Families of planar lattices with arbitrarily high T_{c} for the ferromagnetic Ising model.

Abstract

We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, T_{c}, for the classical uniform and nearest-neighbor ferromagnetic Ising model. Our approach is motivated by recently found exact bounds, which imply that large values of T_{c} require large values of the maximal coordination number of the lattice, q_{max}. We create such lattices through iterative triangulation and derive explicit expressions for their T_{c}. Furthermore, we show that T_{c} for these families scales asymptotically as T_{c}/J∼Alnq_{max}-2lnlnq_{max} with a universal prefactor A=2/ln2. We introduce a function T_{c}^{*}(q_{max}) that we conjecture to be an upper bound on the critical temperature of any periodic tessellation of the plane. We show that the family of so-called Apollonian lattices, which are derived from the triangular lattice through iterative triangulation, saturates this bound. The lattices discussed in this work are relevant for theoretical questions of optimality in network systems and may be realized experimentally in Coherent Ising Machine or topoelectric circuits in the future.