CORE and the Haldane Conjecture

TL;DR

This paper demonstrates that the CORE real-space renormalization group method can accurately analyze complex quantum spin systems and provides a first-principles explanation of the Haldane conjecture for spin chains.

Contribution

The paper shows that simple CORE computations can yield accurate results and offer qualitative insights into topological properties of quantum spin chains, including the Haldane conjecture.

Findings

CORE provides highly accurate results with few states and terms.

Simple CORE captures the physics behind the Haldane conjecture.

CORE offers better qualitative understanding than naive RG methods.

Abstract

The Contractor Renormalization group formalism (CORE) is a real-space renormalization group method which is the Hamiltonian analogue of the Wilson exact renormalization group equations. In an earlier paper\cite{QGAF} 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 despite the folklore which asserts that all real-space renormalization group schemes are necessarily inaccurate, simple Contractor Renormalization group (CORE) computations can give highly accurate results even if one only keeps a small number of states per block and a few terms in the cluster expansion. In addition I argue that even very simple CORE computations give a much better qualitative understanding of the physics than naive renormalization group methods. In particular I show that the simplest CORE computation yields a first principles understanding of how the famous Haldane conjecture works for the case of the spin-1/2 and spin-1 HAF.