Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry

We perform Transition matrix Monte Carlo simulations to evaluate the entropy of rhombus tilings with fixed polygonal boundaries and 2D-fold rotational symmetry. We estimate the large-size limit of this entropy for D=4 to 10. We confirm analytic predictions of N. Destainville et al., J. Stat. Phys. 120, 799 (2005) and M. Widom et al., J. Stat. Phys. 120, 837 (2005), in particular that the large size and large D limits commute, and that entropy becomes insensible to size, phason strain and boundary conditions at large D. We are able to infer finite D and finite size scalings of entropy. We also show that phason elastic constants can be estimated for any D by measuring the relevant perpendicular space fluctuations.Comment: Accepted for publication in Eur. Phys. J.

Free Fermions Violate the Area Law For Entanglement Entropy

We show that the entanglement entropy associated to a region grows faster than the area of its boundary surface. This is done by proving a special case of a conjecture due to Widom that yields a surprisingly simple expression for the leading behaviour of the entanglement entropy.Comment: Proceedings of the 9th Hellenic School on Elementary Particle Physics and Gravity, Corfu 2009. 4 page