Evaluation of low-energy effective Hamiltonian techniques for coupled spin triangles

Abstract

Motivated by recent work on Heisenberg antiferromagnetic spin systems on various lattices made up of triangles, we examine the low-energy properties of a chain of antiferromagnetically coupled triangles of half-odd-integer spins. We derive the low-energy effective Hamiltonian to second order in the ratio of the coupling J₂ between triangles to the coupling J₁ within each triangle. The effective Hamiltonian contains four states for each triangle which are given by the products of spin-½ states with the states of a pseudospin ½. We compare the results obtained by exact diagonalization of the effective Hamiltonian with those obtained for the full Hamiltonian using exact diagonalization and the density-matrix renormalization group method. It is found that the effective Hamiltonian gives an accurate value for the ground-state energy only if the ratio J₂/J₁ is less than about 0.2 and that too for the spin-½ case with linear topology. The chain of spin-½ triangles shows interesting properties like spontaneous dimerization and several singlet and triplet low-energy (possibly gapless) states which lie close to the ground state. We have also studied the spin-3/2 case and find the low-energy effective Hamiltonians (LEH's) to be less accurate there than in the spin-½ case. Finally, we have studied nonlinear topologies where the LEH results deviate further from the exact results.