CORE and the Haldane Conjecture

In an earlier paper[1] I 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 I show 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 = ∑ i[~s(i) · ~s(i + 1) − β (~s(i) · ~s(i + 1))] which has a valence bond ground state when β = −1/3. I then argue 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 operators ~s(i) ·~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)[6], all of which have valence bond ground states. Submitted to Physical Review D. ∗Work supported in part by Department of Energy contracts DE–AC03–76SF00515 and DE– AC02–76ER03069