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Physical review. E

Revisiting the field-driven edge transition of the tricritical two-dimensional Blume-Capel model

Henk W. J. Blöte, Youjin Deng

Published Jun 26, 2019• 15 citations
Publisher

Abstract

We reconsider the tricritical Blume-Capel model on the square lattice with a magnetic field acting on the open boundaries in one direction. Periodic boundary conditions are applied in the other direction. We apply three types of Monte Carlo algorithms, local Metropolis updates, and cluster algorithms of the Wolff and geometric type, adapted to the symmetry properties of the model. Statistical analyses of the bulk magnetization, the bulk Binder ratio, the edge magnetization, and the connected product of the edge and bulk magnetizations lead to new results confirming the presence of a singular edge transition at H_{sc}≈0.68, as we reported earlier [Phys. Rev. E 71, 026109 (2005)PLEEE81539-375510.1103/PhysRevE.71.026109]. We provide a plausible answer concerning a discrepancy between the behavior of the edge Binder ratio reported in that work and our new results.

DOI
10.1103/physreve.99.062133

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We reconsider the tricritical Blume-Capel model on the square lattice with a magnetic field acting on the open boundaries in one direction. Periodic boundary conditions are applied in the other direction. We apply three types of Monte Carlo algorithms, local Metropolis updates, and cluster algorithms of the Wolff and geometric type, adapted to the symmetry properties of the model. Statistical analyses of the bulk magnetization, the bulk Binder ratio, the edge magnetization, and the connected product of the edge and bulk magnetizations lead to new results confirming the presence of a singular edge transition at H_{sc}≈0.68, as we reported earlier [Phys. Rev. E 71, 026109 (2005)PLEEE81539-375510.1103/PhysRevE.71.026109]. We provide a plausible answer concerning a discrepancy between the behavior of the edge Binder ratio reported in that work and our new results.
Using Monte Carlo techniques and finite-size analysis, we investigate several two-dimensional lattice models with open edges, including the Blume-Capel model and the q=1 and 3 Potts models with vacancies. At bulk tricriticality, we find that the open edges are dominated by the vacancies when the surface coupling K(s) and the chemical potential D(s) of the vacancies assume the bulk values. When K(s) and/or D(s) is sufficiently enhanced, an edge phase transition takes place, beyond which spontaneous one-dimensional order occurs on the edges. Edge phase transitions can also be induced by a surface magnetic field H(s) . We numerically determine a number of edge critical exponents and derive phase diagrams in terms of K(s) , D(s) , and H(s) . In the low-temperature region, we observe first-order transitions when K(s) and D(s) are varied; the associated hysteresis loops of surface quantities are remarkably asymmetric. Some further insight into these edge transitions is provided by the exact equivalence of the tricritical q=1 Potts model and the Ising model.
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 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.
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.
The Jx-Jy Blume-Capel model, with and without an external magnetic field, is investigated using the Monte Carlo method. Our aim is to verify the location of the tricritical point, examine the existence of reentrant behavior, and study the magnetocaloric properties of this specifically designed system. In the non-field case, the critical temperature given by the Monte Carlo simulations is lower compared to the conventional antiferromagnetic square lattice. When an external magnetic field is applied, we observed signs of a one-dimensional spontaneous symmetry-breaking phase in the mixed-states regime, which is counterintuitive with secondary specific-heat peak. We take the snapshot of this state and elucidated the underlying mechanisms using mean field theory in the field of spin-1 spinor Bose-Einstein condensate . Regarding reentrant behavior, we confirmed its occurrence as the magnetic field varies, but only when the temperature is kept constant. Temperature-dependent reentrant behavior could not be identified within this system. Finally, we examined the magnetocaloric properties and found no evidence of first-order phase transitions within the region of the considered phase-transition point. Although both inverse and direct magnetocaloric effects are present and related to the mixed state, we identified that the relevant phase transitions are all of the second order.
We investigate several tricritical models on the square lattice by means of Monte Carlo simulations. These include the Blume-Capel model, Baxter's hard-square model, and the q=1 , 3, and 4 Potts models with vacancies. We use a combination of the Wolff and geometric cluster methods, which conserves the total number of vacancies or lattice-gas particles and suppresses critical slowing down. Several quantities are sampled, such as the specific heat C and the structure factor C(s) , which accounts for the large-scale spatial inhomogeneity of the energy fluctuations. We find that the constraint strongly modifies some of the critical singularities. For instance, the specific heat C reaches a finite value at tricriticality, while C(s) remains divergent as in the unconstrained system. We are able to explain these observed constrained phenomena on the basis of the Fisher renormalization mechanism generalized to include a subleading relevant thermal scaling field. In this context, we find that, under the constraint, the leading thermal exponent y(t1) is renormalized to 2- y(t1) , while the subleading exponent y(t2) remains unchanged.