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Valence Bond Calculations for Quantum Spin Chains

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Valence Bond Calculations for Quantum Spin Chains VALENCE BOND CALCULATIONS FOR QUANTUM SPIN CHAINS VALENCE BOND CALCULATIONS FOR QUANTUM SPIN CHAINS: FROM IMPURITY ENTANGLEMENT AND INCOMMENSURATE BEHAVIOUR TO QUANTUM MONTE CARLO By ANDREAS DESCHNER, Dipl.-Phys. A Thesis Submitted to the School of Graduate Studies in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy McMaster University Copyright by Andreas Deschner, 2014. Doctor of Philosophy, Physics and Astronomy, McMaster University (2014), Hamilton, Ontario Title: Valence Bond Calculations for Quantum Spin Chains: From Impurity Entanglement and Incommensurate Behaviour to Quantum Monte Carlo Author: Andreas Deschner, Dipl.-Phys. (Heidelberg University) Supervisor: Dr. Erik S. Sørensen Number of pages: xi, 120 ii Abstract In this thesis I present three publications about the use of valence bonds to gain information about quantum spin systems. Valence bonds are an essential ingredient of low energy states present in many compounds. The first part of this thesis is dedicated to two studies of the antiferromagnetic 퐽1-퐽2 chain with 푆 = 1/2. We show how automated variational calculations based on valence bond states can be performed close to the Majumdar-Ghosh point (MG-point). At this point, the groundstate is a product state of dimers (valence bonds between nearest neighbours). In the dimerized region surrounding the MG-point, we find such variational computations to be reliable. The first publication is about the entanglement properties of an impurity attached to the chain. We show how to use the variational method to calculate the negativity, an entanglement measure between the impurity and a distant part of the chain. We find that increasing the impurity coupling and a minute explicit dimerization, suppress the long-ranged entanglement present in the system for small impurity coupling at the MG-point. The second publication is about a transition from commensurate to incommensurate behaviour and how its characteristics depend on the parity of the length of the chain. The variational technique is used in a parameter regime inaccessible to DMRG. We find that in odd chains, unlike in even chains, a very intricate and interesting pattern of level crossings can be observed. The publication of the second part is about novel worm algorithms for a popular quantum Monte Carlo method called valence bond quantum Monte Carlo (VBQMC). The algorithms are based on the notion of a worm moving through a decision tree. VBQMC is entirely formulated in terms of valence bonds. In this thesis, I explain how the approach of VBQMC can be translated to the 푆푧-basis. The algorithms explained in the publication can be applied to this 푆푧-method. iii Acknowledgements It is done. My Ph.D. thesis is written. What is to do now, is to acknowledge those who have helped me during my years in Hamilton. Those who made it possible that I have grounds to claim that during my stay in Canada I have learned much and grown substantially, as a physicist and as a person. The first person I want to thank is Erik S. Sørensen. He gave me the opportunity to come here, he encouraged me to trust my judgement and put it to use. He guided me throughout these years. He also payed me. Thank you Erik. I also want to thank Catherine Kallin and Sung-Sik Lee, the other members of my com- mittee. They have always been a pleasure to meet with to discuss my work. Sung-Sik Lee has often, and certainly not only in committee meetings, impressed me with his clarity of thought and his willingness to share his knowledge. I am indebted to many people. More than I could be able to express my gratitude to. Also more than would be appropriate to express my gratitude to in this thesis. I nevertheless want to use this opportunity to acknowledge as well as I can. Anyone who goes to a foreign country can only wish to be welcomed by a Joshua D. McGraw; by somebody as open, friendly and genuine. He has seen me at my strongest and my weakest. I could always rely on him. This is also true for Jessica Thornton, who for many years was a source of warmth, trust, support and access to cats. Without her, this undertaking would have been much more difficult than it was already. Having people in my life that I like and trust gave me strength. I have always had great friends around me. Some faces disappeared quickly, some I have had the opportunity to see age a tiny bit, some I have only known for a very short while. Thank you (in approximately alphabetical order) Allan, Anna, Annie, Alex, Andrew, Andrew, Brendan, Brit, Clare, Claire, Jane, Jackie, Jesse, Faiyaz, Laura, Martin, Mark, Mike, Paul, Phil, Prasanna, Patrick, Rafael, Ray, Rory, Rob, Sedigh, Spencer, Wen and all those who I have forgotten to mention here. You all have taught me something profound. You all have made me feel at home in Hamilton. A home that I will now leave to go back to where I came from. To go where Diana Fichtner is, who believed in me when I needed somebody to believe in me while I was writing this thesis and who is giving me something to look forward to after my departure. To go where Joachim Hahn is, my dearest friend who has been there for me during the last 21 years. To go where my family is, who has always (and to my amazement) put up with my shortcomings. Every time I am among them, I know who is most important to me, regardless of how often I see them or speak with them. For my parents this is not merely true. It is due to them and due to my mother, who I know will always do and has always done everything she could to support her children. Thank you both, for giving me the opportunity to live and feel loved, to rejoice and be amazed. v List of Figures 1.1. Illustration of Néel- and dimer-order ..................... 5 1.2. The 퐽1-퐽2 chain ................................ 7 1.3. The phase diagram of the 퐽1-퐽2 chain .................... 8 2.1. A system separated into two subsystems ................... 22 3.1. The classical 퐽1-퐽2 chain in two limiting cases ................ 52 3.2. Condition for the energy of the classical 퐽1-퐽2 chain to be stationary. .... 52 3.3. Schematic generic phase diagram at a Lifshitz point ............. 53 3.4. A generic structure factor at a Lifshitz point ................. 55 3.5. Pole structure of a generic structure factor at a Lifshitz point ........ 56 3.6. The correlation length at a disorder point of the first kind .......... 57 4.1. The effect of the projection operator ..................... 76 4.2. The action of a bond operator for systems with spins of 푆 =1/2 ....... 78 4.3. The action of a bond operator for systems with spins of 푆 =1. ........ 79 4.4. Monte Carlo results for the Heisenberg chain ................ 83 vi List of Tables 4.1. Comparison of 푆푧-projector Monte Carlo and Bethe ansatz results for ground- state energies of the XXZ-model. ....................... 88 vii Preface In this sandwich thesis, I present the major results of my Ph.D.-research. The first chapter is an introduction that shall serve the purpose of setting the stage for the later chapters which contain the publications themselves as well as their own short introductions. • Publication I: Andreas Deschner and Erik S. Sørensen: Impurity Entanglement in the 퐽-퐽2-훿 Quantum Spin Chain; J. Stat. Mech. (2011) P10023; doi:10.1088/1742-5468/2011/10/P10023; ©IOP Publishing Ltd and SISSA Calculations: I performed all variational calculations. DMRG calculations were performed by Erik S. Sørensen. Manuscript: I wrote the bulk of the manuscript and made all figures. The introduction was written in equal parts by Erik S. Sørensen and me. Furthermore, Erik S. Sørensen provided (partly substantial) edits, comments and supervision. • Publication II: Andreas Deschner and Erik S. Sørensen: Incommensurability effects in odd length 퐽1-퐽2 quantum spin chains: On-site magnetization and entanglement; Physical Review B 87, 094415 (2013); doi: 10.1103/PhysRevB.87.094415 Copyright (2013) by the American Physical Society Calculations: I performed all variational calculations. DMRG calculations were performed by Erik S. Sørensen. Manuscript: I wrote the bulk of the manuscript and made all figures. Erik S. Sørensen wrote section IV. He also added several para- graphs throughout the manuscript. Furthermore, Erik S. Sørensen provided (partly substantial) edits, comments and supervision. viii • Publication III: Andreas Deschner and Erik S. Sørensen: Valence Bond Quantum Monte Carlo Algorithms defined on Trees; submitted to Physical Review E; pre-print: arXiv:1401.7592 Calculations: I performed all calculations. Manuscript: I wrote the bulk of the manuscript and made all figures. Erik S. Sørensen provided (partly substantial) edits, comments and supervision. ix Contents 1. Introduction 1 1.1. Magnetic models ............................... 2 1.2. Different orders ................................ 4 1.3. The 퐽1-퐽2 chain ................................ 6 1.4. Numerical approaches ............................. 10 1.5. Outline .................................... 14 I. Variational Calculations in the dimerized 퐽1-퐽2 chain 17 2. Impurity entanglement in a dimerized quantum spin chain 19 2.1. Entanglement ................................. 19 2.2. Entanglement is useful ............................ 21 2.3. Measures of entanglement .......................... 23 2.4. The paper ................................... 26 3. Incommensurability effects in an odd length quantum spin chain 51 3.1. Incommensurate order ............................ 51 3.2. Quantum effects ................................ 53 3.3. Disorder points ................................ 56 3.4. The paper ................................... 57 II. Concerning Valence Bond Quantum Monte Carlo 71 4. Worm algorithms for valence bond quantum Monte Carlo 73 4.1. Valence bond quantum Monte Carlo ..................... 73 4.1.1. How it works ............................. 74 4.1.2. Calculations in the 푆푧-basis ....................
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