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Register a new userPercolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
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K. Malarz
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We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The newly proposed single scalar index $xi=sum_i z_ir_i^2/i$ (depending on the coordination zone number $i$, the neighbourhood coordination number $z$ and the square-distance $r^2$ to sites in $i$-th coordination zone from the central site) allows to differentiate among various neighbourhoods and relate $p_c$ to $xi$. The thresholds roughly follow a power law $p_cproptoxi^{-gamma}$ with $gammaapprox 0.710(19)$.
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We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, Physical Review E 90, 062101 (2014)], of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $p_c(text{4NN})=0.192410(43)$, $p_c(text{3NN+2NN})=0.232008(38)$, $p_c(text{5NN+4NN})=0.140286(5)$, $p_c(text{3NN+2NN+NN})=0.215484(19)$, $p_c(text{5NN+4NN+NN})=0.131792(58)$, $p_c(text{5NN+4NN+3NN+2NN})=0.117579(41)$, $p_c(text{5NN+4NN+3NN+2NN+NN})=0.115847(21)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(text{NN})=p_c(text{2NN})=p_c(text{3NN})=p_c(text{5NN})=frac{1}{2}$ is recovered with five digits accuracy $p_c(text{NN})=0.500029(46)$ by averaging over one thousand lattice realisations only.
In the paper random-site percolation thresholds for simple cubic lattice with sites neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation [Bastas et al., arXiv:1411.5834] is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are $p_C(text{4NN})=0.31160(12)$, $p_C(text{4NN+NN})=0.15040(12)$, $p_C(text{4NN+2NN})=0.15950(12)$, $p_C(text{4NN+3NN})=0.20490(12)$, $p_C(text{4NN+2NN+NN})=0.11440(12)$, $p_C(text{4NN+3NN+NN})=0.11920(12)$, $p_C(text{4NN+3NN+2NN})=0.11330(12)$, $p_C(text{4NN+3NN+2NN+NN})=0.10000(12)$, where 3NN, 2NN, NN stands for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with two times larger lattice constant the percolation threshold $p_C$(4NN) is exactly equal to $p_C$(NN). The simplified Bastas et al. method allows for reaching uncertainty of the percolation threshold value $p_C$ similar to those obtained with classical method but ten times faster.
We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a non-bipartite lattice, dimer-dimer correlations are short-ranged, so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 1/3 ln2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 1/4 in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground state energy of the quantum Heisenberg model on the TKL.
Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate $p$, and predator particles spread only to neighboring sites occupied by prey particles at rate $1$, killing the prey particle that existed at that site. It was found that the prey can survive with non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlo simulations on the square lattice, we estimate the value of $p_c = 0.49451 pm 0.00001$, and the critical exponents are consistent with the undirected percolation universality class. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. For all $p < p_c$ in $D$-dimensions, the number of predators in the absorbing configuration has a stretched-exponential distribution in contrast to the exponential distribution in the standard percolation theory. We also study the problem starting from the line initial condition with predator particles on all lattice points of the line $y=0$ and prey particles on the line $y=1$. In this case, for $p_c<p < 1$, the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of $p$, but with no predators. At $p=1$, the fronts undergo a depinning transition. The fluctuations of the front follow Kardar-Parisi-Zhang scaling both above and below this depinning transition.
The recently fabricated two-dimensional magnetic materials Cu9X2(cpa)6.xH2O (cpa=2-carboxypentonic acid; X=F,Cl,Br) have copper sites which form a triangular kagome lattice (TKL), formed by introducing small triangles (``a-trimers) inside of each kagome triangle (``b-trimer). We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite temperature phase diagram for this XXZ-Ising model, including the residual zero temperature entropies of the seven ground state phases. Whereas the disordered (spin liquid) ground state of the pure Ising TKL model has macroscopic residual entropy ln72=4.2767... per unit cell, the introduction of transverse(quantum) couplings between neighboring $a$-spins reduces this entropy to 2.5258... per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions, and derive the ground state phase diagram. A small (or even infinitesimal) field leads to a new phase that corresponds to a non-intersecting loop gas on the kagome lattice, with entropy 1.4053... per unit cell and a mean magnetization for the b-spins of 0.12(1) per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase which maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.
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