Higher-order field theories: ${\varphi }^6$, ${\varphi }^8$ and beyond

Abstract

The ${\varphi }^4$ model has been the "workhorse" of the classical Ginzburg--Landau phenomenological theory of phase transitions and, furthermore, the foundation for a large amount of the now-classical developments in nonlinear science. However, the ${\varphi }^4$ model, in its usual variant (symmetric double-well potential), can only possess two equilibria. Many complex physical systems possess more than two equilibria and, furthermore, the number of equilibria can change as a system parameter (e.g., the temperature in condensed matter physics) is varied. Thus, "higher-order field theories" come into play. This chapter discusses recent developments of higher-order field theories, specifically the ${\varphi }^6$, ${\varphi }^8$ models and beyond. We first establish their context in the Ginzburg--Landau theory of successive phase transitions, including a detailed discussion of the symmetric triple well ${\varphi }^6$ potential and its properties. We also note connections between field theories in high-energy physics (e.g., "bag models" of quarks within hadrons) and parametric (deformed) ${\varphi }^6$ models. We briefly mention a few salient points about even-higher-order field theories of the ${\varphi }^8$, ${\varphi }^{10}$, etc.\ varieties, including the existence of kinks with power-law tail asymptotics that give rise to long-range interactions. Finally, we conclude with a set of open problems in the context of higher-order scalar fields theories.

24 pages, 7 figures, Springer book class, invited contribution to the upcoming book "A dynamical perspective on the ${\varphi }^4$ model: Past, present and future", eds. P.G. Kevrekidis and J. Cuevas-Maraver; v2: revisions as a result of feedback from book editor; v3: corrects some typos

References in corpus (13)

• Making Sense of Non-Hermitian Hamiltonians
• Collective Coordinates in One-Dimensional Soliton Models Revisited
• Long-range interactions of kinks
• Scattering of the ${\phi }^8$ kinks with power-law asymptotics
• Multi-kink collisions in the ${\varphi }^6$ model
• Deformed defects for scalar fields with polynomial interactions
• Kink Dynamics in a Parametric ${\varphi }^6$ System: A Model With Controllably Many Internal Modes
• Highly interactive kink solutions
• Analytical study of kinklike structures with polynomial tails
• Topological defects with power-law tails
• New scalar field models and their defect solutions
• A Mechanical Analog of the Two-bounce Resonance of Solitary Waves: Modeling and Experiment
• Resonance phenomena in the ${\phi }^8$ kinks scattering

Cited by in corpus (5)

• Scattering of the ${\phi }^8$ kinks with power-law asymptotics
• Family of Potentials with Power-Law Kink Tails
• Scattering of compactlike structures
• Novel Deformation Function Creating or Destroying any Number of Even Kink Solutions
• Thick branes via higher order field theory models with exponential and power-law tails