Universality in the two-dimensional dilute Baxter-Wu model

We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field $\mathrm{\Delta }$. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of $\mathrm{\Delta }$ and a parallelized variant of the multicanonical approach performed at constant temperature $T$. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the $(\mathrm{\Delta },T)$ phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.

Figure 1

Representation of the Baxter-Wu triangular lattice as a superposition of the three sublattices, $A, B$, and $C$. Each sublattice corresponds to spins of a different color. The spins are shown in the ferromagnetic ground state.

Figure 2

Phase diagram of the two-dimensional spin-1 Baxter-Wu model. Several transition points are given, including those obtained in the current work. The black rhombus and black triangle mark the pentacritical point as estimated by Dias et al. [26], $({\mathrm{\Delta }}_{\mathrm{pp}},T_{\mathrm{pp}})\approx (0.8902,1.4)$, and Jorge et al. [27], $({\mathrm{\Delta }}_{\mathrm{pp}},T_{\mathrm{pp}})\approx (1.68288\left(62\right),0.98030\left(10\right))$, respectively. The black dashed and solid lines correspond to first- and second-order phase transitions. The intermediate regime between the two pentacritical point estimations is not crossed by a line as it calls for further investigation. Blue vertical and red horizontal dashed arrows indicate the two numerical approaches used, namely, the Wang-Landau and multicanonical methods, at fixed values of $\mathrm{\Delta }=\{-10,-1\}$ and $T=\{2.2578,1.8503\}$, respectively.

Figure 3

Specific-heat curves of the spin-1 Baxter-Wu model at $\mathrm{\Delta }=-10$ for a system with linear size $L=24$ obtained with Wang-Landau and Metropolis simulations.

Figure 4

Specific-heat (main panel) and magnetic susceptibility (inset) curves corresponding to Eqs. (4) and (5) at $\mathrm{\Delta }=-10$ from Wang-Landau simulations. After an increase in the system size, the location of the peaks shifts to the left.

Figure 5

Specific-heat-like quantity (main panel) and first-order logarithmic derivative of the order parameter (inset) obtained via multicanonical simulations at $T=1.8503$. Like in Fig. 4 the location of the peaks shifts to the left as we increase the linear size of the system.

Figure 6

$P\left(E_{\mathrm{\Delta }}\right)$ for $L=96$ obtained via multicanonical simulations at $T=2.2578$. Results for three adjacent crystal-field values are shown.

Figure 7

(a) Reweighted probability density functions $P\left(E_{\mathrm{\Delta }}\right)$ for various system sizes. With increasing $L$ the distance between the two peaks decreases. (b) Limiting behavior of the corresponding surface tension $\mathrm{\Sigma }\left(L\right)$ (main panel) and latent heat $\mathrm{\Delta }{e}_{\mathrm{\Delta }}\left(L\right)$ (inset). Results were obtained via multicanonical simulations at $T=1.8503$.

Figure 8

Shift behavior of the peak locations of the specific heat and magnetic susceptibility as a function of the inverse linear system size at $\mathrm{\Delta }=-10$ (main panel) and $\mathrm{\Delta }=-1$ (inset). The black dashed line denotes the critical temperature of the model at $\mathrm{\Delta }\to -\infty$, i.e., the critical temperature of the spin-$1/2$ Baxter-Wu model. In both panels the black solid lines are joint fits of the form (16). Data were produced with the Wang-Landau algorithm.

Figure 9

(a) Shift behavior of several pseudocritical fields as a function of the inverse linear system size. (b) Fourth-order Binder cumulant curves of the order parameter. The black vertical dashed line marks the value $\mathrm{\Delta }=-1$. The inset shows the limiting behavior of the crossings $U_m^{*}$ on pairs of lattice sizes $(L,2L)$. Data were produced at $T=1.8503$ via multicanonical simulations.

Figure 10

Finite-size scaling of the logarithmic derivatives of powers $n=1$ and 2 of the order parameter at $T=1.8503$. The solid lines are fits of the form (18). Results were obtained via multicanonical simulations.

Figure 11

(a) Finite-size scaling behavior of $C^{*}$ (main panel) and ${\chi }^{*}$ (inset) at $\mathrm{\Delta }=-10$ and $\mathrm{\Delta }=-1$. Data were produced with the Wang-Landau algorithm. (b) Similar analysis of data produced at $T=1.8503$ via multicanonical simulations.

Figure 12

Main panel: Typical $\xi /L$ curves as a function of the temperature obtained from Wang-Landau simulations for all pairs of system sizes studied and for $\mathrm{\Delta }=-10$. The temperature area of the crossings conforms to the value $T_{\mathrm{c}}=2.2578$ in Fig. 8. Inset: Finite-size scaling of the correlation-length ratios at their crossing points ${(\xi /L)}^{*}$. Results are shown for the largest pairs $(L,2L)$ of system sizes: $(30,60)$, (36,72), $(48,96)$, and $(60,120)$. The solid line shows a linear in $L^{-\omega }$ extrapolation to $L\to \infty$. The black dashed line marks the value of ${(\xi /L)}_{\infty }$ of the four-state Potts model, as taken from Ref. [63].