Information-theoretic approach to ground-state phase transitions for two- and three-dimensional frustrated spin systems

The information-theoretic observables entropy (a measure of disorder), excess entropy (a measure of complexity), and multi-information are used to analyze ground-state spin configurations for disordered and frustrated model systems in two and three dimensions. For both model systems, ground-state spin configurations can be obtained in polynomial time via exact combinatorial optimization algorithms, which allowed us to study large systems with high numerical accuracy. Both model systems exhibit a continuous transition from an ordered to a disordered ground state as a model parameter is varied. By using the above information-theoretic observables it is possible to detect changes in the spatial structure of the ground states as the critical point is approached. It is further possible to quantify the scaling behavior of the information-theoretic observables in the vicinity of the critical point. For both model systems considered, the estimates of critical properties for the ground-state phase transitions are in good agreement with existing results reported in the literature.

Figure 1

GS samples for (a) the $L=64$ 2D RBIM with mixed boundary conditions (periodic $=$ solid line, free $=$ dashed line) at $\mu =1.15,0.97,0.40$ (from left to right), and (b) the $L=48$ 3D RFIM with fully periodic boundary conditions at $b=2.0,2.4,2.9$ (from left to right; only the spins on the $x,y,z=0$ planes are displayed).

Figure 2

Neighborhood templates for the definition of the conditional entropies [see Eq. (7)] for (a) the 2D RBIM, and (b) the 3D RFIM. The target variable is represented by a shaded cell and the numbers within the cells reflect the order in which the sites are picked in order to compute the conditional entropies (see text).

Figure 3

Results for the entropy density of the $2D$ RBIM with side length $L=384$. (a) illustrates the convergence of the average entropy density $\langle h\left(M\right)\rangle$ as a function of the neighborhood size $M$ for three values of the disorder parameter $\mu$. (b) shows the average entropy density for different neighborhood sizes as a function of the disorder parameter $\mu$.

Figure 4

Results for the $2D$ RBIM for different square lattices with side length up to $L=512$. (a) The main plot shows the average excess entropy $\langle E\rangle$ as a function of the mean $\mu$ of the bond disorder distribution (lines are guides to the eyes only). The inset illustrates the finite-size scaling of the peak position, the line shows the result of a fit to Eq. (10). For all data-points error bars are obtained via bootstrap resampling. (b) The main plot shows the finite-size susceptibilities ${\chi }_E\left(L\right)=L^2\mathrm{var}\left(E\right)$ associated to the excess entropy (lines are guides to the eyes only). The inset signifies the finite-size scaling of the peak width obtained from a fit of the data points close to the peak to a Gaussian function, the line shows the result of a fit to Eq. (11). (c), (d) are the same as (a), (b), respectively, but for the multi-information $I$.

Figure 5

Results for the entropy density of the 3D RFIM with side length $L=64$. (a) illustrates the convergence of the average entropy density $\langle h\left(M\right)\rangle$ as a function of the neighborhood size $M$ for three values of the field strength $b$. (b) shows the average entropy density for different neighborhood sizes as a function of the field strength $b$.

Figure 6

Finite-size scaling analysis for the 3D RFIM on cubic lattices with side length up to $L=64$. (a) the main plot shows the average excess entropy $\langle E\rangle$ as a function of the field strength $b$. (For this and the other main plots, lines are guides to the eyes only). The inset illustrates the finite-size scaling of the peak position, the line, as for the other insets, shows the result of a fit to Eq. (12). For all data-points error bars are obtained via bootstrap resampling, (b) the main plot shows the finite-size susceptibilities ${\chi }_E\left(L\right)=L^3\mathrm{var}\left(E\right)$ associated to the excess entropy and the inset indicates the finite-size scaling of the associated peak location. (c), (d) are the same as (a), (b), respectively, but for the multi-information $I$.

Figure 7

Complexity-entropy diagram for the $2D$ RBIM and 3D RFIM for different system sizes $L$. Note that for the complexity-entropy curve for the $2D$ Ising FM, the corresponding peak is located at entropy density $\langle h\rangle \approx 0.57$ with a peak height of $\langle E\rangle \approx 0.4$, see Ref. [6].