Second-order and fluctuation-induced first-order phase transitions with functional renormalization group equations

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We investigate phase transitions in scalar field theories using the functional renormalization group (RG) equation. We analyze a system with $U(2)\times U(2)$ symmetry, in which there is a parameter $\lambda_2$ that controls the strength of the first-order phase transition driven by fluctuations. In the limit of $\lambda_2\to0$, the $U(2)\times U(2)$ theory is reduced to an $O(8)$ scalar theory that exhibits a second-order phase transition in three dimensions. We develop a new insight for the understanding of the fluctuation-induced first-order phase transition as a smooth continuation from the standard RG flow in the $O(8)$ system. In our view from the RG flow diagram on coupling parameter space, the region that favors the first-order transition emerges from the unphysical region to the physical one as $\lambda_2$ increases from zero. We give this interpretation based on the Taylor expansion of the functional RG equations up to the fourth order in terms of the field, which encompasses the $\epsilon$-expansion results. We compare results from the expansion and from the full numerical calculation and find that the fourth-order expansion is only of qualitative use and that the sixth-order expansion improves the quantitative agreement.