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arXiv:0805.1578 · Physical Review B

Critical properties of the edge-cubic spin model on a square lattice

Tasrief Surungan, Naoki Kawashima, Yutaka Okabe

Published Jun 2, 2008• 4 citations
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Abstract

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 ${O}_{h}$ into the group of ${C}_{3h}$, 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-$T$ transitions. We plot the temperature dependence of the moment and correlation ratios of the order parameters and obtain the high- and low-$T$ transitions at ${T}_{h}=0.602(1)$ and ${T}_{l}=0.5422(2)$, 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-$T$ transition as ${\ensuremath{\eta}}_{h}=0.260(1)$. For the low-$T$ transition, ${\ensuremath{\eta}}_{l}=0.267(1)$ is extracted from the finite size scaling of susceptibility. The universality class of the low-$T$ critical point is the same as the three-state Potts model.

DOI
10.1103/physrevb.77.214401

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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 \ensuremathνh=1.50(1){\ensuremath{\nu}}_{h}=1.50(1)\ensuremathνh​=1.50(1) and \ensuremathνl=0.833(1){\ensuremath{\nu}}_{l}=0.833(1)\ensuremathνl​=0.833(1). By using correlation ratio and size dependence of correlation function, we estimate the decay exponent for the high-TTT transition as \ensuremathηh=0.260(1){\ensuremath{\eta}}_{h}=0.260(1)\ensuremathηh​=0.260(1). For the low-TTT transition, \ensuremathηl=0.267(1){\ensuremath{\eta}}_{l}=0.267(1)\ensuremathηl​=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.
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).
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
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 ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc{T}_{c}Tc​ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of qqq-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(nnn) symmetry with nnn=qqq-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(nnn=qqq-1) universality classes.
The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.