Path Integral Quantum Monte Carlo Study of Coupling and Proximity Effects in Superfluid Helium-4 A Thesis Presented by Max T. Graves to The Faculty of the Graduate College of The University of Vermont In Partial Fullfillment of the Requirements for the Degree of Master of Science Specializing in Materials Science October, 2014 Accepted by the Faculty of the Graduate College, The University of Vermont, in partial fulfillment of the requirements for the degree of Master of Science in Materials Science. Thesis Examination Committee: Advisor Adrian Del Maestro, Ph.D. Valeri Kotov, Ph.D. Frederic Sansoz, Ph.D. Chairperson Chris Danforth, Ph.D. Dean, Graduate College Cynthia J. Forehand, Ph.D. Date: August 29, 2014 Abstract When bulk helium-4 is cooled below T = 2.18 K, it undergoes a phase transition to a su- perfluid, characterized by a complex wave function with a macroscopic phase and exhibits inviscid, quantized flow. The macroscopic phase coherence can be probed in a container filled with helium-4, by reducing one or more of its dimensions until they are smaller than the coherence length, the spatial distance over which order propagates. As this dimensional reduction occurs, enhanced thermal and quantum fluctuations push the transition to the su- perfluid state to lower temperatures. However, this trend can be countered via the proximity effect, where a bulk 3-dimensional (3d) superfluid is coupled to a low (2d) dimensional su- perfluid via a weak link producing superfluid correlations in the film at temperatures above the Kosterlitz-Thouless temperature. Recent experiments probing the coupling between 3d and 2d superfluid helium-4 have uncovered an anomalously large proximity effect, leading to an enhanced superfluid density that cannot be explained using the correlation length alone. In this work, we have determined the origin of this enhanced proximity effect via large scale quantum Monte Carlo simulations of helium-4 in a topologically non-trivial ge- ometry that incorporates the important aspects of the experiments. We find that due to the bosonic symmetry of helium-4, identical particle permutations lead to correlations between contiguous spatial regions at a length scale greater than the coherence length. We show that quantum exchange plays a large role in explaining the anomalous experimental results while simultaneously showing how classical arguments fall short of this task. Acknowledgements First and foremost I would like to thank my advisor, Professor Adrian Del Maestro: I learned a great deal while working with you. Our discussions guided me as a scientist, a programmer, and a thinker. You trusted me with parts of this project that I did not believe that I could complete at the time, but through my seemingly endless struggles I have become a better researcher. I feel lucky to have been able to work with you, and for that I cannot thank you enough. I would like to thank Dr. Chris Herdman for all of our conversations about physics and programming. Chris: From my collaboration with you, I have become a more precise and thorough scientist. I enjoyed my time working with you. Professor Dennis Clougherty: You have provided me with guidance and support from the first day that I came to the University of Vermont. I would like to thank you for all of your encouragement and help throughout my time here. I always appreciate my conversations with you, as they impact me in such a positive way. Professor Valeri Kotov: Your support and faith in my abilities has helped me a great deal in my time here. I enjoyed all of our conversations about science and life, and I express my full gratitude to you. This work was enabled by the use of computational resources provided by the Ver- mont Advanced Computing Core supported by NASA (NNX-08AO96G). I would like to specifically thank Jim Lawson for supporting our high-performance computing needs. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. I am in debt to all members of the University of Vermont community who have helped me along my way, both in and out of the physics department. Finally, I would like to thank Dave Hammond for questioning what it meant that I had all of the components to build a ‘useless machine’ but there they sat for years, unassembled. ii To my beautiful wife, family, and friends... for your unwavering support in all that I do. iii Table of Contents Acknowledgements................................... ii Dedication........................................ iii List of Figures..................................... ix 1 Introduction1 1.1 Impact.......................................1 1.2 Outline and Notes to Reader..........................3 2 Helium-4 as a Quantum Fluid6 2.1 General Properties of Helium-IV........................6 2.2 Aziz Potential...................................9 2.2.1 Interatomic Interaction Potentials...................9 2.2.2 1979 Aziz Functional Form....................... 10 2.2.3 Potential Tail Cutoff Correction.................... 12 2.2.4 Pressure Tail Cutoff and Impulse Corrections............. 14 2.3 Bose Einstein Condensates............................ 14 2.3.1 Off-Diagonal Long Range Order.................... 15 2.3.2 BEC in the Ideal Bose Gas....................... 18 2.3.3 On Determining the BEC for Real Systems.............. 21 2.4 Superfluidity................................... 22 2.4.1 Superfluid Transition and Superflow.................. 22 2.4.2 Quantization of Circulation....................... 25 2.4.3 Superfluid Response under Rotation.................. 26 2.5 Confined Helium-4................................ 28 2.5.1 Correlation Length............................ 29 2.5.2 Enhanced Coupling and Proximity Effects............... 31 2.5.3 Classical Proximity Effects....................... 34 3 PIMC Background 37 3.1 Path Integrals................................... 37 3.1.1 Classical Action............................. 37 3.1.2 Entering the Quantum Regime..................... 39 3.1.3 Summing Over All Paths........................ 40 3.1.4 Successive Events............................. 41 3.1.5 Reclaiming the Schrödinger Equation................. 43 3.1.6 Steady State Wave Functions...................... 47 3.2 Statistical Mechanics............................... 49 3.2.1 Partition Function and Density Matrix................. 49 3.2.2 Path Integrals in Statistical Mechanics................. 52 iv 4 Path Integral Quantum Monte Carlo 54 4.1 Monte Carlo.................................... 54 4.1.1 Why QMC?................................ 54 4.1.2 Importance Sampling.......................... 56 4.2 The Action.................................... 57 4.2.1 Trotter Formula............................. 57 4.2.2 Polymer Isomorphism.......................... 59 4.2.3 Primitive Approximation........................ 61 4.2.4 Generalized Suzuki Factorization.................... 64 4.3 Monte Carlo Sampling Updates......................... 67 4.3.1 Diagonal Updates............................ 67 4.3.2 Off-Diagonal Updates.......................... 68 4.4 Thermodynamic Energy Estimator....................... 69 4.4.1 Primitive Thermodynamic Energy................... 71 4.4.2 GSF Thermodynamic Energy...................... 72 4.5 Centroid Virial Energy Estimator........................ 73 4.5.1 Centroid Virial Energy Derivation................... 74 4.5.2 Primitive Centroid Virial Energy.................... 83 4.5.3 GSF Centroid Virial Energy....................... 85 4.6 Comparison of Energy Estimators....................... 88 4.6.1 Convergence of Error Bars........................ 89 4.6.2 Virial Window Scaling.......................... 91 4.7 Specific Heat Estimator............................. 93 4.8 Local Permutation Number Estimator..................... 95 4.9 Winding Estimator for Superfluid Fraction................... 97 4.10 Thermodynamic Pressure Estimator...................... 97 4.10.1 Pressure Derivation........................... 98 4.10.2 Pressure in Engineering Units...................... 100 5 PIMC Studies of Confined He-4 101 5.1 Confinement Simulation Cell.......................... 102 5.1.1 Modified Insert Update......................... 104 5.2 Coupled 3d and 2d Regions........................... 106 5.2.1 Scaling of the Density with Chemical Potential............ 107 5.2.2 Lower Potential in the Film Region................... 108 5.2.3 Scaling of the Density with Film Potential Energy.......... 109 5.3 Non Trivial Topological Winding........................ 112 5.3.1 Normalization of Angular Winding................... 114 5.3.2 Scaling of Angular Winding with Bulk Region Size.......... 116 5.3.3 Scaling of Angular Winding with Bulk Separation.......... 118 5.3.4 Superfluid Fraction of Confined Systems................ 121 v 6 Prospects for Future Research 122 6.1 Specific heat for our confined system...................... 122 6.1.1 Specific Heat of 3d bulk Helium-4................... 122 6.2 Planar Confinement............................... 123 6.2.1 Scaling of the Specific Heat....................... 124 7 Conclusions 126 A Pressure Tail Correction for 1979 Aziz Potential 129 B Relationship between Partition Function and Density Matrix 131 C PIMC Sample Code 133 C.1 Simple PIMC Code................................ 133 C.2 Centroid Virial Energy Method......................... 137 C.3 Comments on Implementation.......................... 143 D Zero Temperature QMC 144 D.1 Trial Functions and Projection Techniques................... 144 D.2 PIGS method..................................