One-dimensional long-range Ising model: Two almost equivalent approximations

Year: 2026

Authors: Pagni V., Giachetti G., Trombettoni A., Defenu N.

Autors Affiliation: Swiss Fed Inst Technol, Inst Theoret Phys, Wolfgang Pauli Str 27, CH-8093 Zurich, Switzerland; Univ Paris Cite, Lab Phys, CNRS, ENS,Ecole Normale Super, F-75005 Paris, France; Univ Paris Cite, PSL Univ, Sorbonne Univ, F-75005 Paris, France; Univ Trieste, Dipartimento Fis, Str Costiera 11, I-34151 Trieste, Italy; SISSA, Via Bonomea 265, I-34136 Trieste, Italy; INFN Sez Trieste, Via Bonomea 265, I-34136 Trieste, Italy; CNR IOM DEMOCRITOS Simulat Ctr, Via Bonomea 265, I-34136 Trieste, Italy; CNR INO, Area Sci Pk, I-34149 Trieste, Italy.

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 v 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 less than or similar to 1. Journal/Review: PHYSICAL REVIEW B

Volume: 113 (1)      Pages from: 14406-1  to: 14406-13

More Information: G.G. acknowledges the support of the MSCA Grant No. 101152898 (DREAMS). This research was funded by the Swiss National Science Foundation (SNSF) Grants No. 200021\u2013207537 and No. 200021\u2013236722, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u2019s Excellence Strategy EXC2181/1- 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and by the European Union under GA No. 101077500\u2013 QLRNet. Partial support by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP) is also acknowledged.
KeyWords: Renormalization-group; Kondo Problem; Critical Exponents; Phase-transition; Dynamics
DOI: 10.1103/2y4k-rvcf