Kink Dynamics in a Topological Phi^4 Lattice

TL;DR

This paper evaluates a topological discretization of the phi^4 lattice, finding it preserves certain static properties but is less accurate for dynamic kink interactions compared to standard methods.

Contribution

It provides a numerical analysis of topological lattice discretizations for kink dynamics, highlighting limitations in their accuracy for dynamic phenomena.

Findings

Topological lattices preserve the Bogomol'nyi bound for static kinks.

Accuracy for dynamical kink problems is limited to fine lattices (~0.1 spacing).

Standard discretizations are simpler and comparably effective for dynamic studies.

Abstract

It was recently proposed a novel discretization for nonlinear Klein-Gordon field theories in which the resulting lattice preserves the topological (Bogomol'nyi) lower bound on the kink energy and, as a consequence, has no Peierls-Nabarro barrier even for large spatial discretizations (h~1.0). It was then suggested that these ``topological discrete systems'' are a natural choice for the numerical study of continuum kink dynamics. Giving particular emphasis to the phi^4 theory, we numerically investigate kink-antikink scattering and breather formation in these topological lattices. Our results indicate that, even though these systems are quite accurate for studying free kinks in coarse lattices, for legitimate dynamical kink problems the accuracy is rather restricted to fine lattices (h~0.1). We suggest that this fact is related to the breaking of the Bogomol'nyi bound during the kink-antikink interaction, where the field profile loses its static property as required by the Bogomol'nyi argument. We conclude, therefore, that these lattices are not suitable for the study of more general kink dynamics, since a standard discretization is simpler and has effectively the same accuracy for such resolutions.