Entropy of diluted antiferromagnetic Ising models on frustrated lattices using the Wang-Landau method

We use a Monte Carlo simulation to study the diluted antiferromagnetic Ising model on frustrated lattices including the pyrochlore lattice to show the dilution effects. Using the Wang-Landau algorithm, which directly calculates the energy density of states, we accurately calculate the entropy of the system. We discuss the nonmonotonic dilution concentration dependence of residual entropy for the antiferromagnetic Ising model on the pyrochlore lattice, and compare it to the generalized Pauling approximation proposed by Ke et al. [Phys. Rev. Lett. 99, 137203 (2007)]. We also investigate other frustrated systems, the antiferromagnetic Ising model on the triangular lattice and the kagome lattice, demonstrating the difference in the dilution effects between the system on the pyrochlore lattice and that on other frustrated lattices.

Figure 1

Illustration of the 16-site cubic unit cell of the pyrochlore lattice ($L=1$).

Figure 2

Illustration of the (a) triangular and the (b) kagome lattice.

Figure 3

Plot of $lng\left(E\right)$ as a function of $E$ of the Ising model on the pyrochlore lattice. The system size is $L=6(N=3456)$.

Figure 4

Temperature dependence of the specific heat per spin of the AFM Ising model on the pyrochlore lattice. The system size is $L=6(N=3456)$.

Figure 5

Plot of the residual entropy per spin of the pure AFM Ising model on the pyrochlore lattice as a function of $1/N$ ($N=N_{\mathrm{spin}}$). The system size is $L=3(N=432), L=4(N=1024), L=5(N=2000)$, and $L=6(N=3456)$. The precise estimate by Nagle [19] (0.205 01) is shown by the solid red arrow, whereas the approximation by Pauling [1] (0.202 73) is shown by the dotted blue arrow.

Figure 6

Temperature dependence of the entropy per spin of the AFM Ising model on the pyrochlore lattice. The system size is $L=6(N=3456)$.

Figure 7

Dilution concentration dependence of the residual entropy per spin of the AFM Ising model on the pyrochlore lattice. The system size is $L=6(N=3456)$. The generalized Pauling approximation [6] is also plotted by a dotted curve. The site-percolation threshold of the pyrochlore lattice is shown by the red arrow.

Figure 8

Plot of $lng\left(E\right)$ as a function of $E$ of the Ising model on the triangular lattice. The system size is $L=48(N=2304)$.

Figure 9

Temperature dependence of the specific heat per spin of the AFM Ising model on the triangular lattice. The system size is $L=48(N=2304)$.

Figure 10

Plot of the residual entropy per spin of the pure Ising model on the triangular lattice as a function of $1/N$ ($N=N_{\mathrm{spin}}$). The system size is $L=24(N=576), L=32(N=1024), L=48(N=2304)$, and $L=64(N=4092)$. The exact value by Wannier [20] (0.323 066) is shown by the red arrow.

Figure 11

Temperature dependence of the entropy per spin of the AFM Ising model on the triangular lattice. The system size is $L=48(N=2304)$.

Figure 12

Dilution concentration dependence of the residual entropy per spin of the AFM Ising model on the triangular lattice and the kagome lattice. The system size is $L=48$ ($N=2304$ for the triangular lattice and $N=3456$ for the kagome lattice). The site-percolation thresholds of the triangular and kagome lattices are shown by the solid blue and dotted orange arrows, respectively.

Figure 13

Plot of $lng\left(E\right)$ as a function of $E$ of the Ising model on the kagome lattice. The system size is $L=48(N=3456)$.

Figure 14

Temperature dependence of the specific heat per spin of the AFM Ising model on the kagome lattice. The system size is $L=48(N=3456)$.

Figure 15

Plot of the residual entropy per spin of the pure Ising model on the kagome lattice as a function of $1/N$ ($N=N_{\mathrm{spin}}$). The system size is $L=24(N=864), L=32(N=1536)$, and $L=48(N=3456)$. The exact value by Kano and Naya [23] (0.501 83) is shown by the red arrow.

Figure 16

Temperature dependence of the entropy per spin of the AFM Ising model on the kagome lattice. The system size is $L=48(N=3456)$.