Blume-Capel model analysis with microcanonical population annealing method

Abstract

We present a modification of the Rose-Machta algorithm (Phys. Rev. E 100 (2019) 063304) 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 results obtained are in good agreement with those obtained previously 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.

Figures (12)

• Figure 1: Phase diagram obtained by different methods: transfer-matrix Beale-1986 (blue circles), Monte-Carlo XALP98 (black triangles), Wang-Landau SCP06 (green squares), high-energy and low-energy expansions Enting-1994Butera-2018 (red diamonds), microcanonical algorithm Zierenberg-2017 (cyan plusses), 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 blue solid squares are calculated using the ceiling algorithm, the red open circles are calculated using the floor algorithm, the right dashed red vertical line marks the rightmost value of $S^c(E)$, the left dashed red vertical line marks the leftmost value of $S^f(E)$. The inner green solid 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 $\epsilon(E)$ 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 $\delta g$ on the number of replicas $R$. The dashed line shows the slop proportional to $1/R^{1/2}$. 2D square lattice Ising model with $L{=}20$.