One-dimensional long-range Ising model: two (almost) equivalent approximations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911328752369664 |
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| author | Pagni, Valerio Giachetti, Guido Trombettoni, Andrea Defenu, Nicolò |
| author_facet | Pagni, Valerio Giachetti, Guido Trombettoni, Andrea Defenu, Nicolò |
| contents | 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 $ν$ between the two methods across the entire range of power-law decays $1/2 < σ< 1$. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, $σ\lesssim 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_02458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-dimensional long-range Ising model: two (almost) equivalent approximations Pagni, Valerio Giachetti, Guido Trombettoni, Andrea Defenu, Nicolò Statistical Mechanics Mathematical Physics 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 $ν$ between the two methods across the entire range of power-law decays $1/2 < σ< 1$. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, $σ\lesssim 1$. |
| title | One-dimensional long-range Ising model: two (almost) equivalent approximations |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2510.02458 |