Higher-Order Field Theories: \(\phi ^6\), \(\phi ^8\) and Beyond

  • Avadh Saxena
  • Ivan C. ChristovEmail author
  • Avinash Khare
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 26)


The \(\phi ^4\) 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 \(\phi ^4\) model, in its usual variant (symmetric double-well potential), can only possess two equilibria. Many complex physical systems possess more than two equilibria and, moreover, 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 \(\phi ^6\), \(\phi ^8\) 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 \(\phi ^6\) 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) \(\phi ^6\) models. We briefly mention a few salient points about even-higher-order field theories of the \(\phi ^8\), \(\phi ^{10}\), 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.



I.C.C. acknowledges the hospitality of the Center for Nonlinear Studies and the Theoretical Division at Los Alamos National Laboratory (LANL), where the authors’ collaboration on higher-order field theory was initiated. We acknowledge the support of the U.S. Department of Energy (DOE): LANL is operated by Triad National Security, L.L.C. for the National Nuclear Security Administration of the U.S. DOE under Contract No. 89233218CNA000001. I.C.C. also thanks V.A. Gani and P.G. Kevrekidis for many insightful discussions on kinks, collisions, collective coordinates, Manton’s method and \(\phi ^8\) field theory. A.K. is grateful to INSA (Indian National Science Academy) for the award of INSA Senior Scientist position.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Avadh Saxena
    • 1
  • Ivan C. Christov
    • 2
    Email author
  • Avinash Khare
    • 3
  1. 1.Los Alamos National LaboratoryTheoretical Division and Center for Nonlinear StudiesLos AlamosUSA
  2. 2.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA
  3. 3.Department of PhysicsSavitribai Phule Pune UniversityPuneIndia

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