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Magnetic Monopoles in Spin Ice Master of Science Thesis Magnetic Monopoles in Spin Ice Axel Nordstr¨om Supervisor: Patrik Henelius Department of Theoretical Physics, School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2014 Typeset in LATEX Examensarbete inom ¨amnet teoretisk fysik f¨or avl¨aggande av civilingenj¨orsexamen inom utbildningstprogrammet Teknisk fysik. Graduation thesis on the subject Theoretical Physics for the degree of Master of Science in Engineering from the School of Engineering Sciences. TRITA-FYS 2014:26 ISSN 0280-316X ISRN KTH/FYS/{14:26{SE © Axel Nordstr¨om,May 2014 Printed in Sweden by Universitetsservice US AB, Stockholm May 2014 Abstract In this thesis, we investigate the behaviour of magnetic monopoles in spin ice when an external magnetic field is applied. We find that steady state direct currents of magnetic monopoles cannot be maintained for long and consider the possibility of alternating magnetic currents by investigating the alternating current susceptibility using both analytical and Monte Carlo techniques. Moreover, we look at the transition that occurs when a magnetic field is ap- plied in a 111 direction. We show that the transition is a continuous crossover rather thanh a phasei transition in the nearest neighbour model and we study the behaviour of the system during the crossover, especially at the critical field where a temperature independent state appears. Using Monte Carlo methods and analyti- cal methods based on the Bethe approximation, we find that the mean monopole density is 0.4 monopoles per tetrahedron in the temperature independent state at the critical field. Keywords: spin ice, magnetic monopoles, phase transitions. iii iv Preface This thesis is the result of my degree project at the Department of Theoretical Physics at the Royal Institute of Technology (KTH) during the spring semester of 2014. The work concerns magnetic monopoles in frustrated pyrochlore magnets { spin ice { and their statics and dynamics in an applied magnetic field. Overview The thesis is divided into six chapters and three appendices. Chapter 1 briefly introduces the subject of magnetism and spin ice, whereas we in Chapter 2 go into some more detail on the generalities of spin ice, introducing the pyrochlore lattice and the model used. Chapter 3 concerns Monte Carlo simulations and we introduce the Metropolis algorithm used in all simulations throughout this thesis. In Chapters 4 and 5, the main results of the work are presented. In Chapter 4, we investigate the dynamics of magnetic monopoles in an applied magnetic field and in Chapter 5 we address the potential phase transition when a magnetic field is applied to spin ice in a 111 direction. Finally, the results are summarised in Chapter 6. Theh firsti appendix contains detailed information regarding the simulations per- formed throughout the thesis. The other two appendices contain brief reviews of rare earth magnetism and complexity classes in computer science, respectively. Knowledge of these topics is not required in order to understand the results pre- sented in the thesis, but may help putting models and problems into context. Conventions It is conventional in the field of frustrated magnetism to use the same unit for energy and temperature since it makes comparison of energy scales convenient.Thus we put kB = 1 throughout the thesis and measure energy in kelvins. Otherwise we use standard SI units. v vi Acknowledgements First and foremost, I would like to thank my supervisor Assoc. Prof. Patrik Henelius for giving me the opportunity to work with this thesis, for valuable guid- ance and discussions as well as for helping me with practical matters surrounding the thesis. I would also like to thank Mikael Twengstr¨omwith whom I have shared office this past semester and with whom I have had many fruitful discussions on coding, simulations and physics. Many thanks to my colleagues at the Department of Theoretical Physics at KTH for welcoming me among them. Finally I want to thank my friends and family for their encouragement and support. vii viii Contents Abstract . iii Preface v Acknowledgements vii Contents ix 1. Introduction 1 2. Background theory 3 2.1. Magnetism . 3 2.1.1. Para-, ferro- and antiferromagnetism . 3 2.1.2. Frustration . 4 2.2. Spin ice . 5 2.2.1. The pyrochlore lattice and the ice rules . 5 2.2.2. Residual entropy . 8 2.2.3. Magnetic monopoles . 8 2.3. Modelling spin ice . 9 2.3.1. The nearest neighbour model . 9 2.3.2. The dipole model . 10 2.3.3. Effect of magnetic fields . 10 3. Monte Carlo simulations 13 3.1. Pseudo-random numbers . 13 3.2. Monte Carlo algorithms . 13 3.2.1. Requirements on Monte Carlo algorithms . 13 3.2.2. Single spin flips . 14 3.2.3. The Metropolis algorithm . 14 3.2.4. Limitations of Metropolis dynamics . 15 ix x Contents 4. Magnetricity 17 4.1. Magnetic monopole current . 17 4.2. Expression for the monopole current . 18 4.2.1. Magnetisation . 21 4.2.2. AC susceptibility . 22 4.2.3. Analogy to an RL-circuit . 22 4.3. Spin-lattice relaxation . 23 4.3.1. Monopole current . 25 4.3.2. Interpretation of χS ...................... 26 4.4. Numerical determination of the susceptibility . 26 4.5. Predicting the monopole current . 28 5. Phase transitions in a 111 magnetic field 31 5.1. Introduction to phaseh transitionsi . 31 5.2. Kagom´eice . 32 5.2.1. Value of the critical field . 33 5.2.2. Magnetisation plateau . 36 5.2.3. Phase diagram of the nearest neighbour model . 37 5.2.4. Low temperature simulations . 40 5.2.5. Phase transition or continuous crossover? . 41 5.3. Temperature independent state at Hc . 42 5.3.1. Monopole density for larger systems . 44 5.3.2. Bethe approximation . 46 5.3.3. Onset of the temperature dependence . 49 5.4. Phase transitions in the dipole model . 50 6. Conclusions 53 6.1. Magnetricity . 53 6.2. Phase transitions . 54 A. Simulations 55 A.1. Details . 55 A.2. Simulation parameters . 55 B. Rare-earth magnetism 59 B.1. Generalities . 59 B.2. Hund's rules . 59 C. Complexity classes 61 C.1. P, NP and #P problems . 61 C.2. NP- and #P-complete problems . 61 C.3. The P versus NP problem . 62 Bibliography 63 Chapter 1 Introduction The phenomenon of magnetism has been known since about 500 BC, possibly even longer. Since the 12th century, ferromagnets have been used as compasses for navigation and much modern technology, such as hard disks and techniques for medical imaging, relies heavily on magnetism. Although magnetism has been used in technology for nearly a millennium, it was not until the 20th century that the microscopic mechanisms of magnetism were properly understood and the discoveries of other types of magnetism than the usual ferromagnetism were made. To this day, magnetism remains a much studied phenomenon and the research into magnetic systems is vast to say the least. One specific kind of systems that have acquired must attention over the past few decades are so-called frustrated magnetic systems. In these systems, it is impossible to minimise the energy of all interactions simultaneously and the ground state is massively degenerate (see Section 2.1.2). When Harris et al. [1] performed susceptibility measurements and neutron scat- tering on the rare earth pyrochlore compound Ho2Ti2O7 in 1997, it turned out that this material was not only frustrated, but also had a net ferromagnetic near- est neighbour interaction. No such system had previously been observed. Due to an analogy to the frustrated structure of water ice, the family of frustrated pyrochlore magnets to which Ho2Ti2O7 belongs was dubbed "spin ice". Since ideal frustrated systems retain their degeneracy even when their temper- ature reach absolute zero, they seem to violate the third law of thermodynamics { which states that the entropy of a perfect crystal, at absolute zero kelvin, is exactly equal to zero. It is, however, widely believed that real frustrated systems have a phase transition to an ordered state at low temperatures, due to small perturba- tions in the system starting to play an important role. In many cases, though, the phase transition would occur at a temperature below what is experimentally attainable today. Electricity and magnetism are intimately related via Maxwell's equations. How- ever, whereas there exist free electric charges which are sources of the electric field, 1 2 CHAPTER 1. INTRODUCTION magnetic fields seem only to be generated by magnetic multipoles. The question thus arises: do magnetic charges, or monopoles, exist? The existence of magnetic monopoles was proposed by Paul Dirac in 1931 [2], but no experiment has been able to detect any magnetic monopoles to this day. However, in 2008 it was suggested that elementary excitations in spin ice materi- als behave much like classical analogues of Dirac's magnetic monopoles [3]. This discovery raises the questions of whether or not it is possible to create currents of magnetic monopoles, as is possible with electric charges, and if it even would be possible to construct magnetic equivalents of electronic circuits in condensed matter systems despite the apparent rareness or lack of natural magnetic monopoles. Chapter 2 Background theory Here we introduce different kinds of magnetism and how these can give rise to frustration. We also introduce the spin ice compounds, the pyrochlore lattice and present the models used to describe spin ice. 2.1 Magnetism 2.1.1 Para-, ferro- and antiferromagnetism Magnetism occurs when the intrinsic magnetic moments of elementary particles, spins, interact { either with a magnetic field or with each other. If the spins do not interact among themselves, but only with external mag- netic fields, we are dealing with paramagnetism.
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