DEFECTS IN NOVEL SUPERFLUIDS: SUPERSOLID HELIUM AND COLD GASES By KINJAL DASBISWAS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 ⃝c 2012 Kinjal Dasbiswas 2 To my parents, both physicians, who have always supported and nurtured my ambitions to be a physicist 3 ACKNOWLEDGMENTS I am grateful to my adviser, Professor Alan T. Dorsey, for the pivotal role he has played as a mentor in my PhD He introduced me to many topics in condensed matter physics, particularly the novel topic of supersolid helium. I have learned a number of techniques through working with him, but equally importantly, he has taught me to choose my problems carefully and to present my results in a coherent manner. I would like to thank Professors Peter J. Hirschfeld, Amlan Biswas, H.-P. Cheng and Susan B. Sinnott for useful discussions and for serving on my committee. I would also like to acknowledge Professors Y. -S. Lee and Pradeep Kumar for their valuable inputs about the supersolid problem, Professors D. L. Maslov, Richard Woodard, S. L. Shabanov, and K. A. Muttalib for their inspiring teaching, and the Max Planck Institute in Dresden for giving me the opportunity to attend a summer school on Bose-Einstein condensates. I am indebted to some of my graduate student colleagues for stimulating discussions and for ensuring a collegial environment in the department. I am grateful to K. Nichola and P. Marlin for their generous assistance with the administrative aspects of my graduate program. I would like to acknowledge the National Science Foundation for the support they have extended to my research, and also the Graduate School and Department of Physics at the University of Florida, who have provided me with an alumni fellowship and other forms of support. Finally I would like to express my gratitude to friends and family, who supported me in these five years. I am especially indebted to my parents who have invested much effort and affection in my upbringing and education. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 LIST OF TABLES ...................................... 7 LIST OF FIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1 INTRODUCTION ................................... 11 1.1 Superfluids: Helium and Dilute BEC ..................... 11 1.2 Topological Defects: Dislocations, Vortices, etc. ............... 18 1.3 The Supersolid Conundrum .......................... 20 1.4 Vortices in Rotating Bose-Einstein Condensates .............. 24 2 DISLOCATION-INDUCED SUPERFLUIDITY ................... 27 2.1 Landau Theory for the Supersolid Phase Transition ............. 28 2.2 Superfluid-Dislocation Coupling ........................ 30 2.3 Effective Landau Theory for Superfluidity along a Dislocation ....... 33 2.4 Quantitative Estimates of the Shift in Critical Temperature ......... 38 2.4.1 The Quantum Dipole Problem in 2D ................. 38 2.4.2 Numerical Methods Used ....................... 39 2.4.3 Variational Calculation ......................... 39 2.4.4 Real Space Diagonalization Method ................. 40 2.4.5 Coulomb Basis Method ........................ 43 2.4.6 Semiclassical Analysis ......................... 45 3 SUPERFLUIDITY IN THE PRESENCE OF MULTIPLE DISLOCATIONS .... 50 3.1 Grain Boundary Superfluidity ......................... 50 3.2 Berezinskii-Kosterlitz-Thouless Superfluid in a Grain Boundary ...... 57 3.3 Network of Dislocation Lines and a Model Supersolid ............ 63 4 VORTICES IN TRAPPED, DILUTE BOSE-EINSTEIN CONDENSATE ..... 69 4.1 Gross-Pitaevskii Formulation ......................... 69 4.1.1 “Weakly nonlinear” Analysis for Small Condensates ......... 72 4.1.2 Thomas Fermi Analysis ........................ 74 4.2 Vortex Energetics and Stability ........................ 75 4.2.1 Energy of a Vortex at the Center of the Trap ............. 77 4.2.2 Energy of an Off-center Vortex .................... 79 4.2.3 Vortex Stabilization in a Rotating Trap ................ 80 4.2.4 Vortex in a Small Condensate ..................... 84 5 4.3 Vortex Dynamics in the Co-rotating Frame .................. 88 4.3.1 The Magnus Force ........................... 92 4.3.2 Vortex Mass ............................... 94 5 MACROSCOPIC QUANTUM TUNNELING OF A VORTEX IN A ROTATING BOSE GAS ...................................... 103 5.1 Classical Mechanics of a Vortex ....................... 104 5.2 Semiclassical Estimates of Vortex Tunneling ................. 110 5.2.1 Schrodinger¨ Equation for a Charge in a Magnetic Field ....... 110 5.2.2 WKB Analysis of Tunneling ...................... 111 5.2.3 The Method of “Instantons” or “Bounce” Trajectories ........ 115 5.3 Experimental Discussion ........................... 118 6 CONCLUSION .................................... 121 APPENDIX A STRAIN FIELD FOR AN EDGE DISLOCATION .................. 125 B ANALYSIS OF A LANDAU MODEL WITH A 1=r POTENTIAL .......... 126 C ANALYSIS OF A TIME-DEPENDENT MODEL .................. 130 D VORTICES IN A TWO-DIMENSIONAL XY MODEL ................ 132 REFERENCES ....................................... 136 BIOGRAPHICAL SKETCH ................................ 143 6 LIST OF TABLES Table page 2-1 Summary of ground state energy estimates of the edge dislocation potential. Energy is given in units of 2mp2=~2. ........................ 39 2-2 Comparison of first few energy eigenvalues obtained from different methods. Energy units: 2mp2=~2. n indicates quantum number of the state. ....... 43 7 LIST OF FIGURES Figure page 1-1 Torsional oscillator setup used by Andronikashvili ................. 14 1-2 The two basic types of dislocations ......................... 19 1-3 Torsional oscillator setup used by Kim and Chan ................. 22 1-4 Period shifts in a torsion oscillator experiment ................... 23 1-5 Shear hardening in solid 4He ............................ 24 1-6 Images of a Bose-Einstein condensate stirred with a laser beam ........ 26 2-1 Single dislocation line with superfluid ........................ 32 2-2 Comparison of eigenvalues of the 2D quantum dipole problem, obtained by different methods ................................... 43 2-3 Fit for the eigenvalue spectrum of 2D quantum dipole potential ......... 46 2-4 Eigenfunctions of the first five states of the 2D quantum dipole problem .... 48 2-5 Eigenfunctions of five higher excited states of the 2D quantum dipole problem . 49 3-1 Symmetric tilt grain boundary ............................ 51 3-2 Network of dislocation lines ............................. 64 3-3 Correlation length vs. temperature ......................... 66 4-1 Condensate wavefunction (r) versus r by different methods in low, and high density situations ................................... 76 4-2 Condensate wavefunction (r) versus r for the l = 1 vortex state in a small condensate ...................................... 86 4-3 Magnus force on a rotating object in a flowing fluid ................ 93 5-1 Energy cost associated with a singly quantized straight vortex in a rotating trap in the TF limit .................................. 107 B-1 Order parameter (r) versus r by different methods ............... 128 B-2 Order parameter amplitude (0) as a function of ϵ, by different methods .... 129 8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEFECTS IN NOVEL SUPERFLUIDS: SUPERSOLID HELIUM AND COLD GASES By Kinjal Dasbiswas August 2012 Chair: Alan T. Dorsey Major: Physics We investigate the role played by various topological defects, especially crystal dislocations and superfluid vortices, in some novel superfluids - such as the putative supersolid phase in solid helium-4 (4He) and in dilute Bose-Einstein condensates (BEC) in traps. The first part of this work addresses recent experimental findings in solid helium, such as the period shift in resonant oscillators that has been interpreted to be a signature of superfluidity coexisting with crystalline order in solid helium. We use Landau’s phenomenological theory for phase transitions to establish that crystal defects such as dislocation lines and grain boundaries can induce local superfluid order and show that a network of dislocation lines can give rise to bulk superfluid order within a crystal. Our findings are also relevant to other phase transitions in the presence of crystal defects. The second part concerns the stability and dynamics of a single vortex in a rotating trap of a Bose-Einstein condensate (BEC) and the possibility of the macroscopic quantum tunneling of such a vortex from a metastable minimum at the trap center. The complete dynamics of such a vortex is derived by integrating out the phonon modes from a hydrodynamic action, and estimates for the tunneling rate are obtained using a variety of semiclassical methods. This is analogous to the problem of tunneling of a charged particle through a potential barrier in the presence of a very high magnetic field, 9 the Magnus force on the vortex being analogous to the Lorentz force on a charge. We conclude that the vortex action has a complicated nonlocal form and further, that the Magnus-dominated dynamics of the vortex tends to suppress tunneling. 10 CHAPTER 1 INTRODUCTION 1.1 Superfluids: Helium and Dilute BEC The history of the modern study of the properties of materials at very low temperatures dates back to the development of