First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach

We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to $48\times 48$ sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, $y_t=1.804\left(5\right)$, $y_g=0.80\left(1\right)$, and $y_h=1.925\left(3\right)$, provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.

Figure 1

Phase diagram of the two-dimensional spin-1 Blume-Capel model around the tricritical point in the plane of temperature $T$ and crystal field $\mathrm{\Delta }$. The transition line and tricritical point are determined from the mixing-field analysis with the calculations using the Wang-Landau density of states. The phase-transition line plotted here is determined in the limit of infinite $L$ from the extrapolation of the pseudotransition points obtained for the systems with different sizes up to $L=48$ (for example, see Fig. 3). The transition points previously available from the literature [19, 33, 42] are given for comparison. The statistical error was unspecified for the data in Ref. [33], and the error bars of the data in Ref. [19] were given in temperature. All our transition points are listed in Table 2.

Figure 2

Specific-heat anomaly appearing with the first-order transition. (a) Double-peak structure observed in the specific heat at $\mathrm{\Delta }=1.994$. While the diverging peak at the lower temperature is associated with the phase transition, the anomalous broad one at the higher temperature does not scale with the system size. The inset indicates that the smaller systems cannot detect the correct structure. (b) Corresponding density of nonzero spins $\langle n\rangle$ and entropy per site $s\left(T\right)$. The zero-spin state is dominant right after the transition and then starts to get thermally excited to the nonzero-spin states distributed with increasing entropy.

Figure 3

Finding the first-order transition points by using the method of field mixing. (a) Symmetric double peaks in the distribution of the field-conjugate variable $\mathcal{Q}$ at the phase coexistence $\mathrm{\Delta }={\mathrm{\Delta }}_L^{*}$ established at temperature $T=0.5$. The distribution is shifted and rescaled for visualization of systems with different sizes. (b) Scaling behavior of ${\mathrm{\Delta }}_L^{*}$ with system size $L$. The extrapolation in the limit of $L\to \infty$ determines the transition point ${\mathrm{\Delta }}_{\infty }^{*}$. The fourth-order cumulant of microcanonical magnetization $U_L^m$ in panel (c) also find its crossing point very close to the estimated transition point.

Figure 4

Probability distribution function of the field-conjugate variable $\mathcal{Q}$ along the transition line. The calculations for the largest available system with $L=48$ are shown.

Figure 5

Location of tricritical point. (a) The tricritical temperature $T_{tc}\simeq 0.6080$ is determined at the crossing point of the fourth-order cumulant of the field-conjugate variable $U_L^{\mathcal{Q}}$ along the transition line $\mathrm{\Delta }={\mathrm{\Delta }}_L^{*}\left(T\right)$. The extrapolation of the transition points ${\mathrm{\Delta }}_L^{*}\left(T_{tc}\right)$ in panel (b) and the crossing point of the fourth-order cumulant of microcanonical magnetization $U_L^m$ in panel (c) provide the estimate of the tricritical crystal field as ${\mathrm{\Delta }}_{tc}\simeq 1.9660\left(1\right)$.

Figure 6

Finite-size-scaling tests of field-conjugate variable $\mathcal{Q}$ for the tricritical thermal exponents. (a) Scaling plots of the probability density distribution $P_L\left(\tilde{\mathcal{Q}}\right)$ with the exponent $y_t=1.8$ at the tricritical temperature $T_{tc}=0.608$. The large systems with $L\ge 40$ provide smooth curves (solid lines) falling on a universal distribution which also fits well with the data points for the smaller systems (symbols). (b) Finite-size scaling of the fourth-order cumulant $U_L^{\mathcal{Q}}$ along the transition line. The data points fall onto each other very well, given the next-to-leading exponent $y_g=0.8$.

Figure 7

Tricritical behavior along the crystal field axis at the tricritical temperature. The finite-size-scaling analysis of (a) number fluctuations ${\langle n\rangle }^2{\kappa }_T$, (b) susceptibility $\chi$, and (c) magnetization $\langle \left|m\right|\rangle$ is performed to determine the tricritical exponents. While the WL method guarantees high-enough resolution to plot the data as continuous curves, the points in low resolution ($L\le 24$) are also given for visualization of finite-size scaling. The ratios ${\alpha }_t/{\nu }_t$ and ${\gamma }_t/{\nu }_t$ are determined from the power-law fits of the maxima of ${\langle n\rangle }^2{\kappa }_T$ and $\chi$, respectively. Each scaling plot with the estimated exponent shows the excellent collapse of the data points falling onto a single curve. The tricritical eigenvalue exponents are deduced as $y_t=1.804$ and $y_h=1.925$ from ${\alpha }_t/{\nu }_t$ and ${\gamma }_t/{\nu }_t$.

Figure 8

Tricritical behavior along the temperature axis. The fugacity is fixed at $lnz\equiv \mathrm{\Delta }/T={\mathrm{\Delta }}_{tc}/T_{tc}$. The finite-size-scaling analysis of (a) specific heat $c$, (b) susceptibility $\chi$, and (c) magnetization $m$ is presented. The tricritical exponents are determined by the same procedures used in Fig. 7. The resulting scaling plots show the excellent collapse of the data points falling onto a single curve, where the corresponding tricritical eigenvalue exponents are deduced to be $y_t=1.809$ and $y_h=1.9275$ from ${\alpha }_t/{\nu }_t$ and ${\gamma }_t/{\nu }_t$.