uni-bielefeld.de · 1 Introduction 1. Introduction Phasetransitionsoccurallovernature....

Bielefeld UniversityFaculty of Physics

Numerical Simulations and Field Theory on the Lattice

Finite size scaling functions of the three-dimensionalO(2) and Z(2) spin models

Supervisor & 1st Referee:Prof. Dr. Frithjof Karsch

Author:Marius Neumann

2nd Referee:Dr. Anirban Lahiri

October 11, 2019

Marius [email protected]

Finite size scaling functions of the three-dimensional O(2) and Z(2) spin models

Master’s thesis in PhysicsBielefeld University

Submitted on October 11, 2019

Contents

1. Introduction 1

2. Theory 22.1. Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1. First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2. Second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3. Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2. The cyclic group Z(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3. The orthogonal group O(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4. A Hamiltonian for the φ4 spin model . . . . . . . . . . . . . . . . . . . . . . 42.5. Universality classes and critical exponents . . . . . . . . . . . . . . . . . . . 42.6. Renormalization group theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.6.1. Finite size scaling functions with an external magnetic field . . . . . . 52.6.2. Finite size scaling functions without an external magnetic field . . . . 7

3. Simulation 93.1. Monte Carlo algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1. Metropolis update algorithm . . . . . . . . . . . . . . . . . . . . . . . 93.1.2. Swendsen-Wang algorithm . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3. Wolff algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2. Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3. Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1. Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2. Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.3. Delete-d-Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.4. Simple-delete-d-Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.5. Keep-d-Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.1. Asymptotic values for the Binder cumulant and Kiskis ratio . . . . . 16

3.5. Calculation of critical exponents and parameters . . . . . . . . . . . . . . . . 17

4. Results 194.1. Measuring the critical exponents and parameters in the XY-model . . . . . . 19

4.1.1. Fixing the magnetization scale . . . . . . . . . . . . . . . . . . . . . . 194.1.2. Fixing the temperature scale . . . . . . . . . . . . . . . . . . . . . . . 20

4.2. Spin models at non-vanishing external magnetic field . . . . . . . . . . . . . 224.2.1. The Ising model at nonzero magnetic field . . . . . . . . . . . . . . . 234.2.2. The XY-model at nonzero magnetic field . . . . . . . . . . . . . . . . 24

4.3. Spin models without an external magnetic field . . . . . . . . . . . . . . . . 264.3.1. The Ising model at vanishing magnetic field . . . . . . . . . . . . . . 284.3.2. The XY-model at vanishing magnetic field . . . . . . . . . . . . . . . 34

4.4. Comparison of ns=48 and ns=96 lattice data . . . . . . . . . . . . . . . . . 37

5. Summary 41

A. List of critical exponents and non-universal parameters 42

B. (E,φ) counting plots 43

C. Padé approximation for fχ for nonzero H 46

D. Fit parameters 47

1 Introduction

1. IntroductionPhase transitions occur all over nature. Like ice melts to water, water evaporates to steam,and steam ionizes to plasma, hadrons - the nuclear matter itself - can also “melt” into quarks.This process is called deconfinement. The phase transition of hadrons has been widely lookedat, as its understanding is crucial to the understanding of the underlying force of the nuclearmatter itself: the strong nuclear force, described by quantum chromodynamics (QCD).Investigating QCD via lattice techniques, several phase diagrams have been developed,

the most famous being the Columbia plot:

Figure 1: The Columbia plot: Location of regions of first order phase transitions and lines ofsecond order phase transitions in the plane of light and strange quark masses. Thecentral “crossover region” marks a region of no phase transitions. Source: [1]

The Columbia plot shows the light quark (up and down combined) mass on the x- andthe strange quark mass on the y-axis. Color coded is the type of phase transition thatis occurring at a particular light/strange mass combination. If a quark mass is infinite orjust very large, it is decoupled from the physical processes relevant for the thermodynamicproperties of a system. On the diagonal all three masses are equal.In the Columbia plot, some transitions lie in the same universality class as simpler Z(2)

or O(4) models, which can be investigated more easily and show the same behavior as QCDclose to their respective transition temperature.In this work, such models (Z(2) and O(2), respectively) have been investigated. O(2) was

chosen instead of O(4), since the staggered formulation of quarks only keeps a remnant ofthe O(4) chiral symmetry, which is O(2). We extracted those scaling functions in a way sothat they can be used to describe the QCD behavior at its phase transition as well. Thiswas done with and without an external magnetic field.

1

2 Theory

2. TheoryIn this chapter, we will first look at the different types of phase transitions, the simulatedsymmetry groups and the Hamiltonian. We also are going to introduce universality classesand critical exponents. After that, we will derive the finite size scaling functions with andwithout an external magnetic field.

2.1. Phase transitions

A phase transition characterizes a point in a system, at which some of the systems propertiesdrastically change. An especially looked at property is the order parameter, which is acommon observable that also measures the degree of order in the system. Usually it iszero in one and nonzero in the other phase, with zero indicating an unordered state. Wedistinguish between two main types of phase transitions, based on how drastic the orderparameter changes.

2.1.1. First order

First order phase transitions are characterized by the coexistence of two phases. At thetransition the energy density is discontinuous, which gives rise to a latent heat that isabsorbed by the system during the transition without changing its temperature. This causesa jump in the order parameter, thus being discontinuous at the transition temperature. Acommon example for a substance featuring a first order phase transition is water evaporatingor ice melting.

2.1.2. Second order

Second order phase transitions are characterized by an order parameter that is continuousas function of temperature. It does not show any discontinuity itself, but its first derivativedoes. A common example is the ferromagnetic transition.The models we are investigating in this work, representing Z(2) and O(2) symmetries,

both feature a second order transition and we will not see any divergences in the orderparameterM . The susceptibility, which is the derivative of the order parameter with respectto the parameter coupled to symmetry breaking term in the Hamiltonian of the system,does diverge. However, this happens only in an infinite system. In any finite system, thesusceptibility does not diverge. The way the susceptibility and other observables approachtheir infinite volume values is characteristic for second order phase transitions in a givenuniversality class. Analyzing this approach to the infinite volume (thermodynamic) limit isthe central topic of this work.

2

2 Theory 2.2 The cyclic group Z(N)

2.1.3. Crossover

A crossover is not canonically considered a phase transition. Although the systems propertiesradically change, this change is smooth and does not show any discontinuities in its orderparameter nor its derivatives, thus not fulfilling the definition of a phase transition, buthaving similar effects. An example is butter transitioning from solid to liquid state [18].

2.2. The cyclic group Z(N)

A cyclic group is a group that is generated by a single element. Thus we can understand itas a group of rotations along the same axis that leave an object invariant. Every finite cyclicgroup is isomorphic to the additive group Z(N), the integers modulo N .Elements of Z(2) now represent an object, that can be put into two different states; in

the Ising model an object of this kind is put on each lattice site and interpreted as a spin ofa particle, that can either be in the up or down state.Originally, the Ising model was formulated to investigate ferromagnetism; the spin would

represent electrons in a chunk of iron. Today, a bunch of other applications has been dis-covered for the Ising model: For example, the liquid-vapor transition in classical fluids canbe described by the Ising phase transition [20].Z(2) is also of relevance in the quark gluon plasma (QGP): There are some Z(2) transition

lines in the Columbia plot shown in Figure 1.

2.3. The orthogonal group O(N)

O(N) is the group of all rotations and reflections in an N -dimensional space. Thus, O(1)corresponds (i.e., is isomorphic) to Z(2), and O(2) features a circular structure. In our case,it represents a spin that can point in any (2D-)direction.An element of O(2) is described by two spin components, where as in the O(1)=Z(2) case

a single component is sufficient.A general O(N) model is referred to as the N -vector model; the N = 2 case is called the

XY-model.Quantum chromodynamics undergoes a chiral phase transition, which is believed to lie

in the O(4) universality class for two degenerate light-quark flavors. In lattice QCD, theWilson action has its chiral symmetry broken and only restored in the continuum limit. Thestaggered formulation on the other hand, keeps a part of the chiral symmetry even in a finitesized lattice. This remaining symmetry is believed to be in the O(2) universality class. Thus,comparisons of QCD with O(2) and O(4) models are made [8].If not directly in the Columbia plot, O(2) indeed does feature physical applications: For

example the superfluid transition of 4He belongs to its universality class [3].

3

2 Theory 2.4 A Hamiltonian for the φ4 spin model

2.4. A Hamiltonian for the φ4 spin modelThe simplest Hamiltonian for the XY-model (O(2)) with an external magnetic field can bewritten as:

βH = −J∑〈x,y〉

φxφy −H∑x

φx,1, (2.4.1)

with the coupling constant J that we also consider to be the inverse temperature 1/T , aspin vector φx = (φx,1, φx,2) at spatial position x arranged in an n-dimensional lattice witha spatial extent ns and an external magnetic field vector H. Here, φx is a two-componentvector of unit length. For H=0, H is invariant under symmetry transformations of the kindφx → φ′x = Oφx, where O ∈ O(2).A study of such a model is not only affected by statistical errors that arise due to the Monte

Carlo simulation, but also by systematical errors from correction to scaling, as introducedin Section 2.6. To reduce these systematical errors, we usually increase the lattice size L.This comes with a higher simulation cost, so we introduce a more elegant method:We try to eliminate the corrections to scaling by choosing another Hamiltonian, which is

in the same universality class as (2.4.1). This improved Hamiltonian [6]

βH = −J∑〈x,y〉

φxφy +∑x

[φ2x + λ

(φ2x − 1

)2]−H

∑x

φx,1 (2.4.2)

features a φ4 spin term with Hasenbusch’s λ-parameter.If a nonzero λ is set, we can allow the spins to vary in length. The λ-term then acts as a

penalty for spins not being normalized to unity, weighted by how much their length differsfrom 1. A well-chosen λ can reduce the effects of correction-to-scaling terms as we will showlater. Changing λ does not effect the critical exponents. For some subtleties, we refer toAppendix A.

2.5. Universality classes and critical exponentsClose to the critical temperature Tc, called the transition temperature of a phase transition,many systems can be described by scaling functions that have the same form with thesame critical exponents α, β, γ, ..., but different constants Tc, T0, H0, ... . Such systems areconsidered to be part of the same universality class. For example, take the susceptibility χand its critical exponent

−γ = limz→0

logχlog z ⇒ χ ∼ z−γ, (2.5.1)

where z is the scaling parameter for H=0 as it will be introduced in Section 2.6.1.

4

2 Theory 2.6 Renormalization group theory

Many critical exponents can be defined, but due to the scaling relations

dν = β(1 + δ) = 2− α, γ = β(δ − 1), νc = ν/βδ and ∆ = βδ, (2.5.2)

all exponents can be computed from any set of two exponents.Critical exponents are universal, i.e., they only depend on broader features instead of

the details of a system. This has been understood through Wilson’s renormalization groupanalysis ([21] and [22]), which gained him the Nobel Prize in physics in 1982.By figuring out the Z(2) and O(2) scaling functions, we hope to be able to make use of

them in fits to the related QCD phase transitions in the Columbia plot by their respectivecritical parameters, keeping our fit parameters and the critical exponents fixed.

2.6. Renormalization group theoryA general renormalization group scaling function fS with a free positive scaling factor b, twoscaling fields ut = ctt and uh = chH with positive exponents yt and yh, rescaled temperaturet, rescaled magnetic field h, inverse length l and an infinite number of irrelevant scaling fieldsu4 with negative exponents y4 looks like [9]:

fS (ut, uh, l, u4, ...) = b−dfS (bytut, byhuh, bl, by4u4, ...) , (2.6.1)

where fS denotes the scaling part of the free energy density f close to the critical point. lis not a scaling field itself, thus there is no ul. Instead, l is treated as correction to leadingscaling behavior and takes the role of an additional relevant scaling field [9].The free energy density is split into two parts

f = fS + fNS, (2.6.2)

with fNS denoting the combined non-singular, regular terms, which are small compared tofS in the scaling region. Since we want to look at scaling functions in the scaling region, wecan omit the regular terms. Later, we will see that the regular terms still contribute to ourmeasurements.

2.6.1. Deriving a finite size scaling function with an external magnetic field

In the case of H 6=0 (which simply gives H>0 since we could always rotate our system forH to be positive) we choose b = u

−1/yhh , such that

fS = ud/yhh fS

(uyt/yhh ut, 1, u−1/yh

h l, uy4/yhh u4, ...

). (2.6.3)

Here the u4 term may be another, subleading relevant term with y4 smaller than yt or yh, oran irrelevant term with y4 < 0 [9]. We ignore these contributions at present, but will comeback to the relevance of correction-to-scaling terms in our later discussion.

5


Since the second argument of fS became constant by carefully choosing b, and the fourthvariable and following are subleading and ignored at present, we can introduce a new scalingfunction Ψ , which only depends on two variables:

fS ≡ ud/yhh Ψ

(uyt/yhh ut, u

−1/yhh l

). (2.6.4)

The remaining scaling fields can now be substituted byH and t by using a Taylor expansionansatz for uh and ut, where we keep only the leading terms [9]:

fS = (chH)d/yh Ψ((chH)yt/yh (ctt) , (chH)−1/yh l

). (2.6.5)

The exponents can now be related to the critical exponents

yt = 1ν

and yh = 1νc

= ∆

νor ∆ = yh

yt, (2.6.6)

which yields

fS = (chH)νcd Ψ((chH)1/∆ (ctt) , (chH)−νc l

). (2.6.7)

Usually, the rescaled temperature variable t̄=ut=ctt= tTc/T0 =(T −Tc)/T0 and magneticfield variable h= uh = chH =H/H0 are used; also we can relate the inverse length scalingfield l=L0/L to L, the number of lattice points in a spatial direction. This gives

fS = hνcdΨ(h1/∆t̄, h−νcl

), (2.6.8)

where the arguments are now denoted by the temperature scaling variable zT and the volumescaling variable zL. Note, that in the literature zT is usually called z, but most papers onlydiscuss scaling either at vanishing or nonzero H and thus do not get in conflict with thescaling variable at H=0, which usually is also called z. This leads us to

fS = hνcdΨ (zT , zL) , with zT ≡ h1/∆t̄ and zL ≡ h−νcl. (2.6.9)

Using the scaling relations, the exponent simplifies to

fS = h1+1/δΨ (zT , zL) . (2.6.10)

We then introduce another function ff to make the following derivative easier:

fS = H0h1+1/δff (zT , zL) . (2.6.11)

The magnetization

M = −∂fS∂H

= − ∂

∂H

(H0h

1+1/δff (zT , zL))

(2.6.12)

6


is now defined as known from thermodynamics.If we plug ff in the equation above (2.6.12), we get

M = −H0

[H−1

0 h1/δ(

1 + 1δ

)ff +H−1

0 h1/δ−1/∆−1∆t̄∂ff∂zT

+H−10 h1/δ−1/νc(−νc)l

∂ff∂zL

](2.6.13)

= −h1/δ(

1 + 1δ

)ff + h1/δ−1/∆ 1

∆t̄∂ff∂zT

+ h1/δ−1/νcνcl∂ff∂zL

(2.6.14)

= −h1/δ(

1 + 1δ

)ff + h1/δ zT

∆

∂ff∂zT

+ h1/δνczL∂ff∂zL

(2.6.15)

= h1/δ[−(

1 + 1δ

)ff + zT

∆

∂ff∂zT

+ νczL∂ff∂zL

](2.6.16)

= h1/δfG (zT , zL) , (2.6.17)

which gives us a relation between the magnetization and its scaling function.The second derivative of fS then gives the (longitudinal) susceptibility

χ = ∂M

∂H= ∂

∂H

(h1/δfG (zT , zL)

). (2.6.18)

This calculation is performed in analogy to the former case and yields:

χ = h1/δ−1

H0

[1δfG(zT , zL)− zT

∆

∂fG(zT , zL)∂zT

− νczL∂fG(zT , zL)

∂zL

](2.6.19)

= h1/δ−1

H0fχ(zT , zL) . (2.6.20)

This also gives us a definition of the finite size scaling function of the susceptibility fχ inrelation to the magnetizations finite size scaling function fG,

fχ(zT , zL) = 1δfG(zT , zL)− zT

∆

∂fG(zT , zL)∂zT


∂zL, (2.6.21)

as well as a relation between fG and ff :

fG (zT , zL) = −(

1 + 1δ

)ff + zT

∆

∂ff∂zT

+ νczL∂ff∂zL

. (2.6.22)

2.6.2. Deriving finite size scaling functions without an external magnetic field

For H = 0 we can set b= l−1 immediately, but have to keep H until after the derivatives.This allows us to fix the third argument of

f̄S = ldf̄S (bytut, byhuh, 1, by4u4, ...) , (2.6.23)

7


where we use a bar to indicate the scaling functions at vanishing magnetic field.Again, we can define a function

Ψ̄(l−ytut, l

−yhuh)≡ l−dfS (2.6.24)

of only two variables, since the irrelevant fields are subleading and may be neglected for ourcurrent discussion again, where we set the exponents and scaling fields just as before:

f̄S ≡ ldΨ̄(l−1/ν t̄, l−1/νch

). (2.6.25)

At this point, we can set the scaling variable z already and introduce a temporary pseudoscaling variable zh for convenience of notation:

f̄S = ldΨ̄ (z, zh) , z ≡ l−1/ν t̄, zh ≡ l−1/νch. (2.6.26)

Before setting H=0, we have to take the first and second derivate of f̄S to get

M = −∂f̄S∂H

∣∣∣∣∣∣H=0

= − ld−1/νc

H0

∂Ψ̄

∂zh

∣∣∣∣∣∣H=0

≡ ld−1/νc f̄G (z) = lβ/ν f̄G (z) (2.6.27)

and

χ = −∂2f̄S∂H2

∣∣∣∣∣∣H=0

= − ld−2/νc

H20

∂2Ψ̄

∂z2h

∣∣∣∣∣∣H=0

≡ ld−2/νc f̄χ (z) = l−γ/ν f̄χ (z) , (2.6.28)

where we have already chosen sensible f̄G and f̄χ, which only depend on z, and then use thescaling relations on the exponents.H0 and H2

0 were absorbed in the definitions of f̄G and f̄χ respectively, in analogy to as itwas also done for the definition of ff in (2.6.11) in the section before. Those functions havealso been stated by Binder [2].Since H is set to zero before f̄G and f̄χ are defined, we cannot extract a relation between

f̄G and f̄χ for the H=0 case, as it would have to include a derivative with respect to H.

8

3 Simulation

3. SimulationIn this chapter, we will introduce some common algorithms used for the simulation of sta-tistical models (Monte Carlo algorithms) and for the analysis of data generated in suchsimulations. In particular, we will discuss Jackknife methods; we then define our observ-ables and show some subtleties in a particular one, the magnetization. We are also going topresent details of the simulation and show how to extract critical parameters and exponents,if those are unknown.

3.1. Monte Carlo algorithms

In statistical physics, Monte Carlo algorithms are used to create a sequence of randomsamples of a non-trivial distribution. Such a sequence is called a Markov chain. On a lattice,the algorithm is performed by changing a part of the lattice’s state and then deciding,whether to accept or discard the change. This yields a chain of configurations.A bunch of acceptance/rejection step methods have been developed to simulate a spin

lattice. We will present the most basic method first and then introduce the advanced algo-rithms.

3.1.1. Metropolis update algorithm

The original and most simple Markov chain Monte Carlo algorithm is the Metropolis algo-rithm, which was first described by Metropolis and Rosenbluth in 1953 [14].When applied to the classic Ising model, the Monte Carlo updates are performed by single

spin flips. A spin is flipped with a flip probability of

p = exp {min [0,−J∆H]} , (3.1.1)

where ∆H is the energy that would be gained by the flip. This means that a proposed spinflip always gets accepted, if the total energy is decreased by the process or stays constant,and occurs with a probability of exp(−J∆H) if the energy increases. The occasional energyincrease lets the algorithm avoid local minima.In a sweep through the lattice, all sites are updated individually, enabling the configuration

to change completely within a single sweep. Nevertheless, one sweep usually does not changemuch, and still has the configuration left highly correlated with the previous one. This makesthe classic algorithm not very efficient.This most simple Monte Carlo algorithm needs to be modified slightly for O(N > 1) or

Z(N > 2) models, of course, since a decision has to be made here, into which state a spinshould be flipped. Suitable generalizations of this local update have been developed.

9

3 Simulation 3.2 Simulation details

3.1.2. Swendsen-Wang algorithm

Instead of sweeping through the lattice and updating each site individually, Swendsen andWang [19] introduced a method of forming clusters of sites with identical state. To form acluster, two sites with the same state are considered bonded with a probability of

p = 1− e−J . (3.1.2)

A set of bonded states is then called a cluster. For each cluster, all spins are in one of theN (in the case of Z(N)) states. Each cluster is then given a new, random spin. Note, thatthis algorithm allows the lattice to change significantly in a single step; there even is a smallpossibility to get from any state to any other. Close to the critical point, this allows for lesscorrelation and shorter run times.This algorithm works for Z(2), but not O(2), since it has no finite number of possible

states, so no cluster would be formed.

3.1.3. Wolff algorithm

The Swendsen-Wang algorithm still has the issue of only working for Z(N) models, since itrequires spins σx to be in an identical state. To circumvent this, the spin flips are redefinedby a reflection vector r with respect to the hyperplane orthogonal to r. Then, a reflection

R(r)σx = σx − 2 (σx · r) r (3.1.3)

is used instead of a traditional spin flip.A bond is formed between neighboring sites with

p = 1− exp {min [0,−J (r · σx) (r · σy)]} . (3.1.4)

The algorithm now requires a random site and reflection vector to be chosen; clustersare then formed like in the Swendsen-Wang case. For the Ising or any other Z(N) case ingeneral, the Wolff algorithm becomes Swendsen-Wang again [23].

3.2. Simulation detailsWe study here statistical models in three spatial dimension based on the Hamiltonian givenin (2.4.2). The calculations were performed with a cluster update Fortran program writtenby Jürgen Engels. As input parameters, we varied the scaling variables and determined Jaccording to

J =(T0zL

−1/ν + Tc)−1

(3.2.1)

10

3 Simulation 3.3 Jackknife

in the H=0 case and J and H according to

J =(T0zT (H/H0)βδ + Tc

)−1, (3.2.2)

H = H0 (zLL/L0)−1/νc (3.2.3)

in the H 6=0 case.Note that the simulation of course only depends on J , H and λ, the other parameters are

only required to calculate those in a way, that it looks like the scaling variables itself werevaried, and can be plotted in a nice looking way. This can be done easily for Z(2) as H0, T0and Tc have been determined previously. For O(2) we will determine H0 and T0 in Section4.1. Find the used input parameters in Tables 1, 2 and 3; already including the in Sections4.1.1 and 4.1.2 measured H0 and T0 for O(2). The values are given as exact, since the Tablespresent the parameters that were used, not the best measurements of those.For the number of measurements made per z in the case of H=0 and for each combination

of zT and zL in the case of H 6=0 we refer to Table 4.

3.3. Jackknife

The on line measurements made on the Markov chain have not only to be averaged as in thecase of E andM , but also the more complex observables χ, K2 and B4 have to be considered.For those cases, several techniques have been developed to extract statistical errors for theseobservables.Those methods are necessary since the measurements made on a Markov chain are usually

correlated, and thus we are required to use specialized statistical tools for their analysis, i.e.,the Jackknife.

3.3.1. Autocorrelation

In a Markov chain, every configuration and thus observable Oi depends on the one simulatedbefore Oi+1. This leads to an autocorrelation [16]

CO(t) = 〈OiOi+t〉 − 〈Oi〉〈Oi+t〉 (3.3.1)

in the measured data, making the use of standard statistical tools impossible or at least letsthem underestimate the statistical errors.To deal with this, we discard intermediate configurations (by always performing multiple

Monte Carlo updates at once) and use improved statistical methods as the Bootstrap orJackknife.

11


H=0 ν z nsZ(2) 0.6304 [-19,19,1], [-10,10,0.2] 16,24,36,48,96O(2) 0.6723 [-19,19,1], [-10,10,0.2] 16,24,36,48,96

Table 1: Simulation parameters used only at H = 0. The scaling variable is given in intervals ofthe form [min,max, step]. Two intervals given indicate a coarse and a fine interval insideit.

H 6=0 H0 β δ νc zT zL nsZ(2) 0.815 0.3258 4.805 0.40269 [-11,8,1], [-2,2,0.5] [0.1,1.3,0.1] 48,96O(2) 1.1941 0.3490 4.69 0.4107 [-11,8,1], [-2,2,0.5] [0.1,1.3,0.1] 96

Table 2: Simulation parameters used only at H 6=0. For detailed sources, see Appendix A. H0 isyet to be measured for O(2), but already presented here.

H=0 and H 6=0 Tc T0 λ L0Z(2) 2.66598 0.8044 1.1 1O(2) 1.9641 0.8462 2.1 1

Table 3: Simulation parameters used at both H=0 and H 6=0. T0 is yet to be measured for O(2).

16 24 36 48 96H=0 500 500 200 300 188

Z(2) H 6=0, zL in 0.1-0.7 0 0 0 100 23-32H 6=0, zL in 0.8-1.3 0 0 0 100 45

H=0 830 245 72 30 16O(2) H 6=0, zL in 0.1-0.7 0 0 0 0 6

H 6=0, zL in 0.8-1.3 0 0 0 0 6

Table 4: Measurements made per data point (measured in thousands) for each symmetry groupand external field. Due to node failures and other errors, the numbers given are in somecases only approximations.

12


3.3.2. Bootstrap

Bootstrapping works by removing randomly chosen configurations from the dataset and cal-culating the observable on the remaining ones. If M random datasets are drawn (repetitionsare allowed), then any observable O can be estimated via

〈O〉 = 1M

M∑m=1

Om, (3.3.2)

while we extract its variance according to:

σ2O = 1

M

M∑m=1

(Om − 〈O〉)2 . (3.3.3)

Bootstrapping is considered a good tool to get rid of correlations between different ob-servables. On the other hand, it cannot be used to reduce the effects of autocorrelation [16].Thus, we used other statistical methods in this work.

3.3.3. Delete-d-Jackknife

In a classic delete-d-Jackknife algorithm, the variance

σ2 = C∑

blocks

(θ̂b − θ̂

)2(3.3.4)

is estimated by removing every subset of d data points from the whole set of n values andcalculating the block observable θ̂b on each remaining subset of n−d points individually.Those subsets are called blocks. σ2 is then computed by looking at the block observable θ̂bcompared to the same observable on the whole dataset θ̂.The constant

C = B

(D −B)N = n− dd(nd

) (3.3.5)

is determined by the size of a block B, size of the whole data set D and number of blocksN . A block consists of the data points that remain, after d have been removed. Note thatin this case D=n.Since all possible combinations have to be considered, this method is very expensive and

we have to introduce cheaper ones.

3.3.4. Simple-delete-d-Jackknife

It has proven to be more useful to only consider d consecutive points instead of each possiblesubset of d points for deletion. This simply cuts the dataset in B=D/N subsets and requires

13

3 Simulation 3.4 Observables

way less computation time. We also get another normalization

C = 1N (N − 1) . (3.3.6)

In exchange for the saved computation time, we usually estimate larger statistical errorscompared to the delete-d-Jackknife, since the number of blocks is reduced drastically.

3.3.5. Keep-d-Jackknife

Instead of averaging over the remaining subsets, one can use the usually left out subsets toperform the calculation. This gives identical results as in the simple-delete-d-Jackknife, butshould be a little bit faster, since the sums are shorter. Again,

C = N − 1N

(3.3.7)

changes. The keep-d-Jackknife was used in this work with usually ten blocks each.

3.4. ObservablesThe energy density

E = − 1V

∑〈x,y〉

φxφy (3.4.1)

on a lattice is given by each spins interaction with its nearest neighbors.We are also interested in a ratio of higher moments of the energy

E3 =

⟨(E − 〈E〉)3

⟩⟨(E − 〈E〉)2

⟩3/2 , (3.4.2)

since it serves as a scaling observable for H=0.The magnetization

M =⟨

1V

∑x

φx

⟩= 〈φ〉 (3.4.3)

on a lattice with spins φx is defined by counting how many spins point in the same directionand subtract the others for Z(2), while for O(N) we have to consider the spin componentparallel to the external field only, and we also introduced the spin’s lattice average φ =V −1∑

x φx.

14


In this work, we are going to look at four different cases: Z(2) and O(2), with and withoutan external magnetic field, respectively. In each of these cases, the magnetization is definedslightly differently:For the simplest case Z(2) and H 6=0, M was already defined above (3.4.3).In the case of Z(2) and H=0, the expectation value of the magnetization is zero (〈φ〉=0),

so we make the approximationM≈〈|φ|〉, which holds for very small H and/or L and returnsnonzero values of M .For O(2), the spins have two components. For H 6= 0 we only take the first components

of the spin’s magnetization, but for H = 0 we need to take the absolute value of the wholemagnetization instead.We present all different magnetizations in Table 5.

M Z(2) O(2)H 6=0 〈φ〉〈φ1〉H=0 〈|φ|〉

⟨√φ2

1 + φ22

⟩Table 5: Different definitions of M depending on the symmetry group and external field. φ1 and

φ2 are the lattice average’s first and second components in the case of O(2), calculatedby averaging over the spin components individually.

The magnetic susceptibility

χ = ∂M

∂H= V 〈φ2 −M2〉 (3.4.4)

is the second moment of the magnetization and also an interesting scaling observable to lookat. It depends on the different definitions ofM , as mentioned before. Note that the meaningof φ2 also depends on the spin group; we show all different susceptibilities in Table 6.

χ/V Z(2) O(2)H 6=0

⟨φ2 − 〈φ〉2

⟩ ⟨φ2

1 − 〈φ1〉2⟩

H=0⟨φ2 − 〈|φ|〉2

⟩ ⟨φ2

1 + φ22 −

⟨√φ2

1 + φ22

⟩2⟩

Table 6: Different definitions of χ, according to Table 5 and (3.4.4)

We are also interested in ratios of higher moments of the energy and magnetization, sincethey serve as scaling observables for H=0. The Binder cumulant

B4 =

⟨(φ− 〈φ〉)4

⟩⟨(φ− 〈φ〉)2

⟩2 = 〈φ4〉〈φ2〉2

(3.4.5)

15


is defined similar to E3, but expectation values that include odd powers of φ vanish.The Kiskis ratio

K2 =χ/V

〈|φ|〉2= 〈φ2〉〈|φ|〉2

− 1 (3.4.6)

looks similar to B4, but is defined differently via the susceptibility.The interpretation of φ2 in the Binder cumulant and Kiskis ratio also changes with the

symmetry group, so we obtain the ratios as shown in Table 7.

H=0 Z(2) O(2)K2

〈φ2〉〈|φ|〉2 − 1 〈φ2

1+φ22〉⟨√

φ21+φ2

2

⟩2 − 1

B4〈φ4〉〈φ2〉2

⟨(φ2

1+φ22)2⟩

〈φ21+φ2

2〉2

Table 7: Different definitions of K2 and B4 for H=0.

3.4.1. Asymptotic values for the Binder cumulant and Kiskis ratio

To get the Binder cumulant’s asymptotics for large z, which means at large T , we can assumeM to be in thermal equilibrium and measure its expectation value [17]

〈f(x)〉 = 1√2π

∫ ∞−∞

f(x)e−x2

2 dx (3.4.7)

like any observable f(x) in quantum mechanics via an integral over a Gaussian distribution.By knowing some variants of the Gauss integral

1√2π

∫ ∞−∞

x2ne−x22 dx = (−2)n dn

dαnα− 1

2

∣∣∣∣∣α=1

(3.4.8)

and1

2π

∫ ∞−∞

∫ ∞−∞

(x2 + y2

)ne−

x22 −

y22 dxdy = 2nn! (3.4.9)

or calculating them via the Feynman-trick, we can give the asymptotic values for O(2) andZ(2).The only difference present between

BZ(2)4 (z →∞) = 〈x4〉

〈x2〉2= 3

12 = 3 (3.4.10)

(3.4.11)

16

3 Simulation 3.5 Calculation of critical exponents and parameters

and

BO(2)4 (z →∞) = 〈(x

2 + y2)2〉〈x2 + y2〉2

= 822 = 2 (3.4.12)

that matters for this case is the different definitions of M .This also works for the Kiskis ratio, where we need some further Gaussian integrals

1√2π

∫ ∞−∞|x|e−

x22 dx =

√2π

(3.4.13)

and1

2π

∫ ∞−∞

∫ ∞−∞

√x2 + y2e−

x22 −

y22 dxdy =

√π

2 , (3.4.14)

which yields

KZ(2)2 (z →∞) = 〈x4〉

〈x2〉2− 1 = 1√

2/π2 − 1 = π

2 − 1 (3.4.15)

for Z(2) and

KO(2)2 (z →∞) = 〈x2 + y2〉⟨√

x2 + y2⟩2 − 1 = 2√

π/22 − 1 = 4

π− 1 (3.4.16)

for O(2).

3.5. Calculation of critical exponents and parametersAlthough we use the parameters given in the Tables 1, 2 and 3 above, we will demonstratehow β and δ could be calculated, if needed, and actually calculate T0 and H0.By simulating at H=0 and T <Tc, we can get T0 by fitting

M(T .Tc, H=0) = B (Tc − T )β [1 + b1 (Tc − T )ων + b2 (Tc − T )] , (3.5.1)with T0 = B−1/β, (3.5.2)

with ω being the correction-to-scaling exponent and b1 being the correction to scaling am-plitudes.H0 can be obtained from calculating

M(T =Tc, H) = dcH1/δ[1 + d1

cHωνc], (3.5.3)

with H0 = d−δc (3.5.4)

17

3 Simulation 3.5 Calculation of critical exponents and parameters

at T =Tc, with d1c again being the correction-to-scaling amplitude.

If a well-suited λ was chosen, the regular terms do not contribute (b1 =b2 =d1c =0), so the

magnetization simplifies in the two cases to

M(T .Tc, H=0) =(Tc − TT0

)β(3.5.5)

and

M(T =Tc, H) =(H

H0

)1/δ. (3.5.6)

Alternatively we may use the asymptotic behavior of the order parameter scaling function,fG, at large negative values of zT and its volume dependence at zT = 0 to extract thenormalization constants T0 and H0, respectively. I.e., we demand

limzL→0

fG(0, zL) = 1, (3.5.7)

to set H0 via zL, and then

limzL→0

fG(zT , zL) = |zT |β, for zT � 0 (3.5.8)

to fix T0 via zT .The remaining critical exponents can be calculated via the scaling relations from any pair

of two exponents.

18

4 Results

4. Results

To discuss our results, we first need to calculate the critical exponents. Afterwards, welook at the scaling functions and ratios for vanishing and nonzero H, while discussing theirasymptotics. We will also show (E, φ)-density plots and some extracted values of the func-tions.In the end, we will compare ns=96 lattices with ns=48 to show that Hasenbusch’s λ-term

actually reduces the required lattice size by suppressing corrections to scaling.

4.1. Measuring the critical exponents and parameters in the XY-model

To be able to look at the finite size scaling functions, we have to figure out the criticalexponents and parameters first. This was done for O(2) only, since H0 and T0 were unknownhere, but not for Z(2).

4.1.1. Fixing the magnetization scale

The first constants we obtain are the magnetizations normalization H0 and δ. Since

M(T =Tc, H) = H−1/δ0 H1/δ (4.1.1)

holds, there only are a few measurements to be made at Tc for small H and the fit via (4.1.1)gives the desired parameters.Since we know δ from the literature, we can fix it and only fit the magnetizations nor-

malization. We expect to obtain a more accurate value for H0 by fixing δ. Using the sameδ as Engels [7] also allows us to produce easier to compare data. From this method weget H0 = 1.3942(26) via the fit shown in Figure 2. This fit returns a reduced chisquare ofχ2/n.d.f. = 9.70811, which already indicates that this method might still have some issuesand that corrections-to-scaling need to be considered here.If we would use that H0 for our analysis of the nonzero H scaling functions, we will notice,

that the normalization obtained here is inconsistent with the nonzero H scaling function’snormalization. We suspect, that regular contributions would have been needed to be takeninto account in the fit before to get better constants, even though, we used Hasenbusch’sλ-term. Thus, we use the nonzero H data directly to figure out a better constant. This isdone by knowing that fG(zT = 0, zL = 0) ≡ 1 (and also fχ(zT = 0, zL = 0) = 1

δ). So we plot

H−1/δ0 fG(zT =0, zLH−νc0 ) and fix H0 so that fG converges to unity for small zL, as it is done

in Figure 3. Via this method, we obtain H0 =1.1941(40), which is the value we are going touse in our further analysis.

19

4 Results 4.1 Measuring the critical exponents and parameters in the XY-model

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.0005 0.001 0.0015 0.002 0.0025

H

M

96120J=Jc

Figure 2: M versus H for small magnetic fields at the critical temperature fitted via (4.1.1) toextract H0. Different lattice sizes were chosen to show that in the given region finitesize effects do not play a role. Only the ns=120 data was part of the fit.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1 1.2

zLH0-νc

H0-1/δ fG(0,zLH0

-νc)H0

-1/δ

Figure 3: H−1/δ0 fG =H−1/δM versus zLH−νc0 =H−νc l. Both axis do not depend on H0, which

allows us to extract the normalization by fitting the plateau at zLH−νc0 ∈ [0.2, 0.4].

4.1.2. Fixing the temperature scale

Knowing H0 and δ, we can extract the temperature normalization T0 and β by simulatingfor T >Tc and H 6=0. Since we know that

M(T .Tc, H=0) = T−β0 (Tc − T )β , (4.1.2)

we can extrapolate the data to H=0 and fit these values to get the desired constants.This can be done for different values of λ, so we can see the impact of the corrections-to-

scaling here, like in Figure 4. If one has to include contributions arising from corrections-to-scaling, (4.1.2) looks like

M(T . Tc, H=0) = T−β0 (Tc − T )β + b1 (Tc − T )ων+β + b2 (Tc − T )β+1 . (4.1.3)

By fitting M(T,H = 0) with contributions arising from corrections-to-scaling and thenplotting without them like in Figure 4 for λ=106, we can see that those contributions must

20

4 Results 4.1 Measuring the critical exponents and parameters in the XY-model

be taken in account here, but not in the case of λ=2.1, since fitting without corrections-to-scaling already works perfectly well here.λ = 106 was chosen as an approximation to the Ising model limit, which is reached by

setting the λ-term to infinity.Since we know β from the literature [7], we can fix it and only fit the temperatures

normalization. From this method we get T0 =0.9165(44).

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.01 0.02 0.03 0.04 0.05

H1/2

M7296

120J=Jc

J=0.51J=0.515

J=0.52J=0.54J=0.56J=0.58J=0.60J=0.62J=0.65

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

T-Tc

M(T,H=0)

λ=106

λ=2.1B(Tc-T)β

Bλ=2.1(Tc-T)β

B((Tc-T)β + b1(Tc-T)ων+β + b2(Tc-T)β+1)

Figure 4: Left: M extrapolated (M(H) = M(T,H = 0) + c1√H + c2H) to H = 0 for different

J and ns = 120. Right: extrapolated values from different λ fitted with corrections toscaling (λ=106) and without (λ=2.1).

Again, this normalization is inconsistent with the nonzero H scaling function’s normal-ization, so we use that data again to find a better T0. In this case, we determine T0 bydemanding that |zT |−βfG(zT , zL = 0) is converging to unity for small zT . So again, we fitT−β0 to the plateau and receive T0 = 0.8462(61) via this method, which again is the value touse further.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-12 -10 -8 -6 -4 -2 0 2 4 6 8

zTT0

T0-β|zT|

-β fG(zTT0,zL=0.2)T0

-β

Figure 5: T−β0 |zT |−βfG =h1/∆|T−Tc|−βfG versus zTT0 =h1/∆(T−Tc). Both axis do not dependon T0, which allows us to extract the normalization by fitting the plateau at zTT0 ∈[−10,−4]. We took zL=0.2 as an approximation for zL→0.

21

4 Results 4.2 Spin models at non-vanishing external magnetic field

4.2. Spin models at non-vanishing external magnetic fieldTo start, we tried to fit the measured scaling functions via a polynomial approach

fG(zT , zL) = fG(zT , 0) +N∑n=0

M∑m=1

anmznT z

mL , (4.2.1)

with

fG(zT , 0) =4∑i=0

biziT +

8∑i=5

b+i z

iT +

7∑i=5

b−i ziT , (4.2.2)

the extrapolation and fit function for nonzero H [9].With the relation between fG and fχ (2.6.21), we can derive a function for

fχ(zT , zL) = 1δfG(zT , zL)− 1

∆

∑i=1

ibiziT −

N∑n=0

M∑m=1

anm

(n

∆+ νcm

)znT z

mL , (4.2.3)

which depends on the same fit parameters.With this relation, we can perform combined fits of both scaling functions. Those try to

minimize the cumulated χ2 of both fG and fχ with the same set of parameters. This can bedone in gnuplot by the multi-fit mechanics, where we introduce a third variable in both fitfunctions and introduce a combined fit function

fC(zT , zL, s) =

fG(zT , zL), if s = 0fχ(zT , zL), if s = 1

. (4.2.4)

Nevertheless, this approach only works close to Tc, especially for small zT and large zL offχ, the functions tend to have wiggles, larger then the expected errors.To resolve this issue, we again try a Padé approximation for the susceptibility

fG(zT , zL) =∑Nn=0

∑Mm=0 anmz

nT z

mL∑I

i=0∑Jj=0 bijz

iT z

jL

, with b00 = 1 (4.2.5)

instead of a polynomial.This gives

fχ(zT , zL) =(anmznT zmL )(( i

∆+ jνc)bijziT z

jL)− (( n

∆+mνc − 1

δ)anmznT zmL )(bijziT z

jL)

(bijziT zjL)2

, (4.2.6)

with sum convention for improved readability (for full derivation, refer to Appendix C).The disadvantage of this method is that instead of two parameters, which we have to guess

(N and M), there are now four. This makes it impossible (read: very expensive) to just try

22


every combination of N , M , I and J . So what we do to get these fits is at first to perform asingle polynomial (the Padé approximation’s nominator) fit to fG, then a full Padé fit withthe polynomials fit parameters as starting values to fG and finally use those fit parametersto perform a multi-Padé-fit of fG and fχ.While fitting, we expect the denominator to not contribute for m= 1, 2, since zL - rep-

resenting the inverse volume - should contribute with z3L in leading order. z0

L is requiredhere, since it represents the nonzero H extrapolation. We also can fix a00 =1, since fG wasnormalized that way.In general, we would expect the nominator polynomial to be larger than the denominator’s,

since the simple polynomial fit was already working quite well.We also expect the fit parameters to stay below a certain threshold; if some of them turn

out to be very large (�10), we would consider the fit to be bad. The appearance of large fitparameters is often paired with an oscillating minus sign between two neighboring parameters(in the same polynomial or with the same indices in nominator and denominator): This oftenis a sign that the parameters mainly work to compensate each other and not describe thescaling function. Thus, those fits are discarded.

4.2.1. The Ising model at nonzero magnetic field

For Z(2), the best fit we could produce is shown in Figures 6 and 7, while Figure 8 providesa magnified view of the region around the critical point at zL=0.Looking at the fG data in Figure 6, we see the results from different zL almost parallel

at the smallest zT , and then start merging around Tc, thus being unified for zT > 6. Thefunction appears to be monotonically decreasing in both scaling variables in the whole rangewe measured.On the other hand fχ shown in Figure 7 appears to have a maximum around Tc for small

zL, which moves to smaller zT with larger zL. This lets the function spread widely for smallzT , while again unifying at large zT .The combined Padé fit worked well, but we could not leave out contributions from j= 1

and 2 in the nominator and denominator. We had to keep j=1 in both polynomials. Also,we had to choose approximately equally sized nominator and denominator; the parametersthis fit yields on the other hand, lay in a well-behaved region. Only three of them are slightlyabove 10.0. Results for zL=0.1, 0.3, 0.4, 0.5 were included in the fit, but are not plotted sincethey are too close to zL=0.2. If we imagine those to be plotted as well, we can see trivially,that the zero H extrapolation works and is already reached by zL=0.2.We consider the region around zT =0 to be the most interesting and thus have increased

the point density in zT in there and include a closer look at the interval zT ∈ [−2, 2] as wellfor both scaling functions in Figure 8.For fG, the estimated errors are small compared to fχ, so the multi-fit reaches the data

points very well and within the errors. The zL→ 0 limit for zT = 0 is known to be 1, sincethe scaling function was normalized to this value.

23


0

0.5

1

1.5

2

-10 -5 0 5

zT

fG(zT,zL)V=963

Z(2)

zL=1.3zL=1.2zL=1.1zL=1.0zL=0.9zL=0.8zL=0.7zL=0.6zL=0.2

Figure 6: fG for Z(2) at nonzero H. A multi-fit of both datasets was performed using a Padéansatz and (4.2.3). The marked point at 1/δ is known for the extrapolation for zL=0.

For fχ, the zL→ 0 limit at zT = 0 is normalized to 1/δ instead and the fit functions doesnot agree with the some of the measurements in the region of small zT and large zL.Although it is not shown, the Padé approximation also works for the zL between the drawn

lines as well.We summarize our fir parameters in Table 11.We can also perform a contour plot of all measurements made in the (E, φ)-plane, where

each pixel represents the number of configurations, that have a particular combination of Eand φ. In the literature, those contour plots indeed do show actual contours of the probabilitydensity, while our plots show the probability color coded. We show the (E, φ)-contour plotsin Appendix B, Figures 34 and 35; each plot was made at the maximum of fχ for each zL.

4.2.2. The XY-model at nonzero magnetic field

For O(2), fG looks quite similar to Z(2); the description of its general shape still applies. Infχ, we can see a difference instead: There are much smaller gaps between the data taken atlarge zL and we can plot data for all the measured zL, without the smallest being redundant.While we could have left out some of the smaller zL in fG, those are actually relevant in fχ.The same Padé fit was used as in the Z(2) case, with the exception that we could not fix

24


0

0.2

0.4

0.6

0.8

1

-10 -5 0 5

zT

fχ(zT,zL)

V=963

Z(2)


Figure 7: fχ for Z(2) at nonzero H. A multi-fit of both datasets was performed using a Padéansatz and (4.2.3). The marked point at unity is known for the extrapolation for zL=0.

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1 0 1 2

zT

fG(zT,zL)V=963

Z(2)


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2 -1 0 1 2

zT

fχ(zT,zL)

V=963

Z(2)


Figure 8: fG and fχ for Z(2) at nonzero H. The region around Tc was magnified in these versionsof Figures 6 and 7.

a00 =1 this time, and we had to rescale the scaling parameters to the improved normalizationsT0 and H0 first. a00 stayed close to unity when used as a fit parameter, but not being ableto set a00 = 1 might indicate that some normalization issues with H0 are still present, butsmall.

25

4 Results 4.3 Spin models without an external magnetic field

0

0.5

1

1.5

2

2.5

-10 -5 0 5

zT

fG(zT,zL)V=963

O(2)

zL=1.279zL=1.181zL=1.082zL=0.984zL=0.885zL=0.787zL=0.689zL=0.590zL=0.492zL=0.394zL=0.295zL=0.197

Figure 9: fG for O(2) at nonzero H. A multi-fit of both datasets was performed using a Padéansatz and (4.2.3). The marked point at 1/δ is known for the extrapolation for zL=0.

Since we used different normalizations during simulations then we use now, we had torescale zT and zL. This lead to the seemingly arbitrary zL we are showing in the Figures 9,10 and 11. During simulation, those had been set to zL ∈ [0.2, 1.3], like we did in the Isingcase. Again, we can see, that the multi-fit worked quite well; with a slight exception in fχfor large zL and small zT , where the curves are a little bit off. This was also the case for theZ(2) model.This time, we constrained the fit parameters to be no larger than 10.0, which resulted in

a nice looking set; only two parameters (a01 and b01) obviously cancel each other. Settingsuch constrains was not possible in Z(2), since our fits generated that way agreed way lesswith the data. We summarize our fit parameters in Table 12.

4.3. Spin models without an external magnetic fieldAfter dealing with H=0, we can now have a look on how those scaling functions behave inan external magnetic field.To look at scaling at a vanishing magnetic field, H was set to zero instead of making ex-

trapolations like before in Figure 4. For exact input parameters and amount of measurements

26


0

0.2

0.4

0.6

0.8

1

-10 -5 0 5

zT

fχ(zT,zL)

V=963

O(2)


Figure 10: fχ for O(2) at nonzero H. A multi-fit of both datasets was performed using a Padéansatz and (4.2.3). The marked point at unity is known for the extrapolation forzL=0.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-2 -1 0 1 2

zT

fG(zT,zL)V=963

O(2)


0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2 -1 0 1 2

zT

fχ(zT,zL)

V=963

O(2)

zL=1.279zL=1.181zL=1.082zL=0.984zL=0.885zL=0.787zL=0.689zL=0.590

zL=0.492zL=0.394zL=0.295zL=0.197

Figure 11: fG and fχ for O(2) at nonzeroH. The region around Tc was magnified in these versionsof Figures 9 and 10

made with the program, see Tables 4 and 10.

27


4.3.1. The Ising model at vanishing magnetic field

If we look at the magnetization M compared to its scaling function f̄G in Figure 12 first,we see how the scaling function unifies the individual magnetizations and that close to thecritical point scaling is very well fulfilled. We still notice the same monotonically decreasingbehavior as we did in Figure 6. Only for z <−15 we notice the scaling functions to differ.Since scaling holds, we can extract the functions value at zero by fitting each lattice sizeindividually via Padé approximations

f̄G(z) =∑Nn=0 anz

n∑Mm=0 bmz

m, with b0 = 1 (4.3.1)

and average their values at the critical temperature so we can also estimate errors for ourvalues given.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-20 -15 -10 -5 0 5 10 15 20

z

M(z,H=0)ns=16, 1σns=24, 1σns=36, 1σns=48, 1σns=96, 1σ

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

z

f‾ G(z,H=0)

f‾ G(z=0,H=0) = 0.8843

ns=16, 1σns=24, 1σns=36, 1σns=48, 1σns=96, 1σ

Figure 12: M (left) and f̄G (right) for Z(2) at H = 0. A perfect scaling can be seen here. 1σdenotes the uncertainty region in transparent colors.

For the susceptibility χ and its scaling function f̄χ, we can see well-defined maxima, andthe rescaling of the individual χs again works very well. In this case, we can already noticedeviations from scaling for positive z; only in z∈ [−5, 5], we cannot see regular contributions.In the plot of the susceptibility, χ was not corrected for the volume as its definition wouldrequire, since the curves fit better in the plot that way.The Kiskis ratio K2 is already a scaling function, since all different volumes fall onto each

other, as it can be seen in Figure 14. If we do not plot versus the scaling variable z, but thetemperature T instead, we can use this observable to extract the critical temperature Tc atthe intersection point, which is shown in Figure 15. That plot also includes a magnified viewof the intersection point. Furthermore, we can see by plotting this way, that the Kiskis ratiobecomes a step function in the infinite volume limit, as the different volumes increase steeperthe larger the volume. The large temperature behavior can be calculated by assuming thesystem to be in a thermalized state, see Section 3.4.1. The value of the intersection pointwas again extracted via Padé approximations.

28


0

0.002

0.004

0.006

0.008

0.01

0.012

-20 -15 -10 -5 0 5 10 15 20

z

χ/V(z,H=0)ns=16, 1σns=24, 1σns=36, 1σns=48, 1σns=96, 1σ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

-20 -15 -10 -5 0 5 10 15 20

z

f‾ χ(z,H=0)

f‾ χ(z=0,H=0) = 0.1868


Figure 13: 1V χ (left) and f̄χ (right) for Z(2) at H=0. A perfect scaling can be seen here.

0

0.1

0.2

0.3

0.4

0.5

0.6

-20 -15 -10 -5 0 5 10 15 20

z

K2(z,H=0)

K2(z=0,H=0)=0.2384(5)

K2(z→∞,H=0)=π/2-1


1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

z

B4(z,H=0)

B4(z=0,H=0)=1.6018(9)

B4(z→∞,H=0)=3


Figure 14: K2 versus z (left) and B4 vs z (right) for Z(2) at H=0. A perfect scaling can be seenin both cases, although the data gets visibly worse for ns=96. The marked points atz=0 are known from literature.

The Binder cumulant behaves similar to the Kiskis ratio and can be used to extract thecritical temperature via the intersection method as well [2], as can be seen in Figure 16. Italso behaves as a scaling function itself if plotted versus the scaling variable, as it was donein Figure 14. For both cumulants, the values given by Engels could be found, but slightlyoutside the estimated errors. The expected asymptotic behaviors on the other hand, arereached very well.For a list of all extracted values so far, refer to Table 8.E3 is also expected to be a scaling observable, but only shows good scaling in z∈ [−4, 0],

see Figure 17. For smaller z, there a slight volume dependence can be seen, while there is alarger dependence for positive z. Since there is still a z-dependence, we cannot simply takean average to extract the E3(z= 0). Instead, we get the values for the individual volumes,again by Padé approximations, and extrapolate those to z = 0 as it can be seen in Figure18. This can be done, since the volume dependence appears to be of the form E3∼1/ns, atleast in leading order.

29


0.2

0.25

0.3

2.662 2.666 2.67

0

0.1

0.2

0.3

0.4

0.5

0.6

2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85

T

K2(T,H=0)

K2(T=Tc,H=0) = 0.2384(5)

K2(T→∞,H=0)=π/2-1

K2(T=0,H=0)=0


Figure 15: K2 versus T for Z(2) at H = 0. A well-defined intersection point at Tc can be seenhere, although the data gets visibly worse for ns=96.

1.5

1.6

1.7

2.662 2.666 2.67

1

1.5

2

2.5

3

2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85

T

B4(T,H=0)

B4(T=Tc,H=0) = 1.6018(9)

B4(T→∞,H=0)=3

B4(T=0,H=0)=1


Figure 16: B4 versus T for Z(2) at H = 0. A well-defined intersection point at Tc can be seenhere, although the data gets visibly worse for ns=96.

From a comparison with the nonzero H case extrapolated to a vanishing magnetic field,we get the asymptotic behaviors of f̄G and f̄χ:

limL→∞

〈|φ|〉∣∣∣∣H=0

= limH→0

limL→∞

〈φ〉 . (4.3.2)

For nonzero H and t<0, L→∞ corresponds to z→−∞. Since it is known (for H=0) thatM= |t|β in the infinite volume limit and this is also true for the normalizations made while

30


ns f̄G(0) f̄χ(0) K2(0) B4(0)16 0.8820 0.1872 0.23832 1.601124 0.8840 0.1872 0.23860 1.602236 0.8854 0.1867 0.23763 1.600848 0.8847 0.1864 0.23848 1.602096 0.8853 0.1867 0.23900 1.6029

average 0.8843(14) 0.1868(4) 0.2384(5) 1.6018(9)literature 0.240(5) [10] 1.604(1) [15]

Table 8: f̄G, f̄χ, K2 and B4 at z = 0 for different volumes and Z(2). Measured, averaged andliterature values are shown; but only K2(0) and B4(0) are of greater interest.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-20 -15 -10 -5 0 5 10 15 20

z

E3(z,H=0)

E3(z=0,H=0)=-0.3539


Figure 17: E3 for Z(2) at H = 0. The marked point at z = 0 is the one extrapolated. Scalingcan be seen, but there still is some kind of finite size scaling of the finite size scalingfunction left.

defining f̄G, we can conclude that

f̄G(z) = |z|β for z � 0. (4.3.3)

Thus, if we plot the scaling functions multiplied by their expected asymptotics, thoseshould converge to a constant. For f̄G, this works well, but for the largest lattice only, sinceit appears to be the only one (whether ns = 48 has a plateau can be argued about) thatdescribes a plateau for large z, as it can be seen in Figure 19.The f̄χ asymptotics again work well for negative z and the largest volume, but do not

converge yet at the measured z=−19. For f̄χ, if an approximation is done like

f̄χ(z) = C+/−|z|−γ, (4.3.4)

31


-0.5

-0.48

-0.46

-0.44

-0.42

-0.4

-0.38

-0.36

-0.34

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1/ns

E3(z=0,H=0,ns)

b=-0.3539(68)

E3(z=0,H=0,ns)=ans-1+b

Figure 18: Interpolated values of E3 at z=0 from Figure 17, that were extrapolated to ns→∞

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

4 6 8 10 12 14 16 18 20

z

|z|γ/2f‾ G(z,H=0)

|z|γ/2f‾ G(ns→∞,z→∞,H=0) = 1.1549


0.97

0.98

0.99

1

1.01

1.02

-20-15-10-5 0

z

|z|-βf‾ G(z,H=0)

f‾ G(ns→∞,z→∞,H=0)=|z|β


Figure 19: Asymptotic behaviors for f̄G for positive (left) and negative (right) z. Both show aplateau for the larger volumes.

the ratio of the amplitudes C+/C− is considered to be universal.While measuring C+ and C−, we have to consider a subtlety: (4.3.2) does not appear to

hold for the symmetry broken phase (i.e., z < 0).Considering the definition of the susceptibility

χ = V(〈φ2〉 − 〈|φ|〉2

), (4.3.5)

where our approximation that was used for the magnetization

M ≈ 〈|φ|〉 (4.3.6)

is also applied, we see that both terms contribute to χ. Usually, this is not the case, since ifthe approximation is not made, the 〈φ〉2-term vanishes. If the second term does contribute,f̄χ gains a contribution proportional to |z|γ, since f̄G ∼ |z|γ/2. That would lead to anadditional constant in C+/C−.To avoid that, only 〈φ2〉 is taken into account here, which leads us to the definition of

f̄χφ(zT , zL) = H0h1−1/δV 〈φ2〉, (4.3.7)

32


a scaling function similar to f̄χ, but missing the susceptibility’s < |φ|>2-component.To sum up, C+ is not extracted from f̄χ, but its equivalent f̄χφ , where everything was

done in the same way, except χφ = V <φ2> was used instead of χ= V (<φ2>−<|φ|>2).The constant shown in Figures 19, 20 and 21 were returned by fits of the form (4.3.4), whichgives the average value of the fit region |z|∈ [10, 19].

0.6

0.65

0.7

0.75

0.8

0.85

4 6 8 10 12 14 16 18 20

z

|z|γf‾ χ(z,H=0)


0.35

0.4

0.45

0.5

0.55

-20-15-10-5 0

z

|z|γf‾ χ(z,H=0)

|z|γf‾ χ(ns→∞,z→-∞,H=0)=0.4359


Figure 20: Asymptotic behaviors for f̄χ for positive (left) and negative (right) z. The latterconverges for ns=96, but not the smallest volumes.

The convergence of f̄χφ can be extracted better than f̄χ at positive z, so we can measurethe amplitude ratio C+/C− very well. We get C+ =2.0748(71) and C−=0.4359(8), and thusC+/C−= 4.736(18), which is within the standard error of the literature value of C+/C−=4.75(3) [4]. We can only use the largest lattice in these calculations, since, as it can beseen in Figures 20 and 21, the largest volume is the only one, which describes a plateau forlarge |z|, but not for positive z in |z|γ f̄χ. We might be able to improve this for the smallervolumes by also considering corrections-to-scaling here, which introduce a correction-to-scaling amplitude D and exponent ξ, and fit

f̄χ(z) = C+/−|z|−γ +D|z|ξ (4.3.8)

to the asymptotic instead, but we are already fine with the value we got by only using thelargest lattice.We can also show that the magnetization at H = 0 is symmetric: We again perform a

contour plot of all measurements made in the (E, φ)-plane, where each pixel represents thenumber of configurations, that have a particular combination of E and φ, as seen in Figure22. Both observables need to be shifted by their expectation value in order to show theirdistributions in the same ranges, so we define

∆E ≡ E − 〈E〉 and ∆φ ≡ φ− 〈φ〉 . (4.3.9)

This is only relevant for E, since again 〈φ〉=0 for H = 0.

33


1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

4 6 8 10 12 14 16 18 20

z

|z|γf‾ χϕ(z,H=0)

|z|γf‾ χϕ(ns→∞,z→∞,H=0)=2.0748


Figure 21: Asymptotic behaviors for f̄χφ for positive z. A plateau can be seen for ns=96 only.

What we see is a well-defined parabolic shape with two elongated maxima in its “arms”. Ifthe variances of the distributions are normalized to unity, the plots’ general shape is believedto be universal within a symmetry group; it is reached in the large volume limit [15].For density plots of all lattice sizes, refer to Appendix B.

-3

-2

-1

0

1

2

-4 -2 0 2 4 ns=16

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

50

100

150

200

250

Figure 22: Z(2) at H=0: density/counting plot of E and φ. A symmetry in φ can be seen.

4.3.2. The XY-model at vanishing magnetic field

The O(2) data was treated in the same way as Z(2) was. Like for H 6= 0, the data had tobe converted to the improved normalization constants. This leads to similar observations as

34


before; the general shape of the scaling functions stay the same. As it can be seen in Figure23, scaling holds, at least for z∈ [−5, 5] and worse compared to Z(2). Additionally, it can benoticed, that the position of the susceptibility’s maximum changed. For Z(2) it was placedleft of Tc; now it is right of it.

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

z

f‾ G(z,H=0)

f‾ G(z=0,H=0)=1.0798(23)


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

-20 -15 -10 -5 0 5 10 15 20

z

f‾ χ(z,H=0)

f‾ χ(z=0,H=0)=0.0933(5)


Figure 23: f̄G (left) and f̄χ (right) for O(2) at H=0. A scaling can be seen here.

K2 and B4 also work very similar to Z(2). The only noticeable changes are the differentasymptotics for large z; as explained in Section 3.4.1. Those different asymptotics are reachedvery well. Again, the intersection of the different volumes can be used to extract Tc. For allmeasured and literature values, refer to Table 9.

0

0.05

0.1

0.15

0.2

0.25

-20 -15 -10 -5 0 5 10 15 20

z

K2(z,H=0)

K2(z=0,H=0)=0.07999(67)

K2(z→∞,H=0)=4/π-1


0.07

0.08

0.09

1.961 1.964 1.967

0

0.05

0.1

0.15

0.2

0.25

1.7 1.8 1.9 2 2.1 2.2

T

K2(T,H=0)

K2(T=Tc,H=0) = 0.07999(67)

K2(T→∞,H=0)=4/π-1

K2(T=0,H=0)=0


Figure 24: K2 for O(2) at H=0. A scaling can be seen here, although the data gets visibly worsefor ns = 96. The marked points at z= 0 are known. Left: Scaling behavior is shownby plotting against z. Right: The intersection point is shown by plotting against T .

For E3, a similar extrapolation as in Z(2) had to be performed, as the scaling functionstill has a volume dependence; only a small region around z =−1 appears to show scalingbehavior. This scaling region appears to be even smaller than the region we saw for Z(2).The 1/ns-dependence of E3(z = 0, H = 0) is also worse compared to the Ising model, seeFigure 26.

35


1

1.2

1.4

1.6

1.8

2

-20 -15 -10 -5 0 5 10 15 20

z

B4(z,H=0)

B4(z=0,H=0)=1.2426(22)

B4(z→∞,H=0)=2


1.2

1.25

1.3

1.961 1.964 1.967

1

1.2

1.4

1.6

1.8

2

1.7 1.8 1.9 2 2.1 2.2

T

B4(T,H=0)

B4(T=Tc,H=0) = 1.2426(22)

B4(T→∞,H=0)=2

B4(T=0,H=0)=1


Figure 25: B4 for O(2) atH=0 against z (left) and T (right). A scaling can be seen here, althoughthe data gets visibly worse for ns=96. The marked points at z=0 are known.

ns f̄G(0) f̄χ(0) K2(0) B4(0)16 1.0811 0.0931 0.07941 1.240524 1.0820 0.0930 0.07965 1.241536 1.0799 0.0931 0.07993 1.242248 1.0800 0.0930 0.07981 1.242696 1.0761 0.0942 0.08114 1.2462

average 1.0798(23) 0.0933(5) 0.07999(67) 1.2426(22)literature 1.242(2) [5]

Table 9: f̄G, f̄χ, K2 and B4 at z = 0 for different volumes and O(2). Measured, averaged andliterature values are shown; a literature value for K2(0) could not be found.

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-20 -15 -10 -5 0 5 10 15 20

z

E3(z,H=0)

E3(ns→∞,z=0,H=0)=-0.0917ns=16, 1σns=24, 1σns=36, 1σns=48, 1σns=96, 1σ

-0.14

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1/ns

E3(z=0,H=0,ns)

b=-0.0917(50)

E3(z=0,H=0,ns)=ans-1+b

Figure 26: Left: E3 for O(2) at H=0. The marked point at z=0 is the one extrapolated. Scalingcan be seen in a region around the critical point at z=0, but outside that region thereis a clear volume dependence left. Right: Interpolated values of E3 at z=0, that wereextrapolated to ns→∞

36

4 Results 4.4 Comparison of ns=48 and ns=96 lattice data

While the ratios and E3 work fine for O(2), its asymptotics are a bit more tedious than forZ(2). For positive z, f̄G goes as expected with |z|γ/2 and reaches the asymptotic region atapproximately z=−7 for the largest lattice, just like for Z(2). For negative z, the functiondoes not scale with |z|−β in the usual region, as we would expect. Instead, by looking atFigure 27, we can see, that the curves are still decreasing at z = −19, but will probablyapproach unity for larger |z|. Also, the dip we saw at z ≈−2 for Z(2) in Figure 19 is notpresent in this case.We might be able to explain this unexpected slower convergence by the non trivial asymp-

totic behavior here due to the presence of Goldstone modes.

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

2 4 6 8 10 12 14 16 18 20

z

|z|γ/2f‾ G(z,H=0)

|z|γ/2f‾ G(ns→∞,z→∞,H=0) = 1.2998


1

1.02

1.04

1.06

1.08

1.1

-20-18-16-14-12-10-8-6-4

z

|z|-βf‾ G(z,H=0)

f‾ G(ns→∞,z→∞,H=0)=|z|β


Figure 27: |z|γ/2f̄G for positive z (left) converges nicely for large volumes, while |z|β f̄G for negativez (right) does not converge in the shown range, yet it looks like the function willapproach unity for larger |z|.

For f̄χ, we face similar problems compared to f̄G. For positive z, |z|γ f̄χ converges aroundz = 10 and even f̄χφ appears to follow that behavior. But in the negative z case, thereis no asymptotic behavior of |z|γ f̄χ to be seen for z < −10, which would be the expectedconvergence region. For O(2), f̄χφ is defined slightly different compared to Z(2), since thesusceptibility also changed:

f̄χφ(zT , zL) = H0h1−1/δV 〈φ2

1 + φ22〉, (4.3.10)

Again, the peak we saw at z≈−4 for Z(2) in Figure 20 is not present in this case.So for z>0, we get the asymptotics’ expected behavior, while for z<0 the scaling functions

feature a more complicated volume dependency. That dependency occurs probably due tothe in this case present Goldstone modes and will not be further looked at in this work, sinceit will not be resolved easily [13].

4.4. Comparison of ns=48 and ns=96 lattice dataTo show that we have used a well-chosen λ, we compare different lattice sizes with enabledmagnetic field for the same scaling variables zT and zL. Since we designed our scaling

37


0.25

0.3

0.35

0.4

0.45

0.5

0.55

0 5 10 15 20

z

|z|γf‾ χ(z,H=0)

|z|γf‾ χ(ns→∞,z→∞,H=0) = 0.4512


0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

-20-15-10-5 0

z

|z|γf‾ χ(z,H=0)


Figure 28: |z|γ f̄χ for positive (left, converging as expected for the largest volume) and negative z(right, not converging in the expected range).

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2 4 6 8 10 12 14 16 18 20

z

|z|γf‾ χϕ(z,H=0)

|z|γf‾ χϕ(ns→∞,z→∞,H=0) = 2.1306


Figure 29: |z|γ f̄χφ converges nicely for the largest lattice.

functions that way, fG and fχ should be independent of the lattice size, except for regularterms. We have done this for Z(2) only.If we have a look at the different volumes simply plotted on top of each other like in

Figure 30 for fG, we notice a very good agreement between the volumes. The scaling holdsin fact well enough, that the only reason for us to be able to see the different volumes arethe statistical errors.Looking at fχ in Figure 31, we can see larger statistical errors and thus some deviations

from the larger lattice, but no substantial differences between the curves. There is a slightdifference at zL=0.2 for large zT visible.To actually be able to see the limits of Hasenbusch’s λ-term, we need to have a look at

fringe cases for low zL. This is done in Figure 32: On the left side, we can see how muchthe scaling functions differ the larger |zT | is even at the smallest zL = 0.1. For zL> 0.3, wedo not see any differences any more, as presented exemplary for zL=0.4.We conclude that the λ-term allows us to safely use the ns = 48 lattice for zL > 0.3 and

zT ∈ [−10, 6].

38


0

0.5

1

1.5

2

2.5

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)

zT

fG(zT,zL) ns=96ns=48zL=1.3zL=1.2zL=1.1zL=1.0zL=0.9zL=0.8zL=0.7zL=0.6zL=0.4zL=0.2

Figure 30: Comparison of fG for ns=48 and 96 for all relevant zL.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)

zT

fχ(zT,zL)

ns=96ns=48zL=1.3zL=1.2zL=1.1zL=1.0zL=0.9zL=0.8zL=0.7zL=0.6zL=0.4zL=0.2

Figure 31: Comparison of fχ for ns=48 and 96 for all relevant zL.

39


0

0.5

1

1.5

2

2.5

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)zL=0.1

zT

fG(zT,zL) ns=96ns=48

0

0.5

1

1.5

2

2.5

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)zL=0.4

zT

fG(zT,zL) ns=96ns=48

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)zL=0.1

zT

fχ(zT,zL) ns=96ns=48

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-12 -10 -8 -6 -4 -2 0 2 4 6 8

Z(2)zL=0.4

zT

fχ(zT,zL) ns=96ns=48

Figure 32: Comparison of different volumes at zL = 0.1 and 0.4 for fG and fχ. At zL = 0.4, thedifferent volumes agree well, while for zL=0.1 the scaling region is left fast.

40

5 Summary

5. SummaryTo sum up, we have been able to successfully analyze the Z(2) scaling functions with andwithout an external magnetic field. We could check the correctness of our calculations bycomparing them with known constants like the amplitude ratio C+/C−, as well as the Kiskisratio and Binder cumulant at z=0. Furthermore, a set of fit parameters for the magnetiza-tion’s and susceptibility’s volume scaling function could be given, which will be applicableunder change of the critical constants for all phase transitions in the Z(2) universality class.That fit simultaneously describes both scaling functions, fG and fχ, at once.We also were able to deduce the effectiveness of Hasenbusch’s λ-term from there by com-

parison of two lattice sizes, that yielded almost identical results for a smaller and a largerlattice.Our analysis of O(2) turned out to be slightly less successful. While the scaling functions

for H= 0 could still be determined, their asymptotic behaviors for negative z could not beanalyzed trivially due to the considerable contributions of Goldstone modes here. Again, wewere able to confirm for O(2) the expectations for the asymptotic values of K2 and B4. Thescaling functions in the H 6=0 case could also be extracted nicely, as well as an appropriateset of fit parameters.Furthermore, a posteriori we would not recommend to use the method we have chosen

to vary the scaling variables when simulating the models, because it makes the calculationsdependent on an unnecessary amount of parameters, which is unfortunate if we want tooptimize these parameters later on.Since O(4) was already analyzed by Engels and Karsch [9], the next steps would now be

to adapt the used program to a combined O(2)×O(4) group, so we can start to look at itsscaling functions.

41

Appendix

A. List of critical exponents and non-universal parameters

Z(2), λ =∞ Z(2), λ = 1.1 O(2), λ =∞ O(2), λ = 2.1Jc 0.2215(12) (Heu) 0.3750966(4) EZ 0.454165(4) EO 0.5019 HaTc 4.515(25) Heu 2.666980 (EZ) 2.20184 (EO) 1.9641 HaT0 0.8044(4) EZ 1.18(2) EO 0.8462(61) mH0 0.8150(56) EZ 1.11(1) EO 1.1941(40) mα 0.1088 (EZ) -0.0169 (EO)β 0.3258(14) EZ 0.3490 EOγ 1.2396 (EZ) 1.3192 EOδ 4.8048 (EZ) 4.7798 EOν 0.6304(13) EZ 0.6723(3) (EO)νc 0.4027 (EZ) 0.4031 EO

Sourcesm measured in this workHa Hasenbusch paper [11]Heu Heuer paper [12]EZ Engels Z(2) paper [6]EO Engels O(2) paper [7](...) calculated from ...

Table 10: Critical parameters and exponents for Z(2) and O(2) spin models. Critical exponentsfor different λ are equal; for the critical parameters this is not the case. The Isingmodels critical amplitudes could not be found in the literature, but were not relevantin this work. The normalization constant L0 for the spatial length L has been set tounity in all our calculations.

42

Appendix

B. (E,φ) counting plots

-3

-2

-1

0

1

2

-4 -2 0 2 4 ns=16

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

50

100

150

200

250

-4

-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4 ns=24

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

20

40

60

80

100

120

140

160

180

-4

-3

-2

-1

0

1

2

3

-4 -2 0 2 4 ns=36

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

10

20

30

40

50

60

70

-5

-4

-3

-2

-1

0

1

2

3

-6 -4 -2 0 2 4 6 ns=48

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

20

40

60

80

100

120

140

160

180

-5

-4

-3

-2

-1

0

1

2

3

-6 -4 -2 0 2 4 6 ns=96

(E-<

E>

)/<Δ

E2>

1/2

ϕ/<Δϕ2>1/2

0

10

20

30

40

50

60

70

Figure 33: Z(2) at H=0: density/counting plots of E and φ. A symmetry in φ can be seen.

43

Appendix

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.1zT=2.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.2zT=2.0

EM

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.3zT=2.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.4zT=2.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.5zT=2.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.6zT=1.5

E

M

0

20

40

60

80

100

Figure 34: Z(2) at H 6= 0: density/counting plots of E and φ. A broken symmetry in φ can beseen.

44

Appendix

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.7zT=1.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.8zT=0.5

EM

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=0.9zT=-1.0

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=1.0zT=-5

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=1.1zT=-7

E

M

0

20

40

60

80

100

-0.23

-0.225

-0.22

-0.215

-0.21

-0.205

-0.2

-0.195

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 zL=1.2zT=-10

E

M

0

20

40

60

80

100

Figure 35: Z(2) at H 6=0: density/counting plots of E and φ, continued.

45

Appendix

C. Padé approximation for fχ for nonzero HTo calculate the dependency of fχ of the fit parameters, we only need to take

fG(z) =∑Nn=0 anz

n∑Mm=0 bmz

m, with b0 = 1 (C.0.1)

and derivate it according to

fχ(zT , zL) = 1δfG(zT , zL)− zT

∆

∂fG(zT , zL)∂zT


∂zL(C.0.2)

= 1δ

anmznT z

mL

bijziT zjL

− 1∆

(nanmznT zmL )(bijziT zjL)− (anmznT zmL )(ibijziT z

jL)

(bijziT zjL)2

(C.0.3)

− νc(manmznT zmL )(bijziT z

jL)− (anmznT zmL )(jbijziT z

jL)

(bijziT zjL)2

(C.0.4)

= 1δ

anmznT z

mL

bijziT zjL

−(( n∆

+mνc)anmznT zmL )(bijziT zjL)− (anmznT zmL )(( i

∆+ jνc)bijziT z

jL)

(bijziT zjL)2

(C.0.5)

= −((−1

δ+ n

∆+mνc)anmznT zmL )(bijziT z

jL)− (anmznT zmL )(( i

∆+ jνc)bijziT z

jL)

(bijziT zjL)2

(C.0.6)

=(anmznT zmL )(( i

∆+ jνc)bijziT z

jL)− (( n

∆+mνc − 1

δ)anmznT zmL )(bijziT z

jL)

(bijziT zjL)2

(C.0.7)

46

Appendix

D. Fit parameters

anm n=0 n=1 n=2 n=3 n=4 n=5m=0 1 -0.3824 0.0623 -0.0019 -0.0004 0.0000m=1 10.1488 -6.0189 2.0437 -0.3181 0.0251 -0.0007m=3 -2.6231 5.8153 -3.2750 0.7373 -0.0780 0.0027m=4 5.7220 -0.6734 -0.0306 -0.2251 0.0663 -0.0036m=5 6.0090 -2.0956 3.1275 -0.5079 -0.0042 0.0021m=6 -1.0352 0.2481 -1.2957 0.2590 -0.0060 -0.0006

bij i=0 i=1 i=2 i=3 i=4j=0 1 -0.1835 0.0381 -0.0003 -0.0003j=1 10.0528 -3.6797 1.4012 -0.1036 0.0135j=3 -0.3608 4.1362 -2.0846 0.2851 -0.0480j=4 -1.3297 2.0916 0.0861 -0.0072 0.0649j=5 4.8457 1.3196 2.6025 -0.2811 -0.0398j=6 13.7155 -1.3280 -1.0777 0.1276 0.0117

Table 11: (Multi-)fit parameters for Z(2) at nonzero H, with (4.2.5) and (4.2.6) as fit functions.

anm n=0 n=1 n=2 n=3 n=4 n=5m=0 0.9729 -0.5484 0.1913 -0.0296 0.0024 -0.0001m=1 9.9450 -5.4875 1.4460 -0.1936 0.0140 -0.0004m=3 1.6762 0.1655 0.4501 0.3721 -0.1106 0.0069m=4 2.0926 0.4595 -0.4628 -1.4710 0.4068 -0.0249m=5 0.9362 0.8652 0.5041 1.3442 -0.4061 0.0257m=6 0.4147 -0.7095 -0.0783 -0.3704 0.1245 -0.0082

bij i=0 i=1 i=2 i=3 i=4j=0 1 -0.3173 0.1413 -0.0093 0.0013j=1 9.8412 -2.9590 0.8817 -0.0540 0.0071j=3 1.8639 0.2467 0.7728 0.1684 -0.0784j=4 0.5610 1.1918 0.1452 -0.4914 0.2911j=5 2.3357 0.6636 -0.6135 0.3735 -0.2937j=6 5.9338 -0.3474 0.9693 -0.0514 0.0931

Table 12: (Multi-)fit parameters for O(2) at nonzero H, with (4.2.5) and (4.2.6) as fit functions.

47

References[1] Collaborative Research Center TransRegio 211. Funding proposal CRC-TR211. page 63,

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[2] Kurt Binder. Monte Carlo Simulation in Statistical Physics. Graduate Texts in Physics.Springer International Publishing, Cham, 6th ed. 2019 edition, 2019.

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[4] M. Caselle and M. Hasenbusch. Universal amplitude ratios in the 3D Ising model.Nuclear Physics B - Proceedings Supplements, 63:613 – 615, 1998. Proceedings of theXVth International Symposium on Lattice Field Theory.

[5] A. Cucchieri, J. Engels, S. Holtmann, T. Mendes, and T. Schulze. Universal amplituderatios from numerical studies of the three-dimensional O(2) model. J. Phys., A35:6517–6544, 2002.

[6] J. Engels, L. Fromme, and M. Seniuch. Numerical equation of state from an improvedthree-dimensional Ising model. Nucl. Phys., B655:277–299, 2003.

[7] J. Engels, S. Holtmann, T. Mendes, and T. Schulze. Equation of state and Goldstonemode effects of the three-dimensional O(2) model. Phys. Lett., B492:219–227, 2000.

[8] J. Engels, S. Holtmann, T. Mendes, and T. Schulze. Finite size scaling functions for 3-dO(4) and O(2) spin models and QCD. Phys. Lett., B514:299–308, 2001.

[9] J. Engels and F. Karsch. Finite size dependence of scaling functions of the three-dimensional O(4) model in an external field. Phys. Rev., D90:014501, 2014.

[10] J. Engels and T. Scheideler. The Calculation of critical amplitudes in SU(2) latticegauge theory. Nucl. Phys., B539:557–576, 1999.

[11] M. Hasenbusch and T. Török. High-precision monte carlo study of the 3D XY-universality class. Journal of Physics A: Mathematical and General, 32:6361–6371,Aug 1999.

[12] Hans-O. Heuer. Critical crossover phenomena in disordered Ising systems. Journal ofPhysics A: Mathematical and General, 26:L333–L339, Mar 1993.

[13] Frithjof Karsch. Personal conversation.

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[14] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H.Teller, and Edward Teller. Equation of state calculations by fast computing machines.J. Chem. Phys., 21:1087–1092, Jun 1953.

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[18] Alessandro Sciarra. The QCD phase diagram at purely imaginary chemical potentialfrom the lattice. page 40, 2016.

[19] Robert H. Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in montecarlo simulations. Phys. Rev. Lett., 58:86–88, Jan 1987.

[20] Wikipedia contributors. Scale invariance — Wikipedia, the free encyclope-dia. https://en.wikipedia.org/w/index.php?title=Scale_invariance&oldid=914282581, 2019. [Online; accessed 17-September-2019].

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49

https://en.wikipedia.org/w/index.php?title=Scale_invariance&oldid=914282581

https://en.wikipedia.org/w/index.php?title=Scale_invariance&oldid=914282581

AcknowledgementsFinally, I want to thank all the people, who made this thesis possible:

• Prof. Dr. Frithjof Karsch for this interesting topic, as well as his open door for anyquestions that came up in the last year,

• Dr. Anirban Lahiri for his help with Engels’ programs and grading this thesis,

• Dr. Jishnu Goswami for his explanations of the data,

• Ferdinand Jünemann and Marco Rozkwitalski for carefully reading the script,

• The FSZ Jülich and Bielefeld Cluster for their computation time,

• and finally my family without their support, this thesis would not have been possible.

Declaration of authorshipI have written this thesis without the assistance of any other person. The parts of my thesisthat were literally or according to its meaning taken out of others work including on linesources have been marked as such.Furthermore, I declare that this thesis has not been submitted to any other examination

office.

Bielefeld, October 11, 2019

Marius Neumann

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