Scalar field theories with $mathbb{Z}_{2}$-symmetry are the traditional playground of critical phenomena. In this work these models are studied using functional renormalization group (FRG) equations at order $partial^2$ of the derivative expansion and, differently from previous approaches, the spike plot technique is employed to find the relative scaling solutions in two and three dimensions. The anomalous dimension of the first few universality classes in $d=2$ is given and the phase structure predicted by conformal field theory is recovered (without the imposition of conformal invariance), while in $d=3$ a refined view of the standard Wilson-Fisher fixed point is found. Our study enlightens the strength of shooting techniques in studying FRG equations, suggesting them as candidates to investigate strongly non-perturbative theories even in more complex cases.
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