects defect shapes, enriches the annihilation process, and reshapes patterns through advection. Together, these results elucidate the fundamental role of activity and non-reciprocity in shaping topological defects and ordering in non-equilibrium systems. Keywords: Topological defects, XY model, Steep XY model, Kuramoto model, Non-reciprocal interactions, Active matter, Phase transitions, Berezinskii-Kosterlitz-Thouless transition, Non-equilibrium statistical mechanics