Monte Carlo study on critical exponents of the classical Heisenberg model in ferromagnetic icosahedral quasicrystal

TL;DR

This work tackles the emergence and critical behavior of ferromagnetic order in a 3D icosahedral quasicrystal by simulating the classical Heisenberg model on the Yb-based i-QC Cd$_{5.7}$Yb. Using Monte Carlo simulations with heat-bath, over-relaxation, and parallel tempering, the authors perform finite-size scaling to extract critical exponents: $\nu=0.792(17)$, $\beta=0.508(30)$, $\gamma=1.361(59)$, and through hyperscaling obtain $\alpha=-0.376(51)$, $\delta=3.68(23)$, and $\eta=0.282(65)$, with a transition temperature $T_c=2.1725$. The results satisfy the hyperscaling relations and reveal a universality class for the i-QC distinct from periodic magnets and spin glasses, with a cooperative development of spin correlations across eight site classes. The analysis of site-class magnetization and correlation lengths shows that local environments drive a coordinated transition to ferromagnetism, providing a theoretical basis for interpreting magnetism in rare-earth i-QCs and their approximants. Overall, the study advances understanding of critical phenomena in aperiodic 3D lattices and suggests experimental routes to observe the predicted exponents in Gd- and Eu-based i-QCs and approximants.

Abstract

Quasicrystals (QCs) lack three-dimensional periodicity of atomic arrangement but possess long-range structural order, which are distinct from periodic crystals and random systems. Here, we show how the ferromagnetic (FM) order arises in the icosahedral QC (i-QC) on the basis of the Monte Carlo simulation of the Heisenberg model on the Yb lattice of Cd$_{5.7}$Yb composed of regular icosahedrons. By finite-size scaling of the Monte Carlo data, we identified the critical exponents of the magnetization, magnetic susceptibility, and spin correlation length, $β=0.508(30)$, $γ=1.361(59)$, and $ν=0.792(17)$, respectively. We confirmed that our data satisfy the hyperscaling relation and estimated the other critical exponents $α=-0.376(51)$, $δ=3.68(23)$, and $η=0.282(65)$. These results show a new universality class inherent in the i-QC, which is different from those in periodic magnets and spin glasses. In the i-QC, each Yb site at vertices of the regular icosahedrons is classified into 8 classes with respect to the coordination numbers of the nearest-neighbor and next-nearest-neighbor bonds. We revealed the FM-transition mechanism by showing that the difference in the local environment of each site is governed by cooperative evolution of spin correlations upon cooling, giving rise to the critical phenomena.

Figures (17)

• Figure 1: (a) The regular icosahedron at 12 vertices of which Yb atom is located. (b) The Yb atoms in the icosahedral QC Cd$_{5.7}$Yb for $N=20364$ viewed from the $z$ axis direction. (c) Regular icosahedrons, at 12 vertices of which the Yb atoms are located, in the QC Cd$_{5.7}$Yb. The N.N. bond (orange) with the length being $a_{\rm ico}$ is along the 5-fold axis direction. The N.N.N. bond (gray) with the length being $5.9821~{\rm \AA}$ is along the 2-fold axis direction. The $z$ axis is set along the 5-fold axis direction.
• Figure 2: Local environment of 8 site class $(\lambda=1, 2, \cdots, 8)$. The number indicates the site class $\lambda$. The site indicated by arrow is the site which belongs to the class $\lambda$. The orange bond denotes the N.N. bond along the 5-fold axis direction and the gray bond denotes the N.N.N. bond along the 2-fold axis direction.
• Figure 3: Each site is classified into 8 classes which are represented by 8 colors. The view is from the $z$ axis (5-fold axis) direction.
• Figure 4: Temperature dependence of Binder parameter $U$ calculated for $J_{\rm 5f}=1$ and $J_{\rm 2f}=1$ in the $N=20364$, 26412, 30048, 39360, 47520 and 62868 systems. Inset is enlargement in the vicinity of $T=2.17$.
• Figure 5: The Yb atoms in the icosahedral QC Cd$_{5.7}$Yb for (a) the outer sites and (b) the inner sites for $N=20364$.