We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.
We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field
h and the dilution
p, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small
h and small
p. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent
β and the average correlation-length exponent
νav are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for
p smaller than the percolation threshold
pc.
The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.
The critical behavior of the Baxter-Wu model belongs to the universality class of the four-state Potts model. While the introduction of annealed vacancies does not alter the criticality of the four-state Potts model, the dilute Baxter-Wu model has remained the subject of several competing scenarios. Here we investigate the phase diagram of the spin-1 Baxter-Wu model in the presence of a crystal field using transfer-matrix calculations and large-scale Monte Carlo simulations. Our results reveal a systematic evolution of the effective critical behavior with increasing crystal field, accompanied by increasingly strong finite-size corrections near the crossover to the first-order regime. Along the line of continuous transitions, the central charge remains close to c=1, while the scaling dimensions systematically deviate from the spin-1/2 limit as the crystal field increases, consistent with either continuously varying effective critical exponents or a slow crossover between competing critical behaviors. The first-order regime is independently characterized through multicanonical simulations, which confirm the expected finite-size scaling and interfacial behavior. Taken together, our results provide a unified picture of the dilute spin-1 Baxter-Wu model, substantially narrowing the range of possible scenarios for the crossover between continuous and first-order phase transitions.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated
M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
The Blume-Capel model on a square lattice and the Baxter-Wu model on a triangular lattice have been studied through extensive Monte Carlo simulations using the single spin update according to the Metropolis algorithm. Both models have been treated with spin
S=1 and spin
S=3/2 in the presence of crystal field interactions. The transition temperature and correlation length critical exponent have been computed, as a function of the crystal field interaction, by employing the recent developed method of the zeros of the energy probability distribution. The important question of the universality class along the second-order transition lines could be resolved by looking at the corresponding universal probability distributions of the energy and magnetizations. Mixing field arguments turned out to be quite important to decide the possible change in the universality class of the models as the crystal field varies. Single histograms techniques have also been utilized in order to obtain simulational data close to the second-order transitions and multicritical points. As expected, the Blume-Capel model has first- and second-order transitions, where a tricritical point occurs for spin
S=1, while an isolated double-critical-endpoint is present for spin
S=3/2. Although the topology of the phase diagram of the Baxte-Wu model for spin
S=1 is similar to that of the Blume-Capel model, with the tricritical point being replaced by a pentacritical point, the Baxter-Wu model has critical exponents that change along the tetracritical line as the crystal field is increased. On the other hand, the Baxter-Wu model with spin
S=3/2 has a phase diagram that is completely different from the one obtained for the Blume-Capel, with the presence of a pentacritical point and a tetracritical endpoint. The present data also show that, despite the change of the probability distributions as the crystal field varies, the models with different spin values belong to the same universality class.