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Physical Review E

Critical behavior of the spin-1 Blume-Capel model on two-dimensional Voronoi-Delaunay random lattices

Francivaldo Pinheiro Fernandes, Douglas F. de Albuquerque, F. W. S. Lima, J. A. Plascak

Published Aug 26, 2015• 9 citations
Publisher

Abstract

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 $J$ 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.

DOI
10.1103/physreve.92.022144

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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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. 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 \ensuremathΔ\ensuremath≤3\mathrm{\ensuremath{\Delta}}\ensuremath{\le}3\ensuremathΔ\ensuremath≤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 \ensuremathΔt=3.05(4){\mathrm{\ensuremath{\Delta}}}_{t}=3.05(4)\ensuremathΔt​=3.05(4) with different critical exponents. For \ensuremathΔt\ensuremath≤\ensuremathΔ<3.4{\mathrm{\ensuremath{\Delta}}}_{t}\ensuremath{\le}\mathrm{\ensuremath{\Delta}}<3.4\ensuremathΔt​\ensuremath≤\ensuremathΔ<3.4 this model undergoes a first-order phase transition. Finally, for \ensuremathΔ\ensuremath≥3.4\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}3.4\ensuremathΔ\ensuremath≥3.4 the system is always in the paramagnetic phase.
The effects of bond randomness on the universality aspects of a two-dimensional (d = 2) Blume-Capel model embedded in the triangular lattice are discussed. The system is studied numerically in both its first- and second-order phase-transition regimes by a comprehensive finite-size scaling analysis for a particularly suitable value of the disorder strength. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the two-dimensional (2D) random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs again to the same universality class. Although the first finding reinforces the scenario of strong universality in the 2D Ising model with quenched disorder, the second is in difference from the critical behavior, emerging under randomness, in the cases of the ex-first-order transitions of the Potts model. Finally, our results verify previous renormalization-group calculations on the Blume-Capel model with disorder in the crystal-field coupling.
We study, via a two-stage Wang–Landau (WL) strategy, the random-bond version of the square lattice ferromagnetic Blume–Capel (BC) model, in both the first- and second-order phase transition regimes of the pure model. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the 2d Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with v=1.30(6) and β/v=0.128(5). These results amount to a strong violation of universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two sets of results supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of 2d systems with and without quenched disorder.
We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model. Phase transition temperatures, thermal and magnetic critical exponents are determined for lattice sizes in the range L=20-100 via a sophisticated two-stage numerical strategy of entropic sampling in dominant energy subspaces, using mainly the Wang-Landau algorithm. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the two-dimensional Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections, our findings thus supporting the marginal irrelevance of quenched bond randomness. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with nu=1.30(6) and beta/nu = 0.128(5) . These results amount to a strong violation of universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two sets of results supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of two-dimensional systems with and without quenched disorder. In the conversion by bond randomness of the first-order transition of the pure system to second order, we detect, by introducing and evaluating connectivity spin densities, a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness.
Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.
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.
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