Corrections to scaling for percolative conduction: anomalous behavior at small L

Recently Grassberger has shown that the correction to scaling for the conductance of a bond percolation network on a square lattice is a nonmonotonic function of the linear lattice dimension with a minimum at $L = 10$, while this anomalous behavior is not present in the site percolation networks. We perform a high precision numerical study of the bond percolation random resistor networks on the square, triangular and honeycomb lattices to further examine this result. We use the arithmetic, geometric and harmonic means to obtain the conductance and find that the qualitative behavior does not change: it is not related to the shape of the conductance distribution for small system sizes. We show that the anomaly at small L is absent on the triangular and honeycomb networks. We suggest that the nonmonotonic behavior is an artifact of approximating the continuous system for which the theory is formulated by a discrete one which can be simulated on a computer. We show that by slightly changing the definition of the linear lattice size we can eliminate the minimum at small L without significantly affecting the large L limit.Comment: 3 pages, 4 figures;slightly expanded, 2 figures added. Accepted for publishing in Phys. Rev.

Anisotropic Scaling in Layered Aperiodic Ising Systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.Comment: 7 pages, 2 eps-figures, Plain TeX and epsf, minor correction

Entanglement entropy of the Q >= 4 quantum Potts chain

The entanglement entropy, ${\cal S}$, is an indicator of quantum correlations in the ground state of a many body quantum system. At a second-order quantum phase-transition point in one dimension ${\cal S}$ generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the $Q$-state quantum Potts chain for $Q>4$ and calculate ${\cal S}$ across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point ${\cal S}$ shows a jump, which is expected to vanish for $Q \to 4^+$. This jump is calculated in leading order as $\Delta {\cal S}=\ln Q[1-4/Q-2/(Q \ln Q)+{\cal O}(1/Q^2)]$