Diagrammatic valence-bond theory for finite model Hamiltonians

Abstract

Valence bond (VB) diagrams form a complete basis for model Hamiltonians that conserve total spin, S, and have one valence state, φ_p, per site. Hubbard and Pariser-Parr-Pople (PPP) models illustrate ionic problems, with zero, one, or two electrons in each φ_p, while isotropic Heisenberg models illustrate spin problems, with only purely covalent VB diagrams. The difficulty of nonorthogonal VB diagrams is by-passed by exploiting the finite dimensionality of the complete basis and working with unsymmetric sparse matrices. We introduce efficient bit manipulations for generating, storing, and handling VB diagrams as integers and describe a new coordinate relaxation method for the ground and lowest excited states of unsymmetric sparse matrices. Antiferromagnetic spin-½ Heisenberg rings and chains of N≤20 spins, or 2^N spin functions, are solved in C₂ symmetry as illustrative examples. The lowest S=1 and 0 excitations are related to domain walls, or spin solitons, and studied for alternations corresponding to polyacetylene. VB diagrams with arbitrary S and nonneighbor interactions are constructed for both spin and ionic problems, thus extending diagrammatic VB theory to other topologies.