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Results in Physics

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

Mouhcine Azhari, Unjong Yu

Published Jun 16, 2023• 5 citations
Publisher

Abstract

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.

DOI
10.1016/j.rinp.2023.106628

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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.
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 employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.
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.
Phase diagram and critical phenomena of a frustrated Blume–Capel model with alternating single-ion anisotropy ±D, nearest- and next-nearest-neighbour exchange interactions with equal magnitude J are studied by the Monte Carlo simulations using Wang–Landau sampling and nonequilibrium relaxation method. The collinear ordered phase due to next-nearest-neighbour interaction and partial ordered phase due to alternating anisotropy are confirmed together with a paramagnetic phase. We obtained the numerical results which strongly suggests that a single phase transition at D=0 splits into two Ising lines as soon as D is switched on. With the increase of D, the critical point of the easy-axis sublattice stays on a usual Ising-like critical line while that of the easy-plane sublattice passes through the tricritical point and the transition becomes of the first order as in the usual Blume–Capel model.
The critical properties of the spin-1 Blume-Capel model in two dimensions is studied on Voronoi-Delaunay random lattices with quenched connectivity disorder. The system is treated by applying Monte Carlo simulations using the heat-bath update algorithm together with single histograms re-weighting techniques. We calculate the critical temperature as well as the critical exponents as a function of the crystal field $\mathrm{\ensuremath{\Delta}}$. It is found that this disordered system exhibits phase transitions of first- and second-order types that depend on the value of the crystal field. For values of $\mathrm{\ensuremath{\Delta}}\ensuremath{\le}3$, where the nearest-neighbor exchange interaction JJJ has been set to unity, the disordered system presents a second-order phase transition. The results suggest that the corresponding exponent ratio belongs to the same universality class as the regular two-dimensional ferromagnetic model. There exists a tricritical point close to ${\mathrm{\ensuremath{\Delta}}}_{t}=3.05(4)$ with different critical exponents. For ${\mathrm{\ensuremath{\Delta}}}_{t}\ensuremath{\le}\mathrm{\ensuremath{\Delta}}<3.4$ this model undergoes a first-order phase transition. Finally, for $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}3.4$ the system is always in the paramagnetic phase.
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.