Matching Kasteleyn cities for spin glass ground states

As spin glass materials have extremely slow dynamics, devious numerical methods are needed to study low-temperature states. A simple and fast optimization version of the classical Kasteleyn treatment of the Ising model is described and applied to two-dimensional Ising spin glasses. The algorithm combines the Pfaffian and matching approaches to directly strip droplet excitations from an excited state. Extended ground states in Ising spin glasses on a torus, which are optimized over all boundary conditions, are used to compute precise values for ground state energy densities.

Figure 1

(Color online) An outline of the steps that convert a spin and bond configuration first to the dual weighted lattice $D$ and then to the weighted graph $G$, in order to compute the extended ground state. (a) The original spin system, here with initial spins $s_i=1$ indicated by white circles, and bond configuration $J_{ij}$ (dashed lines) determine edge weights $w_{ij}=J_{ij}{s}_i{s}_j$ (taking the initial BCs to be periodic) in the dual graph $D$ (solid vertices and edges), with periodic boundary conditions. (b) The vertices in $D$ are replaced by Kasteleyn cities (light lines have zero weight in $G$). (c) An example set $S$ of negative weight loops in $D$ is shown, with two simple loops and one winding loop. (d) Heavy lines indicate the minimum-weight perfect matching $M$ for $G$ (light lines are free edges and solid circles are vertices in $G$). The negative weight loops $S$ (heavy dashed lines) are found by clipping out the Kasteleyn cities and keeping the remaining edges. Finally, spins are assigned by scanning across the sample: each time an odd number of loops is crossed, the spins are flipped (gray circles indicate $s_i=-1$). In this case, the inconsistency at the right side is corrected by changing horizontal boundary conditions from periodic to antiperiodic.

Figure 2

(Color online) Extended ground state energy density $e_0\left(L\right)$ for the 2D Ising spin glass on a torus vs scaled system size $L^{\theta -2}$. Two scales for each disorder type are used, to show the linear fit at large $L$ and the higher-order corrections at small $L$. (a),(b) Assuming $\theta \approx -0.28$ for Gaussian disorder gives $e_0(\infty )=-1.314788\left(4\right)$. (c),(d) A similar plot using $\theta =0$ for discrete values of $J_{ij}=\pm 1$ gives $e_0^{\pm }(\infty )=-1.401925\left(3\right)$.