Catastrophe Theory for $\Gamma$-invariant Unfoldings with Applications to Quantum Many-body Theory

Abstract

The theory of singularities and its broad ramifications, especially catastrophe theory, have found fertile ground in some areas of physics (e.g., caustics, wave optics) for their applications. In the context of quantum many-body theory, however, their results, despite being useful, are not generally known by the scientific community and often require a non-trivial adaptation, mainly due to the effects of symmetries in the associated physical system, which are not encompassed by the original theory. In this article, we provide an extension of the main results of Ren\'e Thom's catastrophe theory for the case of germs and unfoldings possessing special symmetries. In a more mathematically precise language, we provide a proof of the determinacy theorems for germs that are invariant under the action of an arbitrary compact Lie group, and of the transversality and stability theorems for the case of invariant unfoldings. The results obtained can be seen as an extension and adaptation of the works of [7] and [8] on singularity theory of $\mathbb{R}^n$ to $\mathbb{R}^n$ mappings to the particular scenario of catastrophe theory. Finally, we also provide a classification theorem for unfoldings invariant under the action of $\mathbb{Z}_2$, and present some of the possible applications of the theory to the study of phase transitions in quantum many-body systems.