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

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.