Finite-size scaling of Monte Carlo simulations for the fcc Ising antiferromagnet: Effects of the low-temperature phase degeneracy

The Ising antiferromagnet on a face-centered cubic (fcc) lattice with nearest-neighbor interaction only is well known to exhibit a macroscopic (exponential in the system size $L$) ground-state degeneracy. With increasing temperature, this degeneracy is expected to be lifted and the model undergoes a first-order phase transition. For a model with an exponential degeneracy in the whole low-temperature phase, it was recently found that the finite-size scaling behavior is governed by leading correction terms $\sim L^{-2}$ instead of $\sim L^{-3}$ as usual. To test the conjecture that such a transmuted behavior may effectively persist also for the fcc antiferromagnet up to some crossover system size, we have performed parallel multicanonical Monte Carlo simulations for lattices of linear size $L\le 18$ with periodic boundary conditions and determined various inverse pseudo phase transition temperatures, as well as the extremal values of the specific heat and the energetic Binder parameter. We indeed find that, for the simulated lattice sizes, the conjectured transmuted finite-size scaling ansatz fits the data better than the standard ansatz. On this basis, we extrapolate for the transition temperature an estimate of $T_0=1.735047\left(46\right)$.

Figure 1

(a) The linear system size $L$ is defined as the number of the depicted four-spin complexes (black) along the $x$, $y$, or $z$ axis. (b) Exemplary illustration of a ground state. The shaded planes are ordered antiferromagnetically. (c) One of the AB structure ground states.

Figure 2

(a) Part of the time series of the energy per lattice site $e=E/V$ of one process of the production run for $L=16$. The variable $t$ denotes the sweep number. (b) Corresponding estimated multicanonical energy probability distribution from all processes together. The vertical axis is the same as in the time-series plot.

Figure 3

Estimated canonical quantities for the lattice sizes $L=5,...,18$. The distributions $P\left(E\right)$ are here calculated at the respective inverse temperatures ${\beta }_{\text{eqh}}\left(L\right)$. For the sake of clarity, the error bars are not depicted here.

Figure 4

Quality of fit parameter $Q\in [0,1]$ for the fits of ${\beta }_{\text{eqw}}\left(L\right)$ with the different fit functions. Here the lattice size range $[L_{\text{min,fit}},L_{\text{max,fit}}]$, for which the fit is performed, is varied. The color-value-identification of $Q$ is shown above.

Figure 5

Obtained data of the regarded inverse pseudo phase transition temperatures for the lattice sizes $L=5,...,18$ together with their best fits according to the nonstandard ansatz (a) and the standard ansatz (b).

Figure 6

Obtained data of $c_{\text{max}}/V$ [upper plots (a) and (b)] and $B_{\text{min}}$ [lower plots (c) and (d)] for the lattice sizes $L=9,...,18$ together with their best fits according to the nonstandard [plots (a) and (c) on the left] and the standard ansatz [plots (b) and (d) on the right].