A Novel phase in the phase structure of the $(g\phi^4 + h\phi^6)_{1+1}$ field theoretic model

TL;DR

This paper investigates a phase transition in a 1+1 dimensional $g\,\phi^4+h\,\phi^6$ model, revealing a transition from Hermitian to non-Hermitian phases due to quantum effects, and introduces a new renormalization group approach.

Contribution

It is the first to demonstrate a quantum-induced Hermitian to non-Hermitian phase transition in this model and introduces a novel renormalization group equation for vacuum energy invariance.

Findings

Identified a Hermitian to non-Hermitian phase transition.

Quantum corrections can break hermiticity while maintaining physical acceptability.

Developed a new renormalization group equation for vacuum energy invariance.

Abstract

In view of the newly discovered and physically acceptable $PT$ symmetric and non-Hermitian models, we reinvestigated the phase structure of the ($g\phi^{4}+h\phi^{6}$)$_{1+1}$ Hermitian model. The reinvestigation concerns the possibility of a phase transition from the original Hermitian and $PT$ symmetric phase to a non-Hermitian and $PT$ symmetric one. This kind of phase transition, if verified experimentally, will lead to the first proof that non-Hermitian and $PT$ symmetric models are not just a mathematical research framework but are a nature desire. To do the investigation, we calculated the effective potential up to second order in the couplings and found a Hermitian to Non-Hermitian phase transition. This leads us to introduce, for the first time, hermiticity as a symmetry which can be broken due to quantum corrections, \textit{i.e.}, when starting with a model which is Hermitian in the classical level, quantum corrections can break hermiticity while the theory stays physically acceptable. In fact, ignoring this phase will lead to violation of universality when comparing this model predictions with other models in the same class of universality. For instance, in a previous work we obtained a second order phase transition for the $PT$ symmetric and non-Hermitian $(-g\phi^{4})$ and according to universality, this phase should exist in the phase structure of the ($g\phi^{4}+h\phi^{6}$) model for negative $g$. Finally, among the novelties in this letter, in our calculation for the effective potential, we introduced a new renormalization group equation which describes the invariance of the bare vacuum energy under the change of the scale. We showed that without this invariance, the original theory and the effective one are inequivalent.