CORE and the Haldane Conjecture

In an earlier paper, the author showed that the Contractor Renormalization group (CORE) method could be used to map a theory of free quarks, and quarks interacting with gluons, into a generalized frustrated Heisenberg antiferromagnet (HAF) and proposed using CORE methods to study these theories. Since generalizations of HAF's exhibit all sorts of subtle behavior which, from a continuum point of view, are related to topological properties of the theory, it is important to know that CORE can be used to extract this physics. In this paper he shows that simple Contractor Renormalization group (CORE) computations provide a first principles understanding of the famous Haldane conjecture. Explicit range-2 computations for the spin-1/2 and spin-1 Heisenberg antiferromagnet reveal the differences between these theories and show that the mass gap in the spin-1 theory is intimately related to the structure of a more general theory with Hamiltonian H = {summation}{sub i}[s(i) {center{underscore}dot} s(i + 1) {minus} {beta}(s (i) {center{underscore}dot} s(i + 1)){sup 2}] which has a valence bond ground state when {beta}= {minus}1/3. He then argues that the case of a general spin-S HAF works similarly. More specifically, for integer S the renormalized Hamiltonian is described by a polynomial in the operatorsmore » s(i) {center{underscore}dot} s(i + 1) with coefficients which lie near the values for which the Hamiltonian would be of the type introduced by Affleck, Lieb, Kennedy and Tasaki (AKLT), all of which have valence bond ground states.« less