Topics in Cold Atoms Related to Quantum Information Processing and A Machine Learning Approach to Condensed Matter Physics Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jiaxin Wu, B.S., M.S. Graduate Program in Physics The Ohio State University 2019 Dissertation Committee: Professor Tin-Lun Ho, Advisor Professor Nandini Trivedi Professor Rolando Vald´es Aguilar Professor Eric Braaten c Copyright by Jiaxin Wu 2019 Abstract This thesis is mainly focused on three topics: Majorana excitations in a number- conserving model, manipulation of quasi-particle excitations in quantum Hall systems, and a new machine learning algorithm to find the ground states of a general Hamil- tonian. In condensed matter physics, Majorana fermions are emergent excitations which are candidates for quantum memory and topological quantum computation. The first and simplest model revealing these excitations does not conserve particle number. Its experimental realization in solid state materials is difficult and still under debate. In comparison, cold atoms provide an alternative platform to realize these exotic excitations. However, cold atoms experiments require the system to be number- conserving. Theoretically, it is not yet clear whether there is a model realizable in cold atoms that also hosts these exotic excitations. In this thesis, we investigate such a number-conserving model and show that it has the same phase diagram and very similar excitations. Although the ground state degeneracy, as one of the signature properties of the original Majorana model crucial for quantum memory, is not present when the total particle number is fixed, one can recover the degeneracy by allowing tunnelling to change the total particle number. As for the quantum Hall system, we discuss how to control quasi-hole excitations with sharp external potentials where the system has integer filling factor. The eigen ii wavefunctions of the quasiholes are discussed in details. Our motivation is that most discussions or experiments regarding quantum Hall systems mainly focus on trans- port properties, but topological quantum computation may require one to have more precise control over the quasiparticle excitations. Although the ultimate goal is to control the non-Abelian excitations predicted in fractional quantum Hall systems, our results, especially in the situation with contact interactions, pave a way to explore this problem analytically. We also demonstrate some basic ideas of manipulating the quantum states, including non-Abelian exchange, in the integer quantum Hall systems. The last topic is about a new machine learning algorithm to find the unbiased ground states of a general Hamiltonian. Solving ground states of quantum many- body systems has been a long-standing problem in condensed matter physics, since the Hilbert space grows exponentially with the system size. In this thesis, we attempt to address this problem with a new unsupervised machine learning algorithm utilizing the benefits of artificial neural network. Without assuming the specific forms of the eigenvectors, our algorithm can find the eigenvectors in an unbiased way with well controlled accuracy. As examples, we apply this algorithm to 1D Ising and Heisenberg models, where the results match very well with exact diagonalization. Furthermore, we point out how this method can potentially be applied to large systems beyond exact diagonalization. iii To my loving family, and my partner on this journey and the next, Paulo. iv Acknowledgments Many people have helped me through my graduate study. With one way or an- other, they have contributed to my growth and unique experience in the Ohio State University. First of all, I sincerely thank my advisor, Professor Tin-Lun Ho, who generously offered me opportunities to learn and to improve myself in all aspects. Under his guidance, I have not only expanded my knowledge in physics, but also sharpened myself in communication, critical and creative thinking, problem solving, decision making, and so many more. All these skills are going to be invaluable assets for the rest of my life, for which I am forever grateful. I would also like to thank Professor Yuan-Ming Lu for his immense patience in our collaboration. Moreover, I am grateful for the anyon and quantum computation study groups he organized, which have brought great enlightenment to me about topological quantum computation. Lastly, I need to thank him for the activities he constantly organizes regardless of his busy schedule. I have enjoyed hiking with him together with all the other students and postdocs. I am deeply grateful to Professor Nandini Trivedi and Professor Rolando Vald´es Aguilar for their kindness and support. They both have offered helpful advice to my study and future career. Furthermore, I extend my thanks to Professor Mohit v Randeria, Professor Ilya Gruzberg, Professor Eric Braaten, Professor Stuart Raby for questions and discussions in physics. Discussing physics with my friends and colleagues has always been my most en- joyable time during my daily study. For the fruitful discussion, I especially want to thank Bowen Shi, Timothy McCormick, Wenjuan Zhang, Xiangyu Yin, Niravkumar Patel, James Rowland, Kyumin Lee, Tamaghna Hazra, Chris Svoboda, Wayne Zheng, Nathanan Tantivasadakarn, Ruixing Zhang, Sisi Zhou, Jingfang Zhou, Tian Zhang, Guanyu Zhu, Fuyan Lu, Cheng Li, and David Ronquillo. My life in graduate school would be much less colorful without my friends. For all the laughter and joyful moments they have brought, I want to genuinely thank Benjamin Buckman, Ethel Perez, Meron Dibia, Emilio Codecido, and Bruce Barrios. Moreover, I would like to express my earnest gratitude to Paulo Montero Camacho, who acts as both my friend and family, for sharing my happiness and sadness, and for inspiring courage and strength in me. Lastly, but also the most importantly, I am forever in dept to my beloved family, my parents and my sister, who always believe in me and support me to pursue my dreams, even if it is 12,000 kilometers away. I can never repay the love to two of my grandparents who passed away during my graduate study. I wish them rest in peace with my love and memories. vi Vita 2014 . .B.S. Sun Yat-sen University 2016 ........................................M.S. The Ohio State University Fields of Study Major Field: Physics vii Table of Contents Page Abstract....................................... ii Dedication...................................... iv Acknowledgments . .v Vita ......................................... vii ListofTables.................................... xi List of Figures . xii 1. Thesis Overview . .1 2. Majorana Zero Modes from Kitaev's model . .7 2.1 Kitaev's 1D Majorana Model . .8 2.1.1 Bulk Properties and Phase Diagram . .9 2.1.2 Majorana Zero Modes and Edge Operators . 13 2.1.3 Analytic Eigenstates in Spin Chain . 17 2.2 Experimental Realization . 22 2.3 Quantum Memory Based on Majorana Zero Mode . 26 3. Majorana-Like Edge Modes in a Number-Conserving Model . 32 3.1 Motivation ............................... 32 3.2 Model and Analytic Behavior with Two Sites . 36 3.3 Perturbation Theory and Phase Diagram . 40 3.4 Recover Ground State Degeneracy with Different Ntot at\SweetPoint" ............................ 45 viii 3.5 Edge Operators and Edge-Edge Correlation in General Parameters 48 3.6 Decaying Behavior of the Majorana-like Edge Operators . 53 3.7 Conclusion . 59 4. Historical View Points of Integer Quantum Hall Effect . 61 4.1 Introduction to Hall Effect . 63 4.2 Integer Quantum Hall Effect . 66 4.2.1 Macroscopic Quantum Phenomena . 66 4.2.2 LandauGauge ......................... 69 4.2.3 Symmetric Gauge . 75 4.3 Anyon Statistics in Quantum Hall Systems . 79 5. Controlling Quasiparticle Excitation in Integer Quantum Hall States . 86 5.1 Motivation ............................... 86 5.2 Single Trapping Potential . 88 5.2.1 Single Delta Potential . 91 5.2.2 Single Exponential Potential . 91 5.3 Double Trapping Potentials . 92 5.3.1 Double Delta Potentials with One Hole . 93 5.3.2 Double Gaussian Potentials with One Hole . 96 5.3.3 Two Quasiholes with Spin Degree of Freedom . 98 5.4 Contact Interaction . 104 5.5 Conclusion . 109 6. Finding Many-Body Ground States with Artificial Neural Network . 114 6.1 Introduction .............................. 114 6.2 Neural Network Structure . 116 6.2.1 Traditional Deep Learning Structure . 117 6.2.2 Modified Neural Network to Find Eigenvectors . 119 6.2.3 Gain Function for Convergence to Ground States . 121 6.3 Numerical Results . 124 6.4 Conclusion . 127 Appendices 129 A. Perturbation Theory in the Number-Conserving Model . 129 A.1 First Order Correction to Bulk Energy Gap . 129 A.2 Eigenstates at \Sweet Point" under First Order Correction . 133 ix B. Singlet Eigenstates with Two Delta Potentials and Contact Interactions 136 C. Gradient Descent in Neural Network . 140 C.1 Gradient Descent of Cost Function . 140 C.2 Gradient Ascent of Gain Function . 141 Bibliography .................................... 143 x List of Tables Table Page 3.1 Order of magnitude for Ai's ....................... 58 A.1 Order of magnitude for different spin configuration projections. 134 xi List of Figures Figure Page 2.1 (a) Kinetic energy of Kitaev's model ( t cos k) versus chemical poten- − tial µ. The bulk gap closes when µ = 2t. (b) The phase diagram of ± Kitaev's Majorana model. The white regions (µ < 2t and µ > 2t) − are non-topological, while the light blue region excluding the red line ( 2t < µ < 2t with ∆ = 0) corresponds to a parity-protected topolog- − 6 icalphase.................................. 12 2.2 (a) An example of a topologically trivial trajectory given by h^(k) with M = 1. (b) An example of a topologically non-trivial trajectory with M = 1................................... 14 − 2.3 (a) False-color experimental setup to detect ZBP's. Al (green) is the s- wave superconductor. By the proximity effect, its Cooper pairs tunnel into the nano wire. The tunnel-gate (coral red) induces a tunnelling barrier between the left electrical contact (yellow) and the Al shell.