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We revisit the approach to the lower critical dimension $d_{\rm lc}$ in theIsing-like $\varphi^4$ theory within the functional renormalization group bystudying the lowest approximation levels in the derivative expansion of theeffective average action. Our goal is to assess how the latter, which providesa generic approximation scheme valid across dimensions and found to be accuratein $d\geq 2$, is able to capture the long-distance physics associated with theexpected proliferation of localized excitations near $d_{\rm lc}$. We show thatthe convergence of the fixed-point effective potential is nonuniform when $d\tod_{\rm lc}$ with the emergence of a boundary layer around the minimum of thepotential. This allows us to make analytical predictions for the value of thelower critical dimension $d_{\rm lc}$ and for the behavior of the criticaltemperature as $d\to d_{\rm lc}$, which are both found in fair agreement withthe known results. This confirms the versatility of the theoretical approach.

Approach to the lower critical dimension of the $\varphi^4$ theory in the derivative expansion of the Functional Renormalization Group