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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>Cd</a:mi> <a:mrow> <a:mn>5.7</a:mn> </a:mrow> </a:msub> <a:mi>Yb</a:mi> </a:mrow> </a:math> 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, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>β</b:mi> <b:mo>=</b:mo> <b:mn>0.508</b:mn> <b:mo>(</b:mo> <b:mn>30</b:mn> <b:mo>)</b:mo> </b:mrow> <b:mo>,</b:mo> <b:mo> </b:mo> <b:mrow> <b:mi>γ</b:mi> <b:mo>=</b:mo> <b:mn>1.361</b:mn> <b:mo>(</b:mo> <b:mn>59</b:mn> <b:mo>)</b:mo> </b:mrow> </b:math> , and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>ν</c:mi> <c:mo>=</c:mo> <c:mn>0.792</c:mn> <c:mo>(</c:mo> <c:mn>17</c:mn> <c:mo>)</c:mo> </c:mrow> </c:math> , respectively. We confirmed that our data satisfy the hyperscaling relation and estimated the other critical exponents <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>α</d:mi> <d:mo>=</d:mo> <d:mo>−</d:mo> <d:mn>0.376</d:mn> <d:mo>(</d:mo> <d:mn>51</d:mn> <d:mo>)</d:mo> </d:mrow> <d:mo>,</d:mo> <d:mo> </d:mo> <d:mrow> <d:mi>δ</d:mi> <d:mo>=</d:mo> <d:mn>3.68</d:mn> <d:mo>(</d:mo> <d:mn>23</d:mn> <d:mo>)</d:mo> </d:mrow> </d:math> , and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>η</e:mi> <e:mo>=</e:mo> <e:mn>0.282</e:mn> <e:mo>(</e:mo> <e:mn>65</e:mn> <e:mo>)</e:mo> </e:mrow> </e:math> . These results show a 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 eight 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.