One-dimensional long-range Ising model: two (almost) equivalent approximations

Abstract

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 $\hat{I}½$ between the two methods across the entire range of power-law decays $1/2<\ddot{I}ƒ<1$. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, $\ddot{I}ƒ\lesssim 1$.

14 pages, 6 figures. Version of the manuscript accepted for publication in Physical Review B