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 ω. 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 ω 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 superconducting 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 computing 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 experimentally 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. It is not possible simulate the infinite size 3DXY model, instead the finite-size versions of the 3DXY model are simulated. In this thesis we 3n simulate the model on a simple cubic three-dimensional lattice, with Nspins = 2 where n ∈ {2, 3, 4, 5, 6, 7}.

1.3 The λ transition

4 When cooled below the critical temperature Tc ≈ 2.2 K at normal pressure, He undergoes a second order phase transition. The liquid transforms into a state historically known as He-II, but nowadays mostly referred to as a superfluid state. The atoms form a coherent state with a number of interesting properties. Famously, the superfluid can be shown to have zero viscosity and entropy, it will “climb” up the walls and escape if kept in a flask. The λ transition in 4He has been studied extensively both in theory, simulations and experiments. In Ref. [7] Feynman used first principles to show that the transition was analogous to that of an ideal Bose-Einstein gas despite the strong interatomic forces present in 4He. The superfluid transition in helium is called the λ transition due to the shape of the graph of heat capacity versus temperature, which resembles the Greek letter lambda. The transition happens at a temperature of about 2.2 K at atmospheric pressure, where the liquid undergoes a transition into a superfluid state. It is important as a experimental verification of the RG theory, as it is one of few systems where experimental measurement can be performed with high accuracy. The lambda transition is one of the most important phase transitions in the field of condensed matter physics, since it is the universality class of phase transitions into the quantum states of matter, i.e. the superfluid and superconducting states. In Ref. [8] it was shown with MC methods and duality arguments that the phase transition of three-dimensional type-II superconductors also belong to the λ universality class, and suggested an analogy also for the nematic-smectic A transition of liquid crystals which since has been investigated, see Ref. [9]. 4He is unlike the lighter isotope 3He a boson with spin 0, which enables it to achieve a superfluid state at a much higher temperature via Bose-Einstein conden- sation, while 3He must form atom Cooper pairs to be able to attain superfluidity. A 4 Chapter 1. Introduction superfluid acts as a mixture of regular liquid with a superfluid component. The superfluid component has zero viscosity and entropy. Another feature is that angular velocity is quantized as vortex excitations that form when a superfluid is rotated.

1.4 Previous Work

In Tab. 4.10 previous estimations as well as the best estimations based on this work is presented. Notable previous work was done by Lipa[1] in 2003, which is based on very accurate experimental measurements of 4He performed in micro-gravity on the space shuttle Colombia. Gravitational gradients on Earth will induce a pressure gradient in the sample, which will cause transition broadening, which makes high precision measurements impossible. Measurements of the heat capacity as well as the superfluid density was performed, to temperatures in the range of nK of Tc, with no evidence of transition broadening. A value of the critical exponent α was stated to unprecedented accuracy, which stands today. The most recent analytical calculation using a pseudo-ε expansion was pre- formed by Sokolov [10] to give predictions of the critical exponents which to the stated precision agrees with the value seen in experiment. However, combined high-temperature expansions and numerical methods performed by Campostrini et al. [2] predicts values that are incompatible with the experimental predictions. These estimates were obtained by partially similar methods as in this thesis. In addition to the XY model they simulate the φ4 model as well as the ddXY model since these models have a parameter which can be adjusted so that the first order scaling correction term is eliminated. This discrepancy moti- vates revisiting the simulation problem to test the accuracy of the previous results. With a new approach to finite size scaling corrections the aim is to estimate the critical exponents to even higher degree of accuracy.

1.5 Goals of This Thesis

The main goals of this thesis is to test the efficiency of the new approach to scaling corrections for determining the critical exponents of the λ transition. The new method has the advantage over preceding methods that fewer fit parameters need to be determined, and a wider range of simulated system sizes can be included. To obtain MC data for which the method can be tested we simulate the three- dimensional XY model using the Wolff algorithm to avoid the problem of critical slowing down. Various methods of determining the critical point and the first order scaling correction are then employed, and the critical exponents ν and η are finally estimated accurately by a non-linear least-squares fit. Chapter 2

Background

The definition of the three-dimensional XY model and important quantities are presented and discussed. Theoretical background of finite-size scaling of lattice models and the new scaling approach are presented. Definitions of the critical exponents and common scaling laws are given.

2.1 Model

The goal of this thesis is to estimate the critical exponents of the λ transition in 4He by Monte Carlo (MC) simulations. The principle of universality allows us to choose any model belonging to the same universality class. The RG theory guarantees that all models in the same universality class have equal universal critical properties, including scaling exponents and dimensionless ratios of critical scaling amplitudes. The three criteria that deﬁne universality classes are the symmetry group of the order parameter, the physical dimension and the length scale of interaction. The order parameter of the XY model has symmetry group O(2). There are no long- range interactions, only nearest-neighbour interactions are included. The three- dimensional model belongs to the same universality class as 4He. With d = 2 the model describes a system that undergoes the topological Kosterlitz-Thouless transition. The model is simulated on a simple cubic lattice with periodic boundary con- 3 3 3 ditions. The total volume of the lattice is V = a L , where Nspins = L , and a is a lattice constant which we let a = 1 from here on. Spin sites on the lattice are labeled by three indices (x, y, z), where x, y, z denote number of steps in the

5 6 Chapter 2. Background respective Cartesian directions from the origin spin. The six nearest neighbours of any given lattice site (xi, yi, zi) is given by the expressions

( mod[xi + L ± 1,L], yi, zi), (2.1)

( xi, mod[yi + L ± 1,L], zi), (2.2)

( xi, yi, mod[zi + L ± 1,L]) (2.3) where the lattice indices (xi, yi, zi) can assume values [0,L − 1], and mod is the modulo operator. In the XY model, the spins have a single continuous degree of freedom, and can be parametrized by si = (cos(αi), sin(αi)). The Hamiltonian of the 3DXY model is X X H = −K si · sj = −K cos(αi − αj) (2.4) hi,ji hi,ji where K is the coupling constant which we during the rest of the thesis set to unity. Thermal averages of the XY model are deﬁned by integrals which covers all of phase space, of the form

Z 2π Y dαi hAi = P ({α})A({α}) (2.5) 2π i 0 where P ({α}) gives the probability of each microstate. For a system with ﬁxed 1 total energy at temperature T = β , the appropriate distribution is the Boltzmann distribution, P ({α}) = (1/Z)exp(−βH({α})), (2.6) where the normalization factor is the partition function,

Z 2π Y dαi Z = e−βH (2.7) 2π i 0

The integrals have dimensionality proportional to the number of spins, in the limit of inﬁnite system size they become path integrals. The Monte Carlo method solves these integrals by evaluating the desired quantity at randomly selected points in the phase space. Deﬁnitions of some of the computed quantities are now presented. The energy and magnetization are given by X E = − cos(αi − αj), (2.8) hi,ji  !2 !21/2

X X X M = si =  cos(αi) + sin(αi)  . (2.9) i i i 2.1. Model 7

From moments of the energy and magnetization, more interesting quantities can be computed. The magnetic susceptibility per spin is given by hM 2i − hMi2 χ = . (2.10) TL3 In the following sections two such quantities discussed in detail.

Binder cumulants As shown by Binder in Ref. [11] the even cumulants

hm2ni U = (2.11) 2n hm2in

M where m = Ld , have a number of desirable properties. In the critical limit their values are ﬁnite and universal. In the limit L → ∞ the fourth order cumulant will approach U4 → 1 + θ(t) where θ(x) is the Heaviside function, which means that it can be used to get a good estimate of Tc. It is useful to consider the temperature ∂U4 1/ν derivative ∂T . Since it scales as L it can be used to determine the critical exponent ν. One method of computing the derivative is to compute U4 for a range of temperatures and use a numerical diﬀerence rule to approximate ∂U4/∂T . By introducing a few new thermal averages it can also be computed directly. Using ∂hAi hAEi − hAihEi = (2.12) ∂T T 2 we obtain ∂hM ni hM nEi − hM nihEi hM ni ≡ = . (2.13) ∂T ∂T T 2 (2.14)

This gives ∂U ∂ hM 4i 4 = (2.15) ∂T ∂T hM 2i2 hM 4i hM 2i2 − 2hM 4ihM 2ihM 2i = ∂T ∂T (2.16) hM 2i4 1 = hM 4EihM 2i + hM 4ihM 2ihEi T 2hM 2i3 (2.17) − 2hM 4ihM 2Ei.

Superfluid density The spin stiffness or helicity modulus measures the response of the system to introducing a constant phase shift along a boundary of the system. In the normal state, 8 Chapter 2. Background where the phase of the system is at large random, there will not be any change in free energy by such a twist. In the superfluid state, the system is in a coherent state, and applying a twist at the boundary will propagate throughout the system. Thus the magnitude of the shift in free energy should be proportional to the total ν superfluid density. The superfluid density scales as ρs ∼ t , which means that the quantity Lρs, like U4, is expected to be constant at the critical point. For bosonic systems the superfluid density is directly proportional to the helic- 2 4 ity modulus, ρs = (m/~) Υ(T ). He is bosonic, with spin equal to one. In Ref. [12] Fisher showed that ρs can be computed within the framework of equilibrium statistical mechanics, but it is necessary to introduce quantities beyond bulk properties. An operational definition for the spherical model and ideal Bose gas was stated,

2L2 Υ(T ) = lim (F 1/2(T ; L) − F 0(T ; L)), (2.18) L→∞ π2 where F 0 and F 1/2 denote periodic and anti-periodic boundary conditions respec- tively. Rudnick et al. [13] employed the operational deﬁnition

2 ∂ F (T ; k0) Υ(T ) = , (2.19) ∂k2 0 k0=0 and showed that the Josephson relation is exact, using epsilon expansion. The twist at the boundary induce a long wavelength twist of the wave number k0 of the order parameter.

The difference in free energy density by introducing a constant phase twist θ over a plane pθ of the system is defined as 1 ∆f = ρ θ2, (2.20) 2 s where f is given by f = L−dF = −TL−d ln Z (2.21) and Z 2π Y dαi Z = exp (−βH) . (2.22) 2π i 0 Differentiation gives 2 ∂ f ρs = 2 . (2.23) ∂θ θ=0

By introducing a twisted Hamiltonian an expression for ρs in terms of thermal averages can be found. Consider a system where spins in a plane pθ have been 2.1. Model 9 twisted, then the Hamiltonian can be written

0 Hθ = H + H , (2.24) 0 X X H = − cos(αi − αi+x + θ) + cos(αi − αi+x), (2.25)

i∈pθ i∈pθ where H is given by Eq. 2.4. The index αi+x denotes the spin at location (+1, 0, 0) relative to a spin at location (xi, yi, zi). Then T ∆f = f − f = − (ln Z − ln Z) , (2.26) θ Ld θ 2 0 ∂ ∆f T ∂ Zθ 2 = − d ∂θ L ∂θ Zθ 00 0 2 T Zθ (Zθ) = − d − 2 , (2.27) L Zθ (Zθ) where Z 2π Y dαi Z = e−βHθ . (2.28) θ 2π i 0 0 00 With θ = 0 the two derivatives Zθ and Zθ are 0 Z 2π Z 1 ∂Zθ 1 Y dαi ∂ θ ≡ = e−βHθ Z Z ∂θ Z 2π ∂θ i 0   1 Z 2π dα Y i X −βHθ = −β sin(αi − αi+x) e Z 0 2π i i∈pθ

= −βhJx,pθ i = 0, (2.29) and 00 2 Z 2π Z 1 ∂ Zθ 1 Y dαi ∂ θ ≡ = −βJ e−βHθ Z Z ∂θ2 Z 2π ∂θ x,pθ i 0 Z 2π 1 Y dαi X = − β cos(αi − αi+x) Z 0 2π i i∈pθ 2 2 X −βHθ + β sin(αi − αi+x) e

i∈pθ = β hE i + β2 J 2 . (2.30) x,pθ x,pθ The superﬂuid density is then

1 2 ρs = − hEx,p i + β J . (2.31) Ld θ x,pθ 10 Chapter 2. Background

Periodic boundary conditions imply that any plane in the system can be used. In fact, it is allowed to calculate it for every plane in the system to obtain as much data as possible. There are L planes from which it it is possible to compute ρs, so 1 ρ = − hE i + β J 2 . (2.32) s L1+d x x The planes with normaly, ˆ zˆ can be included by symmetry. The ﬁnal expression is 1 ρ = − hEi + β hJ 2i + hJ 2i + hJ 2i . (2.33) s 3L1+d x y z 2.2 Finite Size Scaling

In a physical realization of a thermodynamic system, such as a gas or a liquid, there 23 number of particles are of the order NA ∼ 10 . In this work systems of at most 1283 particles are simulated. To obtain physical predictions from the simulations a precise theory of how the finite systems relate to the practically infinite systems is needed. The theory of finite size scaling (FSS) was established in the early 1970’s. In Ref. [14] Fisher gathered various established ideas and techniques to form a theoretical framework. Specifically for the FSS of periodic systems[15, 16], the following three state- ments are often assumed to hold: (i) 1/L is an exact scaling field without corrections proportional to 1/L2, 1/L3,...

(ii) The non-linear scaling fields (yt, yh, y3 . . . ) have coefficients that are not dependent on L (iii) The analytic background in the free energy depends on L through exponentially small terms. Consider a thermodynamic system with linear finite size L which is coupled to an arbitrary amount of scaling fields, with coupling constants [K] = t, h, K3,K4,... with linear finite size L. Applying a RG transformation, the linear size of the system 0 is rescaled by a factor l > 1, and the coupling constants go to [K ] = Rl([K]). Using the above assumptions, the singular part of the free-energy density scales as

−1 −d −1 fs([K],L ) = l fs(Rl([K]), lL ). (2.34) One of the main result of RG theory is that close to a ﬁxed point of the RG transformation Eq. 2.34 can be written in terms of right eigenvectors of the linearised transforms,

−1 −d yt yh y3 −1 fs(t, h, K3, ··· ,L ) = l fs(tl , hl ,K3l , ··· , lL ), (2.35) where only two of the couplings are so-called relevant couplings. Upon repeated application of the transform, only the couplings with exponent y ≥ 1 will remain, 2.2. Finite Size Scaling 11 which is what deﬁnes a relevant coupling. It is apparent that the inverse size of the system can be treated as a relevant eigenvector with eigenvalue yL = 1, as long as the system size is ﬁnite. If the scaling parameter is set to the system size, l = L, the scaling form of the free-energy density becomes

−1 −d 1/ν (2−η+d)/2 fs(t, L ) = L Ff (tL , hL ), (2.36) where the relations ν = 1/yt and 2 − α = dν have been used. In this work we let the external ﬁeld h vanish. Then scaling forms of any thermodynamic quantity derivable from the free-energy density can be expressed in the form

−1 δA 1/ν A(t, L ) = L FA(tL ). (2.37)

For instance the scaling form of the speciﬁc heat is found to be

∂2f c = s ∼ L−d+2/ν F (tL1/ν ) = Lα/ν F (tL1/ν ). (2.38) V ∂t2 c c

For ﬁnite L the form function FcV will have a maximum at a shifted value of T ,

−1/ν Tc(L) = Tc + a0L , (2.39) where Tc denotes the critical point of the inﬁnite system. These scaling forms are correct only in the limit t = T −Tc → 0. Away from the Tc critical points, the relations obtain correction terms from the irrelevant couplings. Correcting for these contributions is sometimes essential to obtain accurate predictions of the critical exponents. A trivial method is to simulate a system as large as possible and assume that the corrections are small enough to ignore. At the critical point the scaling form functions can be treated as constants, which limits the number of free parameters, and thus requires data from fewer system sizes in total compared to other methods. To quantify the corrections to the scaling relation in Eq. 2.36 we include the leading order irrelevant coupling, so that

−1 −d 1/ν y3 fs(t, L ) = L Ff (tL ,K3L ). (2.40)

y3 Assuming that K3L is a small parameter, Taylor expansion to ﬁrst order gives

∂F (tL1/ν , z) d −1 1/ν y3 f 1/ν 1/ν −ω L fs(t, L ) ≈ Ff (tL , 0)+K3L ≡ af (tL )+bf (tL )L . ∂z z=0 (2.41) 1/ν 1/ν At the critical point the two independent functions af (tL ) and bf (tL ) become constants that do not depend on L. This gives

d −1 −ω −ω L fs(0,L ) = a0(0) + a1(0)L ≡ af + bf L (2.42) 12 Chapter 2. Background

This form for a general quantity A with exponent δA is

δA −ω A|t=0 = L aA + bAL . (2.43)

The constants aA and bA can be determined by suitable subtractions, and then the critical point can be located by an intersection method. The methods will be discussed in detail in later chapters. The new approach is to consider corrections to scaling only at the critical point, in contrast to the common approach of considering a range of temperatures close to 1/ν 1/ν the critical point. This is done by approximating the functions ag(tL ), bg(tL ) by polynomials. The problems with this approach is that the region of temperatures in which this is a valid approximation decreases as L−1/ν , and that correctly limiting the region is crucial to obtaining accurate results. Only considering scaling at the critical point resolves these problems and simpliﬁes the analysis, as the number of unknown parameters is reduced.

2.3 Scaling Relations and Deﬁnitions of Critical Exponents

Common definitions of critical exponents and various scaling relations are stated. The exponents are mostly of historical significance, as RG theory implies that there are only two independent exponents, from which all the others can be derived. The exponents are defined by

c ∼ t−α t → 0, h = 0 (2.44) m ∼ (−t)β t → 0, h = 0 (2.45) χ ∼ |t|−γ t → 0, h = 0 (2.46) ξ ∼ |t|−ν t → 0, h = 0 (2.47) h ∼ mδ t = 0, h → 0 (2.48) hm(r)m(0)i ∼ r−d+2−η t = 0, h → 0 (2.49)

Before the famous RG result, most of the scaling laws were already proposed by inference from experimental results, and some were shown to hold in mean ﬁeld 2.3. Scaling Relations and Deﬁnitions of Critical Exponents 13 theory for certain classes of models. Examples of scaling laws are 1 y = (2.50) t ν 2 − η + d y = (2.51) h 2 ∆ = yh/yt (2.52) γ δ − 1 2 − η = = d (2.53) ν δ + 1 δ + 1 νd = 2 − α = 2β + γ = β(δ + 1) = γ (2.54) δ − 1

The ﬁrst two laws relate the RG exponents yt and yh to the historical exponents. 14 Chapter 3

Method

The Monte Carlo method is introduced, and the Wolﬀ algorithm is presented. Meth- ods of data extraction, error estimation and implementation are discussed.

3.1 Monte Carlo Methods

In the theoretical framework of statistical physics, physical predictions come from thermal averages as defined in Eq. 2.5. For most models, except some famous counter examples, e.g. the Ising model with d = 1, 2 and the Kuramoto model, it is not known how to compute these averages analytically. Other methods of evaluating the averages have been developed, most notable are Monte Carlo and expansion methods. Monte Carlo methods are typically formulated to estimate the value of an arbitrary integral, but can also be applied to solve other problems, e.g. differential equations, which can be reformulated as integral equations. To estimate the integrals random sampling points are used, in contrast to other numerical integration rules, such as Runge-Kutta methods, Simpson’s rule, etc. which evaluate the integrand at predefined discrete points. For such integration rules the error will typically grow exponentially with the dimension of the integral, while for MC methods, the error is proportional to √1 independent of dimension, where N denotes the number of N independent evaluation points. Thus it is often more efficient to use integration rules for problems with low dimension, and MC methods for problems with high dimensionality. When simulating spin systems, it is typically wanted to simulate as large systems as possible, to minimize finite size effects, therefore MC methods are more suited. Markov-Chain Monte-Carlo methods (MCMC) are the most commonly used methods today. A Markov Chain is a stochastic process where the probabilities of entering other states depend only on the current state of the system. By simulating a model, and updating the state of the model randomly with rules that depend only

15 16 Chapter 3. Method on the current state, much faster convergence can be achieved since the rules can be chosen so that the model is more likely to be in states which contribute more to the integral, i.e. the simulated states are Boltzmann distributed. Methods where the sampling is uniformly random are referred to as simple Monte Carlo methods. A notion of time in MCMC simulations can be introduced as the number of proposed updates. Monte Carlo time does not correspond to physical time, as no equations of motion are used. A natural unit is the number of proposed updates equal to to total number of spins on the simulated lattice, commonly known as a Monte Carlo sweep (MCS).

3.1.1 Ergodicity Ergodicity is a property of a dynamical system, which guarantees that time averages are equal to the phase space averages defined in Eq. 2.5. Experimentally, only time averages are measurable. To estimate the phase space averages correctly, it is essential that the implemented model and algorithm have this property. One exam- ple are systems which are restricted to a subset of the phase-space by spontaneous symmetry breaking. The conditions a dynamical system must fulfil to be ergodic was first established by Boltzmann[17] and later put into mathematical rigour by Birkhoff[18] and von Neumann[19]. It must be possible to reach any point in phase space from any other in a finite amount of time. In the context of MCMC, ergodicity is ensured if for any given state of the simulated system, any other state is reachable in a finite number of updates. If a possible update of the algorithm has the sole action of changing a single spin to any angle, it automatically follows that any state can be reached within Nspins updates. As we present the update rules of the Wolff algorithm, it will be clear that this is a possible update.

A Markov chain Monte Carlo method is deﬁned by its update rules, consisting of a set of transition probabilities T (X → X0). By enforcing a set of conditions on these probabilities, it can be ensured that the generated chain of states will have the desired distribution after an initial equilibration time. Let the desired stationary distribution or target distribution be ρ(X), the transition probability be T (X → X0) and the probability of being in state X at Markov time step t be ρ(X, t). The master equation is then [20] X X ρ(X, t + 1) − ρ(X, t) = − T (X → X0)ρ(X, t) + T (X0 → X)ρ(X0, t). (3.1) X0 X Solving for a stationary solution by letting t → ∞ we get X X T (X → X0)ρ(X) = T (X0 → X)ρ(X0). (3.2) X0 X 3.1. Monte Carlo Methods 17

A general solution has not been found, but a famous particular solution called the detailed balance solutions is given by

T (X → X0) ρ(X0) = , ∀X,X0. (3.3) T (X0 → X) ρ(X)

This solution can be decomposed into

0 T (X → X ) = ωXX0 AXX0 , (3.4) where ωXX0 represents a trial probability and AXX0 an acceptance probability. Imposing the following conditions on ωXX0

ωXX0 = ωX0X , (3.5)

0 ≤ ωXX0 ≤ 1, (3.6) X 0 1 = ωXX0 , ∀X,X , (3.7) X0 and inserting the decomposed form into Eq. 3.3 gives

A 0 ρ 0 XX = X . (3.8) AX0X ρX

Since the target distribution is the Boltzmann distribution, the fraction ρX0 re- ρX duces to an expression of the form eβ(HX −HX0 ), which is easy to compute, requiring only partial knowledge of the full state. There is still freedom in how to choose AXX0 and ωXX0 . The trial probability is usually selected to be uniform, and with no exception in this work. Other choices may be beneﬁcial when studying non-trivial geometries or boundary conditions. As we simulate a simple cubic lattice with periodic boundary conditions there is no motivation to chose anything other than a uniform distribution.

3.1.2 Monte Carlo Algorithms For Spin Systems The simplest and most well known MCMC method for solving spin lattice models is the Metropolis-Hastings algorithm[21]. An update of the system consist of selecting a spin at random with trial probability ωXX0 , and flipping or rotating the spin depending on the model, with a probability proportional to the difference in energy. This process is then repeated. The acceptance probability is defined as ( 1 if ρ(X0) ≥ ρ(X) 0 0 AXX = ρ(X ) 0 (3.9) ρ(X) if ρ(X ) < ρ(X). and the trial probability is usually chosen to be uniform. 18 Chapter 3. Method

Averages are estimated by sampling the system at discrete time steps, and taking the average of the sampled quantities. Sampling means that certain quantities of the system, e.g. energy, magnetization, are evaluated for the current state and stored in an array. Sampling after each update does not give the highest efficiency for the Metropolis algorithm, since sequential states will be highly correlated, pro- viding little new information. Instead the system is sampled at discrete time steps, commonly at Nupdates = k · Nspins. In theory, the best quality data per CPU-time- ratio is obtained by choosing sampling frequency equal to the autocorrelation time τrel. This optimization can only be partially implemented for the Wolff algorithm, as the rules imply that sampling must take place after a fixed number of clusters to ensure detailed balance. In this work we chose to sample after each update.

3.1.3 Critical Slowing Down And The Wolﬀ Algorithm

Critical slowing down is a phenomenon appearing for certain phase transitions. It is characterized by the divergence of the time required for a system to recover from a small perturbation. In practice, the amount of CPU time required to obtain uncorrelated samples grows rapidly as the critical point is approached. For the XY model, this is explained by the divergence of the susceptibility. A single fluctuation may propagate throughout the whole lattice, and in such cases the Metropolis- Hastings algorithm becomes inefficient. In the 1970s a number of new methods was developed to combat this problem, notably the Swendsen-Wang method[22] of cluster updates. Later an improved version was presented in Ref. [23], which only updates a single cluster for each update. It is known today as the Wolff algorithm and will be used in this thesis. The Swendsen-Wang algorithm[22] scans all spin bonds, which are then either deleted or frozen. This divides the lattice into clusters, which are then tried to be flipped with certain probabilities. In the Wolff algorithm only a single cluster is generated at each update, with the starting spin chosen with uniform probability. Thus one can expect to hit large clusters with higher probability, since the probability of hitting any given cluster will depend on the size of that cluster, elim- inating most of the unwanted single-spin clusters generated by the Swendsen-Wang method. This means that the Wolff algorithm becomes more efficient than the Swendsen- Wang and local methods in the vicinity of a second order phase transition, where the average cluster size diverges. For the same reason, the Wolff algorithm becomes less efficient away from the critical point where cluster sizes are small, since cluster identification is more work than single spin flip attempts. 3.1. Monte Carlo Methods 19

For the XY model with d = 3, each spin is characterized by an angle α. The Hamiltonian is given by X H = −β cos(αi − αj) (3.10)

hsi,sj i where hsi, sji denote nearest neighbour summation. The Wolﬀ algorithm is deﬁned as follows for the XY model:

(i) Select a starting spin αi with uniform probability. Select a random plane with normal u and angle αu, through which spins are reﬂected. Reﬂect the starting spin, taking it to

αi → Ru(αi) = 2αu + π − αi (3.11) and mark it as part of the cluster.

(ii) Add its nearest neighbours to a list of perimeter spins.

(iii) Pick a spin i from the perimeter list and remove it form the list. Check that it is not part of the cluster already. Try to add it to the cluster with probability

Pu(αi, αj) = 1 − exp(2β cos(αj − αu) cos(αi − αu)), (3.12)

where spin j is the nearest neighbour which caused spin i to be added to the perimeter list. If it is accepted, reﬂect it and mark it as part of the cluster and add any nearest neighbours that are not part of the cluster already to the perimeter list.

(iv) Repeat (iii) until the perimeter list is empty.

Note that as a consequence of these rules, the single spin flip is a possible update. A cluster can contain only a single spin, since it is always possible that all the neighbours of the starting spin are rejected, i.e. the probability in Eq. 3.12 is never unity. Thus any given state can be reached with finite probability in Nspins updates. To show that the Wolff algorithm satisfies detailed balance, we consider two distinct states X and X0, differing by a flip of the cluster c, so that the transition from X → X0 is a possible update of the Wolff algorithm. Then the transition probability obeys

0 T (X → X ) Y 1 − Pu(Ru(αi), αj) 0 = 0 0 (3.13) T (X → X) 1 − Pu(Ru(αi), αj) hsisj i∈∂c where the product is restricted to nearest neighbours where si ∈ c and sj ∈/ c, as all 0 other probabilities are equal and therefore cancel. This implies that Ru(αi) = αi, 0 0 Ru(αi) = αi and αj = αj. 20 Chapter 3. Method

Using Eq. 3.12 we get

0 T (X → X ) Y exp {2β cos(Ru(αi) − αu) cos(αj − αu)} 0 = 0 0 T (X → X) exp 2β cos(Ru(αi) − αu) cos(αj − αu) hsisj i∈∂c 0 0 Y exp 2β cos(αi − αu) cos(αj − αu) = exp {2β cos(αi) − αu) cos(αj − αu)} hsisj i∈∂c ρ(X0) = . (3.14) ρ(X)

3.1.4 Histogram Extrapolation Histogram extrapolation is a method used to extrapolate data from a single run to a wider range of parameter space[24]. In this work it is used to gather data for a range of temperatures. Implementing this method requires that new quantities are calculated each time the system is sampled, but it oﬀers a signiﬁcant reduc- tion in time compared to running simulations at each temperature point separately.

A thermal average of a quantity A at inverse temperature β0 is calculated as

2π Q R dαi −β0H i 0 2π Ae hAiβ0 = 2π (3.15) Q R dαi −β0H i 0 2π e

To get the average of the observable A at some diﬀerent temperature, say β1, we can compute

2π Q R dαi −β1H i 0 2π Ae hAiβ1 = 2π Q R dαi −β1H i 0 2π e 2π Q R dαi −(β1−β0)H −β0H i 0 2π Ae e = 2π Q R dαi −(β1−β0)H −β0H i 0 2π e e hAe−(β1−β0)H i = β0 . (3.16) −(β1−β0)H he iβ0 The range of temperatures to which extrapolation is possible will decrease with system size. Loss in precision will occur when the exponential factor becomes large. To minimize the exponential factor, the energy of the system can be shifted by a constant Hmax, without changing the values of the averages, as

2π 2π βHmax Q R dαi −βH Q R dαi −β(H−Hmax) e i 0 2π Ae i 0 2π Ae hAi = βH 2π = 2π . (3.17) e max Q R dαi −βH Q R dαi −β(H−Hmax) i 0 2π e i 0 2π e Energy ﬂuctuations will grow with system size, making this method increasingly less eﬃcient. 3.2. Error Estimation 21

The accuracy of the extrapolation will decrease as the diﬀerence |β0 − β1| increases, as the overlap of the Boltzmann distributions at temperature β0 and β1 will decrease.

3.2 Error Estimation

The theory of error estimation in Monte Carlo methods is discussed. The jackknife method is presented, which is used to estimate the error of calculated quantities.

3.2.1 Monte Carlo Error Estimation The theoretical basis on which the validity MC estimations rely is the central limit theorem, stated here without proof.

Theorem 1 Let X1,X2,... be independent and identically distributed with E[Xi] = N √ ¯ 2 ¯ 1 P Xi−µ µ, V AR(Xi) = σ , if Xi = N Xi then N σ i=1 converges in distribution to a Gaussian distribution with zero mean and unit variance as N → ∞. Our MC method produces estimations of thermal averages as 1 X hAi = A + δA , (3.18) N i i i 1 X A¯ ≡ A , (3.19) N i i where Ai are stochastically uncorrelated estimates of a given quantity. The error of this estimator is then by Theorem 1 proportional to σ δA ≈ √ A . (3.20) N − 1

Note that σA will also have to be estimated, and that estimation will have a analogous error term, which again can be estimated, and so on ad inﬁnitum. For each iteration, the new error term will contain an additional factor √1 , so higher order N terms can often safely be ignored.

Certain quantities used in the analysis are functions of averages, e.g. the 4th order Binder cumulant, the superﬂuid density, the susceptibility, etc. An estimator for functions of averages is given by 1 X f(hAi, hBi, hCi,... ) ≈ f(A ,B ,C ,... ), (3.21) N i i i i for which Eq. 3.20 can be used to estimate the error. However, if the functions are nonlinear, this estimator can introduce a signiﬁcant bias. For instance, the function 22 Chapter 3. Method f(hXi) = X2 will be estimated by f(X¯ + δX) = (X¯)2 + XδX¯ + (δX)2 where the last term gives a positive bias for each term in the sum. A way to minimize the bias is to increase overall accuracy of each independent MC estimation, but one can also use a better estimator. In this thesis the estimator

f(hAi, hBi, hCi,... ) ≈ f(A,¯ B,¯ C,...¯ ) (3.22) is used, which gives a better estimate, as the bias will not accumulate. The error of this estimator cannot be estimated by Eq. 3.20, instead resampling methods will be used.

3.2.2 The Jackknife Methods The jackknife methods of estimation of bias and variance were developed by Que- nouille[25] during the 1940s, and later named and improved by Tukey[26]. The name reflects that the methods can be used to quickly implement a solution for nearly any problem, although more efficient solutions may exists for specific problems. Efron and Stein showed in Ref. [27] that the jackknife estimate of variance is biased upwards. As suggested in Ref. [28], it can also be used to estimate error of fit parameters, which is done in this work. The jackknife methods are so called resampling methods. The full data set is divided into N subsets to obtain estimations. First, the jackknife average is defined as 1 X xJ = x , (3.23) i N − 1 j j6=i where data from all sets but one is included. Then the jackknife estimate of a function of averages is given by

N 1 X f¯J ≡ f(xJ ). (3.24) N i i=1 and the uncertainty in this estimate is then √ σf(X¯ ) ≡ N − 1σf J (3.25) where 2 ¯J 2 ¯J 2 σf J = (f ) − (f ) . (3.26)

The number of subsets Nblock can be chosen freely, but might inﬂuence the estimation of the error. A downside of this method is that it can be computationally expensive, as it increases the necessary computations by approximately a factor Nblock. Throughout this work the value Nblock = 500 is used. Most error bars shown in this thesis are derived using this method. 3.3. Implementation 23

3.3 Implementation

The MC simulations using the Wolﬀ algorithm was implemented in C++ using the MPI standard for parallelization. The analysis code was written in Python using the libraries SciPy and Numpy. The ﬁgures where produced using the open source program Grace[29].

3.3.1 Parallel Programming Simulation time required grows roughly as L3 with system size. One way to speed up data gathering is to parallelize the simulations. For the problems at hand, we chose to implement the trivial version of parallelization; running multiple instances of the simulation serially on different cores simultaneously. The Wolff algorithm can in fact be parallelized[30], but the gain in efficiency is not enough motivate an implementation in this work. Instead trivial parallelization have been used. By running multiple independent simulations on multiple cores simultaneously, it is possible to obtain considerably more data. If the simulations had been ran sequentially, it would have taken several decades to get the same amount of data. The main part of the simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High Performance Computing. Initial test runs, equilibration and autocorrelation studies were performed on the Octopus cluster provided by the Department of Physics at KTH and a laptop computer with an i5-3320M CPU. The parallelization was implemented using MPI, having a master process which handled data output while the other processes performed simulations. The reason for this is that writing output data to disk storage is a relatively slow operation. The initialization cost of a write operation is quite high, so if output is stored up in memory and then printed in larger chunks, CPU time can be saved. In practice, efficiency improves with chunk size, so the limiting factor is the available memory. Depending on computer system, the program may slow down considerably or even crash should the available memory be depleted during a run, therefore one should be careful when choosing chunk size.

3.3.2 Random Numbers Monte Carlo methods rely on the ability of computers to quickly produce random numbers of high quality, by use of so-called pseudo-Random Number Generators (pRNGs). In the early days of MC, prepared lists of random numbers were used, obtained by some method of rolling die in the real world and writing down the results. That method guarantees that the numbers used are truly random, as long as the die is not loaded. Pseudo-random number generators are, as the name implies, inherently non- random. They are mathematical algorithms that produce sequences of numbers that are seemingly random, but given full knowledge of the state of the generator, 24 Chapter 3. Method one can predict what number will follow any other. Most of them also have a finite period, meaning that after Nperiod generated numbers, the generator will produce the same numbers it did the preceding loop. The main problems for pRNGs are the quality of the random numbers they produced, and the high computational expense. The quality of numbers generated by pRNGs is crucial for the reliability of the simulations. If the generated numbers contain correlations, the MC estimates can obtain systematic errors, which was a large problem for MC methods historically. Today, the available computer power has increased substantially. New algorithms which produce high quality random numbers for relatively low computing cost have been developed. Present MC methods use such a large amount of random numbers that inputting a pre-produced list would not be feasible. There are three conditions that are important to consider when choosing a pRNG. The period length should for practical purposes be infinite, there should be no correlation between sequential output numbers, and it should not be too computationally expensive. In this work we use the Mersenne Twister algorithm first proposed in Ref. [31], and the specific mt 19937 64 implementation in the standard library of C++. This generator has a period length of 219937 − 1 and performs well in various quality tests. Chapter 4

Results

The main results of the work are presented. An equilibation study was performed, and the autocorrelation functions were computed. The analysis methods and the results of the main simulations are presented. Table 4.10 lists the best estimates of this work together with recent results in the literature.

4.1 Equilibration

The Monte Carlo algorithm samples states that in the infinite time limit are Boltz- mann distributed. To eliminate transients from the initial configuration the system is initially updated without sampling. A suitable value for the equilibration time is different for different algorithms and models. To estimate the equilibration time and study how it depends on system size, long runs have been performed for two different starting configurations. The model was sampled during time intervals 0 - 2, 2 - 4, 4 - 8,..., 210 - 211 MCS for both the T = 0 and the T = ∞ initial configurations. In the T = 0 state, all spins point in the same direction. In the T = ∞ state, all spins are assigned a random value. The equilibration time can then be estimated by studying the magnetization as a function of time for the two different initial configurations. It can be chosen as the time after which the two different curves intersect. The resulting curves are plotted in Fig. 4.1. It was found that the equilibration time grows roughly as τeq ∼ L, when measured in units of Monte 4 Carlo sweeps, which means that the cpu time scales as teq ∼ L as the cpu time for a MCS is expected to scale linearly with the total number of spins. To ensure that 5 equilibrium had been reached, an equilibration time of at least Neq = 10 MCS was used in the production runs. To reduce the CPU time used for equilibration, a single equilibration was performed for each simulated lattice size, after which the state of the lattice was stored to disk. For the main productions runs the pre-equilibrated lattice was first loaded and equilibrated for 103 sweeps to reach a state uncorrelated with the saved state.

25 26 Chapter 4. Results

1 L = 4, T = inf L = 8, T = inf 0,8 L = 16, T = inf L = 32, T = inf L = 64, T = inf 0,6 L = 4, T = 0 L = 8, T = 0 L = 16, T = 0 0,4 L = 32, T =0 L = 64, T = 0 Magnetization 0,2

0 1 2 4 8 16 32 64 128 256 512 1024 2048 Nsweeps

Figure 4.1. The magnetization for the T = 0 and T = ∞ initial conﬁgurations are plotted as a functions of time, measured in units of sweeps. We ﬁnd that equilibration time scales roughly as L. Note that the scale on the time axis is logarithmic.

4.2 Autocorrelation

Another interesting property of the algorithm is the relaxation time τrel, which is a measure of the amount of updates that is required to reach an uncorrelated state. z The scaling of the relaxation time τrel ∼ L deﬁnes the dynamical exponent z, and is algorithm dependent. The autocorrelation function of the magnetization is deﬁned by

2 CMM (t) = hM(t)M(0)i − hMi , (4.1)

−t/τ and is expected to decrease exponentially, CMM (t) ∼ e rel . These averages were estimated by performing a run where the magnetization and time were printed after each update. The time of a cluster update is not fixed, so the mean number of MC sweeps per cluster update is computed and used as the time between sequential updated. This introduces an uncertainty in t, and could potentially obscure short timescale effects. However, it is a convenient way of binning the data, which is necessary in one way or another when analysing an algorithm where the time of each update is uncontrollable, and the methods should be accurate for long enough times. The correlation functions are shown in Fig. 4.2. The relaxation time is defined by the time it takes for the autocorrelation function to reach a fraction of its initial value, CMM (0). If the autocorrelation functions are assumed to follow exponential decay, the correlation time can be defined by the 4.3. Analysis and Main Results 27

1 L = 4 L = 8 L = 16 L = 32 L = 64 (0) MM 0,01 (t)/C MM C

0,0001 0 10 20 30 40 50 60 70 Nsweeps

Figure 4.2. Normalized autocorrelation functions CMM (t)/CMM (0) plotted vs. time measured in MCS. In total 107 cluster updates per system size was included. 3 Time diﬀerences up to Nupdates = 5 · 10 clusters were considered. integration rule

∞ X CMM (t) τ = 1 + 2 , (4.2) rel C (0) t=1 MM where the sum can be terminated at the first negative term. As evident in Fig. 4.2 the functions deviate from the expected exponential decay. The expected order of increasing τrel with system size is inverted for short times, which could be explained by effects from a shorter timescale. Later deviations from exponential decay starting at t ≈ 45 is likely caused by the limited amount of data. To account for these deviations from exponential decay a non-linear least squares fit limited to values of the correlation functions in the interval Nsweeps = 15 - 45 was performed. Estimated values of τrel are presented in Fig. 4.3. The dynamical exponent was estimated to z = 0.1.

4.3 Analysis and Main Results

Data statistics and the simulation conﬁguration is presented. The methods used to estimate the critical properties from the MC data are explained and the results are presented. 28 Chapter 4. Results

Integration Exponential fit 0.097 8 y = 5.68x τ

4 8 16 32 64 L

Figure 4.3. The relaxation time estimated by the integrals of the autocorrelation functions and a fit to exponential functions. The dynamical exponent z is found to 0.01 be τrel ∼ L , as expected for the Wolff algorithm. The green curve is given by a non-linear least squares fit.

4.3.1 Statistics and Conﬁguration

Approximately 50 years of CPU time has been used on the Beskow system at PDC to obtain the Monte Carlo results. Total numbers of MC sweeps for individual system sizes as well as an estimate of CPU time is presented in Tab. 4.2. Post-processing and analysis has been performed on the Tegner system at PDC and on a laptop computer with an i5-3320M CPU. During development, initial results were produced on the Octopus cluster at the department of theoretical physics at KTH.

The simulations started from a pre-equilibrated lattice, which was updated for 103 MCS without sampling to ensure that the phase space near the saved equilibrated state is not repeatedly sampled. The system was then sampled for 105 MCS for L = 4 - 64 and for 103 MCS for L = 128. Averages are then computed and stored, 102 MCS are performed to reach an uncorrelated state, after which sampling takes place again. This is then repeated until a desired number of averages has been computed. The reason a shorter sampling period had to be chosen for L = 128 is the limit of 24 hours of runtime per job at the Beskow system, during which only about 6 · 104 MCS could be obtained. To choose the running temperature a series of test runs were performed. By studying the partial results, the simulated temperature and extrapolation interval could be successively narrowed down. The ﬁnal running temperature chosen was T = 2.20184. Data was extrapolated to 101 temperatures in the range T = 2.2016 - 2.2020. 4.3. Analysis and Main Results 29

L CPU seconds 4 92.218 8 718.07 16 5782.3 32 45072

Table 4.1. Times for 106 sweeps per system size were measured. From this CPU time per sweep was estimated to scale as L3.

L Total MCS Est. CPU hours 4 1.08 · 1011 1170 8 7.63 · 1010 6420 16 2.25 · 1010 15200 32 1.76 · 1010 93000 64 1.63 · 109 67200 128 6.91 · 108 242000

Table 4.2. Total amount of MCS used in the estimation of the thermal averages for each system size. The estimates was performed by requiring that runtime scales as found in Tab. 4.1. The estimates were initially high, to obtain more reasonable estimates the total runtime was ﬁxed to 425000 hours, which is the total time logged on the Beskow system discounting initial test runs and some experimental runs which were not included in the ﬁnal averages.

The CPU time was not internally measured, but the Beskow system stores used CPU time for individual users. To estimate the used CPU time per system size, a set of smaller runs were performed to ﬁnd the scaling of CPU time. The results are presented in Tab. 4.1. The total amount of MCS and the found scaling was then used to estimate the total CPU time for each system size, presented in Tab. 4.2.

4.3.2 Estimated Quantities

Both the 4th order Binder cumulant U4 and the rescaled superfluid density Lρs are expected to be system size independent at the critical point. In Fig. 4.4 and 4.5 the MC estimates of these quantities are plotted. The finite size effects causes the intersections to drift. The intersection points of sequentially larger sizes approach Tc, and since Lρs and U4 approaches Tc from opposite directions, we can restrict Tc to lie in the interval T = 2.201843 - 2.201872.

4.3.3 Parameter Fitting The non-linear least square method is extremely useful to determine scaling relations. A ﬁtting function is deﬁned from the assumed scaling form, then the free 30 Chapter 4. Results

1,26 L = 4 L = 8 1,25 L = 16 L = 32 L = 64 L = 128 1,24 4 U 1,23

1,22

1,21 2,2016 2,2017 2,2018 2,2019 2,202 T

4 2 2 Figure 4.4. The fourth order Binder cumulant, U4 = hM i/hM i is plotted against temperature for all simulated system sizes. This quantity is used in several −ω analyses due to its scaling form U4(L, t = 0) = aU4 + bU4 L . The limiting behaviour U4 → 1 for T < Tc, U4 → 2 for T > Tc is not apparent due to the small temperature interval.

L = 4 1,25 L = 8 L = 16 L = 32 L = 64 1,2 L = 128 s ρ L 1,15

1,1

1,05 2,2016 2,2017 2,2018 2,2019 2,202 T

Figure 4.5. The rescaled superfluid density, Lρs, is plotted as a function of temperature for all simulated system sizes. 4.3. Analysis and Main Results 31 parameters of the function is adjusted to fit to the MC values. The parameters are determined by minimizing the function 2 X f(xi,A) − yi χ2 = (4.3) ∆y i i where A are the free parameters and xi, yi are the MC data points. In this work the curve fit function of the SciPy module in Python was used, which also estimates the covariance matrix of the parameters. The variance of the best-fit value of the parameters, i.e. the diagonal elements of the covariance matrix, are related to the error of the fitted parameter. The variance of the best-fit value is given by the value which if added to the best-fit value causes χ2 to increase by unity, if other parameters are re-optimized. The Jackknife method was used to estimate the statistical uncertainty and the variance of the fitted parameter can show how well the data fits the function. The scaling form from Eq. 2.43 is expected to fit better for larger system sizes, as the FSS corrections become smaller, so for some analyses multiple fits were performed omitting 0, 1 or 2 of the smallest sizes. The trade off is that having a smaller data set will increase uncertainty, and that data for small L often has smaller error bars. It was found that the minimum of the variance decreases when omitting smaller sizes, but the statistical uncertainty increases.

Using the derived scaling form in 2.43, the scaling of the 4th order Binder cumulant U4 and the rescaled superﬂuid density Lρ take the form −ω g(L) = ag + bgL (4.4) at the critical point. This function has three free parameters and can be ﬁtted directly to MC values of g(L) to estimate ω. The resulting curves are plotted in Figs. 4.6 to 4.9. Estimates of ω and Tc are listed in Tab. 4.3.

By subtracting U4 and Lρs for diﬀerent system sizes, we get the scaling form −ω −ω ˜ −ω vk(L) ≡ g(kL) − g(L) = bg(k − 1)L ≡ bgL . (4.5)

This eliminates the free parameter ag, but also reduces the number of available data points for each fit. The new subtracted quantities is used to estimate ω and Tc by non-linear least-squares fit and an intersection method. By fitting the MC values for U4 and Lρs to this form using the nonlinear least-squares method, ω and Tc can be estimated. The results of the non-linear least squares fit are are plotted in Figs. 4.10 and 4.11. The variance of the best-fit value of ω is plotted in Figs. 4.12 and 4.13. In Tab. 4.4 estimates of ω and Tc are presented.

Intersection Method Certain quantities can be expected to intersect at the critical point. For such quantities one can vary a parameter to ﬁnd the optimal value which gives the best 32 Chapter 4. Results

1,5 U L = 4 4 min U L = 8 4 min U L = 16 4 min 1 ω

0,5

0 2,20182 2,20184 2,20186 2,20188 2,2019 T

−ω Figure 4.6. Values of ω from ﬁtting U4 to the scaling ansatz g(L) = a0 + a1L .

L L = 8 ρs min L L = 16 1,5 ρs min

1 ω

0,5

0 2,2018 2,20185 2,2019 T

Figure 4.7. Values of ω from fitting U4 and Lρs to the scaling ansatz g(L) = −ω a0 + a1L . When including L = 4 data, the fitting failed, indicating significant contributions from higher order corrections, and the result is therefore not presented. 4.3. Analysis and Main Results 33

Lmin = 4 1 Lmin = 8 Lmin = 16 ) ω Var(

0,0001

2,20184 2,20186 2,20188 T

Figure 4.8. The variance the best-ﬁt value of ω from ﬁtting U4 directly to the −ω scaling ansatz g(L) = a0 + a1L .

Lρs Lmin = 8 1 Lρs Lmin = 16 ) ω Var(

0,0001

2,2018 2,20182 2,20184 2,20186 2,20188 2,2019 T

Figure 4.9. The variance the best-ﬁt value of ω from ﬁtting Lρs directly to the −ω scaling ansatz g(L) = a0 + a1L . The lower error bars were omitted for clarity. 34 Chapter 4. Results

Method ω Tc

Min(Var(ωU4 )), Lmin = 4 0.7699(8) 2.201864(2)

Min(Var(ωU4 )), Lmin = 8 0.895(3) 2.201856(1)

Min(Var(ωU4 )), Lmin = 16 0.663(9) 2.201864(6) Min(Var(ωLρ)), Lmin = 8 0.642(19) 2.201856(61) Min(Var(ωLρ)), Lmin = 16 1.21(35) 2.201864(27)

Int ωU4 ,Lmin = 4, 8 0.752(3) 2.20187369(4)

Int ωU4 ,Lmin = 8, 16 0.902(13) 2.2018552(4) Int ωLρ,Lmin = 8, 16 0.626(22) 2.201853(5)

Table 4.3. Values of ω from non-linear least-squares fit to U4 and Lρs MC values, estimated using two different methods. By locating the minimum of the variance of the fitted parameter one estimate can be obtained for each curve. By finding points of intersection between sequential curves another estimate is found. The error estimate of ω is obtained using the jackknife method. For the minimum-variance method the uncertainty in Tc is estimated from the uncertainty and slope of the variance. For the intersection method the slope and uncertainty in ω is used.

1,5 L = 4 min L = 8 min L = 16 min L = 32 1 min ω

0,5

0 2,20186 2,20188 2,2019 2,20192 T

Figure 4.10. Subtracted U4 values are ﬁtted using the non-linear least-squares method to estimate ω and Tc. 4.3. Analysis and Main Results 35

L = 4 min L = 8 3 min L = 16 min L = 32 2 min

ω 1

-1

2,2018 2,20182 2,20184 2,20186 2,20188 T

Figure 4.11. Subtracted U4 values are ﬁtted using the non-linear least-squares method to estimate ω and Tc. It is clear that the estimates where L = 4 data is included does not ﬁt the scaling form well.

Lmin = 4 Lmin = 8 L = 16 0,01 min ) ω Var(

0,0001

2,20182 2,20184 2,20186 2,20188 2,2019 T

Figure 4.12. The variance the best-ﬁt value of ω from a non-linear least-squares ﬁt of subtracted U4 estimates. 36 Chapter 4. Results

L = 4 1 min Lmin = 8 Lmin = 16 )

ω 0,01 Var(

0,0001

2,2018 2,20182 2,20184 2,20186 2,20188 2,2019 2,20192 T

Figure 4.13. The variance the best-fit value of ω from a non-linear least-squares fit of subtracted Lρs estimates. Note that the scale on the y-axis is logarithmic. The lower error bars were not displayed for clarity. The variance of the fit including L = 4 data does not have the expected minimum around the critical point, which indicates that higher order corrections are needed for such small sizes.

Method ω Tc

Min(Var(U4,Lmin = 4)) 0.7590(8) 2.20187(3) Min(Var(U4,Lmin = 8)) 0.8996(40) 2.201856(6) Min(Var(U4,Lmin = 16)) 0.663(9) 2.201864(6) Min(Var(Lρs,Lmin = 8)) 0.64(2) 2.201856(41) Min(Var(Lρs,Lmin = 16)) 1.22(38) 2.201864(28)

Int. U4,Lmin = 4, 8)) 0.718(4) 2.2019064(8) Int. U4,Lmin = 8, 16)) 0.904(12) 2.2018550(4) Int. U4,Lmin = 16, 32)) 0.68(9) 2.201863(2) Int. Lρs,Lmin = 8, 16)) 0.62(25) 2.201852(6)

Table 4.4. Estimates of ω and Tc from the minimum of the variance and curve intersections from non-linear least-square fitting of subtracting two U4 and Lρs values from different system sizes. 4.3. Analysis and Main Results 37 intersection. A measure of the spread of intersection points is then needed. The function N 1 X C(p ) = p(x − x¯)2 + (y − y¯)2 (4.6) i N i i i=1 was used to quantify the spread of a set of points pi. Note that an arbitrary weight parameter could have been included, since x and y are typically living in different spaces, and there might not be a natural way to compare distances in the respective spaces. An idea is to choose a weight so that importance is proportional to the fraction of relative errors. This method can be unstable. As the parameter is varied, intersection points may travel outside the considered x interval, which often causes a jump of the spread measure in Eq. 4.6. By visually inspecting the intersections, one can judge whether certain system sizes should be omitted.

Constant Subtraction

−ω By variation of the constant parameter ag in the scaling relation g(L) = ag +bgL for phenomenological couplings, it is possible to estimate ω, Tc and ag. We have that

−ω g(L) − ag = bgL , (4.7) g(kL) − a g = k−ω, (4.8) g(L) − ag g(kL) − a ω = − ln g / ln k. (4.9) g(L) − ag

A value for ω is computed for L = 4, 8, 16, 32, 64 and k = 2 for a range of temperatures. Varying the paramter ag to obtain the best intersections then gives an estimate of ag, ω and Tc. A starting estimation for ag can be obtained by ﬁnding intersection points of U4 and Lρs and plotting them vs. 1/L and extrapolating to 1/L = 0. The method was only applicable to U4, as the uncertainty in Lρs causes the term inside the logarithm to be negative for most of the data points. The results are presented in Table 4.5. −ω The valuea ¯U4 ≈ 1.244 is then used to ﬁt U4 to the function f(L) =a ¯U4 +bU4 L . The resulting curves are plotted in Fig. 4.14, the variance is plotted in Fig. 4.15. The estimates of ω from this method is listed in Tab. 4.6.

Rescaling The intersection method can also be used on the subtracted quantities, by rescaling ˜ −ω ω ˜ vk(L) = bgL to L vk(L) = bg, which is then expected to be constant at the ω critical point. The quantity L vk(L) can be computed from MC values for a range 38 Chapter 4. Results

L aU4 ω Tc 4-128 1.24451 0.7746 2.20185269 8-128 1.244465 0.78056 2.20186813 16-128 1.245363 0.68157 2.20186316 4-64 1.243224 0.84621 2.20182008 8-64 1.242352 0.93209 2.20184562

Table 4.5. The values of aU4 which gives the best intersection. The mean the intersection points estimates ω and Tc. The values of ω and Tc are found to be

sensitive to which system sizes are included, while the values of aU4 are more stable. −6 The mean of the estimates isa ¯U4 = 1.243983 with variance 1.1 · 10 .

0,95 L = 4 min L = 8 0,9 min L = 16 min L = 32 0,85 min

ω 0,8

0,75

0,7

0,65 2,20184 2,20185 2,20186 2,20187 2,20188 T

−ω Figure 4.14. Fit of the Binder parameter to the function f(L) = 1.244 + a1L

Method ω Tc

Min(Var(ωU4 )), Lmin = 4 0.7962(2) 2.201856(3)

Min(Var(ωU4 )), Lmin = 8 0.8200(7) 2.201860(3)

Min(Var(ωU4 )), Lmin = 16 0.790(2) 2.2018600(8)

Min(Var(ωU4 )), Lmin = 32 0.824(12) 2.2018600(16)

Int ωU4 ,Lmin = 4, 8 0.7902(7) 2.2018417(4)

Int ωU4 ,Lmin = 8, 16 0.836(2) 2.2018697(4) Int ωLρ,Lmin = 16, 32 0.785(12) 2.2018590(3)

Table 4.6. Values of ω from nonlinear fit of the Binder parameter with the parameter a0 = 1.244, estimated using two different methods. By locating the minimum of the variance of the fitted parameter one estimate can be obtained for each curve. By finding points of intersection points between curves another estimate is found. 4.3. Analysis and Main Results 39

L = 4 0,01 min Lmin = 8 Lmin = 16 L = 32 0,001 min ) ω Var( 0,0001

1e-05

2,20184 2,20186 2,20188 2,2019 2,20192 T

Figure 4.15. The variance of ω from ﬁtting the Binder parameter to f(L) = −ω 1.244 + a1L . The scale on the y-axis is logarithmic.

of values of ω. An estimate of ω and Tc is then obtained by ﬁnding the lowest spread measure of the intersection points as a function of ω. The results are presented in Figs. 4.16 and 4.17 and Tab. 4.7. The estimates from this method proved to be uncertain, but it is useful as a consistency check.

2,2019

2,20185 c T

U L = 4 2,2018 4 min U L = 8 4 min L L = 4 ρs min 0,6 0,7 0,8 0,9 1 1,1 ω

Figure 4.16. The critical point as estimated by an intersection method for the quantities U4(2L) − U(L) and 2Lρs − Lρs for a range of values of ω. The jump for U4,Lmin = 4 is where an intersection point move outside the considered temperature interval. 40 Chapter 4. Results

0,7 U L = 4 0,6 4 min U4 Lmin = 8 Lρ L = 4 0,5 s min

0,4

0,3

0,2

Spread of Intersection Pts. 0,1

0 0,6 0,8 1 1,2 ω

Figure 4.17. The spread of intersection points is plotted for a range of values of ω. The estimated uncertainty is large for Lρs and was hidden for clarity. For Lρs, discarding the smallest system size gave the same curve, meaning that the intersection between Lmin = 4 and Lmin = 8 is not within the considered temperature interval for the considered range of values of ω. Estimation of the critical point and ω is quite imprecise for this method, with the exception of the curve for U4 with Lmin = 4, which has a quite sharp minimum of the spread measure.

Method ω Tc

Min. Cl. U4,Lmin = 4 0.769 2.2018565 Min. Cl. U4,Lmin = 8 0.769 - 0.845 2.2018625 - 2.201874 Min. Cl. Lρs,Lmin = 4 0.783 - 1.031 2.2018643 - 2.2018788

Table 4.7. Estimated values of ω and Tc from the minimum of the spread measure deﬁned in Eq. 4.6. 4.3. Analysis and Main Results 41

Direct Computation of ω From Eq. 4.5 we get

−ω −ω vk2 (L) bg(k − 1)(kL) −ω = −ω −ω = k , (4.10) vk(L) bg(k − 1)L v (L) ln k2 ω = − vk(L) . (4.11) ln k

Computing ω for k = 2,L = 4, 8, 16, 32 using U4 and Lρs is expected to give an intersection for the diﬀerent system sizes at the critical point. The results are plotted in Figs. 4.18 and 4.19 and estimates of ω and Tc are listed in Tab. 4.8.

1,4 L = 4, 8, 16 L = 8, 16, 32 1,2 L = 16, 32, 64 L = 32, 64, 128 1

ω 0,8

0,6

0,4

0,2 2,2018 2,20182 2,20184 2,20186 2,20188 2,2019 2,20192 2,20194 T

Figure 4.18. Using U4 data for three system sizes differing by a constant factor, the first order scaling correction ω has been computed. The curves have L = (4, 8, 16), (8, 16, 32), (16, 32, 64), and (32, 64, 128). The curve with L = (32, 64, 128) has too large uncertainty to be included. The intersection of the curves with Lmin = 8, 16 gives the estimate ω3L = 0.646(25) at T = 2.2018457(23). The curve with Lmin = 4 indicates significant contributions from higher order corrections.

Summary of ω Estimates

The estimated values of ω and Tc are sensitive to which method of parameter fitting was used. The direct-computation method and the non-linear fits give clear indications that higher order corrections needs to be considered for the smallest size L = 4. The non-linear fit of U4 with the constant aU4 = 1.244 gives the most consistent and accurate predictions for different data subsets. The minimum- variance estimates gives the estimates Tc = 2.201860(3) and ω = 0.8200(7) when discarding data for L = 8 which we quote as our best estimates. 42 Chapter 4. Results

2 L = 4, 8, 16 1,5 L = 8, 16, 32 L = 16, 32, 64 L = 32, 64, 128 1

ω 0,5

-0,5

-1 2,20175 2,2018 2,20185 2,2019 T

Figure 4.19. Using Lρs data for three system sizes diﬀering by a constant factor, the ﬁrst order scaling correction ω is plotted vs. the temperature. The curves have L = (4, 8, 16), (8, 16, 32), (16, 32, 64) and (32, 64, 128). The large uncertainty makes the estimates unreliable. The two curves with largest sizes intersect at ω ≈ 0.8, which is consistent with other methods and indicates that the method could be useful with more statistics.

Sizes ω Tc

Int(U4Lmin = 4, 8) 0.4848(26) 2.2019317(16) Int(U4Lmin = 8, 16) 0.646(26) 2.2018457(23) Int(U4Lmin = 16, 32) 0.471(65) 2.2018634(15)

Int(LρsLmin = 8, 16) 0.39(24) 2.201841(16) Int(LρsLmin = 16, 32) 0.86(176) 2.2018642(44)

Table 4.8. Values of ω and Tc estimated from intersections of the ω-curves from direct computation. It is obvious that the curves using L = 4 gives inconsistent results, which can be explained by higher order corrections to the scaling which are not included. The estimates of ω are still not satisfactory, which can be explained by the inherent restriction to using data for only 3 system sizes per curve, compared to other methods of analysis. 4.3. Analysis and Main Results 43

4.3.4 Critical Exponents

dU4 Using the best estimates a non-linear least-squares ﬁt of dT and χ can be performed to estimate the critical exponents ν and η. Using the scaling form from Eq. 2.43 we have

dU4 1/ν −ω = L (aU 0 + bU 0 L ), (4.12) dT 4 4 2−η −ω χ = L (aχ + bχL ). (4.13)

The results of the ﬁts are plotted in Figs. 4.20 and 4.21 where values from the literature have been included. The variance of the ﬁts are plotted in Figs. 4.22 and 4.23. Estimates of the critical exponents are presented in Tab. 4.9.

0,674 Lmin = 4 Lmin = 8 0,673 Lmin = 16 Campostrini et al. 2006 Sokolov et al. 2016 0,672 Lipa et al. 2003 ν 0,671

0,67

0,669

2,20184 2,20185 2,20186 2,20187 2,20188 T

Figure 4.20. The critical exponent ν as estimated from a non-linear least-squares ﬁt of the temperature derivative of U4 for Lmin = 4, 8, 16 and ω = 0.82. Estimates from the literature are included in the plot. In Refs. [1, 10] no estimate of the critical point is stated, so those estimates were placed near our best estimate at T = 2.201860.

We find that our estimate of ν with Lmin = 4 is consistent with the experimental values in Ref. [1]. For η, the estimate with Lmin = 16 is most consistent with previous estimates. This could suggest that higher order corrections are more significant for the scaling of χ than for dU4/dT . The determination of the scaling correction exponent ω was problematic as different methods produced inconsistent results. Therefore it is useful to study how the estimates of ν and η depend on the value of ω. Parameter fits of dU4/dT and χ were performed for Lmin = 4, 816 over a range of values of ω and the mean intersection point was used as a rough estimate of the critical exponents and Tc. The results are plotted in Figs. 4.24 to 4.28. 44 Chapter 4. Results

Lmin = 4 Lmin = 8 0,045 Lmin = 16 Campostrini et al. 2006 Sokolov et al. 2016

η 0,04

0,035

2,20183 2,20184 2,20185 2,20186 2,20187 2,20188 2,20189 T

Figure 4.21. The critical exponent η as estimated from a non-linear least-squares ﬁt of Lρs for Lmin = 4, 8, 16 and ω = 0.82. Estimates from the literature are included in the plot. In Ref. [10] no estimate of the critical point was stated, so the estimate was placed near our best estimate at T = 2.201860.

1e-06 Lmin = 4 Lmin = 8 Lmin = 16 ) ν 1e-08 Var(

1e-10 2,20184 2,20185 2,20186 2,20187 2,20188 2,20189 2,2019 T

Figure 4.22. The variance of the best-ﬁt value of the critical exponent ν. 4.3. Analysis and Main Results 45

1e-06

Lmin = 4 Lmin = 8 Lmin = 16 ) η Var( 1e-08

2,2018 2,20182 2,20184 2,20186 2,20188 2,2019 2,20192 T

Figure 4.23. The variance of the best-ﬁt value of the estimated critical exponent η.

Method Lmin = 4 Lmin = 8 Lmin = 16

ν, Tc = 2.201860 0.67100(5) 0.6723(1) 0.6728(2) ν, Min(Var(ν)) 0.67031(5) 0.6720(1) 0.6701(2) Tc, Min(Var(ν)) 2.20188 2.201864 2.20187

η, Tc = 2.201860 0.04094(3) 0.04087(8) 0.0393(1) η, Min(Var(η)) 0.04166(3) 0.04279(8) 0.0386(2) Tc, Min(Var(η)) 2.201852 2.201840 2.201864

Table 4.9. The critical exponents are estimated from a non-linear least-squares ﬁt of dU4/dT and χ with ω = 0.82. As estimates we take the value at our best estimate of the critical temperature, Tc = 2.20186 and as a consistency check the values at the minimum of the variance is included. 46 Chapter 4. Results

0,672

ν 0,671

0,67

0,6 0,7 0,8 0,9 1 1,1 ω

Figure 4.24. The ω dependence of the exponent ν is estimated by computing the mean intersection point of non-linear least-square ﬁts for Lmin = 4, 8, 16.

0,05

0,045

η 0,04

0,035

0,03 0,7 0,8 0,9 1 1,1 ω

Figure 4.25. The ω dependence of the exponent η is estimated by computing the mean intersection point of non-linear least-square ﬁts for Lmin = 4, 8, 16. 4.3. Analysis and Main Results 47

2,20189 T [ ] c η T [ν] 2,20188 c

2,20187 c T 2,20186

2,20185

2,20184 0,7 0,8 0,9 1 1,1 ω

Figure 4.26. The critical temperature as estimated by the mean intersection point plotted vs. ω.

0,0016

0,0015

0,0014

0,0013

0,0012 Spread of Intersection Pts. 0,0011

0,6 0,7 0,8 0,9 1 ω

Figure 4.27. The spread of intersection points of the ν curves plotted vs. ω. The minimum is found at ω = 0.945. 48 Chapter 4. Results

0,09

0,08

0,07

0,06

0,05

Spread of Intersection Pts. 0,04

0,03 0,7 0,75 0,8 0,85 0,9 0,95 1 ω

Figure 4.28. The spread of intersection points of the η curves plotted vs. ω. The minimum is found at ω = 0.834.

For our ﬁnal estimates of the critical exponents, we state the value of the ﬁts at Tc = 2.201860 for Lmin = 8 and ω = 0.820. We obtain ν = 0.6723(1) and η = 0.04087(8). Using the hyperscaling relation we obtain α = 0.0169(3), which is compared to previous estimates in Fig. 4.29. A compilation of recent estimates in the literature together with the best estimates of this work is presented in Tab. 4.10.

Quantity Lipa[1] Sokolov[10] Campostrini[2] Jeon[32] This work

Tc 2.20184(6) 2.20186(9) 2.201860(3) α -0.0127(3) -0.0117(31) -0.0151(3) -0.0124(30) -0.0169(3) ν 0.6709(2) 0.6706(10) 0.6717(1) 0.6708(5) 0.6723(1) η 0.0376(28) 0.0381(2) 0.04087(8) ω 0.785(20) 0.8200(7) CPU time 20 years 50 years

Table 4.10. Estimates of the critical exponents of the λ transition from MC studies, pseudo-ε expansion and experiment are compared to the estimates of this work. For the MC methods the reported used CPU time is also included. 4.3. Analysis and Main Results 49

-0,008

-0,01 Sokolov et al. 2016 Lipa et al. 2003 -0,012 α -0,014 Campostrini et al. 2006

-0,016 This work

-0,018

Figure 4.29. The estimate of the heat capacity exponent α is compared to previous estimates. Although the determination of Tc and ω is relatively uncertain, the esimate of this work gives further conﬁdence that a discrepancy exist between MC estimates and experimental measurement, which suggests that further experimental investigation is needed. 50 Chapter 5

Summary and discussion

5.1 Summary

A new approach to finite-size scaling corrections has been employed to estimate the critical exponents of the λ transition. The method relies on locating the critical point and the first order scaling correction before estimation of the critical exponents. In our analysis we have used data from 50 years of CPU time, which is roughly 2.5 times more than previous work. The statistical uncertainty of the final estimates of the critical exponents are equal in magnitude, or lower by a factor 2 if data for L = 4 is included. Considering that the MC algorithm used has not been aggressively optimized, this further shows that the new scaling approach is an efficient method of determining critical exponents of lattice models. The critical exponent α has been estimated to α = −0.0169(3), which agrees better with the previous theoretical estimate in Ref. [2] than the experimental values in Ref. [1], giving further indications that a discrepancy exist which may need improved experimental investigations to resolve. However, the estimates of the critical exponents were found to depend strongly on the values of Tc and ω. The various methods used to estimate Tc and ω produces conflicting results and further investigation is necessary to obtain a more accurate determination of the critical exponents. The new scaling method allows for various different methods of estimating the scaling correction exponent ω. Various methods have been tested in this thesis with different levels of success. Of the methods tested in this thesis, it was found that an intersection method to estimate the constant aU4 followed by a non-linear least-squares fit of U4 produced the most consistent results.

51 52 Chapter 5. Summary and discussion

5.2 Outlook

The new approach to scaling corrections has proved to be an efficient method of estimating the critical exponents of the λ transition. However, there are difficulties in determining ω and locating Tc, which we have shown to be crucial in estimation of the critical exponents. To better locate the critical temperature and the first order scaling correction, there are two possible methods. One way is to gather data for more system sizes. Larger system sizes should fit the current scaling ansatz better, so a natural extension would be to obtain MC estimates for L = 256. To achieve comparable uncertainty to our other sizes, an estimate based on Tab. 4.2 gives that about 106 CPU hours would be needed. However, this estimate is based on the algorithm used in this work, which has not been heavily optimized. An optimization related to the data structure of the lattice representation was found, which increased efficiency of the program with a factor 1.5, but it was not implemented since production runs were already in progress. It is also possible to gather data for more system sizes, possibly L = 3 · 2n to give more data points for a lower cost than increasing the size. That could be a more efficient way of reducing the final uncertainty in the critical exponents. Another method is to consider more elaborate scaling forms. A proposed extension of the scaling form is to include an analytical term. The scaling form from Eq. (2.43) is then δg −ω −1 Ag = L (ag + bgL + cgL ). (5.1) This form could potentially give a better fit to data from the smallest sizes. The contribution from the sub-leading irrelevant coupling could also be included, as well as the second term in the Taylor expansion from Eq. (2.41). This work shows that with some further investigation new theoretical predictions with unprecedented accuracy should be attainable with moderate additional simulations. This thesis has shown the efficiency of this new scaling approach in determining critical properties of lattice models. During this work the method has also been tested on the three-dimensional Ising-model with promising initial results, which would be interesting to pursue further. Based on the results of this thesis one can expect to achieve similar efficiency in determining the exponents, as the two models have similar properties. Bibliography

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