Exact ground states of the Kaya-Berker model

Here we study the two-dimensional Kaya-Berker model, with a site occupancy $p$ of one sublattice, by using a polynomial-time exact ground-state algorithm. Thus, we were able to obtain $T=0$ results in exact equilibrium for rather large system sizes up to ${777}^2$ lattice sites. We obtained the sublattice magnetization and the corresponding Binder parameter. We found a critical point $p_c=0.64\left(1\right)$ beyond which the sublattice magnetization vanishes. This is clearly smaller than previous results which were obtained by using nonexact approaches for much smaller systems. For each realization we also created minimum-energy domain walls from two ground-state calculations, for periodic and antiperiodic boundary conditions. The analysis of the mean and the variance of the domain-wall energy shows that there is no thermodynamic stable spin-glass phase at nonzero temperature, in contrast to previous claims about this model. For large values of $p$, the standard deviation of the domain-wall decreases with the system size like a power law with exponent roughly $\theta \simeq -0.1$, which is different from the standard two-dimensional Ising spin glass where $\theta \simeq -0.29$.

Figure 1

$6\times 4$ triangular lattice showing the subdivision into the three sublattices, as indicated by the three colors of the nodes (white, gray, black). Every triangle is frustrated, since not all bonds can be satisfied.

Figure 2

Illustration of steps involved in ground-state calculation. The sample shows a $3\times 4$ system with periodic boundary conditions in the horizontal direction. (a) Initial system with $p=0.5$ occupation on the white sublattice. (b) Construction of the dual graph of the original undiluted system with additional vertices and edges at the top and bottom. Edges that do not correspond to edges in the diluted system carry zero weight and are marked by using stroked lines. (c) Set of minimal weighted loops on the dual graph. These paths separate clusters of aligned spins. (d) Ground state constructed by setting the top left spin to up and assigning all remaining spins using the clusters determined earlier.

Figure 3

Illustration of a domain wall generated by switching the boundary conditions. This is equivalent to changing the signs of all bonds in one (top-down) column (vertical gray-black interface). This leads typically to the flip of a domain of spins (black) with respect to the original ground state (gray). The (left) linear domain wall, located at the column of flipped bonds, is trivial, does not lead to an energy change, and is not shown below.

Figure 4

Magnetization of sublattice $b$ at different fractions $p$ of occupied sites and system sizes $L$. The magnetization was obtained for each disorder realization from a randomly sampled state among the degenerate GSs.

Figure 5

Binder parameter of sublattice $b$ as a function of the sublattice occupancy $p$, for different system sizes.

Figure 6

Rescaled Binder parameter with a rescaled $p$ axis using the appropriate scaling variables $p_c=0.6423\left(3\right)$, $\nu =1.39$.

Figure 7

Exemplary domain walls at different occupancy. The solid and dashed border marks periodic and open boundary conditions, respectively. The occupancy was set to $p=0.2$ (left), $p=0.6$ (center), and $p=0.8$ (right).

Figure 8

Domain-wall length $l$ at different system sizes $L$. The straight lines are power-law fits to the data sets.

Figure 9

Fractal dimension $d_{\text{f}}$ of the domain-wall length at different occupancy $p$. The vertical dashed line marks the critical point $p_{\text{c}}=0.64\left(1\right)$.

Figure 10

Average domain-wall energy at different system sizes. Data points are connected by using straight lines for better visibility.

Figure 11

Standard deviation of domain-wall energy at different system sizes. Data points are connected by using straight lines for better visibility. The straight solid line shows a power law $\sim L^{-0.1}$ for comparison.