Spin-orbit coupling and strong correlations in ultracold Bose gases DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By William S. Cole, Jr., B.S. Graduate Program in Physics The Ohio State University 2014 Dissertation Committee: Professor Nandini Trivedi, Advisor Professor Mohit Randeria Professor P. Chris Hammel Professor Richard J. Furnstahl c Copyright by William S. Cole, Jr. 2014 Abstract The ability to create artificial gauge fields for neutral atoms adds a powerful new dimension to the idea of using ultracold atomic gases as \quantum simulators" of models that arise in conventional solid state physics. At present, cold atom experiments are able to simulate orbital magnetism and a certain kind of spin-orbit coupling at scales which are quite difficult to achieve in the solid state. This takes us beyond the realm of simulation into questions about new states of matter which might only be possible in cold atom experiments. Indeed, as a byproduct of finding methods to simulate traditional gauge potentials for neutral gases, it has been realized that gauge potentials with no solid state analog can also be created and finely tuned to manipulate the few- and many-body physics of bosons and fermions in remarkable new ways. Motivated along these lines, in this dissertation I address several issues related to bosons with spin-orbit coupling. After an introduction to synthetic gauge fields in general I describe Rashba spin-orbit coupling specifically and how it modifies the usual behavior of bosons in the continuum and in a harmonic trapping potential. Following this, I study the effects of interactions induced by combining spin-orbit coupling with an optical lattice potential. I use a variety of theoretical tools to characterize and classify the exotic phases which emerge. In the weakly interacting limit, mean-field theory suggests Bose-Einstein condensed states with the possibility of magnetically ordered phases arising through interference between macroscopically occupied single particle states. In the deep lattice limit, the ground state is a Mott insulator with magnetic order driven by bosonic superexchange. I address the Mott transition, as well as the nature of superfluid states near the Mott transition, and introduce a novel \slave-boson" approach to understand these results. Finally, using some of the ii insight developed for this general case, I restrict my attention to a more experimentally relevant implementation of SOC, where I can use numerically exact calculations to justify the mean-field results. In the final Chapter, I address a problem which perhaps seems, at first glance, to be unrelated to bosons with Rashba SOC. The problem is identifying the ground states of spinless hardcore bosons on a frustrated honeycomb lattice. The connection between these problems is the delicate question of how bosons condense in a flat band. In that section I provide numerical support to a recently proposed scenario of \statistical transmutation," where an approximate ground state of the hard core bosons can be obtained from a mean- field approximation in terms of composite fermions. iii Acknowledgments First and foremost, I am deeply thankful to my advisor Nandini Trivedi for her guidance along my winding Ph.D. path. Her enthusiasm for physics and her breadth of interests are inspiring. I would also like to give special thanks to Mohit Randeria, who often served a similar role, and together the examples they have set as teachers and mentors I can only hope to emulate. I am also particularly indebted to Arun Paramekanti and Shizhong Zhang for their collaboration on much of the work presented in this dissertation, and for enlightening dis- cussions over the past few years. The condensed matter community at Ohio State has been a stimulating, nurturing environment, full of talented post-docs and students from whom I have learned a great deal. Special thanks go to Yen Lee Loh, who helped me to really get my bearings in the folklore of condensed matter theory. Also I must thank Eric Duchon and Weiran Li for being ideal officemates, along with several other fellow students (Onur, Nganba, Mason, Tim, Daniel, Jim, John) who have helped to guide my own development as a physicist. That this thesis was completed at all is a great credit to my wife Alexis. Her unfailing support and encouragement have kept me afloat through every challenge. I have also been blessed with a wonderful, loving family who sparked my initial interest in math and science and continue to fan the flames. iv Vita May, 2008 . B.S. in Physics, University of Central Florida, Orlando, FL September, 2008 { present . Graduate Teaching and Research Assis- tant, and Ohio State Presidential Fellow, the Ohio State University, Columbus, OH Publications William S. Cole, Shizhong Zhang, Arun Paramekanti, Nandini Trivedi, Bose Hubbard Models with Synthetic Spin-Orbit Coupling: Mott Insulators, Spin Textures and Superfluidity, Phys. Rev. Lett. 109 085302 (2012) Anamitra Mukherjee, William S. Cole, Nandini Trivedi, Mohit Randeria, Patrick Wood- ward, Theory of strain controlled magnetotransport and stabilization of the ferromagnetic insulating phase in manganite thin films, Phys. Rev. Lett. 110 157201 (2013) O. Nganba Meetei, William S. Cole, Mohit Randeria, Nandini Trivedi, Novel magnetic state in d4 Mott insulators, arXiv:1311.2823 Zhihao Xu, William Cole, Shizhong Zhang, Mott-superfluid transition for spin-orbit-coupled bosons in one-dimensional optical lattices, Phys. Rev. A 89 051604(R) (2014) William S. Cole, Nandini Trivedi, Statistical transmutation of hard core bosons on the frus- trated honeycomb lattice, in preparation Shizhong Zhang, William S. Cole, Arun Paramekanti, Nandini Trivedi, Synthetic gauge fields in optical lattices, in preparation for Annual Reviews of Cold Atoms and Molecules Fields of Study v Major Field: Physics vi Table of Contents Page Abstract........................................... ii Acknowledgments..................................... iv Vita.............................................v List of Figures ...................................... ix List of Tables .......................................x Chapters 1. Introduction, motivations, and outline .................... 1 1.1 Spin-orbit coupling................................ 2 1.2 Ultracold bosons ................................. 4 1.2.1 The Bose-Hubbard model........................ 5 1.3 Synthetic gauge fields .............................. 6 1.3.1 Berry's connection and the adiabatic principle ............ 8 1.3.2 Raman induced spin-orbit coupling................... 12 1.3.3 \Realistic" Rashba spin-orbit coupling................. 15 2. Exotic Bose matter from nonabelian gauge fields.............. 18 2.1 Bose-Einstein condensation with spin, and with spin-orbit coupling . 18 2.2 The Rashba hamiltonian............................. 21 2.2.1 Cylindrical coordinates ......................... 23 2.2.2 The Rashba hamiltonian with a harmonic trapping potential . 26 2.3 Adding interactions................................ 27 2.4 An exact two-body ground state ........................ 29 3. The Bose-Hubbard model with spin-orbit coupling............. 32 3.1 Spin-orbit coupling in a periodic optical potential............... 34 3.2 Model hamiltonian................................ 36 3.3 Weak-coupling approximation.......................... 36 3.4 Strong-coupling approximation ......................... 42 3.5 Mean-field theory of the superfluid-insulator transition............ 51 3.5.1 Description of the mean-field method.................. 51 3.5.2 Results .................................. 54 3.6 Slave-boson theory................................ 56 3.7 Exact numerics in the one-dimensional limit.................. 60 vii 4. Hard core bosons on the frustrated honeycomb lattice .......... 63 4.1 Hard core bosons and Jordan-Wigner fermions ................ 63 4.2 Model and methods ............................... 67 4.3 Comparing the ground-state energies of hard core bosons and fermions . 70 4.4 Monte Carlo results ............................... 72 4.4.1 Variational bounds on the ground state energy............ 72 4.4.2 One-body density matrix ........................ 72 4.5 Discussion and conclusions............................ 73 Bibliography...................................... 75 Appendices A. Numerical solution of the Rashba hamiltonian in a harmonic trap . 83 A.1 Series expansion in harmonic trap solutions.................. 84 B. Effective spin hamiltonian from two-site perturbation theory . 90 C. The variational principle and variational Monte Carlo........... 94 C.1 Application of VMC to fermionized hamiltonians............... 97 viii List of Figures Figure Page 1.1 Mean-field phase diagram of the Bose-Hubbard model............ 7 1.2 Schematic implementation of the Raman scheme for spin-orbit coupling . 13 1.3 Spectrum of the hamiltonian for Raman-induced one-dimensional spin-orbit coupling...................................... 15 2.1 Energy spectrum of the Rashba hamiltonian.................. 24 2.2 Energy spectrum of the Rashba hamiltonian in a harmonic trap . 27 2.3 Spatial variation of the eigenstates of the Rashba hamiltonian in a harmonic trap ........................................ 28 3.1 Energy spectrum of a single particle in a 2D square optical lattice with spin orbit coupling................................... 38 3.2 Density of states and lower-band spin wavefunction for a Rashba coupled particle in a 2D square