Reduction of two-dimensional dilute Ising spin glasses

The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. In combination with a graph-theoretical matching algorithm, this allows us to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent $y_2$. A value of $y_2=-0.281\left(3\right)$ is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions $d=3,\dots ,7$.

Figure 1

Star-triangle relation to reduce a three-connected spin $x_0$. The new bonds on the right are obtained in Eq. (6).

Figure 2

Illustration of rule VI for “strong” bonds. Top, the local topology of a graph is shown for two spins, $x_0$ and $x_1$, connected by a bond $J_{0,1}$ (thick line). If $J_{0,1}>0$ $(J_{0,1}<0)$ satisfies Eq. (6), $x_0$ and $x_1$ must align (antialign) in the ground state and $x_0$ can be removed. Bottom, the remainder graph is shown after the removal. The other bonds emanating from $x_0$ (dashed lines) are now directly connected to $x_1$ (potentially with a sign change, if $J_{0,1}<0$). This procedure may lead to a double bond (rule III), if $x_1$ was already connected to a neighbor of $x_0$ before.

Figure 3

(Color online) 2D spin glass with all spins up (left, up spins not shown). Straight lines are ferromagnetic, jagged lines are antiferromagnetic bonds. The dotted lines connect frustrated plaquettes (crosses). The bonds crossed by the dotted lines are unsatisfied. In the right part, the ground state with three spins pointing down (all other up) is shown, corresponding to a minimum number of unsatisfied bonds.

Figure 4

Logarithmic plot of ${\sigma }_L$ at $p=p^{*}=1∕2$. An asymptotic power-law regime corresponding to Eq. (9), different from those for $p>p^{*}$, is reached quickly, and an asymptotic fit extrapolates to $y_P=-0.98\left(2\right)$.

Figure 5

Plot on a logarithmic scale of the stiffness $\sigma$ as a function of system size $L$. The data is grouped into sets (connected by lines) parametrized by the bond density $p$. Most sets show a distinct scaling regime as indicated by Eq. (2) for sufficiently large $L$. In particular, the data for $p=0.51$ appears to never reach scaling and will be disregarded entirely.

Figure 6

Scaling plot of the data from Fig. 5 for $\sigma$, fitted to Eq. (10) as a function of the scaling variable $x=L∕\xi \left(p\right)$ where $\xi \left(p\right)={(p-p^{*})}^{-{\nu }^{*}}$. Data below the scaling regime in each set (i.e., below a certain value of $L$ for each $p$) from Fig. 5 was cut. The continuous line represents the power law ${[L∕\xi \left(p\right)]}^{-0.281}$, using the fitted value for $y$.