We investigate the critical behavior of the one-dimensional Ising model with long-range interactions using the functional renormalization group in the local potential approximation (LPA), and compare our findings with Dyson's hierarchical model (DHM). While the DHM lacks translational invariance, it admits a field-theoretical description closely resembling the LPA, up to minor but nontrivial differences. After reviewing the real-space renormalization group approach to the DHM, we demonstrate a remarkable agreement in the critical exponent $\nu$ between the two methods across the entire range of power-law decays $1/2<\sigma <1$. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, $\sigma \lesssim 1$.

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• Critical phenomena
• Long-range interactions
• Classical spin models
• Lattice models in statistical physics
• Renormalization group