Two-dimensional Ising model on random lattices with constant coordination number

We study the two-dimensional Ising model on networks with quenched topological (connectivity) disorder. In particular, we construct random lattices of constant coordination number and perform large-scale Monte Carlo simulations in order to obtain critical exponents using finite-size scaling relations. We find disorder-dependent effective critical exponents, similar to diluted models, showing thus no clear universal behavior. Considering the very recent results for the two-dimensional Ising model on proximity graphs and the coordination number correlation analysis suggested by Barghathi and Vojta [Phys. Rev. Lett. 113, 120602 (2014)], our results indicate that the planarity and connectedness of the lattice play an important role on deciding whether the phase transition is stable against quenched topological disorder.

Figure 1

Coordination number fluctuations on different length scales for several lattices. The curves are obtained using Eq. (2) and found to decay as ${\sigma }_Q\sim L_b^{-3/2}$ for the Voronoi-Delaunay triangulation (VD) and as ${\sigma }_Q\sim L_b^{-1}$ for the others. Measured decay exponent values: Gabriel Graph (GG): $0.999\left(1\right)$, Relative Neighborhood Graph (RNG): $1.001\left(2\right)$, Site-Diluted regular square lattice (SD): $1.001\left(3\right)$, VD: $1.501\left(2\right)$, Random Geometric Graph (RGG): $1.004\left(6\right)$, and symmetrized $q$-Nearest-Neighbor graph (qNNsym) with $q\ge 6$: $1.001\left(2\right)$.

Figure 2

Coordination number correlation functions. (a) First-layer, according to Eq. (3), for the VD triangulation, Gabriel Graph, Relative Neighborhood Graph, Random Geometric Graph, and symmetrized $q$-Nearest-Neighbor graph with $q\ge 6$. For the RGG the radius was chosen such that the average coordination number is $\overline{q}=6$ and $C\left(r\right)$ is rescaled by a factor of 0.1. (b) Second-layer coordination number correlation function (4) for VD and the constant coordination models (CC4, CC6, and CC10). We show a magnification in the inset.

Figure 3

Samples of the considered lattice constructions. Note that all lattices with exception of those in the second line are constructed from the same set of points. For the RGG, the radius is set according to Eq. (5) such that $\overline{q}=6$. The dilution probability for the diluted lattice is 20%. For the symmetrized $q$-Nearest-Neighbor lattice, $q\ge 6$ holds, as all links have been made bidirectional.

Figure 4

Normalized bond lengths histogram, rescaled in units of $R_{\overline{q}=6}$ for the VD triangulation and the CC6 lattices on Poissonian and Poissonian disk-sampled (see Appendix pp2) point distributions. Each color shows collapsing curves for $L=32,64$, and 128. The mean values of the histograms are 0.819 (VD), 0.767 (CC6), and 0.834 (CC6-HCPP).

Figure 5

Residuals of four out of 28 fits for the exponent $\nu$ for the Ising model on a regular square lattice, shifted vertically for convenience. The vertical dotted line separates the region that is excluded in the fits. The curves, listed from top to bottom, are guides to the eye.

Figure 6

Residuals for the fits of the exponents (a) $\gamma /\nu$ and (b) $\beta /\nu$ for the regular lattice. The single lines in each panel correspond to observables (10c) to (11e) from top to bottom and are shifted vertically for convenience. The vertical dotted line separates the region which is excluded from the fitting procedure, and the curves are guides to the eye.

Figure 7

Schematic representation of the local fitting procedure, here for $L=96$. Each circle symbolizes one lattice size $L$, whereas the numbers inside the circle are the corresponding weights.

Figure 8

Linear fit residuals for $\gamma /\nu$ for the VD lattice in the range from $L_{\min }=160$ to $L_{\max }=400$. The single lines in each panel correspond to observables (10c) to (11e) from top to bottom and are shifted for convenience. The region of the dashed black lines show the excluded region. The curves are guides to the eye. Only the green curve (third from the top) shows no systematic deviation and yields $\gamma /\nu =1.7512\left(7\right)$, with a goodness-of-fit value $Q=0.90$.

Figure 9

(a) Estimate of ${\beta }_c$, and its error (shaded region) as a function of fixed $\omega$. (b) Corresponding reduced ${\chi }^2$ values, as described in the text. Employing different fitting algorithms generates qualitatively similar results, which are also insensitive to the choice of lattice size range.

Figure 10

Comparison of critical exponent estimates for the CC lattices with $q=4,6,10$ and VD lattices, as a function of $L$. Dashed lines indicate the clean universal values of the 2D Ising universality class.

Figure 11

Lattice configurations (a) before and (b) after the simulated annealing process. The long bonds, highlighted in dark, have been effectively rearranged.

Figure 12

Effective critical exponent estimates for the CC6-HCPP model as a function of the center of the fitting window, $L$. Dashed lines indicate the clean universal values of the 2D Ising universality class.