Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.08385 · Journal of Statistical Mechanics Theory and Experiment

Monte Carlo studies of the Blume–Capel model on nonregular two- and three-dimensional lattices: phase diagrams, tricriticality, and critical exponents

Mouhcine Azhari, Unjong Yu

Published Mar 1, 2022• 17 citations
arXivHTMLPDF

Abstract

Abstract We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy Δ / J ∈ 0 , 1 , 1.34 ( for LL ) , 1.51 ( for DTL and DSC ) , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t / J = 1.3457(1); k B T t / J = 0.309(2)), (Δ t / J = 1.5766(1); k B T t / J = 0.481(2)), and (Δ t / J = 1.5933(1); k B T t / J = 0.569(4)) for LL, DTL, and DSC, respectively.

DOI
10.1088/1742-5468/ac561b

Related papers

Tricriticality and finite-size scaling in the triangular Blume-Capel ferromagnet

Dimitrios Mataragkas, Alexandros Vasilopoulos, Nikolaos G. Fytas +1 more

We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.

arXiv:2411.11689•Physical Review Research•Feb 27, 2025•13 citations

Static universality of the Ising and Blume–Capel models on two-dimensional Penrose tiles

Mouhcine Azhari, Unjong Yu

We study the phase transitions and the critical behavior of the Ising and Blume–Capel models on two kinds of two-dimensional Penrose tiles employing Monte Carlo methods and detailed finite-size scaling analysis. The Wolff algorithm is used for the Ising model whereas the Wang–Landau and Metropolis algorithms are used for the Blume–Capel model to obtain critical temperatures and exponents. The phase diagram of the Blume–Capel model reveals the presence of the double transition, the reentrant behavior, and the tricriticality in the vicinity of the critical single-ion anisotropy. Our results indicate that continuous phase transitions of the Ising and Blume–Capel models on the Penrose tiles belong to the two-dimensional Ising universality class. However, the scaling functions depend on the spin magnitude, anisotropy, and graph shape. Lastly, we verify that the critical Binder cumulant is also universal but depends on the shape of the graph.

Results in Physics•Jun 16, 2023•5 citations

A Monte Carlo Study of Dynamic Phase Transitions Observed in the Kinetic S = 1 Ising Model on Nonregular Lattices

Yusuf Yüksel

In the present paper, we discuss the thermodynamic and dynamic phase transition properties of the kinetic Blume-Capel model with spin-1, defined on non-regular lattices, namely decorated simple cubic, decorated triangular, and decorated square (Lieb) lattice geometries. Benefiting from the recent results obtained for the thermodynamic phase transitions of the aforementioned lattice topologies [Azhari, M. and Yu, U., J. Stat. Mech. (2022) 033204], we explore the variation of the dynamic order parameter, dynamic scaling variance, and dynamic magnetic susceptibility as functions of the amplitude, bias, and period of the oscillating field sequence. According to the simulations, a second-order dynamic phase transition takes place at a critical field period for the systems with zero bias. A particular emphasis has also been devoted to metamagnetic anomalies emerging in the dynamic paramagnetic phase. In this regard, the generic two-peak symmetric behavior of the dynamic response functions has been found in the slow critical dynamics (i.e. dynamic paramagnetic) regime. Our results yield that the characteristics of the dynamic phase transitions observed in the kinetic Ising model on regular lattices can be extended to such non-regular lattices with a larger spin value.

Entropy•May 16, 2025•5 citations

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

Martin Hasenbusch B

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 ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.

arXiv:2005.04448•Physical review. B./Physical review. B•Jul 6, 2020•52 citations

Critical and tricritical singularities from small-scale Monte Carlo simulations: the Blume–Capel model in two dimensions

Leïla Moueddene, Nikolaos G. Fytas, Yurij V. Holovatch +2 more

Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.

arXiv:2401.02720•Journal of Statistical Mechanics Theory and Experiment•Feb 9, 2024•30 citations

Critical and tricritical behavior of the d = 3 Blume-Capel model: Results from small-scale Monte Carlo simulations

Leïla Moueddene, Nikolaos G. Fytas, Bertrand Berche

We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.

arXiv:2410.01710•Physical review. E•Dec 23, 2024•12 citations
Abstract We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy Δ / J ∈ 0 , 1 , 1.34 ( for LL ) , 1.51 ( for DTL and DSC ) , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t / J = 1.3457(1); k B T t / J = 0.309(2)), (Δ t / J = 1.5766(1); k B T t / J = 0.481(2)), and (Δ t / J = 1.5933(1); k B T t / J = 0.569(4)) for LL, DTL, and DSC, respectively.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
We study the phase transitions and the critical behavior of the Ising and Blume–Capel models on two kinds of two-dimensional Penrose tiles employing Monte Carlo methods and detailed finite-size scaling analysis. The Wolff algorithm is used for the Ising model whereas the Wang–Landau and Metropolis algorithms are used for the Blume–Capel model to obtain critical temperatures and exponents. The phase diagram of the Blume–Capel model reveals the presence of the double transition, the reentrant behavior, and the tricriticality in the vicinity of the critical single-ion anisotropy. Our results indicate that continuous phase transitions of the Ising and Blume–Capel models on the Penrose tiles belong to the two-dimensional Ising universality class. However, the scaling functions depend on the spin magnitude, anisotropy, and graph shape. Lastly, we verify that the critical Binder cumulant is also universal but depends on the shape of the graph.
In the present paper, we discuss the thermodynamic and dynamic phase transition properties of the kinetic Blume-Capel model with spin-1, defined on non-regular lattices, namely decorated simple cubic, decorated triangular, and decorated square (Lieb) lattice geometries. Benefiting from the recent results obtained for the thermodynamic phase transitions of the aforementioned lattice topologies [Azhari, M. and Yu, U., J. Stat. Mech. (2022) 033204], we explore the variation of the dynamic order parameter, dynamic scaling variance, and dynamic magnetic susceptibility as functions of the amplitude, bias, and period of the oscillating field sequence. According to the simulations, a second-order dynamic phase transition takes place at a critical field period for the systems with zero bias. A particular emphasis has also been devoted to metamagnetic anomalies emerging in the dynamic paramagnetic phase. In this regard, the generic two-peak symmetric behavior of the dynamic response functions has been found in the slow critical dynamics (i.e. dynamic paramagnetic) regime. Our results yield that the characteristics of the dynamic phase transitions observed in the kinetic Ising model on regular lattices can be extended to such non-regular lattices with a larger spin value.
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 ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.
Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.
We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.