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Novel Quantum Monte Carlo Approaches for Quantum Liquids Brenda M. Rubenstein Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 c 2013 Brenda M. Rubenstein All Rights Reserved ABSTRACT Novel Quantum Monte Carlo Approaches for Quantum Liquids Brenda M. Rubenstein Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum proper- ties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconduc- tivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previ- ously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my re- search by first reviewing the physics of quantum liquids in Chapter 2 and the mathematics behind Quantum Monte Carlo algorithms in Chapter 3. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling. I next present the Bose-Fermi Auxiliary-Field Quan- tum Monte Carlo algorithm. This algorithm generalizes the well-known Auxiliary-Field Quantum Monte Carlo algorithm for fermions to bosons and Bose-Fermi mixtures. Despite some shortcomings, the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm rep- resents the first exact technique capable of studying Bose-Fermi mixtures of any size in any dimension. In Chapter 6, I describe a new Constant Stress Path Integral Monte Carlo algorithm for the study of quantum mechanical systems under high pressures. While the eventual hope is to apply this algorithm to the exploration of yet unidentified high-pressure, low-temperature phases of hydrogen, I employ this algorithm to determine whether or not quantum hard spheres can form a low-temperature bcc solid if exchange is not taken into account. In the final chapter of this thesis, I use Path Integral Monte Carlo once again to explore whether glassy para-hydrogen exhibits superfluidity. Physicists have long searched for ways to coax hydrogen into becoming a superfluid. I present evidence that, while glassy hydrogen does not crystallize at the temperatures at which hydrogen might become a su- perfluid, it nevertheless does not exhibit superfluidity. This is because the average binding energy per p-H2 molecule poses a severe barrier to exchange regardless of whether the sys- tem is crystalline. All in all, this work extends the reach of Quantum Monte Carlo methods to new systems and brings the power of existing methods to bear on new problems. Portions of this work have been published in Rubenstein, PRE (2010) and Rubenstein, PRA (2012) [167; 169]. Other papers not discussed here published during my Ph.D. include Rubenstein, BPJ (2008) and Rubenstein, PRL (2012) [166; 168]. The work in Chapters 6 and 7 is currently unpublished. Table of Contents I Introduction: Quantum Liquids and Their Modeling 1 1 Introduction 2 2 Quantum Liquids 4 2.1 Properties of Quantum Liquids . 4 2.1.1 Quantum Mechanical Effects ...................... 5 2.1.2 Quantum Statistical Effects....................... 6 2.2 Example Quantum Liquids . 13 2.2.1 Helium . 13 2.2.2 Hydrogen . 16 2.2.3 Ultracold Alkali Gases . 18 2.3 Bose-Einstein Condensation . 20 2.4 Superfluidity ................................... 24 2.5 Outlook...................................... 28 3 Quantum Monte Carlo 29 3.1 Why Quantum Monte Carlo . 30 3.2 Quantum Monte Carlo Methods . 34 3.2.1 The Monte Carlo Method . 34 3.2.2 Quantum Monte Carlo . 38 3.3 Outlook...................................... 67 i II Novel Quantum Monte Carlo Algorithms 68 4 The Monte Carlo Power Method 69 4.1 Introduction.................................... 70 4.2 Background . 73 4.3 ModelsandMethods............................... 76 4.3.1 IsingModels ............................... 76 4.3.2 Harmonic Trap . 79 4.4 Results....................................... 81 4.4.1 IsingModels ............................... 81 4.4.2 Harmonic Trap . 85 4.5 ConcludingRemarks............................... 89 5 Bose-Fermi Auxiliary-Field Quantum Monte Carlo 92 5.1 Introduction: Bose-Fermi Mixtures in Optical Lattices . 93 5.2 Preliminaries................................... 97 5.2.1 Generic Mixture Hamiltonian and Definitions . 97 5.3 Methods ..................................... 98 5.3.1 Finite-Temperature AFQMC for Bosons . 98 5.3.2 Bose-Fermi AFQMC . 99 5.3.3 Importance Sampling . 101 5.3.4 The Constrained Path and Phaseless Approximations . 104 5.4 Results ...................................... 106 5.4.1 Bose-Hubbard Model . 107 5.4.2 Spin-Polarized Bose-Fermi-Hubbard Model . 111 5.5 Discussion .................................... 115 5.5.1 Challenges . 115 5.5.2 Conclusions . 119 ii III Applications of Quantum Monte Carlo to Condensed Phases 121 6 Constant Stress Quantum Monte Carlo and the BCC Phase of Quantum Hard Spheres 122 6.1 Introduction.................................... 123 6.2 Constant Stress Quantum Monte Carlo . 125 6.2.1 Classical Parrinello-Rahman Molecular Dynamics Algorithm . 126 6.2.2 Classical Monte Carlo Algorithm . 128 6.2.3 Constant Stress Quantum Monte Carlo . 129 6.3 Computational Details . 131 6.3.1 Potential . 131 6.3.2 Propagator . 132 6.3.3 Simulations . 133 6.4 Results....................................... 134 6.4.1 Benchmarks . 134 6.4.2 BCC Phase of Quantum Hard Spheres . 136 6.5 Conclusions . 143 6.5.1 Summary . 143 6.5.2 Future Work . 143 7 (Super?)glassy Hydrogen 145 7.1 Introduction.................................... 146 7.2 Methods...................................... 148 7.2.1 Path Integral Monte Carlo . 149 7.2.2 Ring Polymer Molecular Dynamics . 150 7.3 Computational Details . 151 7.3.1 Intermolecular Potential . 151 7.3.2 Implementation Specifics . 153 7.4 Results....................................... 155 7.4.1 Diffusion in Three Dimensions . 155 7.4.2 Exchange . 162 iii 7.5 Conclusions . 165 IV Conclusions 168 8 Conclusions 169 8.1 Overview ..................................... 169 8.2 FutureDirections................................. 170 V Bibliography 173 Bibliography 174 VI Appendices 195 A Scaling of Transition Matrix Eigenvalues 196 B Derivation of the Boson Partition Function 199 C Working with Boson Green’s Functions 203 D Variational Path Integral Monte Carlo 206 iv List of Figures 2.1 The 3He and 4He phase diagrams. Bcc denotes the body-centered cubic phase. The 3He-A and 3He-B phases are superfluid 3He phases, the 4He-I phase is the normal liquid 4He phase, and the 4He-II phase is the superfluid 4He phase. Because of their large zero-point energies, both helium isotopes remain fluids down to T = 0 K. As a result, both possess superfluid phases in which exchange is prevalent. Figure is prepared from phase diagrams given in Enss and Hunklinger [60]............................ 15 2.2 The low-temperature p-H2 phase diagram. Liquid p-H2 crystallizes at 13.8 K at zero pressure. The superfluid critical temperature, Tc, of ideal p-H2 is predicted to be 6.6 K, as depicted by the red arrow above [68]. Previous re- ∼ search suggests that the T of interacting p-H is 1K[182]. p-H therefore c 2 ∼ 2 crystalizes at a temperature several degrees greater than the temperature at which it might otherwise become a superfluid. Figure is adapted from the phase diagram given in Annett [11]. ...................... 17 2.3 The change in the superfluid and normal component densities of 4He as the temperature is lowered below Tc. Figure is taken from Enss and Hunklinger [60]. ........................................ 26 2.4 The excitation spectrum of He-II. Because no excitations may be formed if He-II moves with a velocity less than the critical velocity, cmin,thereisno dissipation in He-II below this speed. Figure is taken from Annett [11] . 27 v 3.1 Comparison of interactions between classical and quantum polymers. Beads on quantum polymers (top) only interact with beads on other quantum poly- mers at the same time slice. Atoms in classical polymers (bottom) interact with all other atoms in other classical polymers. Note that, in the bottom panel, I only illustrate the interactions between one bead on the first polymer and all other beads on the other polymer for clarity. The remaining beads of the first classical polymer interact similarly. Colored circles represent poly- mer beads at different time slices. The black lines represent the interbead polymersprings. ................................. 48 3.2 An illustration of the bisection algorithm. In the bisection algorithm, a segment of a quantum polymer is regrown in stages (2). During the first stage (3), the position of a bead is sampled at the midpoint time slice between the two end time slices of the segment. If that new position is accepted, the algorithm then samples bead positions between each end and the new midpoint (4). This sampling process is continued until the full segment is regrown (5). In the above, the red circles denote polymer beads at different time slices. The black lines denote the interbead polymer springs. 50 3.3 An illustration of the permutation-bisection algorithm. In the permutation- bisection algorithm, permutations of a set of quantum polymers are sampled (1). First, a permutation of the particle labels is sampled (2). Then, segments are regrown between time slices k and k + s among the particles according to the permutation sampled (3).
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