Domain-wall excitations in the two-dimensional Ising spin glass

The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10000\times 10000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta =-0.2793\left(3\right)$ and $d_{\mathrm{f}}=1.27319\left(9\right)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_{\mathrm{f}}=1.279\left(2\right)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.

Figure 1

Mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect matching. An auxiliary graph is constructed by expanding each plaquette of the dual lattice into a complete graph $K_4$ of nodes (left). Additional rows and columns of $K_4$ nodes are added instead of the outer plaquette to make the auxiliary graph more regular. Edge weights on the auxiliary graph are $J_{ij}$ for each bond that crosses a bond $(i,j)$ of the original graph and zero otherwise. Then, a minimum-weight perfect matching is determined on the auxiliary graph (middle). By contracting the $K_4$ vertices again, the matching reduces to a minimum cut on the spin lattice, i.e., a set of closed loops surrounding islands of down spins in a sea of up spins or vice versa (right). Dashed bonds on the spin lattice correspond to antiferromagnetic couplings $J_{ij}<0$, solid bonds to ferromagnetic ones, $J_{ij}>0$.

Figure 2

Schematic representation of the windowing technique to determine ground states for toroidal systems. The dashed square shows the window and the blue diamonds represent the sites whose spins will be updated next by the windowing technique, as they are contained within the current window. Red squares indicate sites whose spins are fixed in their current orientation with strong bonds, indicated by the thick black lines. As a result, the MWPM problem will be solved for the system of red and blue spins with using free boundary conditions.

Figure 3

Application of the windowing method to find a ground state of a sample with toroidal boundaries. Spins on white lattice sites are consistent with the ground-state orientation $s_i^0$, i.e., $s_i{s}_i^0=+1$, black spins are oppositely oriented, i.e., $s_i{s}_i^0=-1$. In a random initial configuration, the spins have $s_i{s}_i^0=\pm 1$ uniformly at random (top left). Exact ground states are found in windows of size $(L-2)\times (L-2)$ placed at a random location (red dotted lines), with the remaining spins acting as fixed boundaries. After a few iterations, all spins have the ground-state orientation (bottom right).

Figure 4

Average time $t$ per sample to determine ground states of systems with PFBC and PPBC for $L\times L$ samples using the minimum-weight perfect matching (MWPM) method for periodic-free samples (PFBC), the windowing technique (WT) for periodic-periodic samples (PPBC), and the spin-glass server (SGS), respectively. The straight lines are fits of the form (7) to the data, whereas the lines for the SGS data are just interpolations to guide the eye.

Figure 5

(a) Disorder-averaged ground-state energy per site ${\langle e\rangle }_J={\langle \overline{E}/L^2\rangle }_J$ for PFBC and Gaussian couplings together with a fit of the form (10) to the data in the range $L=10,...,10000$. (b) Average defect energies ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J$ for the same system as calculated from the difference in ground-state energies between periodic and antiperiodic boundary conditions in the $x$ direction. The points show our data for $8\le L\le 10000$ and the solid line represents a fit of the form ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J\left(L\right)=A_{\theta }{L}^{\theta }+C_{\theta }/L^2$ to the data. The inset shows the correction ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J\left(L\right)-A_{\theta }{L}^{\theta }$ plotted against $1/L^2$ illustrating that this single term describes the corrections very well. (c) Average length $\ell$ of the domain-wall in the overlap of ground states for periodic and antiperiodic boundaries in $x$ direction and free boundaries in $y$ direction (PFBC boundaries). The line shows a fit of the functional form ${\langle \ell \rangle }_J=A_{\ell }{L}^{d_{\mathrm{f}}}$ to the data for $L\ge L_{\min }=40$. The inset shows a blow-up of the deviations for small $L$.

Figure 6

Overlap configuration of the ground states for P and AP boundaries for a $L=10000$ disorder realization of the PFBC Gaussian system. The red line demarcates the domain wall which traverses $\ell =233141$ dual links.

Figure 7

(a) Average ground-state energies for PPBC and Gaussian couplings together with a fit of the form (16) to the data in the range $L\ge L_{\min }=16$. (b) Scaling of defect energies for the Gaussian model with fully periodic boundary conditions. The solid line shows a fit of the form ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J\left(L\right)=A_{\theta }{L}^{\theta }+C_{\theta }/L^2$ to the data. The inset shows the correction ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J\left(L\right)-A_{\theta }{L}^{\theta }$ plotted against $1/L^2$ illustrating that this single term describes the corrections very well. (c) Scaling of the length of the domain wall between P and AP ground states for the Gaussian PPBC case. The solid line shows a fit of the form ${\langle \ell \rangle }_J=A_{\ell }{L}^{d_{\mathrm{f}}}$ to the data with $L_{\min }=40$. The inset shows a detail of the main plot for small $L$.

Figure 8

(a) Scaling of the kurtosis $Kurt\left[e\right]$ of the distribution of ground-state energies per spin for the Gaussian model with PFBC as a function of system size. For $L\ge 20$, it is consistent with the value $Kurt\left[e\right]=3$ of a normal distribution. (b) Distribution of defect energies $\left|\mathrm{\Delta }E\right|$ for the same model, rescaled with the expected asymptotic behavior $\propto L^{\theta }$ with $\theta =0.2793$. The solid line shows a Gaussian distribution of the same mean and variance. (c) Distribution of domain-wall lengths $\ell$ for the PFBC Gaussian case, rescaled according to the limiting form $\propto L^{d_{\mathrm{f}}}$ with $d_{\mathrm{f}}=1.27319$. The solid line represents a lognormal distribution fitted to the empirical data.

Figure 9

(a) Average ground-state energies for bimodal couplings and PFBC boundaries, together with a fit of the functional form (10) with $\theta =0$ to the data for the range $L=20,...,3000$. (b) Defect energies for systems with bimodal couplings and PFBC boundaries. Clearly, ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J$ converges to a nonzero value as $L\to \infty$, indicating that $\theta =0$. The line shows a fit of the form ${\langle \left|\mathrm{\Delta }E\right|\rangle }_J=\mathrm{\Delta }{E}_{\infty }+B_{\theta }{L}^{-\omega }$ to the data with $L\ge L_{\min }=10$ yielding $\mathrm{\Delta }{E}_{\infty }=0.960\left(5\right)$. (c) Probability distribution over disorder of the defect energies for the PFBC $\pm J$ model and different system sizes. For $L\to \infty$ the distribution approaches a limiting shape close to the $L=1000$ case shown here.

Figure 10

(Left) Schematic representation of the set of dual bonds satisfying the condition (14) for the case of bimodal couplings. Besides the domain wall, it contains isolated loops enclosing free clusters of spins as well as bubbles of free spins attached to the domain wall. Removing the isolated free loops one arrives at the set ${\mathcal{D}}_{\mathrm{long}}$, which we denote as the “long” domain wall. (Right) After the additional removal of bubbles one arrives at the set ${\mathcal{D}}_{\mathrm{short}}$ of dual bonds comprising the “short” domain wall of the configuration.

Figure 11

(a) Average length ${\langle {\ell }_{\mathrm{short}}\rangle }_J$ of the short domain wall for the bimodal model as a function of linear system size $L$ for the three different algorithms employed. The inset shows the deviation of each data set from the fit of the power law ${\langle \ell \rangle }_J=A_{\ell }{L}^{d_{\mathrm{f}}}$ to the uniform sampling data for $L\ge L_{\min }=16$, which results in $d_{\mathrm{f}}=1.279\left(2\right)$ ($Q=0.33$). (b) Average length ${\langle {\ell }_{\mathrm{long}}\rangle }_J$ of the long domain walls for the different algorithms. The inset shows the deviation of each data set from the fit of a pure power law to the uniform data, yielding $d_{\mathrm{f}}=1.281\left(3\right)$ for $L_{\min }=16$ ($Q=0.97$). (c) Ratio of the average lengths of long and short domain walls as estimated from the different algorithms. In all cases, the ratio approaches a constant, in line with the identical estimates of fractal dimension for ${\ell }_{\mathrm{short}}$ and ${\ell }_{\mathrm{long}}$.

Figure 12

Distribution of the lengths ${\ell }_{\mathrm{long}}$ of the long domain walls for $\pm J$ couplings and PFBC boundaries as resulting from the uniform sampling approach. The rescaling of the axes is with respect to the fractal dimension $d_{\mathrm{f}}=1.281\left(3\right)$ estimated from the data in Fig. 11.