Zigzag Phase Transition in Quantum Wires and Localization in the Inhomogeneous One-Dimensional Electron Gas by Abhijit C. Mehta Department of Physics Duke University Date: Approved: Harold U. Baranger, Supervisor Shailesh Chandrasekharan Gleb Finkelstein Ashutosh Kotwal Thomas Mehen Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2013 Abstract Zigzag Phase Transition in Quantum Wires and Localization in the Inhomogeneous One-Dimensional Electron Gas by Abhijit C. Mehta Department of Physics Duke University Date: Approved: Harold U. Baranger, Supervisor Shailesh Chandrasekharan Gleb Finkelstein Ashutosh Kotwal Thomas Mehen An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2013 Copyright c 2013 by Abhijit C. Mehta All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial License Abstract In this work, we study two important themes in the physics of the interacting one- dimensional (1D) electron gas: the transition from one-dimensional to higher di- mensional behavior, and the role of inhomogeneity. The interplay between inter- actions, reduced dimensionality, and inhomogeneity drives a rich variety of phe- nomena in mesoscopic physics. In 1D, interactions fundamentally alter the nature of the electron gas, and the homogeneous 1D electron gas is described by Luttinger Liquid theory. We use Quantum Monte Carlo methods to study two situations that are beyond Luttinger Liquid theory — the quantum phase transition from a linear 1D electron system to a quasi-1D zigzag arrangement, and electron localization in quantum point contacts. Since the interacting electron gas has fundamentally different behavior in one dimension than in higher dimensions, the transition from 1D to higher dimen- sional behavior is of both practical and theoretical interest. We study the first stage in such a transition; the quantum phase transition from a 1D linear arrangement of electrons in a quantum wire to a quasi-1D zigzag configuration, and then to a liquid-like phase at higher densities. As the density increases from its lowest val- ues, first, the electrons form a linear Wigner crystal; then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase; and, finally, the electrons form a disordered liquid-like phase. We show that the linear to zigzag phase transition occurs even in narrow wires with strong quantum fluc- iv tuations, and that it has characteristics which are qualitatively different from the classical transition. Experiments in quantum point contacts (QPC’s) show an unexplained feature in the conductance known as the “0.7 Effect”. The presence of the 0.7 effect is an indication of the rich physics present in inhomogeneous systems, and we study electron localization in quantum point contacts to evaluate several different pro- posed mechanisms for the 0.7 effect. We show that electrons form a Wigner crystal in a 1D constriction; for sharp constriction potentials the localized electrons are separated from the leads by a gap in the density, while for smoother potentials, the Wigner crystal is smoothly connected to the leads. Isolated bound states can also form in smooth constrictions if they are sufficiently long. We thus show that localization can occur in QPC’s for a variety of potential shapes and at a variety of electron densities. These results are consistent with the idea that the 0.7 effect and bound states observed in quantum point contacts are two distinct phenomena. v for Karen vi Contents Abstract iv List of Tablesx List of Figures xi List of Abbreviations and Symbols xiv Acknowledgements xv 1 Introduction: The One-Dimensional Electron Gas1 1.1 The Interacting Electron Gas.......................3 1.1.1 The Wigner-Seitz Radius.....................4 1.2 Electrons in One Dimension: Luttinger Liquid Theory........5 1.2.1 The Wigner Crystal in Quantum Wires............8 1.3 Experimental Realizations of the 1D Electron Gas.......... 10 1.4 Quantum Point Contacts......................... 12 1.4.1 The 0.7 Effect........................... 13 2 Quantum Monte Carlo 17 2.1 Other Methods for Interacting Electrons................ 18 2.1.1 Hartree-Fock........................... 18 2.1.2 Density Functional Theory................... 19 2.2 Introduction to Monte Carlo....................... 21 vii 2.2.1 The Metropolis Algorithm.................... 21 2.3 Variational Monte Carlo......................... 23 2.3.1 Trial Wave Function....................... 26 2.4 Diffusion Monte Carlo.......................... 30 2.4.1 The Fixed-Node Approximation................ 32 2.4.2 Importance Sampling...................... 33 2.4.3 Measuring Observables..................... 34 3 Model 38 3.1 Quantum Rings.............................. 39 3.1.1 Single-Particle Orbitals...................... 41 3.2 Quantum Wires.............................. 46 3.2.1 The Coulomb Interaction in Periodic Systems......... 47 3.2.2 Trial Wavefunction........................ 50 3.3 Comparison of Rings and Wires..................... 52 3.4 Comparison to Luttinger Liquid..................... 56 4 Zigzag Quantum Phase Transition 61 4.1 Introduction................................ 62 4.1.1 The Length Scale r0 ........................ 64 4.2 Visualizing the Transition to a Quasi-1D Zigzag: Pair Densities... 65 4.2.1 Linear Regime........................... 68 4.2.2 Zigzag Regime.......................... 68 4.2.3 Liquid Regime.......................... 69 4.2.4 Power Spectrum......................... 72 4.2.5 1D Pair Density and Structure Factor.............. 76 4.3 Probing Zigzag Order: Transverse Correlation Functions...... 81 viii 4.3.1 The y y Correlation Function................. 82 − 4.3.2 The Zigzag Correlation Function................ 86 4.4 Order Parameter: The Zigzag Amplitude............... 92 4.4.1 Quantum Wires.......................... 95 4.5 Addition energies............................. 97 4.6 Summary and Conclusions....................... 100 5 Localization in Quantum Point Contacts 103 5.1 Modeling a QPC.............................. 104 5.2 Localization in Sharp QPC’s....................... 107 5.2.1 The QMC Interaction Potential................. 111 5.2.2 High-Density Leads....................... 113 5.3 Localization in Smooth QPC’s...................... 116 5.4 Parabolic Constrictions.......................... 127 5.5 Summary and Conclusions....................... 130 6 Conclusions 133 A The Lieb Mattis Theorem 136 B Path Integral Monte Carlo 143 C Notes on Laplacian for Jastrow in Periodic Wires 146 D Choice of Single-Particle Orbitals for the Linear, Zigzag, and Liquid Regimes in Quantum Rings 149 E Finite Size Even-Odd Effect in Noninteracting Rings 153 F List of Supplemental Files: Pair Density Animations 155 Bibliography 157 Biography 166 ix List of Tables 3.1 Luttinger Liquid parameters from fit of pair density, ! = 0:1 .... 57 3.2 Luttinger Liquid parameters from fit of pair density, ! = 0:6 .... 58 D.1 Table of single particle orbitals used for runs at ! = 0:1 ....... 151 D.2 Table of single particle orbitals used for runs at ! = 0:6 ....... 152 x List of Figures 1.1 Comparison of particle-hole spectrum in 1D to higher dimensions.5 1.2 The cleaved edge overgrowth process................. 11 1.3 Localization in Cleaved Edge Overgrowth Wires........... 12 1.4 A Quantum Point Contact........................ 13 1.5 The 0.7 Effect................................ 14 1.6 An 0.5 Effect at B = 0........................... 15 2.1 Flowchart illustrating the VMC algorithm............... 25 2.2 An illustration of DMC walker evolution................ 31 3.1 Confinement potential for a quantum ring............... 40 3.2 Pair Density calculated in QMC for a ring............... 41 3.3 A single-particle Gaussian orbital in a ring.............. 43 3.4 Comparison of 1D pair densities for wavefunctions using different types of single-particle orbitals..................... 45 3.5 Effective harmonic ! for interacting electrons in rings......... 46 3.6 Comparison of 1D pair densities in a ring for wavefunctions using different types of interaction potentials................. 53 3.7 Comparison of 1D pair densities in a ring and in a wire....... 54 3.8 Fit of the amplitude of pair density oscillations to the Luttinger Liq- uid power-law decay, ! = 0:1 ...................... 57 xi 3.9 Fit of the amplitude of pair density oscillations to the Luttinger Liq- uid power-law decay, ! = 0:6 ...................... 58 3.10 Sensitivity of fitting procedure for LL power-law decay....... 59 3.11 Amplitude of pair density oscillations in a wire, ! = 0:1 ....... 60 4.1 Illustration of the zigzag transition................... 62 4.2 Pair Densities at ! = 0:1 ......................... 66 4.3 Pair Densities at ! = 0:6 ......................... 67 4.4 Pair Density at ! = 0:1; rs = 2:0;N = 60 ................ 70 4.5 Pair Density at ! = 0:1; rs = 1:0;N = 60 ................ 71 4.6 Power Spectra for ! = 0:1 ........................ 73 4.7 Power Spectra for ! = 0:6 ........................ 74 4.8 1D Pair Densities and Sρρ(kθ) at low density for ! = 0:1 ....... 76 4.9 1D Pair Densities and Sρρ(kθ) at higher density for ! = 0:1 ..... 77 4.10 1D Pair Densities and Sρρ(kθ) at low density for ! = 0:6 ....... 78 4.11 1D Pair Densities and Sρρ(kθ) at higher density for ! = 0:6 ..... 79 4.12 A measure of the one-dimensionality, ρ^(0)^ρ(0) ........... 80 h i 4.13 Cyy(θ) and Syy(kθ) at low density for ! = 0:1 .............. 82 4.14 Cyy(θ) and Syy(kθ) at high density for ! = 0:1 ............. 83 4.15 Cyy(θ) and Syy(kθ) at low density for ! = 0:6 .............. 84 4.16 Cyy(θ) and Syy(kθ) at high density for ! = 0:6 ............. 85 2 4.17 Czz( i j ), normalized by y at ! = 0:1 and ! = 0:6......... 87 j − j h i 4.18 Czz( i j ), in units of r0 at ! = 0:1 and ! = 0:6............ 88 j − j 4.19 Comparison of Czz( i j ) at N = 30 and 60 in the Liquid Regime. 91 j − j 4.20 The order parameter Mzz: the “Zigzag Amplitude”........