Violation of the Harris-Barghathi-Vojta Criterion

In 1974, Harris proposed his celebrated criterion: Continuous phase transitions in $d$-dimensional systems are stable against quenched spatial randomness whenever $d\nu >2$, where $\nu$ is the clean critical exponent of the correlation length. Forty years later, motivated by violations of the Harris criterion for certain lattices such as Voronoi-Delaunay triangulations of random point clouds, Barghathi and Vojta put forth a modified criterion for topologically disordered systems: $a\nu >1$, where $a$ is the disorder decay exponent, which measures how fast coordination number fluctuations decay with increasing length scale. Here we present a topologically disordered lattice with coordination number fluctuations that decay as slowly as those of conventional uncorrelated randomness, but for which the clean universal behavior is preserved, hence violating even the modified criterion.

Figure 1

(a) Sample of a usual VD triangulation of a Poissonian point cloud. (b) The lattice we introduce in this Letter, ${\mathrm{VD}}^{+}$, constructed from (a) by adding random local bonds.

Figure 2

Coordination number fluctuations on different length scales for ${\mathrm{VD}}^{+}$, ordinary VD, the set of additional bonds that distinguishes both ($+$), and a site-diluted square lattice (SD). The curves are found to decay as ${\sigma }_Q\sim L_b^{-3/2}$ for the Voronoi-Delaunay (VD) triangulation, and as ${\sigma }_Q\sim L_b^{-1}$ for the other lattices. For the analysis, 100 independent realizations of size $L=5000$ have been used for each lattice. Note that the data points for ${\mathrm{VD}}^{+}$ and ($+$) almost coincide.

Figure 3

Quasistationary density (a) and lifetime (b) as a function of the linear lattice size $L$, ranging from 32 to 1024. The slope of $\ln \rho$ yields $\beta /\nu$, whereas $\ln \tau$ yields $z$. The dots represent averages over 320 independent disorder realizations. The error bars are smaller than the symbol sizes.

Figure 4

Finite size data collapse of simulations starting from a fully occupied ${\mathrm{VD}}^{+}$ lattice at the critical point $p_c=0.589775$, using the critical exponent estimates stated in the figure. All curves are averages over 500 disorder realizations with 5 runs per realization. $L$ denotes the linear system size. The inset shows the nonrescaled density as a function of time.

Figure 5

Data collapse in the off-critical region, starting from a fully occupied ${\mathrm{VD}}^{+}$ lattice of size $L=2048$ using the critical exponent estimates given in the figure. All curves are averages over 250 disorder realizations, with 5 runs per realization. The symbol $\mathrm{\Delta }$ denotes the distance from the critical point $p_c=0.589775$. The inset shows the nonrescaled density as a function of time.