Valence Bond Calculations for Quantum Spin Chains: From Impurity Entanglement and Incommensurate Behaviour to Quantum Monte Carlo

Abstract

<p>In this thesis I present three publications about the use of<br />valence bonds to gain information about quantum spin systems.<br />Valence bonds are an essential ingredient of low energy states present<br />in many compounds.<br /><br />The first part of this thesis is dedicated to<br />two studies of the antiferromagnetic J₁-J₂ chain with<br />S=1/2. We show how automated variational calculations based on<br />valence bond states can be performed close to the Majumdar-Ghosh point<br />(MG-point). At this point, the groundstate is a product state of<br />dimers (valence bonds between nearest neighbours). In the dimerized<br />region surrounding the MG-point, we find such variational computations<br />to be reliable.<br /><br />The first publication is about<br />the entanglement properties of an impurity attached to the chain. We show<br />how to use the variational method to calculate the negativity, an<br />entanglement measure between the impurity and a distant part of the<br />chain. We find that increasing the impurity coupling and a<br />minute explicit dimerization, suppress the long-ranged entanglement<br />present in the system for small impurity coupling at the MG-point. <br /><br />The second publication is about a<br />transition from commensurate to incommensurate behaviour and how its<br />characteristics depend on the parity of the length of the chain. The<br />variational technique is used in a parameter regime inaccessible to<br />DMRG. We find that in odd chains, unlike in even chains, a very<br />intricate and interesting pattern of level crossings can be observed. <br /><br />The publication of the second part is about novel worm algorithms for<br />a popular quantum Monte Carlo method called valence bond quantum Monte<br />Carlo (VBQMC). The algorithms are based on the notion of a worm<br />moving through a decision tree. VBQMC is entirely formulated in<br />terms of valence bonds. In this thesis, I explain how the approach<br />of VBQMC can be translated to the S_z-basis. The algorithms explained<br />in the publication can be applied to this S_z-method.</p>