$O(N)\times O(2)$ scalar models: including $\mathcal{O}(\partial^2)$ corrections in the Functional Renormalization Group analysis

The study of phase transitions in frustrated magnetic systems with $O(N)\times O(2)$ symmetry has been the subject of controversy for more than twenty years, with theoretical, numerical and experimental results in disagreement. Even theoretical studies lead to different results, with some predicting a first-order phase transition while others find it to be second-order. Recently, a series of results from both numerical simulations and theoretical analyses, in particular those based on the Conformal Bootstrap, have rekindled interest in this controversy, especially as they are still not in agreement with each other. Studies based on the functional renormalization group have played a major role in this controversy in the past, and we revisit these studies, taking them a step further by adding non-trivial second order derivative terms to the derivative expansion of the effective action. We confirm the first-order nature of the phase transition for physical values of $N$, i.e. for $N=2$ and $N=3$ in agreement with the latest results obtained with the Conformal Bootstrap. We also study an other phase of the $O(N)\times O(2)$ models, called the sinusoidal phase, qualitatively confirming earlier perturbative results.