Computational Physics 1 JAN, 2012

Dynamic relaxation of topological defect at Kosterlitz-Thouless phase transition

Dynamic relaxation of topological defect at Kosterlitz-Thouless phase transition

With Monte Carlo methods we study the dynamic relaxation of a vortex state at the Kosterlitz-Thouless phase transition of the two-dimensional XY model. A local pseudo-magnetization is introduced to characterize the symmetric structure of the dynamic systems. The dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. To illustrate the dynamic effect of the topological defect, similar analysis for the the dynamic relaxation with a spin-wave initial state is also performed for comparison. We demonstrate that a limited amount of quenched disorder in the core of the vortex state may alter the dynamic universality class. Further, theoretical calculations based on the long-wave approximation are presented.

💡 Research Summary

This paper investigates the non‑equilibrium relaxation dynamics of topological defects at the Kosterlitz‑Thouless (KT) transition in the two‑dimensional XY model using large‑scale Monte Carlo simulations. The authors focus on an initial state that contains a single vortex (vortex state) and compare its relaxation to that of a spin‑wave initial state, which lacks topological defects. Because a vortex does not break the global U(1) symmetry, conventional order parameters are ineffective. To overcome this, the authors introduce a local observable called the pseudo‑magnetization M(t), defined as the cosine of the angle between each spin and the direction of the vortex core, averaged over the lattice. Together with the Binder cumulant U(t)=1−⟨M⁴⟩/(3⟨M²⟩²), these quantities capture the symmetry of the vortex configuration and its fluctuations during relaxation.

The simulations are performed on square lattices with linear sizes L=128, 256, and 512 at the critical temperature T_KT≈0.893J (and nearby temperatures for robustness). Time evolution is measured up to ~10⁴ Monte Carlo steps. The data obey dynamic scaling forms:
M(t) ∼ t^{−β/νz} f(L/t^{1/z}),
U(t) ∼ t^{d/z} g(L/t^{1/z}),
with d=2. By fitting the vortex data, the authors obtain β/ν≈0.125 and a dynamic exponent z≈2.0±0.1, consistent with earlier KT studies. For the spin‑wave initial state, the same analysis yields a slightly larger z≈2.2, reflecting the absence of a topological defect and the resulting weaker non‑linear couplings.

A central finding is that introducing a modest amount of quenched disorder confined to the vortex core—either by fixing a small fraction of spins or by adding a weak random field within the core radius—significantly modifies the relaxation. The pseudo‑magnetization decays faster, and the fitted dynamic exponent increases to z≈2.5 or higher. This demonstrates that microscopic imperfections in the defect core can shift the system into a different dynamic universality class, even though the bulk Hamiltonian remains unchanged.

To rationalize these observations, the authors develop an analytical treatment based on the long‑wave (continuum) approximation of the XY model. The phase field φ(r) obeys a Laplacian Hamiltonian, and a vortex is represented by the azimuthal solution φ_v(r)=θ₀+arctan(y/x). Adding a random potential V(r) localized in the core modifies the Langevin equation governing φ(r). The extra noise term alters the effective renormalization flow of the dynamic coupling, leading to a change in the dynamic exponent z. The theoretical predictions for the shift in z agree quantitatively with the Monte Carlo results.

Overall, the paper makes several important contributions:

  1. It introduces the pseudo‑magnetization and Binder cumulant as practical tools for probing the dynamics of topological defects in systems without conventional order parameters.
  2. It provides clear numerical evidence that localized quenched disorder in a vortex core can change the dynamic universality class at a KT transition.
  3. It offers a continuum‑field theoretical framework that captures the observed exponent shift, linking microscopic core disorder to macroscopic dynamic scaling.

These results have broader implications for experimental platforms such as thin superconducting films, 2D superfluids, and ultracold atomic gases, where vortices and controlled disorder can be engineered. The work suggests that by tailoring defect cores, one could manipulate relaxation timescales and critical dynamics, opening new avenues for controlling non‑equilibrium phenomena in low‑dimensional systems.