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

TL;DR

This study investigates the critical behavior of the one-dimensional long-range Ising model with couplings decaying as $|i-j|^{-(1+σ)}$ for $1/2<σ<1$ using a nonperturbative functional RG in the local potential approximation (LPA) and compares results to Dyson's hierarchical model (DHM). The authors show that the correlation-length exponent $ν$ computed via the FRG in the LPA and via real-space RG in the DHM agree to within about $10^{-3}$ across the full range, highlighting a deep connection between these two approaches: both retain a fixed gradient term while allowing the local potential to flow, with the DHM providing a constructive realization of this truncation. They benchmark these results against perturbative expansions near $σ=1/2$ and $σ=1$, and against Monte Carlo data, finding good agreement in the intermediate regime (up to $σ\approx 0.8$) and identifying limitations near the BKT-like regime as $σ\to 1$. The work suggests that the DHM can serve as an effective surrogate for the LPA in studying long-range critical phenomena and motivates extending the FRG framework to the DHM’s hierarchical field theory to capture topological effects and short-range contributions more accurately.

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 $ν$ 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, $σ\lesssim 1$.

Figures (6)

• Figure 1: Inverse of the critical exponent $\nu$ of the 1D LRI model in the region $1/2 < \sigma < 1$. The red dots are the results coming from a non-perturbative FRG calculation at the LPA level, described in Sec. \ref{['sec2']}. The blue crosses represent the estimates obtained from the DHM via a real-space RG, see Sec. \ref{['sec3']}. The green triangles at $\sigma = 0.5$, $\sigma \approx 0.65$ and $\sigma = 0.875$ are obtained via the effective dimension approach, Eq. \ref{['eq:effdim']}. The dotted line is the two-loop $\epsilon$ expansion \ref{['eq: epsilon expansion']} of fisher1972critical, while the pink stars are the three-loop Padé-Borel resummed exponents of benedetti2025addendum. The black dashed and olive dash-dotted lines are the results coming from expansions around $\sigma = 1$, respectively from kosterlitz1976phase and benedetti2025one. More details about these are provided in Sec. \ref{['sec4']}. A detailed comparison of LPA and DHM results is presented in Fig. \ref{['fig:differencelpahm']}. Finally, Monte Carlo points are taken from Refs. tomita2008monte and uzelac2001critical.
• Figure 2: Schematic representation of the hierarchical structure of interactions in the DHM with $N=3$ levels and $L=8$ sites. The weakest interaction corresponds to the top-level $p=N$. A possible external field $h$ couples linearly to the spins $s_i$ at the sites $i \in \{1,\dots,8\}$.
• Figure 3: One-dimensional chains of $L=8$ Ising spins with either (a) hierarchical or (b) long-range interactions. (a) The interaction matrix elements \ref{['eq: matrix elements adjacency']} of the hierarchical model exhibit ultrametric structure, since they depend on the distance $d_{ij}$, measuring the depth of the lowest common ancestor of two 'leaves' (spins). Arranging distances as a triangle (shown for sites $1$, $2$ and $7$), the ultrametric inequality forces an isosceles configuration with a short base. (b) The intensity of matrix elements \ref{['eq:LR_Hamiltonian']} in the 1D LRI decays as a power law $J_{ij}\sim |i-j|^{-(1+\sigma)}$ of the Euclidean distance $d^{\rm E}_{ij} = |i-j|$. Contrary to (a), Euclidean distances form a generic scalene triangle, for which the usual triangle inequality holds. In these figures, lighter colors correspond to larger numerical values of the matrix elements, while the dark diagonal spots in (b) correspond to $J_{ii}=0$.
• Figure 4: Comparison between the long-range dispersion $\omega(q) = c_\sigma q^\sigma$ and its hierarchical counterpart $\omega_k$ from Eq. \ref{['eq:weddingcake']}. For this plot $\sigma = 0.5$ and $N=20$ have been chosen.
• Figure 5: In the upper panel we show the thermal eigenvalue $y = \nu^{-1}$ obtained as solution of \ref{['eq: eigenperturbation equation']} in the LPA and \ref{['eq: thermal eigenvalue DHM']} for the DHM. The points overlap almost exactly, as in Fig. \ref{['fig:exponentnu']}. Their relative difference, where $\Delta y = y_{\rm LPA}-y_{\rm DHM}$ and $y_{\rm mean} = (y_{\rm LPA}+y_{\rm DHM})/2$, is shown in the lower panel. The error is peaked in the region of large $\sigma$, but it stays below $10^{-3}$ throughout the interval $1/2<\sigma<1$.