Blume-Capel model analysis with a microcanonical population annealing method

We present a modification of the Rose-Machta algorithm [N. Rose and J. Machta, Phys. Rev. E 100, 063304 (2019)] and estimate the density of states for a two-dimensional Blume-Capel model, simulating ${10}^5$ replicas in parallel for each set of parameters. We perform a finite-size analysis of the specific heat and Binder cumulant, determine the critical temperature along the critical line, and evaluate the critical exponents. The obtained results are in good agreement with those previously obtained using various methods—Markov chain Monte Carlo simulation, Wang-Landau simulation, transfer matrix, and series expansion. The simulation results clearly illustrate the typical behavior of specific heat along the critical lines and through the tricritical point.

Figure 1

Phase diagram obtained by different methods: transfer matrix [30] (blue circles), Monte Carlo [31] (black triangles), Wang-Landau [32] (green squares), high-energy and low-energy expansions [5, 33] (red diamonds), microcanonical algorithm [34] (cyan pluses), and microcanonical population annealing (current work, violet crosses). The error bars are much smaller than the symbols.

Figure 2

Example entropy estimate for Blume-Capel model with $D=1.5$ and linear system size $L=32$. The orange points are calculated using the ceiling algorithm, the blue points are calculated using the floor algorithm, the right red vertical line marks the rightmost value of $S^c\left(E\right)$, and the left red vertical line marks the leftmost value of $S^f\left(E\right)$. The inner green vertical lines mark the stitching region. Details are given in the text.

Figure 3

Number of replicas with energy $E$: initial random configuration of 2D Ising model with square lattice size $L=20$.

Figure 4

Culling factor $\epsilon \left(E\right)$ of a two-dimensional Ising model with linear lattice size $L=40$. Inset: Absolute difference between the calculated and exact culling factor, multiplied by a factor of 1000.

Figure 5

Variation of the relative DoS error $\delta R$ on the number of replicas, $R$. The dashed line shows the slop proportional to $1/R^{1/2}$. 2D square lattice Ising model with $L=20$.

Figure 6

The entropy $S\left(E\right)$ is calculated for the ratios $\mathrm{\Delta }/J=0$ and 1.5 in the top row and $\mathrm{\Delta }/J=1.966$ and 1.99 in the bottom row. The number of replicas is $2^{17}$ and the system size is $L=64$.

Figure 7

Change of coefficient $C_0$, expression (12), with anisotropy parameter $D$. The solid line is the heuristic fitting described in the text.

Figure 8

Maximum value of the specific heat of the Blume-Capel model. Red circles: $\mathrm{\Delta }/J=1.95$; black squares: $\mathrm{\Delta }/J=1.96$; purple stars: $\mathrm{\Delta }/J=1.962$; magenta circles: $\mathrm{\Delta }/J=1.966$.

Figure 9

Variation in the effective exponent of the specific heat maximum.

Figure 10

Specific heat in the critical region. $\mathrm{\Delta }/J=1.965$.

Figure 11

Binder cumulant in the critical region. $\mathrm{\Delta }/J=1.965$.

Figure 12

Entropy density for several values of $D=\mathrm{\Delta }/J$.