The Ammann-Beenker tilings revisited

This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann-Beenker tilings as the solution of a simple optimization problem.Comment: 7 pages, 4 figure

Jack on a Devil’s Staircase

We review a simple mechanism for the formation of plateaux in the fractional quantum Hall effect. It arises from a map of the microscopic Hamilto- nian in the thin torus limit to a lattice gas model, solved by Hubbard. The map suggests a Devil\u2019s staircase pattern, and explains the observed asymmetries in the widths. Each plateau is a new ground state of the system: a periodic Slater state in the thin torus limit. We provide the unitary operator that maps such limit states to the full, effective ground states with same filling fraction. These Jack polynomials generalise Laughlin\u2019s ansatz, and are exact eigenstates of the Laplace-Beltrami operator. Why are Jacks sitting on the Devil\u2019s staircase? This is yet an intriguing problem. Talk given in Milan, Congresso di Dipartimento 2017 (L.G.M.)