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arXiv:1709.03720

Study of spin models with polyhedral symmetry on square lattice

Tasrief Surungan, Yutaka Okabe

Published Sep 12, 2017• 2 citations
arXivHTMLPDF

Abstract

Anisotropy is important for the existence of true long range order in two dimensional (2D) systems. This is firmly exemplified by the $q$-state clock models in which discreteness drives the quasi-long range order into a true long range order at low temperature for $q> 4$. Previously we studied 2D edge-cubic spin model, which is one of the discrete counterpart of the continuous Heisenberg model, and observed two finite temperature phase transitions, each corresponds to the breakdown of octahedral ($O_h$) symmetry and $C_{3h}$ symmetry, which finally freezes into ground state configuration. The present study investigates discret models with polyhedral symmetry, obtained by e equally partioning the $4π$ of the solid angle of a sphere. There are five types of models if spins are only allowed to point to the vertices of the polyhedral structures such as Tetrahedron, Octahedron, Hexahedron, Icosahedron and Dodecahedron. By using Monte Carlo simulation with cluster algorithm we calculate order parameters and estimate the critical temperatures exponents of each model. We found a systematic decrease in critical temperatures as the number of spin states increases (from the Tetrahedral to Dodecahedral spin model).

DOI
10.48550/arxiv.1709.03720

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Anisotropy is important for the existence of true long range order in two dimensional (2D) systems. This is firmly exemplified by the qqq-state clock models in which discreteness drives the quasi-long range order into a true long range order at low temperature for q>4q> 4q>4. Previously we studied 2D edge-cubic spin model, which is one of the discrete counterpart of the continuous Heisenberg model, and observed two finite temperature phase transitions, each corresponds to the breakdown of octahedral (OhO_hOh​) symmetry and C3hC_{3h}C3h​ symmetry, which finally freezes into ground state configuration. The present study investigates discret models with polyhedral symmetry, obtained by e equally partioning the 4π4π4π of the solid angle of a sphere. There are five types of models if spins are only allowed to point to the vertices of the polyhedral structures such as Tetrahedron, Octahedron, Hexahedron, Icosahedron and Dodecahedron. By using Monte Carlo simulation with cluster algorithm we calculate order parameters and estimate the critical temperatures exponents of each model. We found a systematic decrease in critical temperatures as the number of spin states increases (from the Tetrahedral to Dodecahedral spin model).
The study of phase transition is an important subject in condensed matter physics as some materials are more versatile when they are at certain phase. The low temperature ferromagnetic phase of magnetic material is a rm example of these. Passing by the Curie temperature, the paramagnetic phase of magnetic materials will change to a ferromagnetic phase. There are three main variables controlling a magnetic phase transition, namely the symmetry of spins, coupling interaction and the lattice structure. It is known that the discreteness can create low temper ature magnetic order, exempli ed by the existence of a true- long range order for the 2D 6-state clock model at lower temperature [1, 2] In this talk we present our study of the ferromagnetic polyhedral spins model on square lattice with bond dilution. Polyhedral spins are the discrete counterpart of the continuous Heisenberg spin model. Two polyhedral spin models, i.e., the icosahedoron and dodecahedron model are discussed. As previously reported, the non-diluted (pure) ferromagnetic case of these models exhibits nite temperature phase transition [3]. We use Monte Carlo simulation with a newly introduced algorithm, two-size PCC algorithm [4]; and probe whether the existing phase transitions are signi cantly a ected by the bond dilution. We simulate the models with various bond concentrations and calculate their corresponding critical temperatures. The phase diagram of the models are presented as a plot of the obtained critical temperatures with respect to their corresponding bond concentrations
The edge-cubic spin model on square lattice is studied via Monte Carlo simulation with cluster algorithm. By cooling the system, we found two successive symmetry breakings, i.e., the breakdown of Oh{O}_{h}Oh​ into the group of C3h{C}_{3h}C3h​, which then freezes into ground-state configuration. To characterize the existing phase transitions, we consider the magnetization and the population number as order parameters. We observe that the magnetization does well at probing the high-temperature transition but fails in the analysis of the low-temperature transition. In contrast, the population number performs well in probing the low- and the high-TTT transitions. We plot the temperature dependence of the moment and correlation ratios of the order parameters and obtain the high- and low-TTT transitions at Th=0.602(1){T}_{h}=0.602(1)Th​=0.602(1) and Tl=0.5422(2){T}_{l}=0.5422(2)Tl​=0.5422(2), respectively, with the corresponding exponents of correlation length ${\ensuremath{\nu}}_{h}=1.50(1)$ and ${\ensuremath{\nu}}_{l}=0.833(1)$. By using correlation ratio and size dependence of correlation function, we estimate the decay exponent for the high-TTT transition as ${\ensuremath{\eta}}_{h}=0.260(1)$. For the low-TTT transition, ${\ensuremath{\eta}}_{l}=0.267(1)$ is extracted from the finite size scaling of susceptibility. The universality class of the low-TTT critical point is the same as the three-state Potts model.
Phase diagram of a discrete counterpart of the classical Heisenberg model, the truncated tetrahedral model, is analyzed on the square lattice, when the interaction is ferromagnetic. Each spin is represented by a unit vector that can point to one of the 12 vertices of the truncated tetrahedron, which is a continuous interpolation between the tetrahedron and the octahedron. Phase diagram of the model is determined by means of the statistical analog of the entanglement entropy, which is numerically calculated by the corner transfer matrix renormalization group method. The obtained phase diagram consists of four different phases, which are separated by five transition lines. In the parameter region, where the octahedral anisotropy is dominant, a weak first-order phase transition is observed.
Abstract We study the effect of bond dilution of a dodecahedron spin model on ferromagnetic square lattice. The dodecahedron spin constitutes a discrete (polyhedral) counterpart of the continuous Heisenberg spin. Previous work found that, in the absence of dilution, the ferromagnetic dodecahedron system exhibits a finite temperature phase transition. We use Monte Carlo simulation with Wolff cluster algorithm to monitor how phase transitions are impacted by the bond dilution. We simulate the model with various bond concentrations and investigate the resulting phase diagram and critical behavior.
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 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 $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, 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 O(3)\text{O}(3)O(3) symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.