Splitting of the ground state manifold of classical Heisenberg spins as couplings are varied

We construct clusters of classical Heisenberg spins with two-spin $vec{S}_i.vec{S}_j$-type interactions for which the ground state manifold consists of disconnected pieces. We extend the construction to lattices and couplings for which the ground state manifold splits into an exponentially large number of disconnected pieces at a sharp point as the interaction strengths are varied with respect to each other. In one such lattice we construct, the number of disconnected pieces in the ground state manifold can be counted exactly.

Pattern formation in growing sandpiles

Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.

Exact entropy of dimer coverings for a class of lattices in three or more dimensions

We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the method also works for graphs without translational symmetry. The partition function for dimer coverings on these lattices can be determined also for a class of assignments of different activities to different edges.

Dependence of ground state energy of classical n-vector spins on n

We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n > n_{max}(N) = floor((sqrt(8N+1)-1) / 2) . We show that this bound is the best possible. We also derive an upper bound for E_G(m) in terms of E_G(n), for m<n. We obtain an upper bound on the frustration in the system, as measured by F(n), which is defined to be (sum_{i>j} |J_ij| + E_G(n)) / (sum_{i>j} |J_ij|). We describe a procedure for constructing a set of J_ijs such that an arbitrary given state, {S_i}, is the ground state.