Critical exponents of the Ising model with quenched structural disorder and long-range interactions at spatial dimension $d=3$

TL;DR

The paper investigates the critical behavior of a three-dimensional Ising system with quenched structural disorder and long-range interactions decaying as $|x|^{-(d+\sigma)}$. Using a field-theoretic renormalization group framework at three loops in the massless minimal subtraction scheme with two couplings $u$ and $v$, the authors obtain the RG functions and analyze the disorder-induced fixed point to extract the correlation-length exponent $\nu$ via fixed-dimension resummation. They find that $\nu(\sigma)$ exhibits a nonmonotonic dependence in the range $d/2<\sigma<2-\eta_{\rm SR}$, with significant deviations from the undiluted long-range Ising model and notable instability in the three-loop results for larger $\sigma$, suggesting that the two-loop truncation may be more reliable in this regime. The results emphasize a distinct random LR universality class for the diluted LR Ising model and motivate Monte Carlo studies to validate the perturbative predictions and map the $\sigma$-dependence of the critical exponents.

Abstract

We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance $x$ as $\sim x^{-d-σ}$ in a $d$-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter $σ$. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent $ν$ characterising the new universality class for $d=3$ and for those values of $σ$ for which long-range interactions are relevant for the critical behaviour.

Figures (6)

• Figure 1: (Colour online) Structurally disordered two-dimensional ($d = 2$) diluted Ising model: yellow disks represent lattice sites ${\bf x}$ occupied by Ising spins ${S}_{\bf x}$ (blue arrows), while dark gray sites correspond to non-magnetic impurities.
• Figure 2: (Colour online) Schematic RG flows (denoted by red arrows) for the LR interacting random Ising model at $d<2\sigma$ in the parametric space of coupling constants $(u,v)$. Unstable FPs $G$ and $P$ are shown by blue discs, while stable FPs $U$and $R$ are depicted with yellow squares.
• Figure 3: (Colour online) Correlation length exponent $\nu$ for the Ising model with LR interactions and quenched disorder at $d = 3$ as a function of $\sigma$. The data in the region $d/2<\sigma<2-\eta_{\rm SR}$ (denoted as RLR) are obtained within the two-loop approximation with the help of the Padé--Borel--Leroy procedure at $b_{\mathrm{opt}} = 27$ (red solid curve) and within the three-loop approximation with the help of the "truncated" version of the subsequent resummation procedure (solid black curve). The values of $\nu$ for the mean field (MF) region ($\sigma{ \leqslant} d/2$) are shown by thin dashed line. Estimates of $\nu$ for the random SR (RSR) model, valid for $\sigma<2-\eta_{\rm SR}$, obtained within different approximations, are also shown: two-loop approximation value $\nu_{\rm RSR} = 0.665$ (red dashed line), three-loop approximation value $\nu_{\rm RSR} = 0.668$ (black dashed line), six loop approximation value $\nu_{\rm RSR} = 0.675$Kompaniets2021. The light red dot-dashed curve presents two-loop data for the critical exponent $\nu$ of the undiluted LR Ising model.
• Figure A.1: (Colour online) Numerical evaluation of $H_\sigma$ in equation (\ref{['f4sigma']}) with respect to the control parameter $\sigma$. The numerical integration error is estimated as the difference between the results obtained at different levels of computational refinement and is determined by the extent to which the outcome stabilizes upon adding additional sampling points. The resulting uncertainty remains within the size of the plotted markers and does not affect the qualitative behaviour of the curves.
• Figure B.1: (Colour online) Correlation length exponent $\nu$ for the Ising model at $d = 3$ as a function of $\sigma$ in two-loop approximation. Dark purple dashed line: critical exponent before resummation; blue dot-dashed line: resummation using the Padé approximant; blue solid line: the Padé--Borel resummation; while red solid line: the Padé--Borel--Leroy resummation.