New Scaling Approach to Critical Exponents of the Λ Transition

Master Thesis New scaling approach to critical exponents of the λ transition Simon Sandell Condensed Matter Theory, Department of Physics, School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2018 Typeset in LATEX Scientific thesis for the degree of Master in Science of Engineering (MSc) in the subject area of Theoretical physics. TRITA-SCI-GRU 2018:351 c Simon Sandell, August 2018 Printed in Sweden by Universitetsservice US AB, Stockholm August 2018 Abstract The λ transistion of 4He is one of the few phase transitions where extremely accurate measurement of critical properties is possible. Measurements in a micro-gravity space shuttle[1] have produced a value of the critical exponent of the heat capacity which is inconsistent with the current best theoretical estimates found in Ref. [2]. This has called for further theoretical investigations to determine if the discrepancy can be resolved with higher precision or if the validity of universality between finite lattice models and the λ universality class needs to be questioned. In this thesis a new approach to scaling corrections in finite lattice studies is investigated. Large scale MC simulations using the Wolff algorithm are used to solve the three-dimensional XY-model for L = 4 - 128 close to the critical point. The new approach allows various methods of locating the critical point Tc and estimating the first order correction to scaling exponent !. In this work several of these methods are tested, and the critical exponents of the λ transition are determined. The new method is shown to be effective for estimating critical exponents, yeilding statistical uncertainties equal in magnitude compared to the previous best estimates. We find that the estimated values of the critical exponents depend sensitively on the values of ! and Tc, which in turn proved hard to determine. Our best estimate of the critical exponent α = −0:0169(3) is in better agreement with previous theoretical predictions than with experimental values. Key words: Monte Carlo simulation, phase transition, finite size scaling, critical exponent, λ transition, three-dimensional XY model , universality iii iv Preface This thesis is the result of 8 months of work between January and August 2018, for the degree of Master in Science in Engineering Physics, Teknisk Fyisk, in the Physics Department of KTH Royal Institute of Technology, Sweden. Acknowledgements Firstly, I offer my sincere gratitude to my supervisor Prof. Mats Wallin for giving me the opportunity to work on this project, which has been a very rewarding experience. Without his guidance, support and encouragement this thesis would not exist. I want to thank all my fellow thesis workers, in particular, Mattias Jönsson, Robert Vedin, Julia Hannukainen, Gunnar Bollmark, Simon Rydell and Kristoffer Aronsen for interesting discussions, banter, and for contributing to a very pleasant work environment. I also thank the PhD candidates and Postdocs at the department for helpful guidance and exciting lunch discussions. Finally, I thank all my family and friends for their constant love and support, which has been very helpful during this project. v vi Contents Abstract . iii Preface v Contents vii 1 Introduction 1 1.1 Scaling . 2 1.2 Universality . 3 1.3 The λ transition . 3 1.4 Previous Work . 4 1.5 Goals of This Thesis . 4 2 Background 5 2.1 Model . 5 2.2 Finite Size Scaling . 10 2.3 Scaling Relations and Definitions of Critical Exponents . 12 3 Method 15 3.1 Monte Carlo Methods . 15 3.1.1 Ergodicity . 16 3.1.2 Monte Carlo Algorithms For Spin Systems . 17 3.1.3 Critical Slowing Down And The Wolff Algorithm . 18 3.1.4 Histogram Extrapolation . 20 3.2 Error Estimation . 21 3.2.1 Monte Carlo Error Estimation . 21 3.2.2 The Jackknife Methods . 22 3.3 Implementation . 23 3.3.1 Parallel Programming . 23 3.3.2 Random Numbers . 23 vii viii Contents 4 Results 25 4.1 Equilibration . 25 4.2 Autocorrelation . 26 4.3 Analysis and Main Results . 27 4.3.1 Statistics and Configuration . 28 4.3.2 Estimated Quantities . 29 4.3.3 Parameter Fitting . 29 4.3.4 Critical Exponents . 43 5 Summary and discussion 51 5.1 Summary . 51 5.2 Outlook . 52 Bibliography 52 Chapter 1 Introduction Phase transitions are quite neat. They have been part of the human experience since the start, with such manifestations as ice melting, water evaporating, etc. and understanding them has proved very useful to us. Take for instance the forging of metals to produce tools, which is not possible without knowing that the metals will undergo a transition when heated, into a liquid which can be molded to derised shapes and then cooled to solidify again. There are many more such examples. Since the beginning of the 2000th century, a new special kind of phase transition has been investigated, where materials enter so called superfluid and superconduct- ing phases. These exotic transitions happen only at very low temperatures, and are characterized by the particle constituents of the material entering a single coherent quantum state, so that the bulk is no longer describable in terms of its compo- nents. When in these states, the materials may exhibit facinating phenomena such as zero resistance conduction, zero viscosity, expulsion of magnetic fields etc, which of course have various useful and interesting applications. The most accurate theory describing phase transitions of thermodynamical systems is called Renormalization Group theory (RG) and has been profoundly success- ful in explaining critical phenomena. The theory makes predictions of the precise behaviour of physical quantities near and at the transtition temperature. For various systems the predictions are both hard to calculate from theory and to measure in experiments. The numerical estimation technique we use is a Monte Carlo method. By com- puting thermal quantities from a randomly drawn selection of states distributed by the Boltzmann distribution, one can estimate the thermal averages. The algorithm is rather straightforward to implement. The difficulty lies in getting estimates accurate enough to draw conclutions about physical quantities. Simulating small system sizes leads to faster convergence, the CPU time required to get accurate estimations grows quickly with the number of spins, and it is not an easy task to 1 2 Chapter 1. Introduction extrapolate data from a simulated system consiting of at most 106 spins to exper- imentally measured values obtained from samples consiting of about 1023 atoms, which practically coincides with the thermodynamic limit of an infinite system. 1.1 Scaling At a second order phase transition, relations between physical quantities can be described by so called power laws, expressions of the form f = a · xp, where the quantity f is said to scale with exponent p in relation to the quantity x. In statistical physics, such laws are fundamental to understanding phase transitions. When studying a specific system, the fundamental quantity is some thermodynamic potential, e.g. Gibbs and Helmholtz. From this potential, several other physical quantities are derived, such as the energy and magnetization. When a system undergoes a phase transition, i.e. a change of order, physical quantities will often diverge. These divergencies are described by power laws. It is worth stating that this is not true for all phase transitions, as phase transitions include a broad class of phenomena that can arise in widely different physical systems. A precise definition of what constitutes a phase transition than includes all known phase transitions does not exist. Using mean field theory to calculate these laws always gives a fractional exponent, but experiments show evidence of non-fractional exponents. Kadanoff realized that a diverging correlation length implied that there was a relation between the length scale at which the order parameter was defined and the coupling constants of an effective Hamiltonian. Although his block-spin approach does not enable computation of the critical exponents, it was an important step. The full theory of Renormalization Group was invented by Wilson [3{6]. The core concept is the RG transformation which takes a Hamiltonian and by some method/rule of coarse-graining clumps together short wave-length degrees of freedom, and defines a new effective Hamiltonian describing the long wave-length degrees of freedom with new coupling parameters for the new length scale. The name \renormalization group" is not entirely appropriate since these transformations are in general complicated and non-linear, thus are not guaranteed to have an inverse transformation. But the transformations do have the associative property of groups. Rescaling the system by some scale factor l1 and then rescaling again by some other factor l2 is equivalent to performing the rescaling in the other order. But so far all we did was remove a finite number of degrees of freedom from out system, how can that explain the singular behaviour at phase transitions? By repeating the transformations an infinite number of times, singular behaviour can be introduced. The partition function is what we really want to compute to know everything about a physical system, but that task is most often simply unachiev- able. The RG transformation is also not easy to compute, but the transformation of the coupling constants can be approximated. 1.3. The λ transition 3 1.2 Universality The thermodynamics of any model, i.e. the phase diagram, correlation functions and other quantities, may depend on the specific values of coupling parameters in the Hamiltonian, symmetries, dimensionality, type of lattice. But it turns out that the critical phenomena only depend on three things, the symmetries of the Hamiltonian, the dimensionality and the range of interactions. We study the critical behaviour of 4He which we know is in the O(2) universality class, by calculation of thermodynamic averages directly using a Monte-Carlo method for solving the 3DXY model, which should have precisely the same exponents by universality.

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