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Monte Carlo Simulation Study of 1D Josephson Junction Arrays

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Monte Carlo Simulation Study of 1D Josephson Junction Arrays Monte Carlo Simulation Study of 1D Josephson Junction Arrays OLIVER NIKOLIC Master Thesis Stockholm, Sweden 2016 Typeset in LATEX TRITA-FYS 2016:78 ISSN 0280-316X ISRN KTH/FYS/-16:78-SE Scientific thesis for the degree of Master of Engineering in the subject of theoretical physics. © Oliver Nikolic, December 2016 Printed in Sweden by Universitetsservice US AB iii Abstract The focus in this thesis is on small scale 1D Josephson junction arrays in the quantum regime where the charging energy is comparable to the Joseph- son energy. Starting from the general Bose-Hubbard model for interacting Cooper pairs on a 1D lattice the problem is mapped through a path integral transformation to a classical statistical mechanical (1+1)D model of diver- gence free space-time currents which is suitable for numerical simulations. This model is studied using periodic boundary conditions, in the absence of external current sources and in a statistical canonical ensemble. It is explained how Monte Carlo simulations of this model can be constructed based on the worm algorithm with efficiency comparable to the best cluster algorithms. A description of how the effective energy Ek, voltage Vk and inverse capacitance −1 Ck associated with an externally imposed charge displacement k can be ex- tracted from simulations is presented. Furthermore, the effect of inducing charge by connecting an external voltage source to one island in the chain via the ground capacitance is investigated. Simulation results display periodic −1 response functions Ek, Vk and Ck as the charge displacement is varied. The shape of these functions are largely influenced by the system parameters and evolves from sinusoidal with exponentially small amplitude deep inside the superconducting regime towards more intriguing shapes closer to the insulat- ing regime where interactions among quantum phase slips are present. When −1 an external voltage source is applied Ck takes on damped oscillatory behav- ior with respect to the induced charge and is vanishing completely when an integer number of Cooper pairs is induced, leading to alternating stability in the potential landscape Ek as the induced charge is varied. iv Sammanfattning Fokus i detta examensarbete är riktat mot småskaliga 1D kedjor av Josep- hsonövergångar i kvantmekaniska beskrivningar där laddningsenergin är jäm- förbar med Josephson-energin. Med utgångspunkt från den allmänna Bose- Hubbard-modellen för interagerande Cooper-par på ett 1D gitter omformu- leras problemet genom en vägintegralformulering till en klassisk statistisk mekanisk (1+1)D modell av länk-strömmar som är lämplig för numeriska simuleringar. Denna modell studeras med periodiska randvillkor, i frånvaro av externa strömkällor och i en kanonisk sammansättning. Det förklaras hur effektiva Monte Carlo-simuleringar av denna modell kan konstrueras baserat på en mask algoritm med effektivitet jämförbar med de bästa klusteralgo- ritmerna. En beskrivning av hur effektiva energyn Ek, spänningen Vk och −1 inversa kapacitansen Ck relaterade till en externt påtvingade laddningsför- skjutning k kan extraheras från simuleringar presenteras. Vidare så unersöks effekten av att inducera laddning genom att en extern spänningskälla kopp- las via en kondensator till en ö i kedjan. Resultat från simuleringar visar att −1 Ek, Vk och Ck är periodiska när laddningsförskjutningen varieras. Former- na på dessa funktioner beror till stor del på systemparameterinställningar och antar sinusform djupt inne i det supraledande tillståndet medans mera intressanta kurvformer erhålles närmare det insulerande tillståndet där in- teraktioner mellan kvantmekaniska fas-hopp-processer (QPS) är närvarande. −1 När en extern spänningskälla ansluts till en av öarna så utför Ck dämpade oscillationer med avssende på den inducerade laddningen och försvinner helt när ett heltal Cooper-par induceras, vilket leder till alternerande stabilitet i energilandskapet som beskrivs av Ek när den inducerade laddningen varieras. vi Preface This master thesis was carried out by me during the spring and autumn of 2016 at the department of theoretical physics in AlbaNova. My supervisor for this project was Jack Lidmar, whom I especially would like to thank for introducing me to this field of physics and for all the helpful discussions during this time. The paper is organized as follows: chapter1 starts with a brief historical in- troduction to superconductivity and highlights some of the important milestones achieved over the past decade which have been crucial for the understanding of the subjects in this paper. Chapter2 gives a review of the classical phase dynamics in Josephson junctions in the limit of infinite junction capacitance coupling where the phase is treated as a classical well defined variable, and address the quantum mechanical description of the Josephson junction in the limit of small capacitive coupling where charging effects are important. Chapter3 is devoted to 1D Joseph- son junction array models in the quantum regime. Starting from the very general Bose-Hubbard for interacting Cooper pairs several simplifications are made leading to a quantum phase model which is then transformed using a path integral ap- proach which allows us to map the original 1D quantum mechanical problem into a (1+1)D classical statistical mechanical problem similar to the 2D XY model where the condensate phases are pseudospins, and to a 2D link-current model of closed loop space-time currents. The latter descriptions of the problem forms the foun- dation to the work in this paper. Chapter4 reviews some of the basic concepts of Monte Carlo simulations in statistical physics and explains how simulations of the models discussed in chapter3 can be constructed in a canonical ensemble, as well as how interesting observables can be measured. Chapter5 present the outcome of the simulations which are then being discussed in chapter6. Contents Preface vi Contents viii 1 Introduction1 2 Josephson Junctions5 2.1 The Josephson equations......................... 5 2.2 The RCSJ model............................. 7 2.3 Characteristic energies.......................... 9 2.4 Gauge invariance............................. 11 Flux quantization............................. 13 The SQUID................................ 13 2.5 Energy band picture........................... 15 Quantization ............................... 15 Coulomb blockade ............................ 16 Dual Josephson equations........................ 17 2.6 Phase slips ................................ 18 The QPS junction ............................ 18 3 1D Josephson Junction Array Models 21 3.1 The Bose-Hubbard model........................ 21 3.2 Quantum phase model.......................... 22 Long-range Coulomb interactions.................... 22 Harmonic approximation......................... 24 3.3 Path integral transformation ...................... 26 Mapping to the (1+1)D XY model................... 28 Superconductor-insulator phase transition............... 29 Mapping to a link-current model .................... 30 Mapping to a charge field model .................... 32 4 Monte Carlo Methods 35 4.1 Basic Monte Carlo principles ...................... 35 viii CONTENTS ix Importance sampling........................... 36 Markov chains .............................. 36 Detailed balance ............................. 37 Relaxation effects............................. 37 4.2 The worm algorithm........................... 38 Metropolis-type updating scheme.................... 38 Simple link-current action........................ 39 Helicity modulus............................. 40 Complete link-current action ...................... 41 4.3 Response to charge displacements.................... 43 5 Numerical Results 47 5.1 Phase transition ............................. 47 5.2 Charging effects.............................. 48 6 Discussion 59 Bibliography 61 Chapter 1 Introduction About one third of the elements in the periodic table are classified as superconduc- tors. This remarkable property was first discovered in 1911 by the dutch physicist Heike Kamerling Onnes in Leiden who noticed how the electrical resistivity in some materials, such as mercury, tin and lead, vanished as the material was cooled down below a material dependent critical temperature Tc [35]. These low temperature investigation were made possible due to his successful invented methods to liquefy Helium in 1908 which served as refrigerant. When traveling down the road towards absolute zero, new physics was expected to emerge since temperature is one of the most universal variables at which states of matter is defined. In fact, the liquid Helium Kamerling Onnes used as refrigerant was later used in the discovery of su- perfluids in 1937, something that Kamerling Onnes already had created in 1908 but never noticed. The electromagnetic response of a superconductor has ever since the discovery of the Meissner effect by Meissner and Ochenfeld in 1933 [31] gained a lot of attention. They were able to demonstrate that a superconductor actively expels magnetic field lines from its interior. This property could not be a consequence of the zero resistivity only and hence a distinction could be made between a perfect conductor and a superconductor, which led scientists to think of superconductivity from a thermodynamic point of view. A phenomenological explanation of the Meissner effect was given by the German brothers Heinz and Fritz London in 1935 [29] with the London equations ∂ E = ΛJ, ∇ × J = −ΛB (1.1) ∂t where E is the electric field, B
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