Comparative Study of Different Discretizations of the ${\varphi }^4$ Model

Abstract

We examine various recently proposed discretizations of the well-known ${\varphi }^4$ field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing $h$, as one deviates from the continuum limit of $h\to 0$. We then proceed to a more ``stringent'' comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well-known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness and attempt to extract qualitative trends and issue pertinent warnings about some of the surprising resulting properties.

21 pages, 16 figures. Submitted for publication

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• Fermion spectrum and localization on kinks in a deconstructed dimension
• On the kinks in discrete systems