We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime, as well as a clear verification of the expected Ising universality in the respective second-order regime. Finally, we discuss the scaling behavior in the vicinity of the tricritical point.
We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime, as well as a clear verification of the expected Ising universality in the respective second-order regime. Finally, we discuss the scaling behavior in the vicinity of the tricritical point.
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 investigate the thermodynamic properties of the zero-field Blume-Capel model in the vicinity of its tricritical point (TCP). We calculate the quadrupole moment, internal energy, and entropy densities employing an exact numerical recursion procedure for the model defined on a hierarchical lattice of fractal dimension d. We explore the scaling behavior of the isothermal quadrupolar susceptibility and, the isothermal and constant crystal-field specific heat as a function of the temperature and the reduced crystal-field parameter along the ferromagnetic and the \emph{ordered paramagnetic} phase frontier. Results achieved for systems with dimensions d=2 and 3 exhibit the main features of the continuous and first-order transitions in the TCP neighborhoods. We also probe the phase coexistence in the λ-diagram and the latent heat in the vicinity of the tricritical point locus.
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.
We present a numerical study of the Blume-Capel model with quenched disorder in three dimensions. The phase diagram is characterized by spin-glass/paramagnet phase transitions of both first and second order in the thermodynamic sense. Numerical simulations are performed using the exchange Monte Carlo algorithm, providing clear evidence for inverse freezing. The main features at criticality and in the phase-coexistence region are investigated. The whole inverse freezing transition appears to be first order. The second-order transition appears to be in the same universality class as the Edwards-Anderson model. The nature of the spin-glass phase is analyzed by means of the finite-size scaling behavior of the overlap distribution functions and the four-spin real-space correlation functions. Evidence for a replica-symmetry-breaking-like organization of states is provided.
The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.