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arXiv:2005.04448 · Physical review. B./Physical review. B

Monte Carlo study of a generalized icosahedral model on the simple cubic lattice

Martin Hasenbusch B

Published Jul 6, 2020• 52 citations
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Abstract

We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to $\ensuremath{-}1$ and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter $D$ that controls the density of these voids. For a certain range of $D$, the model undergoes a second-order phase transition. On the critical line, $\text{O}(3)$ symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of $D$, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to $L=400$ by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with $\text{O}(3)$ symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, and $\ensuremath{\omega}=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the $\text{O}(3)$ symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.

DOI
10.1103/physrevb.102.024406

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We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to \ensuremath−1\ensuremath{-}1\ensuremath−1 and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter DDD that controls the density of these voids. For a certain range of DDD, the model undergoes a second-order phase transition. On the critical line, O(3)\text{O}(3)O(3) symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of DDD, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to L=400L=400L=400 by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with O(3)\text{O}(3)O(3) symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain \ensuremathν=0.71164(10)\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)\ensuremathν=0.71164(10), \ensuremathη=0.03784(5)\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)\ensuremathη=0.03784(5), and \ensuremathω=0.759(2)\ensuremath{\omega}=0.759(2)\ensuremathω=0.759(2) for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the O(3)\text{O}(3)O(3) symmetry is yico=\ensuremath−2.19(2){y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)yico​=\ensuremath−2.19(2).
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