Information theoretic aspects of the two-dimensional Ising model

We present numerical results for various information theoretic properties of the square lattice Ising model. First, using a bond propagation algorithm, we find the difference $2H_L\left(w\right)-H_{2L}\left(w\right)$ between entropies on cylinders of finite lengths $L$ and $2L$ with open end cap boundaries, in the limit $L\to \infty$. This essentially quantifies how the finite length correction for the entropy scales with the cylinder circumference $w$. Secondly, using the transfer matrix, we obtain precise estimates for the information needed to specify the spin state on a ring encircling an infinitely long cylinder. Combining both results, we obtain the mutual information between the two halves of a cylinder (the “excess entropy” for the cylinder), where we confirm with higher precision but for smaller systems the results recently obtained by Wilms et al., and we show that the mutual information between the two halves of the ring diverges at the critical point logarithmically with $w$. Finally, we use the second result together with Monte Carlo simulations to show that also the excess entropy of a straight line of $n$ spins in an infinite lattice diverges at criticality logarithmically with $n$. We conjecture that such logarithmic divergence happens generically for any one-dimensional subset of sites at any two-dimensional second-order phase transition. Comparing straight lines on square and triangular lattices with square loops and with lines of thickness 2, we discuss questions of universality.

Figure 1

The quantity $\Delta (w,\beta )$ defined in Eq. (16), for critical Ising systems of finite widths $w$, plotted against $\beta$. The vertical dotted line is the critical ${\beta }_c$.

Figure 2

Log-log plot of the peak heights of Fig. 1, i.e., of the maximal values of $\Delta (w,\beta )$, vs $w$. The straight line has slope $1.18$.

Figure 3

Log-log plot showing the scaling of the peak positions of Fig. 1 vs $w$. The straight line has slope $-0.98$, indicating that ${\beta }_{w,\max }-{\beta }_c\sim w^{-0.98}$.

Figure 4

Shannon entropies (in nats) for a ring encircling a cylinder of width $w$, after subtracting the leading term $\propto w$.

Figure 5

Excess entropy $\mathcal{E}(w,\beta )$ (in bits per unit width) between two halves of a cylinder. From top to bottom (at the critical temperature), the curves correspond to $w=12,14,16,18,20$. The inset shows the global behavior for $w=6,8,10,12,14$. For $w<6$, the curves are monotonically increasing.

Figure 6

Peak heights (in bits) of the excess entropy in Fig. 5. A minimum and an inflection point are located at $w=15$ and 21, respectively. The inset shows the derivative of the height with respect to $w$.

Figure 7

Peak positions of the excess entropy in Fig. 5. Again the inset shows the slope of the curve.

Figure 8

Slope of the excess entropy $d\mathcal{E}(w,\beta )/d\beta$ per width in Fig. 5.

Figure 9

Log-log plot of the positions of the minimum in Fig. 8. The straight line has slope $-1.39$. The inset shows the value of the minimum in Fig. 8 on a linear plot. The straight line indicates a power.

Figure 10

Mutual information between two parts of a ring of $w=28$ spins encircling a cylinder (in natural units), plotted against the length $m$ of one of the parts. The continuous line corresponds to Eq. (25).

Figure 11

Mutual information between two equal halves of a ring of $w=2m$ spins (in natural units), plotted against $\log w$. The detailed form of the fit is presumably not to be relied upon, but the coefficient of the term linear in $\log w$ seems to be robust.

Figure 12

Convergence of $h_n$, the entropy per spin in a line of length $n$ embedded in lattice of size $L\times L$ with helical (i.e., practically periodic) b.c. On the $x$ axis is plotted $1/n$, so that a straight line indicates a convergence $h\approx h+b/n$. The straight line has $h=0.54378...$ as obtained for rings encircling cylinders. The inset shows the data after subtraction of this linear part.

Figure 13

Convergence of $h_n$ for a line of spins embedded in a triangular lattice. In contrast to Fig. 12, now the slope of the straight line fit is fixed (to the value $0.085$ obtained for the square lattice), but its intercept is fitted. From the latter we obtain $h=0.5839\left(3\right)$ for the triangular lattice.

Figure 14

Convergence of $h_n$ for an $n\times 2$ rectangular block of spins embedded in a square lattice. The straight line has the same slope as in Fig. 12, and its intercept is also given by the results of the previous sections. Thus it involves no new fitting parameter.

Figure 15

Analogous to Fig. 12, but for square loops instead of straight lines. Here, $n=4k$ is the length of the loop ($k=1,2,...,6$). As in Fig. 12, the intercept of the straight line is fixed while its slope is fitted to the large-$n$ behavior for the largest value of $L$.

Figure 16

Residues in fitting $w^{-1}{H}_1(w,{\beta }_c)-f\left(w\right)$ where the fit $f\left(w\right)$ is a polynomial in $1/w$ [see Eq. (B1)], for the 1D Ising chain in a transverse magnetic field, using the data from Ref. [8]. Data are plotted against $1/w$. Thus, if both the data and the fit have simple asymptotics, the residues should be smooth for small $1/w$ and their extrapolation should pass through zero. The continuous curve is such an extrapolation.

Figure 17

Convergence of entropy estimators for the critical 2D Ising model.