[Paper Review] Kinks and realistic impurity models in $φ^4$-theory

Mariya A. Lizunova, Jasper Kager+2 more|UvA-DARE (University of Amsterdam)|Jul 9, 2020

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper investigates kink-impurity interactions in 1+1-dimensional φ⁴-theory using realistic Gaussian and Lorentzian impurity profiles, showing qualitative agreement with idealized delta-function impurities but significant quantitative differences in impurity mode properties and collision dynamics. It identifies a regime where kinks lose all kinetic energy upon collision with strong impurities, leading to complete localization.

ABSTRACT

The $φ^4$-theory is ubiquitous as a low-energy effective description of processes in all fields of physics ranging from cosmology and particle physics to biophysics and condensed matter theory. The topological defects, or kinks, in this theory describe stable, particle-like excitations. In practice, these excitations will necessarily encounter impurities or imperfections in the background potential as they propagate. Here, we describe the interaction between kinks and various types of realistic impurity models. We find that realistic impurities behave qualitatively like the well-studied, idealized delta function impurities, but that significant quantitative differences appear in both the characteristics of localized impurity modes, and in the collision dynamics. We also identify a particular regime of kink-impurity interactions, in which kinks loose all of their kinetic energy upon colliding with an impurity.

Motivation & Objective

• To model realistic impurity effects in φ⁴-theory beyond idealized delta-function impurities.
• To analyze how Gaussian and Lorentzian impurity profiles affect kink scattering, capture, and resonance dynamics.
• To determine whether the kink-impurity interaction exhibits a regime of complete kinetic energy loss upon collision.
• To compare the characteristics of localized impurity modes and critical velocities between realistic and idealized impurity models.

Proposed method

• The study uses a (1+1)-dimensional φ⁴-theory Lagrangian with a self-interaction potential modified by a spatially localized impurity function γ(x).
• Impurity profiles are modeled using Gaussian and Lorentzian functions with adjustable width and strength, replacing the idealized Dirac delta function.
• The equation of motion is numerically solved using a second-order finite difference scheme on a spatial and temporal lattice with stability constraints (τ < h).
• Energy conservation is monitored throughout simulations by tracking energy flux through spatial boundaries to ensure numerical accuracy.
• Impurity mode frequency and amplitude are extracted via discrete Fourier transform or min-max averaging of φ(x₀,t) for t > 170, ensuring post-collision analysis.
• The critical velocity for transmission vs. reflection is determined by varying initial kink velocity and observing long-term dynamics.

Experimental results

Research questions

• RQ1 How do Gaussian and Lorentzian impurity profiles quantitatively alter the shape and frequency of localized impurity modes compared to idealized delta-function impurities?
• RQ2 What is the effect of impurity width and strength on the critical velocity for kink transmission or reflection?
• RQ3 Does a regime exist where a kink loses all its kinetic energy upon colliding with a strong impurity, resulting in complete localization?
• RQ4 How do resonance windows in kink-impurity scattering depend on the impurity profile shape and parameters?

Key findings

• The kink-impurity interaction dynamics show qualitative agreement with idealized delta-function impurities, including reflection, transmission, and capture, but with significant quantitative differences in mode properties.
• The amplitude and frequency of the localized impurity mode depend explicitly on the impurity width and strength, deviating from the delta-impurity limit.
• For strong impurities, kinks can lose all their kinetic energy and remain localized at the impurity site, indicating a complete capture mechanism.
• The critical velocity for transmission increases with impurity width, and resonance windows in scattering dynamics shift compared to the delta-impurity case.
• Energy conservation in simulations is maintained within 0.74% deviation, validating the numerical accuracy of the results.
• The Gaussian impurity profile reduces to the Dirac delta limit as its width approaches zero, recovering known results from idealized models.