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Quantum Monte Carlo Method for Boson Ground States: Application to Trapped Bosons with Attractive and Repulsive Interactions

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Quantum Monte Carlo Method for Boson Ground States: Application to Trapped Bosons with Attractive and Repulsive Interactions W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2005 Quantum Monte Carlo method for boson ground states: Application to trapped bosons with attractive and repulsive interactions Wirawan Purwanto College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Atomic, Molecular and Optical Physics Commons, and the Condensed Matter Physics Commons Recommended Citation Purwanto, Wirawan, "Quantum Monte Carlo method for boson ground states: Application to trapped bosons with attractive and repulsive interactions" (2005). Dissertations, Theses, and Masters Projects. Paper 1539623468. https://dx.doi.org/doi:10.21220/s2-mynw-ys14 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. NOTE TO USERS This reproduction is the best copy available. ® UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission ofof the the copyright copyright owner. owner. FurtherFurther reproduction reproduction prohibited prohibited without without permission. permission. QUANTUM MONTE CARLO METHOD FOR BOSON GROUND STATES Application to Trapped Bosons with Attractive and Repulsive Interactions A Dissertation Presented to The Faculty of the Department of Physics The College of William and Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Wirawan Purwanto 2005 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3172894 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3172894 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Wirawan Purwanto Approved by the Committee, January 2005 Shiwei Zhang, Chair William E. Cooke Henry Krakauer Andreas Stathopoulos Department of Computer Science ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission DEDICATION I present this dissertation in honor of my beloved parents, Benjamin Purwanto and Linda Kristiani, who have loved and nurtured me in the Lord since my youth. Their support has been indispensable in bringing me through many difficult times in my study. I also dedicate this work to Yuliana Salim, my beloved in Christ, whose fervent love and care have really cheered me up in the last year of my graduate study. I am deeply indebted to her for all her patience and unconditional support while I was preparing this dissertation and its defense, during which 1 was often unavailable for her. Above all, I commit this completed work to my Lord and God, who through His church has supplied me with His infinite grace and mercy through the years of my study in the College of William and Mary. “Because out from Him and through Him and to Him are all things. To Him be the glory forever. Amen.” (Romans 11:36) iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknolwedgements .......................................................................................................viii List of T ables................................................................................................................ ix List of F ig u r e s............................................................................................................. x Abstract .............................................................................................................................xiii CHAPTER 1 Introduction.................................................................................................... 2 2 A Model for Interacting Bose G a s e s............................... 10 2.1 Quantum Theory of Many-Body S y s te m s .............................................. 11 2.1.1 Many-Body Hamiltonian in Second Quantization ..................... 12 2.2 Model Hamiltonian for Bose G as .............................................................. 13 2.2.1 Atom-Atom Interaction ................................................................... 13 2.2.2 Bose-Hubbard Model ...................................................................... 15 2.2.3 Regularization of the Hubbard U .................................................. 17 2.2.4 On-Site Versus S Potentials ............................................................ 18 2.3 Mean-Field Approach: Gross-Pitaveskii Equation ................................ 19 3 Quantum Monte Carlo Method for the Ground State of Boson Systems 22 3.1 Ground-State Projection ........................................................................... 23 3.2 Basic Auxiliary-Field Method ................................................................. 23 3.2.1 Wave Function Representation ...................................................... 25 3.2.2 M e t r o p o l i s AF Q M C ............................................................... 26 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Branching Random-Walk for Boson Ground S tates ............................... 27 3.4 Importance Sampling 1 . 28 3.4.1 Importance-Sampled Random Walkers ..................................... 29 3.4.2 Modified Auxiliary-Field Transformation .................................. 29 3.4.3 The Optimal Choice for Auxiliary-Field Shift x ........................ 31 3.4.4 Local-Energy Approximation ...................................................... 32 3.5 Measurement of O bservables ...................................................................... 33 3.5.1 “Brute-Force” and Mixed Estimators ......................................... 33 3.5.2 Back-Propagation Estimator ......................................................... 36 3.6 Phaseless Approximation for QM C ............................................................ 41 3.6.1 Phaseless Approximation and Back-Propagation ..................... 43 3.7 Implementation to Interacting Bose G a s .................................................. 43 3.7.1 Implementation of Q M C ................................................................ 44 3.7.2 Implementation of the Gross-Pitaevskii Equation ..................... 46 3.7.3 Trial Wave Function for Q M C ...................................................... 47 3.8 Benchmark Results ....................................................................................... 48 3.8.1 Characteristics of QMC M e th o d ................................................... 50 3.8.2 Comparison With Analytic Results in ID: as < 0 ..................... 57 3.8.3 Comparison With Exact Diagonalization: as > 0 ..................... 62 3.8.4 Benchmarking Phaseless Approximation: Large Systems and Realistic Param eters..................................................................... 65 3.9 Connection Between QMC and Gross-Pitaevskii Projections .............. 68 4 Repulsive Interactions: Study of Correlation E ffects................................ 71 4.1 Homogenous Bose G a s ................................................................................ 72 4.1.1 Equation of S ta te ............................................................................. 73 4.1.2 Effect of Interaction Strength ......................................................... 76 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.3 Momentum Distribution ............................................................... 78 4.2 Trapped Bose G a s ........................................................................................ 80 4.2.1 Dependence on Number of Particles N ..................................... 80 4.2.2 Momentum Distribution ............................................................... 89 4.3 Effect of Potential’s D etail ....................................................................... 94 5 Attractive Interactions: Collapse of the C ondensate .............................. 99 6 Discussions........................................................................................................ 104 6.1 Finite-Size Effect Analysis .......................................................................... 104 6.2 Limitations of the Simple On-Site Potential .............................................107
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