Parallel multicanonical study of the three-dimensional Blume-Capel model

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.

Figure 1

General sketch of the phase diagram of the 3D BC model in the temperature–crystal-field plane. The phase boundary separates the ferromagnetic (F) phase from the paramagnetic (P) phase, in which the solid line indicates continuous phase transitions and the dotted line first-order phase transitions. The two lines merge at the tricritical point (TP), as highlighted by the black diamond. The limiting cases of $T=0$ and $\Delta =0$ are marked on the relevant axis with ${\Delta }_0=3$ and $T_0=3.195\left(1\right)$ [22], respectively. The horizontal arrows illustrate the direction of crossing the phase boundary at fixed temperatures, studied in this work, in both the second-order ($T_1=2.0$) and the first-order ($T_2=1.0$ and $T_3=0.9$) regimes, as well as in the vicinity of the tricritical point ($T_{\mathrm{t}}=1.4182$) [14].

Figure 2

Illustration of the reweighted probability distribution at the transition field ${\Delta }_{\mathrm{eqh}}$ with respect to $e_{\Delta }=E_{\Delta }/V$ at (a) $T=0.9$, which is well inside the first-order regime of the system (note the strongly increasing barrier with the system size), exhibiting a major suppression of intermediate states which is common for first-order transitions, and at (b) $T=T_{\mathrm{t}}=1.4182$, which is in the vicinity of the tricritical point.

Figure 3

Specific-heat curves as a function of the crystal field $\Delta$ for $T=2.0$ and several system sizes in the second-order regime. Smooth curves typical of continuous transitions with a clear shift behavior are observed.

Figure 4

Shift behavior (8) of the pseudocritical fields ${\Delta }_L^{*}$ at $T=2.0$ obtained from the peak location of the specific-heat curves shown in Fig. 3. The solid line is a fit for linear sizes $L\ge L_{\min }=18$, giving a stable, under $L_{\min }$ changes, critical field value ${\Delta }_{\mathrm{c}}=2.523\left(6\right)$. The inset illustrates the extrapolation of the effective exponent $\nu$, obtained by varying the cutoff $L_{\min }$ of the fits, to the thermodynamic limit.

Figure 5

Specific-heat curves as a function of the crystal field $\Delta$ at $T=1.0$ and two system sizes $L=8$ (dashed line) and 12 (solid line). The inset illustrates a typical first-order-like specific-heat curve for a system with linear size $L=28$ on a double logarithmic scale.

Figure 6

Shift behavior (9) of the pseudocritical fields ${\Delta }_L^{*}$ obtained from the peak location of the specific-heat curves (a sample of which is shown in Fig. 5) for both temperatures $T=1.0$ and $T=0.9$ in the first-order regime of the phase diagram. The obtained transition fields ${\Delta }^{*}$ are given in the main text. The inset is a mere enlargement of the $\sim L^{-d}$ approach of the pseudotransition points ${\Delta }_L^{*}$ to $L\to \infty$ for the temperature $T=1.0$.

Figure 7

Simultaneous fitting of the specific-heat maxima at both temperatures in the first-order regime. The expected $\sim L^d$ scaling behavior is obtained as can be clearly seen.

Figure 8

Scaling of the barrier height $B$ for all considered temperatures $T\le T_{\mathrm{t}}$. This barrier may be associated with the transition from a spin-0 dominated to a spin-$\pm 1$ dominated regime (see Fig. 2). Then $B/L^2$ plays the role of an interface tension for spin-0 strips.

Figure 9

Critical aspects of the 3D BC model at the tricritical point proposed in Ref. [14]: $T_{\mathrm{t}}=1.4182$. (a) Shift behavior of ${\Delta }_L^{*}$ obtained from the location of the specific-heat peaks. The solid line is a power-law fit of the form (8) for $L\ge L_{\min }=20$. The inset illustrates the infinite-volume extrapolation of the correlation-length effective exponent by varying the lower cutoff $L_{\min }$ of the fits. (b) Scaling behavior of the specific-heat peaks again for the larger system sizes. The corresponding inset shows an infinite-volume extrapolation of the effective exponent ratio $\alpha /\nu$ using the same procedure as in (a).