New Scaling Method for the Kosterlitz-Thouless Transition

Master of Science Thesis

New Scaling Method for theKosterlitz-Thouless Transition

Hannes Lindstrom

Department of Theoretical Physics,School of Engineering Sciences

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2016

Typeset in LATEX

Examensarbete inom amnet teoretisk fysik for avlaggande av civilingenjorsexameninom utbildningsprogrammet Teknisk fysik.

Graduation thesis on the subject Theoretical Physics for the degree of Master ofScience in Engineering from the School of Engineering Sciences.

TRITA-FYS 2016:76ISSN 0280-316XISRN KTH/FYS/--16:76—SE

c© Hannes Lindstrom, December 2016Printed in Sweden by Universitetsservice US AB, Stockholm December 2016

Abstract

The Kosterlitz-Thouless transition is studied from the perspective of two differentnon-neutral 2D Coulomb gas models by Monte Carlo simulations using a Metropolis-Hastings algorithm. A new approach of allowing charge fluctuations allows us tocompare the results to scaling formulas linked to the magnetic permeability that arewell suited to uncover the critical behavior. We focus on pinpointing the transitiontemperature Tc for the models by least squares optimization of the inverse squarednet vorticity 1/m2 to its expected critical and low temperature forms. The parame-ters varied include the system size L, the chemical potential µ and a model-specificparameter λ related to the magnetic susceptibility.

We find that the method accurately portrays the phase transition and can be usedto calculate the Kosterlitz-Thouless temperature and related quantities in a directmanner. The results are roughly the same for the two models and different valuesof λ, but there are some differences in the efficiency of the simulations. We alsooutline some future applications of the method and how a connection to experi-ments can be made.

Key words: Kosterlitz-Thouless transition, 2D Coulomb gas, charge fluctuations,magnetic permeability, renormalization group, finite-size scaling, Monte Carlo sim-ulation, Metropolis-Hastings algorithm.

iii

iv

Preface

This thesis is the result of my Master’s degree project at the Department of The-oretical Physics of the Royal Institute of Technology, which lasted throughout thespring and fall of 2016. The project concerns Monte Carlo simulations of the non-neutral 2D Coulomb gas near the critical temperature and comparing the resultsto formulas obtained from a scaling argument.

The thesis is divided into five chapters. Chapter 1 provides an overview of theKosterlitz-Thouless transition in two-dimensional systems and introduces the prob-lem at hand. Chapter 2 is dedicated to deriving the Coulomb gas model fromGinzburg-Landau theory and introducing two proposed non-neutral models. Theconcepts are more thoroughly explained in Ch. 3, which describes the phase tran-sition and its treatment with renormalization group theory and finite-size scalinganalysis. Chapter 4 is about the Monte Carlo method used and also gives an ac-count of the technical details of the simulation. The results of the simulation anddata analysis are presented in Ch. 5. Finally, Ch. 6 summarizes the findings, at-tempts to make some general conclusions and concludes with an outlook for futurework.

v

vi

Acknowledgements

I would like to thank my supervisor Prof. Mats Wallin and additional advisor Assoc.Prof. Jack Lidmar for giving me the opportunity to work on this project. Theirconstant availability to answer my questions and provide me with feedback hasbeen invaluable. Without their patience and positive attitude, this thesis could nothave been finished.

My thanks also go out to my friends and family for their support. The years I havespent as a student have been tough at times, but they have always been there tohelp me pull through.

vii

viii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Preface v

Contents ix

1 Introduction 1

2 2D Coulomb Gas 3

2.1 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Coulomb Gas Picture . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Magnetic Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.2 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Kosterlitz-Thouless Transition 11

3.1 Nature of the Transition . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Scaling Formulas . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Simulation Method 17

4.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . 18

4.1.2 Main Program . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Pre- and Post-Processing . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

ix

x Contents

5 Results 235.1 Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Configuration Examples . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Basic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Critical Point and Scaling Properties . . . . . . . . . . . . . . . . . 335.6 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Summary, Conclusions and Outlook 39

Bibliography 40

Chapter 1

Introduction

One of the great breakthroughs of classical physics was the development of elec-trostatics by Charles Augustin de Coulomb in the late 18th century. The law ofCoulomb allowed physicists for the first time to make accurate predictions concern-ing the attraction and repulsion of electric charges. Because of the simplicity of themathematical formulation, it is not surprising that the model would be rediscoveredin all kinds of different fields. Indeed, the old gravitation theory of Newton is justanother instance of the Coulomb gas. For its historical importance and straight-forward interpretation, the model is also well suited to picturesque explanations ofmore complicated phenomena.

A current cause for study of the Coulomb gas is connected to a phase transitionknown as the Kosterlitz-Thouless (KT) transition, discovered in the early 1970sby Berezinskii [1], Kosterlitz and Thouless [2]. The KT transition can be foundin two-dimensional systems containing thermal excitations in the form of vortices,which includes thin films of superconductors and superfluids. It is characterized bya shift between a low temperature phase of bound vortex-antivortex pairs and ahigh temperature phase that contains free vortices.

The analogy to the Coulomb gas model is that vortices are represented by chargedparticles, with their winding numbers acting as charge quanta [3]. Under certainconditions, these virtual particles interact with one another according to the Pois-son equation of electrostatics. The comparatively high complexity of the model liesin part with the statistical treatment of a large number of particles and in partwith the logarithmic potential that follows from the restriction to two dimensions.

Recent breakthroughs have been made in the construction of thin materials in gen-eral, such as the single-layered graphene [4]. In particular, the feasibility of thinsuperconductors has contributed to the importance of better understanding the KT

1

2 Chapter 1. Introduction

transition. Not only are there a wide range of possible applications for such mate-rials, but they also serve as an ideal setting for experiments at a single-atom levelthat can test out theories and simulation methods of condensed matter physics. Thegreat potential of these thin films is reflected in the decision of the Royal SwedishAcademy of Sciences to award the Nobel Prize of 2016 to Kosterlitz, Thouless andHaldane, in part due to the discovery of the KT transition.

The KT transition has been subject to a great deal of experimental and theoreticalstudies, but there still remain some open questions. The focus of this thesis liesin testing a new scaling method for the Coulomb gas based on a finite-size scalingargument made by Andersson and Lidmar [5]. Most previous simulation studiesexamine the neutral Coulomb gas, where the charges only occur in dipole pairs.We here take a different approach and study the fluctuations of single chargesby altering the original model to allow for non-neutral configurations. This leadsto interesting physics and a new promising method for future studies of the KTtransition.

Chapter 2

2D Coulomb Gas

In this chapter, we give a brief overview of the 2D Coulomb gas, beginning witha derivation from the Ginzburg-Landau theory of superconductivity. We show theexistence of vortices carrying quantized flux and the presence of phases containingthese. The definition of the Coulomb gas and some immediate properties are thenpresented. Finally, we describe the discretization of the model and the two proposedmodifications that allow a non-zero net vorticity.

2.1 Ginzburg-Landau Theory

The Ginzburg-Landau theory of superconductivity [6] is described by the free energyfunctional

F = Fn+

∫ddr

[α|Ψ(r)|2 +

β

2|Ψ(r)|4 +

1

4me|(−i~∇− 2eA)Ψ(r)|2 +

B2

2µ0

], (2.1)

where Fn is the normal phase free energy, α and β are phenomenological parameters,me is the electron mass, e is the electron charge and B = ∇ ×A is the magneticfield. The Cooper pair density is given by |Ψ|2, where Ψ is the order parameter ofthe system. By introducing a phase θ, we get the decomposition

Ψ(r) = |Ψ(r)|eiθ(r). (2.2)

Minimizing the free energy of Eq. (2.1) with respect to the order parameter leadsto the famous Ginzburg-Landau equations

αΨ + β|Ψ|2Ψ +1

4me(−i~∇− 2eA)2Ψ = 0 (2.3)

J =∇×B

µ0=

e

me|Ψ|2(~∇θ − 2eA), (2.4)

3

4 Chapter 2. 2D Coulomb Gas

where J is the superconducting current density. We will ignore magnetic fields inthe derivation that follows, meaning that B = A = J = 0, and reintroduce themin Ch. 2.4.

2.1.1 Spin Waves

In mean field theory, the superconducting order parameter is a constant and Eqs.(2.3) and (2.4) give the solutions

√−α/β and 0. By extension from the Landau

theory of phase transitions [7], it must be that α/β is only negative for temperaturesT below the critical temperature Tc. Hence, the mean field solution is

Ψ0 =

{√−α/β if T < Tc

0 if T > Tc. (2.5)

Mean field theory neglects the importance of fluctuations in the system, which isnot always a valid assumption. Indeed, the Ginzburg criterion [8] tells us that themean field solution is accurate to great precision only near the critical point for adimensionality d ≥ 4. The fluctuations of the order parameter become increasinglyimportant for the physics of the phase transition as d decreases.

To better account for these fluctuations, suppose that the order parameter has thefull form

Ψ(r) = Ψ0eiθ(r), (2.6)

where Ψ0 is the mean field solution. Up to a constant, the free energy of Eq. (2.1)is then

F = const. +J0

2

∫ddr(∇θ(r))2, (2.7)

where the (bare) superfluid stiffness is defined by

J0 =~2|Ψ0|2

4me. (2.8)

Assuming small fluctuations of θ, the integration of phase field configurations in thepartition function Z =

∫Dθe−F/(kBT ) can be extended to the real line. The system

described by Eq. (2.7) then leads to a Gaussian integral for the expectation valueof the order parameter. Using equipartition and assuming translational invariance,this leads to the correlation function

〈Ψ(r)Ψ∗(0)〉 ∼ r−kBT

2πJ0 (2.9)

The correlation function has an algebraic dependence on the distance r, character-izing a quasi-long range order. The assumption of smoothly variating θ correspondsto so-called spin wave fluctuations and is typically valid for low temperatures. Forhigh temperatures, ignoring the periodic nature of θ does not accurately reproducethe behavior of the system.

2.2. Coulomb Gas Picture 5

2.1.2 Flux Quantization

There are two length scales that characterize superconductors in Ginzburg-Landautheory. The first of these is obtained by noting that the quotient of α and ~/4me

has the dimension of a squared length according to Eq. (2.3). We can thereforedefine the superconducting coherence length as

ξ =

√~2

4me|α|. (2.10)

The second length scale is defined from Eq. (2.4) in a similar manner and is calledthe London penetration depth,

λ =

√me

2µ0e2|Ψ|2. (2.11)

The quotient κ = λ/ξ is known as the Ginzburg-Landau parameter. Superconduc-tors for which κ < 1/2 are called type-I and those for which κ > 1/2 are calledtype-II. Ginzburg and Landau showed that the energy of an interface between anormal and a superconductive region can only be negative for type-II superconduc-tors. Hence, such materials can display a mixed phase for which there are bothnormal and superconductive regions. These normal regions are known as vortices.

Consider now a type-II superconductor with a spatially fluctuating phase θ(r).According to Stokes’ law and Eq. (2.4), the magnetic flux through a vortex, wherethe current vanishes, is

Φ =

∫B · dS =

∮A · dr =

~2e

∮∇θ(r) · dr. (2.12)

The order parameter is single valued, so its phase must change by a multiple of 2πalong the contour of the region. For some q ∈ Z, it follows that

Φ = qΦ0, (2.13)

where the flux quantum is defined as

Φ0 =h

2e≈ 2 · 10−15 Tm2. (2.14)

This shows that vortices carry quantized magnetic flux. The index q is known asthe winding number or vorticity of the vortex.

2.2 Coulomb Gas Picture

Spin wave fluctuations are not always sufficient to describe a system like the 2Dtype-II superconductor, because the presence of vortices is not accounted for. To


understand the role these flux quanta play, it is useful to rewrite the system interms of a classical Coulomb gas. In doing so, we will assume from the start thatthe vortices are located at discrete points, although the essence of the argumentalso holds for a continuous density of vortices.

By using Stokes’ law and setting the vortex normals along the z-axis, the definingflux relations of Eqs. (2.12) and (2.13) show that

∇×∇θ(r) = 2πz∑r′

qr′δ(r− r′), (2.15)

where the system vortices are located at positions r and have winding numbers qr.Only the part of ∇θ with a non-zero curl contributes to the flux quantization andit is therefore useful to decompose the vector as

∇θ(r) = ∇φ(r) +∇× (zψ(r)), (2.16)

where ∇φ is curl free. The quantization condition of Eq. (2.15) then reads

∇2ψ(r) = −2π∑r′

qr′δ(r− r′). (2.17)

The system Hamiltonian corresponding to Eq. (2.7) can with this decompositionbe written in two dimensions as

Htot =J0

2

∫d2r

[(∇φ(r))2 + (∇× (zψ(r)))2 − 2∇φ(r) · ∇ × (zψ(r))

]. (2.18)

The cross term of Eq. (2.18) cancels through partial integration because the phasefactor φ · ∇ × (zψ(r)) vanishes along the system contour and ∇ × (zψ(r)) is di-vergence free. Furthermore, the first term corresponds to the model mentionedearlier for spin waves. Since the term is Gaussian, its free energy is analytic forany temperature and it can therefore not be the cause of a phase transition. Thespin waves completely decouple from the rest of the system, so it will be sufficientto only consider the vortex contribution to study the critical phenomena.

By partial integration and the use of Eq. (2.17), the vortex contribution to Eq.(2.18) simplifies to

H0 = πJ0

∑r, r′

qrqr′V (r− r′), (2.19)

where the sum runs over all pairs of vortices and the potential V (r) is the solutionto the Poisson equation

∇2V (r) = −2πδ(r). (2.20)

Equations (2.19) and (2.20) define the Coulomb gas. The model is exactly that ofa system of charged particles according to electrostatics.

2.3. Regularization 7

The creation of vortices involves thermal excitations that are not part of the elec-trostatic potential derived. The process can be included with a local contributionof a constant Ec known as the core energy, which is added to the model with achemical potential µ. We introduce this according to Ec = −µL2 and make the fullHamiltonian

H = H0 −µ

L2N, (2.21)

where N is the total number of vortices. The choice to make µ scale with the sizeof the system does not follow the usual convention or understanding of a chemicalpotential, which can lead to some unexpected behavior. We will in the followingalmost exclusively focus on the case where µ = 0.

We must also account for the spatial extension of the charges by some lower cutoffin length a and for the size of the system by an upper cutoff L. The potential ofself-interaction according to Eq. (2.20) with these length scales in mind is

V (0) = 2π

∫d2k

(2π)2

eik·0

k2=

∫ 2π/a

2π/L

dk

k= ln

(L

a

). (2.22)

The divergence for a → 0 indicates that the model must be regularized, e.g. bybeing transferred to a lattice. The potential also diverges for N →∞ and this maybe avoided by introducing additional constraints. One option is to require chargeneutrality, which nullifies all self-interactions in Eq. (2.19).

We will not force charge neutrality in what follows. This makes it possible for thesystem to have a non-zero net vorticity

m =∑r

qr. (2.23)

2.3 Regularization

The model regularization adopted here is to use an L×L square lattice with a cellwidth a. To approximate the behavior of a large system in a direct manner, werequire the lattice to have periodic boundaries.

On the periodic lattice, the potential of Eq. (2.20) can be written as a Fourier seriesaccording to

V (r) =2π

L2

∑k

V (k) cos(r · k) (2.24)

V (k) =1

4− 2 cos(kx)− 2 cos(ky), (2.25)


where the positions r = (rx, ry) are all pairs of integers between 0 and L − 1 andk = (kx, ky) are the corresponding phase space combinations, kx, y = 2πrx, y/L.

To facilitate numeric treatment, we also introduce the local function

U(r) =∑r′

qr′V (r− r′) (2.26)

The system Hamiltonian of Eqs. (2.19) and (2.21) then takes the form

H = πJ0

∑r

qrU(r)− µ

L2N. (2.27)

2.4 Magnetic Field Fluctuations

Since we are considering a non-neutral system, Eq. (2.13) tells us that there will bea total flux mΦ0 that is potentially non-zero. The flux corresponds to a magneticfield B = ∇×A that we neglected in deriving the Hamiltonian of Eq. (2.19).

To reintroduce the magnetic field, we add the last term of Eq. (2.1) to Eq. (2.21)and modify the remaining expression to be gauge invariant. The only factor thatneeds to be changed is the charge q, since ~∇θ−2eA is an invariant and q ∼ ∇×∇θ.Hence, the Hamiltonian with a magnetic field is

HB = πJ0

∑r, r′

(qr −B(r))(qr′ −B(r′))V (r− r′)− µ

L2N +

λL2

∑r

B(r)2. (2.28)

The constant λL is the magnetic susceptibility (for a system of size L) and measuresthe degree of magnetization in response to the magnetic field.

Since we do not consider any external magnetic fields, we will allow all variationsof B and can remove their explicit mention in the Hamiltonian by integrating outtheir contribution to the partition function as

∫DBe−βHB . This is most easily

done using the phase space expression

HB =πJ0

2L2

∑k

V (k)|qk −B(k)|2 − λ2L

2L2

∑k

|B(k)|2. (2.29)

The result is exactly the Hamiltonian of Eq. (2.27), but with the phase spacepotential of Eq. (2.25) slightly modified depending on the local properties of thefield.

2.4. Magnetic Field Fluctuations 9

2.4.1 Model A

In model A, we assume the magnetic field B is constant over the entire system.The resulting modified phase space potential is

V (k) =

λ2L if k = 0

1

4− 2 cos(kx)− 2 cos(ky)if k 6= 0

. (2.30)

The modification does the bare minimum of lifting the self-interaction divergencein Eq. (2.22) and makes the magnetic fluctuations uniform across the system in asense.

2.4.2 Model B

Model B allows B to have any combination of values at the lattice sites, which leadsto

V (k) =1

4− 2 cos(kx)− 2 cos(ky) + λ−2L

. (2.31)

This also corresponds to using a modified Poisson equation

(∇2 − λ−2L )V (r) = −2πδ(r), (2.32)

which means that λL fits the definition of a screening length.

10

Chapter 3

Kosterlitz-ThoulessTransition

This chapter introduces the basics of the Kosterlitz-Thouless transition. We beginwith a simple argument that shows why the transition should occur and its generalclassification. We then present some thermodynamic properties that will be of usein the data analysis as well as the major results of renormalization group theory.The last part of the chapter is dedicated to introducing a finite-size scaling analysisof the magnetic permeability that will be exploited to interpret the results.

3.1 Nature of the Transition

In the early theoretical study of phase transitions, their only known cause wasspontaneous symmetry breaking. To this day, the vast majority of known criticalbehaviors can be explained by such a mechanism. A system with dimensional-ity d ≤ 2 and short-range interactions, however, can not exhibit a continuousphase transition within this framework. This is because the continuous symmetriesthat they have simply cannot be spontaneously broken, as rigorously stated by theMermin-Wagner theorem [9]. The 2D Coulomb gas is an example of such a system.The net vorticity m averages to 0 even in the algebraically ordered phase and thereis thus no broken symmetry.

A different kind of phase transition can still be found when including the contribu-tion of vortex excitations. The transition takes the system from the low temperaturequasi-ordered phase to a high temperature disordered phase and is known as theKosterlitz-Thouless (KT) transition. Instead of relying on symmetry breaking, theKT transition is a direct cause of the topological nature of its available vortex con-figurations.

11

12 Chapter 3. Kosterlitz-Thouless Transition

A simple thermodynamic argument allows us to understand the essence of the KTtransition. The energy of a single vortex according to Eqs. (2.19) and (2.22) is

E1 = πJ0 ln(L/a). (3.1)

The number of places to insert such a vortex for a cutoff length scale a in a systemof size L is (L/a)2, so the entropy of a single vortex is

S1 = kB ln(L/a)2. (3.2)

The free energy E1 − TS1 is therefore positive only for temperatures below

TKT =πJ0

2kB. (3.3)

Above this critical temperature, it is favorable to have single vortices. On theother hand, vortex-antivortex pairs separated by a distance r have an energy cost2πJ0 ln(r/a). These pairs are favorable at any temperature if r is sufficiently small.The transition that occurs is therefore one that unbinds pairs of vortices as thetemperature exceeds a critical value.

3.2 Thermodynamics

There are a number of thermodynamic properties that showcase the critical behav-ior of the model, but we will here focus on a few that are of use to us in checkingthe agreement between simulation and theory. The basic quantities extracted fromsimulations are the system energy E = H and the net vorticity m =

∑r qr as well

as their powers and we therefore only consider derivatives of these.

The heat capacity per lattice point is defined by

CV =1

L2T 2(〈E2〉 − 〈E〉2). (3.4)

This generally has an infinite peak at the critical point in an unbounded system.For a bounded system, the peak has a finite value that is shifted away from thetemperature at which the transition occurs.

A key quantity to our investigation is the magnetic permeability. This effectivelymeasures the degree of magnetization the system can obtain in response to anapplied magnetic field. For the model at hand, it is given by

µV =L2

T(〈B2〉 − 〈B〉2). (3.5)

The magnetic field B is connected to m by the vortex quantization of Eq. (2.13).Since 〈m〉 = 0 always holds, we are left with

µV =Φ2

0

L2T〈m2〉. (3.6)

3.3. Renormalization Group 13

For completeness, it is worth also mentioning the dielectric response function ε,which can be defined in phase space as

ε−1(k) = 1− V (k)

L2T〈qkq−k〉, (3.7)

Because of the correlation function involved, ε cannot be calculated as efficientlyas any quantity related only to m and E, but it has the advantage of a clearlydefined discontinuity at the critical point. The so-called universal jump of the KTtransition is given by

ε−1(0)

T=

{4 if T = Tc−

0 if T = Tc+. (3.8)

Beyond the general idea used in this thesis, the universal jump is the main methodto detect the KT transition by measurements. Through a finite-size version of Eq.(3.8) with a logarithmic correction, Monte Carlo simulations estimate the criticaltemperature to Tc = 0.2115 · 2πJ0/kB , which is just below that of Eq. (3.3).

3.3 Renormalization Group

One of the main results in the theoretical study of the KT transition is the renor-malization group (RG) analysis of Kosterlitz [10]. Although we will not go intodetail here, the main idea is to integrate a renormalized form of superfluid stiffnessby gradually increasing the lower cut-off a with a scale factor b and hence excludingshort-range fluctuations.

The main variables involved are the superfluid stiffness

J =~2ρR2me

, (3.9)

where ρR is the fully renormalized superfluid areal density, and the vortex fugacity

ζ = e−ERc /(kBT ), (3.10)

where ERc is the renormalized core energy.

The final results of the analysis are the RG flow equations. In terms of the reducedvariables x = 1− πJ β/2 and y = 2πζ, the equations are to lowest order

dx

dl= 2y2 (3.11)

dy

dl= 2xy, (3.12)


where l = ln b. For a constant C depending on the initial conditions, the equationsobey the simple relation

x2 − y2 = C2. (3.13)

The RG flow is plotted from Eq. (3.13) in Fig. 3.1 with the flow direction indicatedby the arrows. The curves are mirrored around x = 0 and we will focus on x < 0,which is fulfilled for the systems explored by our simulations.

Figure 3.1. Flow diagram of the Kosterlitz RG equations.

For negative x, we have the presence of two distinct regions divided by a separatrixy = −x flowing to the critical point x = y = 0. The separatrix corresponds to C = 0and serves as the critical line of the model, where T = Tc. Below the line, C2 > 0and the RG flow ends at x = −C on the line of fixed points x < 0, y = 0. Since theflow tends to zero fugacity, this region corresponds to the low temperature phasewith T < Tc. The region above the critical line, where C2 < 0, must therefore corre-spond to T > Tc. Its flow continues through x = 0 on to diverge in positive x and y.

The critical point emerges from the RG treatment by taking x = −y to get thetemperature

TRKT =πJ

2kB(1 + y0), (3.14)

where y0 corresponds to the bare fugacity. By neglecting the small y0, the tem-perature agrees with that of Eq. (3.3), but with J instead of its bare counterpart J0.

3.4. Finite-Size Scaling 15

The solutions to Eqs. (3.11) and (3.12) below and on the critical line are

y(b) =

2C(b/b0)−2C

1− (b/b0)−4Cif T < Tc

1

2 ln(b/b0)if T = Tc

, (3.15)

where the scaling constant b0 depends on the initial values. Furthermore, theconstant C can be related for T < Tc to the fully renormalized superfluid stiffnessJR according to

C =πJR2kBT

− 1. (3.16)

3.4 Finite-Size Scaling

Since the net vorticity m averages to zero for all temperatures, it is not a mean-ingful property to be used in interpreting the simulation data. However, its secondpower m2 can easily be calculated and contains information about the system fluc-tuations. From Eq. (3.6), we also know that m2 is proportional to the magneticpermeability µV , which can be measured on a macroscopic scale and is therefore ofinterest in order to get a connection to experiments. This leads us to the questionof what form µV takes near the critical point for systems of finite size.

Let b be a scaling factor and suppose µV depends on x, y, λL and L. We knowthat λL and L scale as lengths and we have the scaling of x and y from RG theory.According to Eq. (3.5), µV itself scales as an area. When we rescale the system byb, we therefore end up with

µV (x, y, λ, L) = b−2µV

(x(b), y(b),

λLb,L

b

). (3.17)

We now choose L = b, which sets the effective system size to L′ = 1. We also take

λL = λL, (3.18)

by introducing a new dimensionless quantity λ. This eliminates two of the variablesinvolved in Eq. (3.17) and the resulting relation is

µV (x, y, λ, L) = L−2µV (x(L), y(L), λ, 1). (3.19)

We also have an explicit solution of y(L) in terms of x(L), so we are left with µVscaling only in relation to y. However, the fugacity tends to zero with the vortexflow for large scales when T ≤ Tc. This is not what we expect of µV ∼ 〈m2〉/L2 forany temperatures and y is therefore dangerously irrelevant.


To conclude the scaling argument we must proceed with systems slightly larger thanjust one vortex. When we allow two vortices, the partition function is Z = 1+ ζ+ ζand to first order we get

µV ∼ 〈B2〉 =1 · 0 + ζ · 12 + ζ · (−1)2

1 + 2ζ2≈ 2ζ ∼ y(L). (3.20)

Hence, it holds for large systems that µV ∼ L−2y when T ≤ Tc. By applying Eq.(3.6), we finally obtain the simple scaling relation m2 ∼ y.

3.4.1 Scaling Formulas

With the scaling of m2 established, we are now in the position to predict how themodels at hand act near the critical point. Equation (3.15) give us the expressions

1

m2= Al

1− (L/L0)−4C

(L/L0)−2Cif T < Tc (3.21)

1

m2= Ac ln(L/L0) if T = Tc, (3.22)

where Al and Ac are unknown proportionality constants and L0 is a scaling con-stant. The formulas obtained are well suited for comparison to the values of m2

sampled from simulations. We could in practice also relate the simulations to µV ,which is a more physical property, but this requires an additional factor of L−2 tobe added.

Equation (3.21) can be used to see the general agreement of simulations with theexpected small temperature behavior of the system and supplies the constant Cconnected to JR by Eq. (3.16). Since m2 has been inversed, the formula has thenice feature of roughly being a straight line in logarithmic scale of L near Tc (whereC is small). The distinct logarithmic form of Eq. (3.22) is also of great interest.Since it should only hold to good accuracy at precisely the critical point, it allowssimulations of multiple system sizes to pinpoint the location of Tc.

The main aim of this project is to explore how well these two formulas fit simulationsof the modified Coulomb gas models. If there is a good agreement, we also want tosee if it can be exploited to efficiently obtain values of Tc, C and L0.

Chapter 4

Simulation Method

We dedicate this chapter to the main aspects of our approach to the simulationof a non-neutral 2D Coulomb gas. The first section introduces the basics of theMonte Carlo method and the Metropolis-Hastings algorithm as well as the specificsfor our simulation program. We then detail the processing performed before andafter the simulations and conclude with some insight into the implementation ofthe programs.

Natural units are used in this chapter and throughout the chapters that follow,which means that ~ = kB = e = 1. The bare superfluid density is also assigned thevalue J0 = 1/(2π).

4.1 Monte Carlo Method

The naive approach to determine the properties of a physical system is to calculatethe contributions of all possible configurations. For a system with a large amountof coupled degrees of freedom, this is not feasible. A commonly used alternative isthat of Monte Carlo methods, which utilize random sampling to reduce the need forcomputations. The class of techniques we focus on here are based on importancesampling, which means that the samples are generated in a way that reflects howstrongly they contribute to the calculation of physical quantities.

For a system such as ours, the states Γ are connected to a distribution ρ(Γ) suchthat the expectation value of a quantity A(Γ) is

〈A〉 =

∫dΓρ(Γ)A(Γ). (4.1)

Instead of performing this integration, we can use ρ as a probability distribution togenerate a number of samples Γ(i), where i = 1, ..., n. These samples produce an

17

18 Chapter 4. Simulation Method

estimate

〈A〉n =1

n

n∑i=1

A(Γ(i)), (4.2)

which by the law of large numbers obeys limn→∞〈A〉n = 〈A〉.

By utilizing estimates as in Eq. (4.2), an efficient simulation method can be foundif we can create samples distributed according to ρ with a small number of compu-tations. One way of doing so is to set the samples in a Markov chain, defined byhaving the probability of generating a new sample only dependent on the previouslygenerated sample. We denote the transition probability from the state Γs to the

state Γt as πst and the probability of the state Γs after m steps as ρ(m)s (with some

given initial values of ρ(1)s ).

The rows and columns defined by the indices in πst and ρ(m)s give us a stochastic

matrix π with∑t πst = 1 and a set of probability distributions ρ(m) with

∑s ρ

(m)s =

1. According to the Markov chain recipe of transitions, the probability distributionchanges after one step according to ρ(m+1) = ρ(m)π. To get the correct distributionafter a long chain of samples, we must therefore require that

ρ = limm→∞

ρ(1)πm. (4.3)

In particular, the limiting distribution ρ must fulfill the eigenvalue equation ρπ = ρ,which in index form reads ∑

t

ρsπst = ρs. (4.4)

A sufficient condition for Eq. (4.4) to hold is that of detailed balance,

ρsπst = ρtπts. (4.5)

This has a number of solutions, but we will here focus on the most common one.

4.1.1 Metropolis-Hastings Algorithm

We can decompose the transitional probability as

πst = gstαst, (4.6)

where gst is the probability of proposing Γt to be the new state given that Γs is thecurrent state and αst is the corresponding probability of accepting the new state.The Metropolis choice of acceptance distribution [11] is

αst = min

(1,ρtρs

gstgts

), (4.7)

which clearly obeys the detailed balance condition of Eq. (4.5) for any choice of gand any given ρ.

4.1. Monte Carlo Method 19

If we take the acceptance of Eq. (4.7), one step in the algorithm can be summarizedas

1. Generate a state Γt at random according to gst, where Γs is the current state.

2. Generate a number r at random from a uniform distribution of numbersbetween 0 and 1.

3. Accept Γt as the current state if αst ≥ r.

4. Keep Γs as the current state if αst < r.

Before the process begins, the system must be initialized with some state. Thechoice does not affect the limit probability distribution, but can influence the rateof convergence. Furthermore, the states generated by the early steps of the Markovchain will not be distributed as wanted. We therefore need to discard the firstnwarm states as a warmup before generating the final nsamp states to be sampled.

4.1.2 Main Program

The simulation program made for the purposes of this thesis is based on theMetropolis-Hastings algorithm. The choices left free from the general recipe ofthe algorithm have been tailored to obtain a decent convergence to the limit prob-ability distribution and a low overall complexity. The algorithm is based on acommon choice for 2D Coulomb gases [12].

The simulation starts from a cold state, where the vorticity is null at all sites, andproceeds with nwarm + nsamp steps in total. For each step, two updates to thesystem are proposed:

1. A single lattice site x is selected at random (from a uniform distribution ofall possible lattice sites). The corresponding vorticity is changed at randomby ∆q = ±1.

2. A lattice site x and one of its four nearest neighbors y are selected at random.The vorticity at x is changed by +1 and the vorticity at y is changed by -1.

The transition probability g of the proposed state does not depend on the currentstate and is uniform in terms of both the lattice sites and the vorticity changes. Thefactor gst/gts of Eq. (4.7) therefore cancels and we are left with a quotient ρt/ρs ofthe state probability distribution ρ to determine the acceptance probability α. Forour system of particles, we have the Boltzmann factors

ρ(Γ) = e−βH(Γ), (4.8)

where β = 1/T is the inverse temperature and H(Γ) is the system Hamiltonian.Suppose ∆E is the energy change in the system as a result of the two proposed


updates. Equation (4.7) then tells us that the update is accepted if either ∆E ≤ 0or e−β∆E is larger than a uniformly generated random number between 0 and 1.

To calculate ∆E, we must add the contributions of both updates. We leave out thedetails of the calculations, but these values are found from Eq. (2.27). The energychange corresponding to update 1 is

∆E1 = ∆q(U(x) + V (0)∆q/2)− µ

L2(|q(x) + ∆q| − |q(x)|). (4.9)

For update 2, we have

∆E2 =U(x)− U(y) + V (0)− V (x− y)+

+µ

L2(|q(x) + 1| − |q(x)|+ |q(y)− 1| − |q(y)|)

. (4.10)

It should here be noted that the actual program used in the simulation included a

minor error. The implementation of Eq. (4.9) contained an extra term ∆qm+∆q/2L2 ,

related to a magnetic field as in Eq. (2.28). This roughly corresponds to a valueλL = 1 that has not been integrated to be part of the potential. In all but the verysmallest systems considered, the value of the the term is extremely negligible andshould not have any impact on the physics.

Once an update is accepted, the algorithm updates the properties of the system.The process includes changing all values of the function U of Eq. (2.26), whichrequires a loop through all L2 sites of the lattice. This corresponds to the bulk ofthe work from the entire algorithm and its contribution to the complexity of theprogram is O((nwarm + nsamp)L2). The full complexity depends on how often theupdates are performed, which has a complicated dependence on all input parame-ters.

The program saves the observables required to estimate the thermodynamics, whichis the system energy E and its power E2 as well as the net vorticity m and its pow-ers m2 and m4. These are sampled even if both update attempts fail, but only ifthe warmup phase has concluded. There is also an option to save the full systeminformation in terms of the vorticities of all sites after all steps of the algorithm,which allows us to make snapshots and animations of the entire configuration.

4.2 Pre- and Post-Processing

Beyond the main simulation algorithm described, the program consists of two ad-ditional parts. These can be run independently and take care of pre-processing theinput parameters and post-processing the output data.

4.3. Implementation 21

Calculating all values of the potential V given by Eqs. (2.24), (2.30) and (2.31)has a complexity of O(L4), which is relatively large for the system sizes considered.To save time on multiple calls for simulations with the same set of potential pa-rameters, V is calculated and saved ahead of the main simulation program. Thevalues are loaded when a simulation is initialized and used to update U whenevera proposed lattice update is accepted.

The statistics are calculated by specifying a number of bins nbins for error analysisand dividing the gathered samples from many separate simulation runs accordingly.The error bars of the thermodynamic properties are estimated according to a schemeknown as Jackknife resampling [13]. If we suppose a quantity A has a set of valuesAi with a mean 〈A〉, its Jackknife means are defined as

〈A〉i =1

nbins − 1

∑j 6=i

Aj . (4.11)

The Jackknife deviance of A is then given by

∆A =

√nbins − 1

nbins

∑i

(〈A〉i − 〈A〉)2. (4.12)

The advantage of using Jackknife resampling is that it assures that all thermody-namic properties derived from the simulation data are calculated correctly. It isnot the most efficient method available, but the complexity of the calculations arestill very low compared to the other parts of the program.

Some additional processing is required to fit the sampled inverse square net vorticity1/m2 to the scaling formulas of Eqs. (3.21) and (3.22). To begin with, we linearlyinterpolate the values of 1/m2 to get a better resolution in β than that gatheredfrom the simulations. At any of the values of β obtained, we then calculate thesum of squared errors scaled by the error estimates as

χ2(β) =∑L

((1

〈m2〉

)β, L

−(

1

m2

)scaling

β, L

)2

/

(∆

1

m2

)2

β, L

, (4.13)

where the sum runs over all system sizes L for which data is available. The minimaof these χ2 values indicate the critical point β = βc. The fitting process also suppliesvalues of the constants Al, Ac and C.

4.3 Implementation

The main program used for simulations, pre-processing and most of the post-processing was written in C11 and compiled with GCC 4.8.4. All calculations


made with this were run on the computational cluster Octopus at the departmentof Theoretical Physics at KTH using Intel Xeon ES-2620 processors. The final partsof the post-processing and all plotting was done with Python 3.5.2 and run locallyon a laptop. The curve fitting was handled by the scipy package of Python.

The C part of the code is written with a largely modular approach and has beenthoroughly unit tested. The results of each function were compared to those ofcalculations made by hand for a number of different scenarios. The program usedto collect the data presented in this thesis has been accepted by all such tests. Arudimentary comparison was also made between the major thermodynamic prop-erties obtained by this program and those of a program independently developedby Mats Wallin, which resulted in a satisfactory agreement.

The random number generator used is PCG [14] and seeding is done with the systemtime. The PCG generator was developed relatively recently and is therefore not aswidely tested as alternatives like the Mersenne Twister, but it does have advantageslike a very low complexity, an arbitrary period and simple implementation.

Chapter 5

Results

The results of the Coulomb gas simulations are presented here. A brief explanationis given for the parameters chosen and the rest of the chapter is devoted to com-paring the data obtained to the expected behavior of the system. The aim is toverify that models A and B of Eqs. (2.30) and (2.31) display a KT transition andto determine which model is most effective in deriving its properties.

The plots shown indicate the model used by its corresponding letter in the legend.If nothing is explicitly stated, the chemical potential µ is set to 0.

5.1 Choice of Parameters

The parameters involved in the simulation are the inverse temperature β, the sys-tem size L, the number of warmup samples nwarm, the number of gathered samplesnsamp, the number of bins for error analysis nbins, the potential parameter λ and thechemical potential µ. For the purposes of finding interesting physics and makingefficient simulations, certain combinations of parameters can be excluded. We herediscuss the choices that have been made and the reasons behind the decisions.

One aspect of choosing appropriate parameters is tuning the acceptance rate αsim

of the simulations, i.e. the fraction of proposed lattice updates that are accepted.If updates are more easily accepted, the variance of the samples is greater. On theother hand, the computation involved in generating vortex changes is wasted to agreater degree if a large amount of updates are rejected. For our system, the abso-lute majority of the work is updating the state is changed and we should thereforeexpect a low acceptance rate to be more efficient.

Figure 5.1 shows how αsim varies as a function of the inverse temperatures for afew fixed parameters and a system size of L = 128. The corresponding plot for a

23

24 Chapter 5. Results

function of the system size with β = 4.7 fixed is given in Fig. 5.2. The rates aregenerally higher for higher temperatures, which should indeed be the case since asystem that allows more unbound vortices will accept a greater amount of states.Smaller systems have higher acceptance rates, unless a negative chemical potentialis included. In terms of the parameter λ, smaller values generally lead to moreacceptance.

Figure 5.1. Acceptance rate for nwarm = 108, nsamp = 109 and L = 128.

As mentioned in Ch. 3.2, simulations using the universal jump put the critical pointaround βc = 4.728 and we mainly want to investigate β in the vicinity of this value.To see how the system acts for some displacement in the temperature, we havechosen a range of β = 4 to β = 5.5 as the main field of investigation.

It is beneficial to use as many and as high L as possible, but we are limited bythe algorithm complexity of L2. To get a logarithmic scale suitable to the scalingexpression, L was chosen in powers of 2 from L = 4 up to L = 128. A system sizeof 256 was used for one case of parameters and doing so required several days ofcomputation to get acceptable error estimations.

A sufficient number of warmup samples from a cold start was determined from trialand error. For all extreme cases of parameters, multiple simulations with differentnumbers of warmup runs were made and compared to check at which point no sys-tematic error could be observed. This was found to be about at about 10 million

5.1. Choice of Parameters 25

Figure 5.2. Acceptance rate for nwarm = 108, nsamp = 109 and β = 4.7.

updates and nwarm was set to 108 to be certain the limit distribution would alwaysbe attained.

As a base case, nsamp at 1010 does well enough for any parameters considered in or-der assure a hardly noticeable error margin. The size of the error should scale withthe system size, but we found this relation to be at most linear in L. We thereforedid not make the number of gathered samples scale with full sweeps of the system,which has the advantage of an increased accuracy for small system samples. Thelow acceptance rate for large β makes collecting enough samples more troublesomeand nsamp was therefore increased to 4 · 1010 for β above 5.

The data binning requires each bin of samples to be large enough for there to benext to no correlation between the series and small enough for noise to be can-celed. Accurately calculating correlation times is a difficult task for this system,but some basic trials were run for a small number of updates. The results indicatethat nbins = 100 is sufficiently small to remove correlation effects and to provideacceptable estimations for the errors bars.

With the other parameters set, some appropriate values for λ were determined byinitial guesses and experimentation. Small values of λ lead to cumbersomely largeacceptance rates and large values seem to shift the critical point from its expectedvalue, which leads to λ between about 0.1 and 0.8 to be most appropriate. The


influence of a chemical potential µ could be relevant, but has not been consideredin detail. The value chosen is µ = 0 in all cases, except for one that instead uses arelatively large value µ = −10.

5.2 Potential

The potential depends on the system size L and the scaled magnetic susceptibilityλ according to Eqs. (2.30) and (2.31). As a function of of one dimension x, thepotential V (x, 0) is plotted in Fig. 5.3 for L = 128 at a few different λ. Othervalues of L produce identical plots because of the periodicity.

Figure 5.3. System potential V (x, 0) in one dimension x.

It can be noted that the λ of model A merely shifts the potential by a constantvalue in all positions and relatively quickly converges to its form at λ = 0. Formodel B, the dominating impact of λ is the zero point of the potential in phasespace, where the potential has the same form as in model A. The two models aretherefore almost indistinguishable for λ above 0.6.

5.3 Configuration Examples

Figures 5.4, 5.5 and 5.6 show four sample configurations taken at nsamp = 1000updates after the warmup phase for model B with λ = 0.4 and µ = 0. Since the

5.3. Configuration Examples 27

configurations are chosen ergodically, each plot is representative of the parametersetup used to produce it. It can be noted that the relative frequency of boundpairs increases as the temperature does and it seems that a phase transition occurssomewhere between β = 4 and β = 5. In all of the cases, large vorticities are heavilysuppressed and almost all of the sites have vorticity 0 or ±1. Furthermore, thenumber of vortices N changes as expected with temperature and the net vorticitym is near 0 in all cases, despite this not being required by the model.

Figure 5.4. Sample configuration from model B after nsamp = 1000 updates withλ = 0.4, µ = 0, L = 64 and β = 3. Colors indicate the site vorticity.




5.4. Basic Quantities 29

5.4 Basic Quantities

In this section we show the main results of a simulation with λ = 0.4 and µ = 0using model B. There are notable differences with other parameters and with modelA, but these are most easily seen in the analysis of Ch. 5.5.

Figure 5.7 shows the system energy per lattice site, E/L2. Fewer configurationsare accepted at lower temperatures, which can be seen from the decreasing energy.There is also a convergence to a fixed curve for large systems, indicating the behaviorfor an infinite lattice.

Figure 5.7. Energy per lattice site of model B for λ = 0.4 and µ = 0.

The heat capacity of Eq. (3.4) is displayed in Fig. 5.8. Peaks can be seen for veryhigh temperatures, but they are shifted from the critical one due to the finite sys-tem size and should not be seen as an actual indicator of the critical point.

Although not directly essential to our analysis, we show the magnetic permeabilityof Eq. (3.6) in Fig. 5.9. In contrast to the squared net vorticity m2, to which µV isproportional, we note that the quantity is heavily shifted for different system sizes.


Figure 5.8. Heat capacity of model B for λ = 0.4 and µ = 0.

Figure 5.9. Magnetic permeability of model B for λ = 0.4 and µ = 0.

5.4. Basic Quantities 31

The inverse squared net vorticity 1/m2 is shown in Fig. 5.10 as a function of β. Itcan be seen that the curves nearly cross each other at some temperature slightlyhigher than the critical. Such a crossing would have been seen if y had not beendangerously irrelevant for the scaling argument of Ch. 3.4, because the result wouldhave been a value of m2 independent of L at the critical temperature.

Figure 5.10. Inverse squared net vorticity of model B for λ = 0.4 and µ = 0.

To make use of the scaling formulas, we also want to look at 1/m2 as a functionof L. Figure 5.11 displays the sampled values of 1/m2 in full lines along withfits obtained from the low temperature formula of Eq. (3.21) in dashed lines. Thecorresponding plot for fits to the critical temperature formula of Eq. (3.22) is shownin Fig. 5.12. In both cases, the data fits well in the regimes for which the formulasare meant and the critical point seems to be at about β = 4.7.


Figure 5.11. Inverse squared net vorticity of model B for λ = 0.4 and µ = 0. Fulllines indicate the simulation data and dashed lines show a fit to Eq. (3.21)

Figure 5.12. Inverse squared net vorticity of model B for λ = 0.4 and µ = 0. Fulllines indicate the simulation data and dashed lines show a fit to Eq. (3.22)

.

5.5. Critical Point and Scaling Properties 33

5.5 Critical Point and Scaling Properties

Finally, we present the full set of fits for 1/m2 to the scaling formulas of Eqs. (3.21)and (3.22) using both model A and B as well as a number of values for λ and µ.

Since the scaling formulas break down when L is small, it is beneficial to excludethe smallest systems up to some minimum L = Lmin. On the other hand, gettingrid of data points makes a good fit more difficult to obtain. The best results forthe data gathered are gained when L < 16 are omitted, leaving L = 16, L = 32,L = 64 and L = 128 left to be used.

Of the two formulas, the critical temperature form given by Eq. (3.21) appearsto be most suitable for accurately determining the critical point. The χ2 valuescalculated according to Eq. (4.13) for the fits are displayed in Fig. 5.13. For mostparameters a clear minimum can be seen for β spread around the expected valueof 4.728. The system for which µ = −10 shows a very indistinct minimal point,indicating that such a major chemical potential is not suitable.

Figure 5.13. χ2 for 1/m2 fit to L at linearly interpolated β for the critical tem-perature scaling formula of Eq. (3.22) with Lmin = 16.

The low temperature formula of Eq. (3.22) should show the critical point by thelowest β at which a good fit can be found, but the data is difficult to adapt withhigh precision since three unknowns L0, Al and C are included. The χ2 plot for


the formula is shown in Fig. 5.14. There is a great deal of noise, but the first majorminima seem to agree with those of Fig. 5.13. The following slumps in the curve aremostly located at the non-interpolated β, which indicates a sensitivity to the crudelinear interpolation performed. It can appear as if the low temperature values of χ2

are too large to indicate an agreement with the formula, but this can be explainedby the values of 1/m2 generally being greater at higher β.

Figure 5.14. χ2 for 1/m2 fit to L at linearly interpolated β for the low temperaturescaling formula of Eq. (3.21) with Lmin = 16.

From the low temperature fit we can also extract the temperature-dependent quan-tity C, which is related to the fully renormalized superfluid stiffness JR accordingto Eq. (3.16) when β > βc. The values of C obtained are plotted in Fig. 5.15. Wenote that C is set to 0 for β < βc and thereafter mostly agrees with the expectedform if JR has a value near its bare version J0 = 1/π.

Table 5.1 summarizes the critical inverse temperatures βc found from fitting to Eq.(3.21). The values are seen as part of a function βc(Lmin) for three options ofthe smallest system size. The table also supplies the fitting constants at βc(16),where L0 and Ac are taken from the critical temperature fit and Al is from the lowtemperature fit. The values of L0 from the low temperature fit are almost exactlythe same as those shown.

5.5. Critical Point and Scaling Properties 35

Figure 5.15. C from 1/m2 fitted to L at linearly interpolated β for the low tem-perature scaling formula.

Model λ µ βc(4) βc(8) βc(16) L0 Ac AlA 0.0 0 4.51 4.65 4.70 0.408 0.418 176A 0.2 0 4.59 4.69 4.71 0.0465 0.807 508A 0.4 0 4.61 4.70 4.71 0.0564 4.68 54.6A 0.6 0 4.61 4.69 4.68 0.0259 70.9 5000B 0.1 0 - 4.31 4.76 0.473 0.141 1.71B 0.2 0 4.55 4.75 4.76 0.262 0.583 21.5B 0.4 0 4.69 4.73 4.72 0.148 4.32 2150B 0.6 0 4.66 4.72 4.73 0.173 100 1540B 0.8 0 4.47 4.49 4.51 0 67.1 61.6B 0.4 -10 - - 4.61 0 0.0277 2.43

Table 5.1. Results from the data analysis for the simulations of models A andB at different λ and µ. The inverse temperature βc is seen as a function of thesmallest system size Lmin included in the fit. The constants L0, Al and Ac from thescaling formulas are provided for Lmin = 16. Dashed values indicate that a fit witha minimum could not be found.

A convergence towards the expected βc = 4.728 can be observed for all parameterconfigurations studied and the strength of this seems to be the greatest for modelB with λ in the vicinity of 0.4. Model B appears worse off for relatively small andlarge values of λ. Model A is indistinguishable from B for large λ, but does givebetter results when λ is small. The addition of a non-zero µ of this magnitude


makes good fits difficult to obtain and does not produce the correct critical point.

5.6 Efficiency

Beyond the capability of the models to find the critical properties, we can alsoconsider how efficient the simulations are to run. A simulation is efficient if itrequires few updates of the system and has a small relative error for 1/m2. Areasonable metric for efficiency is therefore

εsim =1/〈m2〉

∆(1/m2)

1

αsim. (5.1)

To see the overall trend, εsim is presented in Fig. 5.16 as a function of β whenaveraged over all L and in Fig. 5.17 as a function of L when averaged over allβ. We can see that the simulations are more efficient at low temperatures, whichagrees with the prediction made from only looking at the acceptance rate αsim. Interms of the system size, the efficiency reaches a peak at some value of L. Thelocation of this peak is shifted to smaller L for larger values of λ.

Figure 5.16. Efficiency of sampling 1/m2 averaged over L between 4 and 128.

Since simulations need to be run for a number of different temperatures and sys-tem sizes for a full picture of the transition, we can view the overall efficiency byaveraging over all β and L. The result of this is shown in Tab. 5.2. Model A is

5.6. Efficiency 37

Figure 5.17. Efficiency of sampling of 1/m2 averaged over β between 4 and 5.5.

slightly more efficient than model B and there is a value of λ in the vicinity of 0.2that leads to the greatest efficiency. Including a negative chemical potential seemsto have potential for improving the efficiency of the simulation.

Model λ µ εsim/104

A 0.0 0 15.8A 0.2 0 24.5A 0.4 0 22.9A 0.6 0 15.6B 0.1 0 15.8B 0.2 0 23.4B 0.4 0 21.1B 0.6 0 14.5B 0.8 0 4.7B 0.4 -10 49.8

Table 5.2. Efficiency of sampling 1/m2 averaged over L between 4 and 128 as wellas β between 4 and 5.5.

38

Chapter 6

Summary, Conclusions andOutlook

In this thesis we have studied the KT transition through two models of a non-charged 2D Coulomb gas with a potential modified by a parameter λ. This wasdone by Monte Carlo simulations on a L×L lattice at various inverse temperaturesβ. The data obtained has been analyzed by fitting the inverse squared net vorticity1/m2 to two scaling formulas valid at temperatures below and on the critical point.Minimizing the fitting errors allowed us to find the critical temperature and thefitting constants provided quantities like the length scale L0 of the RG analysis andthe fully renormalized superfluid stiffness JR.

The approach to KT simulations studied appears well suited to locating the criti-cal point in a direct manner by simply using a Metropolis-Hastings algorithm andsampling the values of m2. A good fit to the scaling formulas was found for mostvalues of λ with the chemical potential µ set to 0. By using values of L between16 and 128, we found the critical point to settle around βc ≈ 4.7, which is close tothe value βc = 4.728 known from other simulations.

The study suggests how to construct an efficient model for establishing βc from vor-ticity fluctuations. Model A allows for the field fluctuations to vary independentlyacross the system with a susceptibility λL, whereas model B forces the fluctuationsto be uniform. We have found that model B with λ ≈ 0.4 is most suitable to accu-rately pinpoint the critical temperature, but that the simulations are more efficientto run with model A and λ ≈ 0.2.

For future work it would be interesting to pursue the method further in order todetermine the details of the transition in the new picture of charge fluctuations.In addition to the investigation presented here, it could be fruitful to consider

39

40 Chapter 6. Summary, Conclusions and Outlook

larger systems and different chemical potentials to get a better understanding ofthe parameter dependence. There is also a great number of additional transitionproperties that could be explored and it remains to be seen how well the simulationmethod performs in comparison to other procedures.

It would also be interesting to look for experiments studying charge fluctuations,which are expressed in superconductors as variations of the magnetic field. Theseare closely related to the magnetic permeability µV , on which our scaling methodis based. Since µV is a physically measurable quantity, it should be possible toestablish a close link between experiments and simulations.

Bibliography

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[2] J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transi-tions in two-dimensional systems, J. Phys. C 6, 1181 (1973).

[3] P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, andsuperfluid-superconducting films, Rev. Mod. Phys. 59, 1001 (1987).

[4] A. K. Geim and K. S. Novoselov, The rise of graphene, Nat. Mater. 6, 183(2007).

[5] A. Andersson and J. Lidmar, Scaling, finite size effects, and crossovers of theresistivity and current-voltage characteristics in two-dimensional superconduc-tors, Phys. Rev. B 87, 224506 (2013).

[6] V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh.Eksp. Teor. Fiz. 20, 1064 (1950).

[7] L. D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz. 7, 19(1937).

[8] V. L. Ginzburg, Some remarks on phase transitions of the 2nd kind and themicroscopic theory of ferroelectric materials, Fiz. Tverd. Tela 2, 2031 (1960).

[9] N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromag-netism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev.Lett. 17, 1133 (1966).

[10] J. M. Kosterlitz, The critical properties of the two-dimensional xy model, J.Phys. C 7, 1046 (1974).

[11] W. K. Hastings, Monte Carlo sampling methods using Markov chains and theirapplications, Biometrika 57, 97 (1970).

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42 BIBLIOGRAPHY

[12] J. Lidmar and M. Wallin, Monte Carlo simulation of a two-dimensional con-tinuum Coulomb gas, Phys. Rev. B 55, 552 (1997).

[13] J. W. Tukey, Bias and confidence in not quite large samples, Ann. Math. Stat.29, 614 (1958).

[14] M. E. O’Neill, PCG: A Family of Simple Fast Space-Efficient StatisticallyGood Algorithms for Random Number Generation, ACM Trans. Math. Soft.(submitted).

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New Scaling Method for the Kosterlitz-Thouless Transition