geometricdependenceofinterfaceﬂuctuations€¦ · contents 1. introduction 1 2. theory of phase...

WESTFÄLISCHEWILHELMS-UNIVERSITÄTMÜNSTER

MasterThesis

Geometric dependence of interface fluctuations

August 25, 2015

Friedrich Bach

Supervisors: Prof. Dr. Gernot MünsterPD Dr. Jochen Heitger

living knowledgeWWU Münster

Corpora sunt porro partim primordia rerum,partim concilio quae constant principiorum.

Lucretius – De rerum natura

This version is a slightly corrected version of the thesis submitted on August 25, 2015.

Irvine, CA September 25, 2015

CONTENTS

1. Introduction 1

2. Theory of phase transitions and critical phenomena 52.1. Basic characteristics of phase transitions and statistical mechanics . . . . 52.2. Microscopic description of phase transitions . . . . . . . . . . . . . . . . . 82.3. Phenomenological theories for phase transition . . . . . . . . . . . . . . . 122.4. Critical behaviour in field theories . . . . . . . . . . . . . . . . . . . . . . 17

3. Interfaces in φ4 theory 233.1. Boundary conditions for the interface . . . . . . . . . . . . . . . . . . . . . 243.2. The kink solution obtained from Landau approximation . . . . . . . . . . 253.3. Perturbing the Hamiltonian with fluctuations . . . . . . . . . . . . . . . . 273.4. The energy splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4. Expressing the energy splitting by integrals 374.1. Zeta function regularization and heat kernel methods . . . . . . . . . . . . 384.2. Transforming the determinants into integrals . . . . . . . . . . . . . . . . 414.3. Calculating the interface tension . . . . . . . . . . . . . . . . . . . . . . . 50

5. The determinant of the Laplace operator 575.1. Kronecker limit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2. Sommerfeld-Watson transformation . . . . . . . . . . . . . . . . . . . . . . 645.3. Massless Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4. Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6. Conclusion and outlook 81

Appendices 83A. Form and energy of the kink-profile . . . . . . . . . . . . . . . . . . . . . . 85B. Properties of special functions . . . . . . . . . . . . . . . . . . . . . . . . . 89C. The error function integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 93D. Calculation of ζ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Bibliography 103

iii

NOMENCLATURE

Dφ Measure of the path integral, page 16

det’ Determinant omitting the zero mode, page 32

∆E Energy splitting, page 20

erf(x) Gaussian error function, page 43

F Free energy, page 6

γ Euler-Mascheroni constant, page 48

Γ(z) Γ-function, page 39

G(r) Correlation function, page 18

H[φ] Hamiltonian of the order parameter field, page 16

k Boltzmann constant, page 7

Kt(A) Heat kernel of the operator A, page 40

∆ Two-dimensional Laplacian: ∆ = −∂1∂1 − ∂2∂

2 , page 38

M Fluctuation operator, page 28

φ Order parameter, page 12

P Reflection operator, page 20

q Square of the nome q = e2πiτ , page 76

(·, ·) Scalar product, page 28

τ Modular ration., page 59

Θ(x) Step function, page 47

θ(z) Jacobi theta-function, page 50

T (a) Translation operator, page 29

TC Critical temperature, page 6

ζ(z) Riemann zeta-function, page 39

ζA(z) Spectral zeta-function of a heat kernel Kt(A), page 40

v

1INTRODUCTION

Interfaces are ubiquitous phenomena in nature, partitioning a system into regions withdisjoined thermodynamic properties like the particle densities in binary-liquid systemsor in liquid-gas transitions. This phase separation prohibits the system to establish aglobal equilibrium, so that the interface vanishes, is not sharp, but rather characterisedby a parting region, whose spatial length is indicated by the correlation length. In thisregion both phases are coexisting in the equilibrium featuring fluctuations permittingthe exchange of particles or molecules. Therefore, from the view of applied sciences,interfaces are significant in fields reaching from material sciences to neurophysiology.But they are also important from the point of view of theoretical physics, because theuniversal behaviour of phase transitions can be applied in many fields of physics. In allphysical systems, the geometrical structure as well as the boundary conditions play a role.The simplest and mostly employed geometrical structure in physics is a square, but itshows, that changing the geometric structure (to rectangular or circular structures) oftenenforces dramatic changes of researched quantities. Obviously, the geometric dependenceof interface fluctuations that shall be investigateded in this thesis is an evident task notexplicitly calculated yet.Theoretical models for continuous phase transitions in different dimensions have been

proposed for a lot of experimental evidences in particular fields of physics, like the liquid-vapour transition, the ferromagnetic transition or the superconducting transition offeringnew applications because of the universal behaviour of the principal models. The mostpromising model for phase transitions set up by elementary physical laws is the Isingmodel, where interface fluctuations have been studied since it’s first analytic solutionin two dimensions [Ons44]. However, as in most applications, the physical nature ischaracterised by three spatial dimensions. Consequently, the most interesting systemsfor investigating interfaces in everyday life are three dimensional, but until now thereis no analytic solution of the Ising model in three dimensions. In contrast, there aresuitable approximate solutions obtained from phenomenological considerations of phasetransitions, like the Landau theory. From the Landau theory, the first profile of an

1

INTRODUCTION

interface, the kink profile, could be derived by J.W. Cahn and J.E. Hilliard [CH58].As the Landau theory does not involve interface fluctuations, the Ginzburg-Landautheory [LG50] has been used for instance by V. Privman and M.E. Fisher [PF83] to offeran expression proposing the structural behaviour of the correlation length for arbitrarydimensions. Making use of the structural analogy between statistical mechanics andquantum field theory, a more accurate calculation in four dimensions by G. Münster[Mün89] offered an explicit result in first loop order approximation of the Ginzburg-Landau theory. Later, the same methods were applied to calculate these expression inhigher orders as well as in the three dimensional setup [Mün90], still restricted on aquadratic basal area L× L of the underlying box.Apparently, the use of field theoretical methods has been fruitful for phase transitions.

The analogy between statistical mechanics and quantum field theory is identified by theWick rotation as well as by the similar structural behaviour – the scale invariance – ofthe statistical mechanical systems and field theories. As the correlation length tendsto diverge near the critical point, where the phase transition takes place, wide rangefluctuations dominate the system. Additionally, in quantum field theory this large-scalebehaviour is identified with massless field theories. Therefore, in this thesis the samefield theoretical methods will be used in order to investigate interface fluctuations inthree dimensions imposing new geometric structures.Firstly, in chapter 2, the Ising model will be motivated by a microscopic analysis of

elementary physical laws on atomic scales. Moreover, recognising that the behaviourof macroscopic physical quantities can be observed in the thermodynamic limit, in thesucceeding, a phenomenological theory, the Landau theory will be proposed, motivatedby a mean field approximation near the critical point. Deduced only from macroscopicsymmetry considerations, the Landau theory, the so-called symmetry breaking will beobserved, occurring when the system passes through the critical point. While this theorydoes not involve fluctuations, it will be expanded to the Ginzburg-Landau theory. In thederivation of this theory, the component that is responsible for the geometric dependencewill be identified. Furthermore, the relationship of the parameters used in the Ginzburg-Landau theory to physical parameters from statistical mechanics and quantum fieldtheory is examined. The correlation length of a system performing a phase transition isidentified by the energy gap of a particle tunneling from one quantum state to another.The relationship between statistical theories and quantum field theories will be explainedin the end of the chapter, so that conclusively, a massless field theory is encouraged forphase transitions by statistical mechanical considerations in this chapter.Secondly, in chapter 3, this analogy will be used to present a φ4 field theory of phase

transitions that satisfies the boundary conditions on a new geometric structure, e.g.L1 × L2 × T . Moreover, using the path integral formalism introduced by R.P. Feyn-man [Fey48], the above mentioned kink profile established in the T direction will beidentified with the most probable path (see figure 1.1). Imposing fluctuations aroundthis classical trajectory, the system can be studied in a Gaussian approximation, result-ing in determinants. As the massless case is studied, certain divergences are treated withthe method of collective coordinates [GS75] effecting in a ghost contribution, compara-

2

L1T

L2

Figure 1.1.: A three dimensional box. The indicated interface can be seen as densitytransition from one liquid to another.

ble to the Faddeev-Popov ghost in quantum field theories. Consequently, this masslessparticle is identified with a quasi-particle called instanton in quantum chromo dynamics,thus the tunneling amplitude for these instantons is calculated in a dilute gas approxi-mation, described by S. Coleman [Col85]. Finally, this tunneling amplitude delivers anexpression for the energy splitting, so that for the behaviour of the correlation length, aφ4 theory will be achieved.The arising expression is evaluated in detail in chapter 4. This chapter is more of

technical nature, transforming the involved determinants into integrals applying zetaregularization techniques. Therefore, the zeta regularization and heat kernel methods,firstly described and systematically used in theoretical physics by S. Hawking [Haw77]are introduced in the beginning of the chapter. As a result of these methods, the ex-pression for the energy splitting will be split off into a normal mode corresponding tothe T direction and a transversal mode corresponding to the L1 × L2 plane. As thecalculation of the normal mode is universal for different geometries as long as T is large,the achieved expressions will be compared to the results obtained by similar methodsin [Mün90,Hop93] in order to identify the part that has to be recalculated imposing othergeometric structures. The transversal mode is identified by the Laplacian introduced byimposing fluctuations in the derivation of the Ginzburg-Landau theory. Consequently,the evaluation of the critical fluctuations is reduced to the evaluation of the determinantof the Laplacian on different geometries.

3

INTRODUCTION

Finally, in chapter 5, the determinant of the two dimensional Laplacian is calculatedexplicitly for a box with rectangular basal area L1 × L2. For interface fluctuations, thisdeterminant has been proposed to relate to the Dedekind eta function in [Mün97]. Thisproposal was motivated by calculations of the two dimensional Laplacian in Bosonicstring theories (see e.g. [DFMS97]) that involve conformal field theories. The Dedekindeta function is recognized to fulfil a certain modular invariance that is also needed inthe rectangular case, because the solution must be invariant under the transformationL1/L2 → L2/L1. Successively, the solution involving the eta function needs to satisfy thefunctional equation f(τ) = f(1/τ). The calculation of the determinant of the Laplacianwill be done using three different methods. As the calculation is older, known as themodular discriminant, calculated by D.B. Ray and I.M. Singer [RS73] in mathematics,applied by C. Itzykson and J.-B. Zuber [IZ86] for the Bosonic string originally knownas the Kronecker limit formula. This method was developed in the 18th century by L.Kronecker [Sie61]. This thesis will, in the first part of chapter 5, follow this approach.The second method, using the Sommerfeld-Watson transformation, was motivated byJ. Polchinski [Pol86] for a similar problem. As the calculation in this paper is veryshortly described, this method will be evaluated in detail. In a third method, proposedby J. Baez in his weekly blog [Bae98a,Bae98b], the action of the Bosonic string will berecalculated as an infinite setup of harmonic oscillators, because of the structural analogybetween the Laplacian and the Klein-Gordon equation for scalar fields. All methods aretherefore expected to give the same solution. Subsequently, the obtained solution willbe inserted for calculating the correlation length and compared with the previous resultsfor a quadratic setup, investigating the change of the correlation length in the explicitcase of a box with rectangular basal area.

4

2THEORY OF PHASE TRANSITIONS AND CRITICAL

PHENOMENA

In the introduction, the theory of interfaces was proposed to be a theory of phase transi-tions. The aim of this thesis is to investigate a theory of interfaces in a cuboid of VolumeV , given a certain geometric structure L1×L2×T , see figure 1.1. Interfaces can be seenas limit behaviour between different phases of a material. The following chapter willdevelop the theory of phase transitions basing on theories of phase transitions like thegas-liquid transition or the ferromagnetic transition known from statistical mechanics.First basic characteristics of phase transitions will be given and then two main as-

pects of statistical mechanics will be outlined: a microscopic description founded onthe elementary physical laws of the model and a macroscopic description. Therefore, athree dimensional model can be given by some macroscopic theory adapted only fromsymmetry considerations inferred from the macroscopic world. The macroscopic descrip-tion will be given in a continuous theory, the Ginzburg-Landau theory, which includesfluctuations around the phase transition as well. In both categories, the microscopic andthe macroscopic models, the correlation length determines these fluctuations. It will beshown, that there is certain universality between different theories due to the similarbehaviour of the correlation length and therefore a connection of the microscopic andmacroscopic theories will be outlined in the last section of this chapter. Finally the struc-tural analogy between fluctuations of physical quantities and fluctuations of observablesin quantum field theory (QFT) will be outlined. In this chapter a continuous field theoryof phase transitions is developed.

2.1. Basic characteristics of phase transitions and statisticalmechanics

Phase transitions are characterized by a drastic change of certain macroscopic valuescalled order parameters in a thermodynamic system when another quantity, the control

5

THEORY OF PHASE TRANSITIONS AND CRITICAL PHENOMENA

parameter, is varied. According to the value of the order parameters in a thermodynamicsystem, the different phases can be discriminated. Heuristically, a thermodynamic phasetransition can be described as a reason of a competition between the internal energy Uand the entropy S of the system [NO11] in the following definition of the free energy

F = U − TS. (2.1)

When the temperature rises, the free energy changes its sign at a certain critical tem-perature TC. A physical interpretation could be, that this last term favours disorderdepending on the value of T , whereas the internal energy favours order [NO11]. Con-clusively, the value of the external control parameter T can be varied to control thecompetition between the internal energy and the entropy so that the order parameter,here determined by the free energy, discriminates between the phases of the system.The graduate thesis [vdW73] of Johannes Diderik van der Waals is often identified

as the genesis of modern theory of phase transition [Nol14]. It is commonly seen asthe first thermodynamic argumentation because later on, van der Waals formulated hisfamous theory of the realistic gas, which mentions a phase transition between phases atdifferent temperatures. The transition between different phases could be described andinterpreted with the Maxwell construction [vdW12] by a coexistence of the two phases,which cannot be distinguished. In 1893, van der Waals transferred his work from thedescription of two phase systems to interfaces, where he could predict the thickness of asurface [Row79]. As the van der Waals theory is a continuous theory, interfaces can bedescribed by a phase transition.At about the same time, in 1872, Ludwig Boltzmann developed a first statistical

interpretation of an ideal gas following Maxwell’s heuristic construction of the speeddistribution of the gas particles [Bol72]. In 1877, Boltzmann then re-derived the prob-ability distribution in the framework of statistical mechanics [Bol77] by evaluating therelationship between the Second Law of Thermodynamics and probability calculus. Instatistical mechanics, the behaviour at a microscopic level is described by classical (orlater quantum physical) laws. In contrast to the macroscopic world, on the microscopiclevel observables like the temperature or entropy do not exist. The main argument ofthe statistical description is the equal a priori argument:

“By this we mean that the phase point for a given system is just as likelyto be in one region of the phase space as in any other region of the sameextent which corresponds equally well with what knowledge we do have asto the condition of the system” [Tol50].

This means, that a system can be found with equal probability for any (micro-) statefor an isolated system with a known energy and composition. All these microstates areequally probable in the system where the thermodynamics limit leads to the measurablemacroscopic observables the thermodynamic limit, i.e. the infinite volume limit N/V =const. is taken. In other words, the thermodynamic properties of a system can bedetermined by the partition function Z, which is the sum over all possible free energies

6

2.1. Basic characteristics of phase transitions and statistical mechanics

weighted with the factor e−βF

Z = Tr e−βH , F = − 1β

lnZ, β = (kT )−1. (2.2)

Here H is the Hamiltonian and k ≈ 1.381 · 10−23 J/K is the Boltzmann constant. Thisapproach is a very idealized method to regain some properties of physical systems, likethe ideal gas. Under less strict assumptions, when particles can interact and have a smallvolume, the van der Waals gas can be constructed out of the framework of statisticalmechanics.Important for such systems, in which phase transitions occur, is that this is only pos-

sible in systems of infinite volume V . This means, there is an infinite number of degreesof freedom N . In simple models, where different phases occur, the thermodynamic limitis taken in each phase. Depending on the value of the control parameter specific ways ofthermodynamic limits are taken. Thus for different phases the value of some propertiesof the system, described by the order parameter, depend on the way the thermodynamiclimit is taken [ZJ07]. After the thermodynamic limit is taken, measurable quantitieslike the magnetization M, the specific heat C, the magnetic susceptibility χ or thecorrelation length ξ can be calculated. This connects the microscopic theories to themacroscopic world.According to Paul Ehrenfest, phase transitions can be classified by considering ther-

modynamic parameters such as volume or temperature. Under this scheme, phase tran-sitions are labelled by the lowest derivative of the free energy that is discontinuous atthe transition. In the modern classification scheme, phase transitions are divided intotwo broad categories [Nol14].When the first-order derivative in the order parameter shows a discontinuity, these

phase transitions are called first-order phase transitions. When, by contrast, the first-order derivative is continuous, but the second- or higher-order derivative shows a discon-tinuity or divergence, these phase transitions are called second-order or continuous phasetransitions. Observing the behaviour around the point, where two or more phases coex-ist and become indistinguishable, anomalous phenomena appear, because the correlationlength diverges. As this is universal for different models, continuous phase transitionsare often called critical phenomena [NO11], where the behaviour of the order parametersaround the critical point can be described by universal scaling laws.The most common example for phase transitions is the ferromagnetic transition. A

simple approach, which involves only the symmetry of the spins in the magnetic material,is the Ising model. More generally, a mean field approximation can be applied to phasetransitions, where the phenomenological Landau theory serves equivalent results as theIsing model after the thermodynamic limit is taken. All these models have in common,that they inhibit a phase transition. The critical exponents describe similar behaviourof the system close to the critical point. In a ferromagnet, the above mentioned physicalquantities can be measured experimentally.

7


−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++−−−−−−−−− +++++++++

A B C

Figure 2.1.: Ising model for interfaces, where the spins are localized at lattice points. Inregion A all spin values are uniformly −1, in region B all spin values are +1.Region C is the transition.

2.2. Microscopic description of phase transitions

The simplest way to construct an interface is to impose a model, in which microscopicsymmetries are used in order to describe the system. This model is based upon the ideathat in a ferromagnet the magnetic moments of the atoms can have only two opposingorientations Si ∈ −1, 1, i = 1, . . . N and a potential energy exist only between next-neighbours, which favours a parallel position. An interface can be constructed by thismodel by assuming, that in two distinct regions ((A) and (C) in figure 2.1) these spinsare uniformly orientated. Therefore an interface is established in the region (C) between(A) and (B), when all spins in (A) are in the opposite orientation to the spins in (B).The main reason to formulate a continuous theory later is, that there is no analyticalsolution of the three dimensional Ising model. In the 1920ies, W. Lenz constructed thismodel to explain ferromagnetic transition, which is a transition between an ordered anda disordered phase in the material. Lenz proposed this model to his student E. Ising assubject for his graduate thesis. Ising examined the model in one dimension, but there isalso an analytical solution for a two dimensional model.

2.2.1. Ising model

In the Ising model each spin (indexed by i = 1, . . . , N) in the ferromagnet is localizedon symmetric crystal lattice, like an equidistant chain of spins in one dimension or aquadratic lattice of spins in two dimensions. The simplest Hamiltonian representing thespins with the coupling constant J between next-neighbouring spins in presence of an

8


external magnetic field h can be written as

H = −J∑〈i,j〉

SiSj − h∑i

Si, J > 0. (2.3)

The brackets indicate that the sum is only taken over nearest neighbouring spins. Thepartition function for a free system, where the external magnetic field is switched off(h = 0), is given by

Z = Tr e−βH =

∑S1=±1

· · ·∑

SN=±1

eβJ∑

(i,j) SiSj .

In one dimension this partition function can be evaluated by taking the sum over Nneighbouring spins (i, i+ 1). The result of Ising [Isi25] is

Z = 2N coshN−1(βJ). (2.4)

From the knowledge of the partition function, the free energy per spin can be determinedby (2.2)

F ≡ F

N= 1β

ln(1

2 cosh(βJ)).

According to the Ehrenfest classification, the derivatives of the free energy could becalculated now in order to search for phase transitions. In the introduction the role ofthe correlation length for phase transitions was proposed, since the correlation lengthdiverges as the critical point due to wide range fluctuations. The correlation function

G(r) = 〈SiSi+r〉

can be calculated after some basic transformations [NO11]. The result is

G(r) = tanhr(βJ).

This indicates some exponential behaviour of the correlation. A value to determine thefluctuations could be a correlation length, defined as the exponential decreasing of thecorrelation function. The correlation length of the one dimensional Ising model can begiven now

ξ = − 1ln(tanh(βJ)) . (2.5)

In the low temperature limit (β → ∞) the correlation lenght diverges exponentiallyξ ≈ exp(2βJ)/2. Since ξ diverges at the absolute minimum T = 0, there is no phasetransition observed, thus there is no spontaneuous magnetisation in the one dimensionalIsing model.This observation was generalised by Peierl’s, who proved, “that for sufficiently low

temperatures the Ising model in two dimensions shows ferromagnetism and the same

9


holds a fortiori also for the three-dimensional model” [Pei36]. This argumentation alsomaintains for higher dimensions.This leads to the need for a solution of the two-dimensional Ising model, which is a

much more ambitious and subtle endeavour. The first solution was given by L. Onsagerin 1944 with the transfer matrix method [Ons44]. His solution for the spontaneousmagnetizationM of the two-dimensional Ising model on a quadratic lattice in absenceof an external magnetic field is given by

M =[1− sinh−4(2βJ)

]1/8. (2.6)

In general, the spontaneous magnetisation is given by

M = 〈Si〉 = limh→0

dFdh . (2.7)

As a discontinuity can be found in the derivatives of the free energy at

sinh(2βJC) = 1 ⇔ kTC = 2Jln(1 +

√2)≈ 2.27J,

this temperature is the critical temperature of the two dimensional Ising model.Onsager’s solution is considered to be “one of the landmarks in theoretical physics”

and was consequently awarded with the Nobel Prize in 1968 [BK95]. In general, phasetransitions occur only in discrete models above dimension one (D ≥ 2) for discrete sym-metry groups and above dimension two (D ≥ 3) for continuous symmetry groups [ZJ07].There has yet not been any analytical solution of the three dimensional Ising model,but the following mean field theory of the Ising model can be evaluated in arbitrarydimensions.

2.2.2. Mean field theory of the Ising model

Before Onsager presented a solution of the two dimensional Ising model, a differenttheory, the mean field (or molecular field theory) was proposed by Pierre-Ernest Weiss in1907. This theory can be applied to the D-dimensional Ising model for the Hamiltonian(2.3). The value of a single spin Sj can be decomposed in the mean value and theirfluctuations according to

Sj = 〈Sj〉+ (Sj − 〈Sj〉).

The local energy of spin Si then reads

Ei = −Si

J∑j 6=i

Sj + h

= −Si(

∑j

〈Sj〉+ h)− JSi∑j 6=i

(Sj − 〈Sj〉)

10


The mean field approximation is the negation of wide range fluctuations

(Sj − 〈Sj〉) = 0

which are set to zero. Then the interaction is restricted on a local field. Only the nearest-neighbouring spins are taken into account. For the two possible configurations Si = ±1there are two possible energy levels

E+i = −J

∑j

〈Sj〉 − h and E−i = J∑j

〈Sj〉+ h,

which by the definition of the local magnetisationM = 〈Sj〉 give

E±i = ∓(qJM+ h),

where q denotes the number of nearest neighbouring spins. The local energy Ei is the

i

Figure 2.2.: Two dimensional Ising model with spin Si and the q = 4nearest neighbouring spins.

energy of the local field at the lattice point i. In two dimensions, q = 4 spins Sj are inequal distance to the spin Si (see Figure 2.2). Thus, the partition function becomes

Z = Tr e−βH = eβ(qJM+h) + e−β(qJM+h) = 2 cosh (β(qJM+ h)) . (2.8)

From here on the self-consitent equation

M = tanh (β(qJM+ h)) (2.9)

for the magnetisation is reached, since the magnetisation was also defined as the deriva-tive of the free energy with respect to the external magnetic field (2.7). This equationcould be evaluated now in the limit h → 0. As the above equation serves no analytic

11


expression forM, this expression is now evaluated for small values ofM =M0, whenβ → βC as the series expansion forM→ 0 of the hyperbolic tangent is

M = tanh(q(β − βC)JM) ≈ qJ(β − βC)M0 −13(qJ(β − βC)M0)3 +O(M5

0)

and because the left hand side is also close to zero, it’s value is set toM = 0. Then

M0 = ±√

3qJ(β − βC) (2.10)

can be found, implying, that the spontaneous magnetisation diverges according to

M0 ∼ |β − βC|−1/2 (2.11)

when the critical temperature is reached from below. Obviously, this result is wrong inone dimension, since in the corresponding Ising model no critical temperature TC > 0could be found. The expression (2.9) indicates, that the critical temperature βC can befound in any dimension

βCqJ = 1 ⇒ TC = qJ

k

when q = 2 in one dimension. Therefore, the mean field expansion is qualitatively wrongin one dimension, but in two dimensions it is qualitatively right, but quantitativelywrong. Because in higher dimensions, the interactions between more nearest-neigbouringspins become more important, the deviation from the analytical or numerical solutiondecreases.

2.3. Phenomenological theories for phase transition

The previous mean field approximation indicated that there could be a continuous de-scription of phase transition near the critical point. The Landau theory is a phenomeno-logical description of phase transitions, which is independent from the elementary degreesof freedom, like the lattice structure or the spins. It will be shown, that the mean fieldapproximation and the Landau theory, named after L.D. Landau [Lan37], are equivalentto each other near the critical point. Because the mean field approximation neglectsfluctuations, the theory will be expanded to the Ginzburg-Landau theory that includeswide range interactions as well.

In a phenomenological theory, the thermodynamic potential is written as a function ofan order parameter φ only from symmetric considerations. In the ferromagnetic case, thefree energy is chosen as the thermodynamic potential. The free energy then is a functionof the magnetisation, which is the order parameter φ =M. In absence of the externalmagnetic field, the magnetisationM = 〈Si〉 changes its sign under a transformation thatchanges the sign of all spins, so the order parameter transforms as

φ→ −φ as Si → −Si for all i.

12


βC β

M

Figure 2.3.: The ferromagnetic transition, where the red line indicatedthe stable branches, whereas the blue line indicates theunstable branch. Above βC there are two branches cor-responding to a magnetic material, whereas below βC nomagnetisation is observed.

This transformation leaves the Hamiltonian of the Ising-model invariant. Furthermorethis transformation represents a global symmetry and the free energy must be an evenfunction of the magnetisation, so that the invariance of the thermodynamic potential(the free energy F) is preserved

F(φ) = F(−φ).

2.3.1. Landau-Theory

As the rigorous calculation of the Ising model shows thatM is small when T is near itscritical value TC, the free energy of the system can be assumed keeping only the smallestvalues in M. By the above argument, that the free energy is invariant under paritytransformation, the free energy expansion with arbitrary parameters α, β ∈ R can bewritten as1

F(φ) = F0 + 12!αφ

2 + 14!βφ

4,

where the smallest orders preserved insure that all relevant symmetries of the microscopiclevel are preserved at a coarse-grained level [NO11]. Graphically, the location of theminima can be understood by the potential theory. A negative factor β ≤ 0 impliesinstability for φ → ±∞ because the function decreases infinitely, this parameter is set

1Not to be confused with the Boltzmann factor β−1 = kT .

13


−v v φ

F(φ)

α = 0

α > 0

α < 0

Figure 2.4.: Symmetry breaking of the Landau theory where differentsigns of α force different fixed points.

β > 0. For the coefficient α there are three possible regions in which F(φ) showsdifferent behaviour. In figure 2.4 it is shown, that for α = 0 the expansion starts infourth order, implying that F(φ) is flat in the origin. For α > 0 the quadratic termsupports the increasing of F(φ) so that the curve is similar to a curve of a secondorder function. Interestingly for α < 0, there are two minima and the minimum frombefore is inverted to a maximum. In words of potential theory there are two stable fixedpoints at F(φ) = F(±v) and one unstable fixed point at F(0) when α < 0, whereasthere is only one stable fixed point at F(0) when α ≤ 0. This dramatic change of thebehaviour of the system in the two different phases is called symmetry-breaking, becausethe global behaviour of the system changes. Every particle in the potential approachesthe minimum at F(0) when α > 0 and when α < 0 the particles of the positive siteapproach the right minimum while the others approach the left minimum. Therefore,this global symmetry is broken.This implies that the critical temperature TC can be identified with α

α ∼ T − TCT

=: t,

because in the critical point the value of this so-called reduced temperature is t = 0. Inthe equilibrium, the free energy depends only on the temperature and on the magneticfield. This is obtained by minimizing the above equation of the free energy according tothe order parameter. Direct minimisation leads to

∂F∂φ

∣∣∣∣φ=φ

= 0 ⇒ φ ≡ ±v ±√−6αβ

∨ φ = 0. (2.12)

14


The second derivative shows that for α < 0 the minimum is approached at φ = ±vwhereas the maximum is found in F(0).

2.3.2. Ginzburg-Landau theory

The last section showed, that the Landau theory is equivalent to the mean field ap-proximation near the critical point. Since in the mean field approximation fluctuationswere neglected, there will be no fluctuations in the Landau theory. This theory, theGinzburg-Landau theory [LG50], is named after V.L. Ginzburg und L.D. Landau, whofound a phenomenological description of superconductivity in 1950. For this theory V.L.Ginzburg was awarded with the Nobel Prize in 2003. Here the theory will be formulatedin three (real) space dimensions following [LBB91].The aim is now to reformulate the Landau theory as a continuous field theory in

three dimensions. Obviously, the Landau theory must be reproduced by the new theoryusing an approximation, because the behaviour near the critical point was qualitativelycorrectly described by the mean field approximation in three dimensions. In order toreproduce the Landau theory by means of a HamiltonianH(φ) depending on a continuousrandom variable φ with values in φ ∈ R, the expectation value 〈φ〉 = φ = M is fixed,because the magnetisation of the whole system shall be finite. The invariances underparity H(φ) = H(−φ) preserve the imposed symmetry, so that the polynomial structureof the free energy expansion can be used. This means, the equation depends on twocoefficients and the following Hamiltonian is constructed

H(φ) = 12αφ

2 + 14!βφ

4.

The partition function can than be written as

Z = Tr e−H(φ) =∫

dφe−H(φ),

where the Boltzmann factor is set to β = 1 as it is assumed to be constant in the criticalpoint. The factor β in the expansion again must be positive, because this integral shallconverge.The Landau theory can be regained by a saddle point approximation. In this case,

this means replacing the integral by the minimum value φ =M of H

Z ≈ e−H(φ).

With equation (2.2), the free energy is given by

F = H(φ) = 12αM

2 + 14!βM

4.

Using the saddle point approximation on the Ginzburg-Landau Hamiltonian regains theLandau Free energy expansion. Thus this procedure will be called Landau approximation.In order to formulate a theory that involves interactions, a step back is necessary.

These interactions in the Ising model were defined by the interactions between spins, an

15


elementary degree of freedom. Generalising the above Hamiltonian to N sites (elemen-tary degrees of freedom), one imposes periodicity conditions

φ(xi + aN) = φ(xi),

where a is the lattice spacing. Defining a discretization

∂φ(xi) = φ(xi + a)− φ(xi)a

which is the differential quotient and for small a, the derivative with respect to x. Inter-actions between next-neighbouring spins can be expressed as

∑i

1a2 (φ(xi + a)− φ(xi))2 =

∑i

(∂φ(xi))2

The Hamiltonian can be postulated in analogy to the free energy expansion

HGL = aN∑i=1

[12(∂φi)2 + 1

2αφ2i + 1

4!βφ4i

],

where φi ≡ φ(xi). Going over to a continuum, when N → ∞ and a → 0 leads to acontinuous variable xi → x. Furthermore, in three dimensions, the continuous variableis x → x. This means the order parameter φi in the discrete setup is replaced by theorder parameter φ(x) in three dimensions. In the continuum limit, the Hamiltonianreads

HGL[φ] =∫

d3x

[12(∇φ(x))2 + 1

2αφ2(x) + 1

4!βφ4(x)

].

Moreover the integral in the partition function becomes a path integral

Z =∫ N∏

i=1dφi exp

(−a3

∫d3x

[12(∇φi)2 + 1

2αφ2i + 1

4!βφ4i

])=∫Dφ(x) exp(−HGL[φ(x)]), (2.13)

where the square brackets indicate, that HGL is a functional of φ. In the above limit themeasure is a measure of a path integral, which indicates that over all possible configu-rations dφi is integrated. In words of quantum field theory, this is the integral over allpossible paths. Introducing a potential V(φ), the Hamiltonian is rewritten

H[φ] =∫

d3x

[12(∇φ)2 + V(φ)

](2.14)

=∫

d3xH[φ]

16

2.4. Critical behaviour in field theories

where H[φ] is the Hamiltonian density and the potential

V(φ) = 12αφ

2 + 14!βφ

4

is used. Considering the minimum of this potential being at φ = ±v = ±√−6αβ (see

equation (2.12)), the factor β can be identified by the mean field approximation withthe coupling constant J in (2.10), so that β will be given by a dimensionless couplingconstant g of field theories. The factor α will be identified with a "mass" according to

α = −m2

2 ,

which will be explained in the next section. Then the potential is

V(φ) = −m2

2 φ2 + g

4!βφ4.

The value of the potential in the minimum

φ = ±v = ±√

3m2

g(2.15)

is then given by

V(φ) = −m2

4 v2 + g

4!v4 = −3

4m4

g+ 3

8m4

g

= −38m4

g= − g4!v

4.

In a last step, as the energy integral does not change under addition of a constant factor,the potential is rewritten in order to ensure that the minima of the potential are zero inthe minima

V (φ) := V(φ) + g

4!v4 = −1

2m2φ2 + g

4!φ4 + g

4!v4

= g

4!(φ2 − v2)2. (2.16)

In the formulation of this theory, the interactions were defined over a derivative. Sincethis derivative will lead to a Laplacian operator, this fact is a first hint, that in thecalculation of the critical fluctuations, the Laplacian will be responsible for the geometricdependence of these critical fluctuations.


All studied theories in the last sections showed a certain critical behaviour near thecritical point. This behaviour is characterised by the divergence of the macroscopicparameters, like the magnetisation.

17


The degree of divergence can be described by some power-law, so that an exponentdescribes the total behaviour of the system [NO11]. These exponents are called criticalexponents that define a certain degree of divergence in the critical point. Moreover,the critical exponents of different physical quantities are linked by scaling laws [Kad66,Wil75]. Experiments confirm these power-law singularities as a function of the controlparameter and its critical value for macroscopic observables, like the susceptibility χ,the magnetisationM, the specific heat C or the correlation length ξ. Using the abovedefined reduced temperature, the following critical exponents can be found approachingthe critical temperature [NO11]:

C ∼ |t|−α t > 0M∼ |t|β t < 0χ ∼ |t|−γ t > 0ξ ∼ |t|ν′ t < 0.

Here the exponents α, β, γ, ν ′ are the critical exponents. In the one dimensional Isingmodel, the correlation length was found to to have the critical exponent ν ′ = 1, (2.5).For the mean field theory and also for the Landau theory, the value β = 1/2 was foundin any dimension (2.11).

2.4.1. The interface tension and the correlation length

The reason for the divergence near the critical point is given by the divergence of thecorrelation length. Whereas far away from the critical point interactions play only a roleon microscopic length scales, the divergence of the correlation length ξ is related to widerange thermic fluctuations. Therefore, microscopic details of the system play no roleclose to the critical point and physical quantities of different system are universal witha characteristic large-scale behaviour. Furthermore, the behaviour of systems within acertain universality class can be identified by the global characteristics like symmetryand dimension of the system [NO11]. This fact has been used in the last sections, wherethe behaviour of the system was just given by a macroscopic value, the order parameterφ.The correlation function, defined by the covariance of the system, is a measure for

fluctuations around the expectation value of a random variable. It has been used beforein order to evaluate the correlation length of the one dimensional Ising model (2.5).Defining

G(r) := 〈φ(r)φ(0)〉

for a continuous random variable φ(x), the fluctuations of the continuous theories canbe evaluated. As indicated in section 2.2, the correlation length can be given as anexponential function. Indeed, according to the Ornstein-Zernike theory [OZ14], thecorrelation function is given by

G(r) ≈ e−r/ξ.

18


According to this expression the correlation length defines the spatial length of fluctua-tions. In the literature this correlation length is often called bulk-correlation length.Another definition of a correlation length that linked to the interface tension is the

longitudinal or tunneling correlation length (see e.g. [PF83])

ξ−1|| = C exp(−σ∞L1L2) (2.17)

where σ∞ is the interface tension defined for large systems (L1 → ∞, L2 → ∞). Theinterface tension is equivalently given for the Ising model by the free energy. Accordingto figure 2.1, the free energy of the system with interface is F−+, while the free energyof the system without an interface is F++. The gap in the free energy then defines theinterface tension

σ∞ = limL1→∞L2→∞

(−F−+ − F++

βL1L2

), (2.18)

where β is again the Boltzmann factor. This definition is equivalent to the definition ofan energy gap ∆E between the two states. Using the partition function, the followingcan be written

σ∞ = limL1→∞L2→∞

(− 1L1L2

ln Z−+Z++

).

2.4.2. Quantum fluctuations

In quantum field theories (QFT) fluctuations are known as well. One example is theLamb shift, originating from fluctuations of the electron around its mean value in thehydrogen atom. As a result a shift of the spectral lines by some energy ∆E takes place.In QFT these diverging quantities depend on a large momentum scale, the ultraviolet cut-off. With the Heisenberg uncertainty principle this means that the divergent quantitiesdepend on small distance scale in position space. This can be seen analogue to thecontinuum description of the Ginzburg-Landau theory above, using a continuous field φas order parameter. In QFT, this cut-off is typical at the scale of atomic distances [PS95].By understanding the physics at these small scales, the parameters for systems on largerscales can be determined. Nevertheless, in QFT, the nature of the physics on atomicscale cannot be determined by elementary physical laws, as all quantities of observablesonly have statistical meaning. Large distance physics in QFT involve particles whosemasses are very small in comparison to the cut-off scale. Therefore, it is useful to describethe statistical theory of critical fluctuation as a quantum field theory of particles withzero mass m. With the above assumptions, this would mean

ξ|| ∼ m−1. (2.19)

19


Considering a time evolved system with a Hamiltonian H, one finds for the correlationfunction [DFMS97]

〈φ(r, 0)φ(r, τ)〉 = 〈0|φ(r, 0)e−Hτφ(r, 0)eHτ |0〉

=∑n

〈0|φ(r, 0)e−Hτ |n〉〈n|φ(r, 0)eHτ |0〉

=∑n

e−(En−E0)τ |〈0|φ(r, 0)|n〉|2.

For a system with two eigenstates, the ground state and the excited state Ea, where∆E = Ea − E0 reads in the limit of large times T

〈φ(r, 0)φ(r, τ)〉 ∼ e−∆ET (2.20)

In the previously defined statistical theory of phase transitions, two vacuum expectationvalues can be found in the broken phase: |+〉 that corresponds to φ = +v and |−〉corresponding to φ = −v. The Hamiltonian H is associated with the quantum mechan-ical Hamiltonian with the eigenstates |+〉 and |−〉. Furthermore, there is a symmetricground state |0s〉 with energy Es via the symmetry transformation φ → −φ performedby an operator P and an anti-symmetric ground state |0a〉, whose energy Ea is exactly∆E higher than Es.

This can be imagined as the transition of a quantum mechanical particle from onepotential well into the other. This energy splitting has been introduced in the lastsection, in order to define the interface tension by the free energy of the anti-symmetricsystem F+− corresponding to the state |0a〉 and the symmetric system system F++corresponding to |0s〉.The reflection symmetry can be performed by the reflection operator P whose action

on a state |ψ〉 = ψ(x) is given by

Pψ(x) = ψ(−x)

implying for the ground states Pφ = −φ. The commutator [H, P ] = 0 vanishes, be-cause the potential (2.16) is even under this symmetry. This means, H and P can bediagonalised simultaneously. The two eigenstates of the reflection operator will be

|0s〉 = 1√2

(|+〉+ |−〉)

|0a〉 = 1√2

(|+〉 − |−〉) .

The linear combination of these states deliver the vacuum expectation values

|+〉 = 1√2

(|0s〉+ |0a〉) (2.21)

|−〉 = 1√2

(|0s〉 − |0a〉) . (2.22)

20


−v v

(a)

φ

ψ

−v v

(b)

φ

ψ

−v v

(c)

φ

ψ

−v v

(d)

φ

ψ

Figure 2.5.: Schematic representation of the symmetric (a) and anti-symmetric (b) state and their linear combinations (c), (d),motivated by [Mün10].

These states and their linear combinations are shown schematically in figure 2.5. Thesymmetric and anti-symmetric state give the demanded symmetry

P |0s〉 = |0s〉, P |0a〉 = −|0a〉.

In the path integral formalism, the time evolution (2.20) can be represented by [Das93]

〈+|e−HT |−〉 =∫Dφe−SE [φ]

=∫Dφe−H[φ],

where H is the Hamiltonian (2.14) of statistical field theory, associated with the Eu-clidean action via Wick rotation [LBB91]. Analogous to the Boltzmann factor, thePlanck constant ~ is set to ~ = 1, since the theory is evaluated near the critical point.The z-direction is identified with the Euclidean time in the field theory. Using this struc-tural analogy, the transition amplitude can be defined equivalently to the previouslydefined partition function of the Ginzburg-Landau theory (2.13)

Z = 〈+|e−HT |−〉 =∫Dφe−H[φ]. (2.23)

This is analogous to the transfer matrix of a particle tunneling from the initial state|−〉 at time −T/2 to the final state |+〉 at time T/2. If the quantum field theoretical

21


Hamiltonian H can be expanded into a complete set of eigenstates, the left hand side of(2.23) with the states (2.21) and (2.22) give the transition amplitude

〈+|e−HT |−〉 = 12(e−EaT + 0 + 0− e−EsT

)= 1

2(e−EaT − e−(Ea+∆E)T

)= 1

2e−EaT(1− e−∆ET

). (2.24)

The mixed states vanish, because in this Hilbert space of symmetric and anti-symmetricfunctions, the states |0s〉 and |0a〉 are orthogonal.This gives a good connection to derive the tunneling correlation length by the energy

splitting

ξ−1|| = ∆E. (2.25)

Moreover, the energy gap ∆E is the mass m of the quantum field, as it is the mass of theparticle in rest. Because the correlation length diverges at the critical point, the theoryof critical phenomena is equivalent to a massless field theory. Therefore, the mass m canbe used as the control parameter deviated from the Landau theory, since the zero valueof that control parameter α was reason for the symmetry-breaking.

22

3INTERFACES IN φ4 THEORY

As shown, the Ginzburg-Landau functional serves as a universal approach on phasetransitions as well as on interfaces. In the end of the last chapter, the structural analogyof statistical mechanics and quantum field theory was outlined. Given a field φ(x),the following chapter drafts out the theory in the setup of a quantum field theory andpath integrals, often called statistical field theory [LBB91]. The theory corresponds toa φ4-theory in QFT. Therefore, reasonable boundary conditions for the field will bedetermined first in order to apply this Hamiltonian to the interface in a cuboid box L1×L2×T , see figure 1.1. This setup will be investigated in the Landau approximation, whichmeans, the vacuum expectation value and the statistically most probable configurationof a field φ0 satisfying the demanded boundary conditions will be calculated. This willlead to a classical solution, the kink-solution, which was first given by J.W. Cahn and J.E.Hilliard in 1958 [CH58]. Additionally the transition energy of the kink will be derived.It will be shown, that the system is invariant under translations in z-direction from aphysical point of view because the form and transition energy of the interface shall beinvariant under different positions of the interfaces in the box.For the field φ(x) there is no general analytic expression. Therefore, a more suitable

approximation, involving fluctuations will be necessary. One way is to perturb theHamiltonian (2.14)

H[φ] =∫

L1×L2×T

d3x

12(∇φ)2 + V (φ)

(3.1)

=∫

d3xH[φ] (3.2)

with the φ4 potential

V [φ] = g

4!(φ2 − v2)2 (3.3)

by a small fluctuation η around the classical trajectory φ0. Then a series expansion will

23

INTERFACES IN φ4 THEORY

allow a reconsideration of the path, involving small fluctuations in a Gaussian approxi-mation. These fluctuations will be identified by a fluctuation operator.In the last chapter the correlation length was observed corresponding to the energy

splitting between the symmetric and anti-symmetric state. The transition through theinterface will be formulated as the tunneling of a particle from an initial state to afinal state through the potential barrier. Under some quantum field theoretical con-siderations involving pseudo particles called instantons, this treatment goes back to S.Coleman [Col85]. The upcoming zero mode will be handled with the method of collectivecoordinates, where the translational invariance of the system is used to split off the zero-mode. The system is translationary invariant from a physical point of view because theform and transition energy of the interface shall be invariant under different positionsof the interfaces in the box.Conclusively, this chapter will lead to an expression for the interface energy of the

system, which can be evaluated by methods of zeta-regularization later on.

3.1. Boundary conditions for the interface

The system that is going to be modeled must be restricted by the following constraints:

• There must be at least one interface. This means the energetic more favourableconstant solution of the system must be avoided and the phase transition is en-forced.

• It must be finite in the plane of the interface, so that the boundary conditionsestablish an interface. In the other direction the system must be infinite, so that thephase transition can take place and the proposed kink solution can be established.

An interface can, as described in the last chapter, only exist in the broken phase ofthe Ginzburg-Landau theory where the two potential minima exist. Furthermore, thegeometry must be fixed by the following conditions to enhance the restrictions on thevolumina stated in the previous chapter: one direction must be able to be extended toinfinity, because for finite volumes, symmetry breaking cannot occur, and the furthertwo dimensions are restricted to a finite spatial volume A = L1L2. To make this clearthe following conventions will be used:

• The site with infinitely extended length will be referred to by the z-coordinate,with z ∈ R, the length of the site will be named T , while later on the limit T →∞will be taken.

• The finite special volume will be referred to by A = L1×L2 and it will be describedby the vector ~x = (x1, x2) ∈ R2 with 0 < x1 < L1 and 0 < x2 < L2.

Summarizing these geometric restrictions, a volume of L1×L2×T has been constructedfor a field φ(x) = φ(~x, z) = φ(x1, x2, z), where T → ∞. Moreover, realistic boundaryconditions are demanded.

24

3.2. The kink solution obtained from Landau approximation

Remembering that the Ginzburg-Landau theory is a field theory, a field φ(x) plays therole of the order parameter. For this field, the interface, established in (x1, x2)-direction,takes place in the infinitely large z direction when fixed boundary conditions are used.This can equally be described by anti periodic boundary conditions

φ(~x, z) = −φ(~x, z + T ). (3.4)

The two ground states of the infinite volume system in broken symmetry phase systemcan be characterized by a vacuum expectation value v > 0 of the field

φ ≡ 〈±|φ(x)|±〉 = ±v. (3.5)

To exclude surface phenomena the boundary conditions in the ~x directions are chosento be periodic:

φ(x1, x2, z) = φ(x1 + L1, x2, z) = φ(x1, x2 + L2, z). (3.6)

These conditions establish at least one transition between φ = −v and φ = v. To beaccurately, more than one transitions satisfy these boundary conditions as long as thenumber of transitions is odd. However, these transitions will be dominated by the one-transition setup, because in every interface, a transition of the potential barrier must beovercome. The energetic most favourable states that are dominated by their statisticalweight, the Boltzmann probability, will establish only one interface. This interface profileoriginates due to the most probable or – in words of quantum field theory – due to theclassical path. To begin with, the vacuum expectation value needs to be calculated. Thishas been already done in the last chapter (2.15), resulting in φ = ±v = ±

√3m2

g , and canbe also achieved by minimizing the Hamiltonian in the framework of variational calculus(see appendix A.1).

3.2. The kink solution obtained from Landau approximation

In the Landau approximation, the system contains merely the state of maximum prob-ability, which is the state of minimum transition energy over the potential barrier. Theinterface profile is therefore a field φ0 which minimizes the transition energy, where theHamiltonian H[φ(x)] is minimized. Consequently, the variational calculus leads to theclassical Euler-Lagrange equation

δH[φ(x)]δφ(x)

∣∣∣∣φ=φ0

= 12(−2∇2φ0) + g

4!4φ30 −

m2

4 2φ0 = 0. (3.7)

The derivative used here is the Fréchet derivative, serving as the functional derivative ofan operator (see e.g. [LBB91,PS95]). Taking the above mentioned boundary conditionsinto consideration, which imply the appearance of one or more interfaces that gives

25


−v

v

a z

Figure 3.1.: The kink profile.

the most probable path through the double well potential, these conditions lead to theCahn-Hilliard [CH58] profile

φ(a)0 (z) = v tanh

[m

2 (z − a)]

(3.8)

for one kink (see appendix A.1), where the parameter a ∈ R can be chosen arbitrarilybecause it is an integral constant and manifests the position of the profile on the z-axis.Solutions of the form (3.8) are called kink solutions. It shows the form presented in figure3.1. The energy of the transition is an integral over the Hamiltonian density accordingto equation (3.2), where the explicit profile, the kink-profile (3.8) is inserted. Since theintegration in z has to be performed over whole R, the arbitrary parameter a can beeliminated by substitution.Furthermore, this means, the energy is independent from the position of the kink.

This translational invariance will be used later in detail in order to handle the zeromode using the method of collective coordinates. Let for now be φ0 ≡ φ

(0)0 , because a

can be chosen arbitrarily.Changing the sign of the boundary conditions φ0 =

z→±∞∓v leads to the anti-kink

φ0(z) = −v tanh(mz/2). The energy of one kink thus is calculated by integrating theHamiltonian density (see appendix A.2) and results in

H0 ≡ H [φ0(x)] = 2L1L2m3

g. (3.9)

26

3.3. Perturbing the Hamiltonian with fluctuations

1m

a z

Figure 3.2.: Localization of the kinetic energy of a kink, obtained in appendix A.2.


In quantum field theory, the Euler-Lagrange equation gives the most probable classicaltrajectory satisfying the Hamiltonian principle of the least action. Small fluctuationsη(x) are added to the classical trajectory (3.8) φ0(x)

φ(x) = φ0(x) + η(x). (3.10)

The Hamiltonian therefore is given by

H [φ(x)] = H [φ0(x) + η(x)] .

The Taylor series of the Hamiltonian H [φ] around the kink solution with fluctuations ηis given by

H [φ(x)] = H [φ0(x) + η(x)]

= H [φ0(x)] +∫

d3xδH [φ]δφ(x)

∣∣∣∣φ0

η(x)

+ 12

∫d3xd3x′

δ2H [φ]δφ(x)δφ(x′)

∣∣∣∣φ0

η(x)η(x′) + . . . .

The potential has the maximal order O(φ4), thus all functional derivatives of the func-tional H above the fourth order vanish. As discussed before, the classical trajectorysatisfies the classical Euler Lagrange equation (3.7), so the second term in the expan-sion also vanishes. The contribution of orders above two, the so-called interactions, will

27


be ignored, as they do not contribute to the Gaussian approximation. This leaves toevaluate the second order of the expression

δ2H [φ]δφ(x)δφ(x′)

∣∣∣∣φ=φ0

= −[∇2 − V ′′(φ0)

]δ(x− x′) ≡Mxx′ [φ0]. (3.11)

So, the expanded Hamiltonian is

H [φ0 + η] = H0 + 12 (η,M [φ0]η) , (3.12)

where (·, ·) denotes the scalar product, here defined by

(f, g) =∫

d3xf(x)g(x)

for square integrable functions f, g : R3 → R. In this case, the operator M [φ0], de-fined above, represents the contribution of the fluctuations around the classical solution.Therefore, it is called fluctuation operator and will be given explicitly later. The corre-sponding partition function reads

Z =∫Dφe−H[φ] = e−H0

∫Dηe−

12 (η,M [φ0]η). (3.13)

This partition function contains a Gaussian integral. A naive way to evaluate this integralwould be

Z = N√det(M [φ0])

e−H0 (3.14)

with a normalizing constant N due to the Gaussian integral. The Gaussian integral ind dimensions is given by ∫

dyde−12∑d

n=1 y2n = (2π)d/2. (3.15)

The partition function in (3.13) can only be determined this way, when all eigevaluesare strictly positive. Because of the eigenvalue λ0 = 0 the Gaussian integral diverges.In other words, as the determinant is a product over all eigenvalues, the determinantin (3.13) delivers a zero value and the partition function diverges. There is a connec-tion between the previously mentioned translational invariance of the kink and the zeromode. This will be evaluated in detail in the next section with the method of collectivecoordinates.In order to evaluate the fluctuations, the operator M [φ0] is studied in detail now.

It is specially relevant that the operator possesses explicitly one eigenmode with zeroeigenvalue called zero mode. This zero mode is described by the function Ψ0. Observingthe fluctuations again, the same extremal condition (3.7) delivers the following expressionfor the fluctuation operator.

28


Varying the extrema of the Hamiltonian by a small fluctuation around a classicaltrajectory, imposing the same extremal condition (3.7) as before

δH [φ]δφ(x)

∣∣∣∣φ=φ0+η

= 0.

Whereas this means

δH [φ]δφ(x)

∣∣∣∣φ=φ0+η

= −∇2(φ0 + η)− m2

2 (φ0 + η) + g

3!(φ0 + η)3

= δH [φ]δφ(x)

∣∣∣∣φ=φ0

−∇2η − m2

2 η + g

2!φ20η +O(η2)

= 0 +[−∇2 − m2

2 + g

2φ20

]η +O(η2).

Recalling, that the classical trajectory satisfies condition (3.7), the first term equals zero.By neglecting higher orders of η, the calculation leads to[

−∇2 − m2

2 + g

2φ20

]η = 0, (3.16)

with the fluctuation operator which is defined by

M [φ0] ≡ −∇2 − m2

2 + g

2φ20 = −

[∇2 − V ′′(φ0)

]. (3.17)

The last step can be shown by calculating the second derivative of the potential (3.3)with respect to φ0

V ′′(φ0) = g

2

(φ2

0 −13v

2)

= g

2φ20 −

m2

2 .

That is exactly the operator defined in (3.11). The fluctuation η solving the condition(3.16) is identified with the zero-mode Ψ0, since the Gaussian integral in the partitionfunction (3.14) diverges by this mode. The connection of this zero-mode to the invarianceunder translations of the kink φ0(z) on the z-axis is outlined now. The translationaloperator T (a) is defined by small translations a:

φ0(z + a) = T (a)φ0(z) =(

1 + a∂

∂z

)φ0(z) = φ0(z) + aφ′0(z) +O(a2). (3.18)

The prime here indicates a derivative with respect to z. By omitting the higher ordersof a, this can be written as

T (a)φ0(z) = φ0(z) + c(a)Ψ0(z),

29


where Ψ0 corresponds to a translation in functional space induced by the translation ofthe position in z-direction. For the classical solution

c(a)Ψ0(z) = a∂zφ0(z) = amv

2 sech2(m

2 z)

(3.19)

is obtained. Assuming Ψ0 to be normalized, the scalar product provides

(c(a)Ψ0(z), c(a)Ψ0(z)) = c2(a)||Ψ0||2 = c2(a).

From (3.19), the constant can be derived

c2(a) = (c(a)Ψ0(z), c(a)Ψ0(z)) = a2(φ′0(z), φ′0(z))

= a2∫

d3x(φ′0(z))2 = a2H0

according to the calculations of the kink energy in appendix A.2. Consequently, in thelimit of infinitesimal a, the translation is given by

T (a)φ0(z) = φ0(z + a) = φ0(z) + a√H0Ψ0(z). (3.20)

Comparing (3.18) with (3.20) the following zero mode can be found

Ψ0(z) = φ′0(z)√H0

.

Inserting the zero-mode Ψ0 of the operator M [φ0] for η in the left hand equation (3.16)gives zero. Because of the above connection between the zero mode and the transla-tion along the z-axis, the zero mode delivers a contribution to all one-kink solutions.The following treatment applied here is called method of collective coordinates [GS75].Defining

ξ(a) :=∫

d3x [φ(x)− φ0(z − a)] φ′0(z − a)√H0

for a field φ(x). By the properties of the Dirac delta function follows

1 =∫

dξδ(ξ) =∫

dadξdaδ(ξ(a)),

where a shift and the derivative give

dξda = d

da

∫d3x [φ(~x, z + a)− φ0(z)] φ

′0(z)√H0

=∫

d3xdφ(~x, z + a)

daφ′0(z)√H0

,

30


because there φ0(z) and φ′0(z) do not depend on a any more. Furthermore, the derivativewith respect to a in φ(~x, z + a) can as well be written as a derivative with respect to z.Therefore the path integral over an arbitrary configuration φ(x) is given by∫

Dφe−H[φ] =∫Dφ

[∫dadξ

daδ(ξ(a))]

e−H[φ]

=∫

da∫Dφ

∫d3x

[φ′(~x, z + a)φ

′0(z)√H0

× δ(∫

d3x [φ(~x, z + a)− φ0(z)] φ′0(z)√H0

)]e−H[φ].

As there is no difference in evaluating all configurations φ(~x, z + a) or φ(~x, z), the ex-pression becomes

∫Dφe−H[φ] =

∫da∫Dφ

∫d3x

[φ′(x)φ

′0(z)√H0

× δ(∫

d3x [φ(x)− φ0(z)] φ′0(z)√H0

)]e−H[φ]

On a symmetric support a ∈ [−T/2, T/2], the first integral becomes T , because the restof the integral is independent of a. Setting now φ = φ0 + η, the integral becomes

∫Dφe−H[φ] = T

∫Dφ

∫d3x

[(φ′0(x) + η′(x))φ

′0(z)√H0

× δ(∫

d3x [φ0(x) + η(x)− φ0(z)] φ′0(z)√H0

)]e−H[φ0(x)+η(x)],

where the last expression can be substituted by the series expansion (3.11) because onlysmall fluctuations are considered. The expression inside the delta function simplifies and

δ

((η,

φ′0√H0

))(3.21)

is gained. This means, the integral has only a meaning, when(η,

φ′0√H0

)= 0. (3.22)

Because of the energy integral∫

d3x(φ′0)2 = H0 the whole integral simplifies to∫Dφe−H[φ] = T

√H0e−H0

∫Dη

[[1 + 1

H0(η, φ′′0)

]δ

((η,

φ′0√H0

))]e

12 (η,Mη)+O(η3).

31


Evaluating the derivatives of the classical trajectory φ0(z) deliver

φ′0(z) = mv

2 sech2(m

2 z)

φ′′0(z) = m2v

4 tanh(m

2 z)

sech2(m

2 z)

= m

2 tanh(m

2 z)φ′0(z).

This means, the scalar product (3.22) vanishes, if∫d3xη(x)φ′(z) = 0.

Explicitly this means (η, φ′′0

)= m

2(η, tanh(mz/2)φ′0(z)

)= 0.

Then the above equation simplifies to∫Dφe−H[φ] ≈ T

√H0e−H0

∫Dηe

12 (η,Mη) (3.23)

in second order expansion. By condition (3.21) follows, that only the fluctuations orthog-onal to the zero mode are included in the integral (3.23). To be explicit, the eigenvaluesλ0 < λ1 < . . . < λn of the eigenfunctions η in (3.16) the integral is evaluated accordingto ∫

Dηe−(η,Mη) =∫ ∏

n≥0dcne−

12∑

n≥0 λnc2n

=∫

dc0

∫ ∏n>0

dcne−12∑

n>0 λnc2n

=

√H [φ0]

2π

∫da∫ ∏

n>0dcne−

12∑

n>0 λnc2n

=

√H [φ0]

2πN∫

da√det’(M(φ0))

,

where the apostrophe denotes that the determinant is evaluated omitting the zero modeλ0 = 0. The factor

√2π is inserted due to the collective coordinate c0 in the Gaussian

integral ∫Dη = N

∫ ∏i

dci√2π∫ dc0√

2πδ(c0) = (2π)−1/2.

32

3.4. The energy splitting

The partition function (3.13) reads

Z = N

√H [φ0]

2πe−H[φ0] ∫ da√det’(M [φ0])

, (3.24)

with H[φ0] given in (3.9).


From statistical field theory it is known, that the partition function for the broken phaseof the Ginzburg Landau theory corresponds to a transition amplitude in quantum fieldtheory induced by a time-evolution operator e−HT . In the broken phase, there is nodegeneracy at finite volume. Furthermore, there is a symmetric ground state |0s〉 withenergy Es via the symmetry transformation φ0 → −φ0 performed by an operator P andan anti-symmetric ground state |0a〉Using the analogy between the partition function and the particle tunneling from

one state to the other, the following contribution for one kink on symmetric support[−T/2, T/2] is followed by (2.23) and (3.24)

〈+|e−HT |−〉 = NT√H0( det′M)−1/2e−H0 (3.25)

with M ≡M [φ0].

3.4.1. Multi kink contribution

A next step is the introduction of a sharp kink approximation, often called dilute gasapproximation. From equation (A.7) it can be inferred, that the kinetic energy of kinksare precisely localized with a size for order 1/m for large T around z = a, see as wellFigure 3.2. This form is often called soliton in nonlinear-physics and resembles a pseudo-particle in quantum field theory, called instanton. This means, there are, as statedpreviously, more than only one kink. There are n kinks playing a role on a large Tscale. If the time is large T → ∞, the characteristic intervals become T m−1 andthe corresponding Hamiltonian corresponding to such a configuration is nH0, where H0is the Hamiltonian of one kink. This can be seen in Figure 3.3. This means, mostcontribution of the multi kink energy splitting result from decreases and increases onthe states with φ0 = ±v, which is due to

M0 = −∇2 +m2.

This expression is comparable to the harmonic oscillator in quantum mechanics. Fromquantum mechanics is known, that the spectrum of the harmonic oscillator is strictlypositive. Therefore, no correction due to a zero mode will be necessary. The pathintegral in this case is similar to the previous path integral, giving

〈+|e−HT |−〉 = N (detM0)−1/2e−H0 ,

33


−v

v

t1 t2 t3 t4 t5

Figure 3.3.: Multi kinks in a sharp kink approximation.

where H0 corresponds to the symmetric functions. If the kinks are in sharp kink approx-imation and the time is large enough, the functional integral can be evaluated summingover all these configurations for widely separated objects go to H ≈ nH0, which givesaccording to [Col85]

e−H = Kne−nH0 (3.26)

with a constant K to be determined later, with n objects centered at times t1, t2, . . . , tn,where

−T/2 < t1 < t2 < . . . < tn < T/2.

All multi kink configurations which fulfill the boundary conditions are the configurationswith an odd number of kink and anti kinks following each other. This means the integralover all these configurations gives

∫da =

T/2∫−T/2

dtn · · ·t3∫

−T/2

dt2t2∫

−T/2

dt1 = Tn/n!,

which can be easily computed and proven by induction. Likewise, n must be odd bydefinition of the boundary conditions. The determinant then is considered being onlythe determinant over a product of separated harmonic oscillators, resembling the wells

34


of the double well potential. For n widely separated kinks, the result is

〈+|e−HT/~|−〉 = N (detM0)−1/2 ∑n odd

(Ke−H0T

)nn! (3.27)

= N (detM0)−1/2exp

(Ke−H0T

)− exp

(−Ke−H0T

)2

= N (detM0)−1/2 12 exp

(Ke−H0T

) [1− exp

(−2Ke−H0T

)],

where the last expression can be compared with the initial equation (2.24), so that

∆E = 2Ke−H0 .

The degeneracy of the two energy eigenvalues is broken by tunneling amplitude, thedifference is given by a Gamov factor e−H0 . This can be compared with the priorstatement, giving (3.25) and (3.27) for n = 1

N (detM0)−1/2Ke−H0T = NT

√H02π ( det′M)−1/2e−H0 ,

so that

K =

√H02π

(det′MdetM0

)−1/2

,

where the factor√

2π is due to the zero mode.Conclusively

∆E = 2Ke−H0

= 2

√H02π

∣∣∣∣∣ det′MdetM0

∣∣∣∣∣−1/2

e−H0 , (3.28)

where the absolute value is used because the determinants are positive.This solution indicates, that the energy splitting can be given in the proposed form

using the interface tension (2.18)

∆E = C exp(−σ(L1, L2)L1L2).

Then, the interface tension is given by

σ(L1, L2)L1L2 = − ln ∆E + lnC.

35


36

4EXPRESSING THE ENERGY SPLITTING BY INTEGRALS

The aim of this chapter is to find an expression to calculate the previously assessmentfor the energy splitting

∆E = 2e−H0

√H02π

∣∣∣∣∣det′MdetM0

∣∣∣∣∣−1/2

. (4.1)

Riemann zeta functions and heat kernels can express the determinants involved in (4.1).By taking into account, that there is a zero mode in det′M , asymptotic behaviours ofheat kernels can be used which go back to H. Weyl in 1911 [Wey11]. In this approach,the divergences can be handled. Especially, the heat kernels transform the problem intoa problem of zeta-functions, which can be calculated via zeta-function regularizationmethods. It will be shown, that the expression

ln∣∣∣∣∣det′MdetM0

∣∣∣∣∣ (4.2)

can be transformed into integrals, following [Mün89]. The calculation is explicitly givenfor L×L in the diploma thesis of P. Hoppe [Hop93]. This calculation will be redone, inorder to ensure, that the structure of these integrals also hold in the geometric dependentsetup. Finally, this chapter will show that only one of the three integrals must berecalculated in order to satisfy the geometric dependence for L1 × L2, L1 6= L2. Thiscalculation is brought up in chapter 5.In the preceding chapters, the expression for the fluctuation operator (3.17) was ob-

37

EXPRESSING THE ENERGY SPLITTING BY INTEGRALS

tained:

M = −∇2 − m2

2 + g

2φ20

= −∇2 − m2

2 + gv2

2 tanh2(m

2 z)

= −∇2 − m2

2 + 32m

2 tanh2(m

2 z)

= −∇2 +m2 − 32m

2 + 32m

2 tanh2(m

2 z)

= −∇2 +m2 − 32m

2 sech2(m

2 z).

The operator M0 corresponds to the previously mentioned dilute gas approximation

M0 = −∇2 +m2.

Defining the two dimensional Laplacian for the transverse modes

∆ := −∂1∂1 − ∂2∂

2,

the fluctuation operator can be expressed as

M = ∆ +Q, M0 = ∆ +Q0

with the z-component (in QFT time-like component) of the fluctuation operator

Q = −∂z∂z +m2 − 32m

2 sech2[m

2 z],

Q0 = −∂z∂z +m2.

First of all, the methods for the following chapters shall be introduced.

4.1. Zeta function regularization and heat kernel methods

In order to evaluate the determinant of an operator, the spectrum of the operator canbe used. For a matrix A the determinant is given in the following manner

detA =n∏i=1

λi, (4.3)

where λi ∈ R, i ∈ 1, 2, . . . , n is the discrete spectrum of A containing a finite numberof eigenvalues λi > 0. The Riemann zeta function is defined as

ζ(z) =∞∑k=1

1kz

38

4.1. Zeta function regularization and heat kernel methods

for <(z) > 1. An integral representation of the zeta-function is

ζ(z) =∑k

k−z = 1Γ(z)

∞∫0

dttz−1

et − 1 = 1Γ(z)

∑k

∞∫0

dttz−1e−kt,

where the Γ-Function is defined as

Γ(z) =∞∫0

dttz−1e−t.

Via analytic continuation (see reflection formula in appendix B.3) the zeta function forz ∈ C\1 can as well be defined. The derivative of the zeta function provides

ddz ζ(z) = −

∞∑k=1

ln kkz−→z→0−∞∑k=1

ln k,

which means the zeta function can be used to evaluate the determinant of a matrix Adefining a spectral version of the zeta function, in short spectral zeta function

ζA(z) :=n∑i=1

1λzi

by

ddz ζA(z)

∣∣∣∣z=0

= −∞∑i=1

lnλi = − ln∞∏i=1

λi = − ln detA (4.4)

using (4.3) in the last step because A is self-adjoint and thus independent from thechosen base.By analogy with the handling of the matrix A, the same approach can be used for

self-adjoint operators O. The product of the eigenvalues gives the determinant of thisoperator

detO =∏i

λi,

where λi, i ∈ N is the spectrum with λi > 0 for all i ∈ N, which is independent from thechosen base, because O was defined to be self-adjoint. Often the spectrum of an operatoris neither discrete nor finite, the main idea in order to evaluate the determinant, is torewrite this expression in terms of an operator zeta function. This zeta function isdefined by analogy with (4.3) for the operator O, where for a continuous spectrum thespectral density will be used

ζO(z) := Tr(O−z) =∞∑i=1

1λzi. (4.5)

39


If the spectrum is not strictly positive, a shift can be done by adding a regularizingmass µ2 to the eigenvalues. Nevertheless, the trace is also defined for continuous spectra.The last step in (4.5) is only defined for discrete finite spectra and to be understoodsymbolically, or as a definition, in other cases.By taking the derivative at z = 0, the previous expression (4.4) can be regained for

operators

dζO(z)dz

∣∣∣∣s=0

= limz→0−∞∑i=1

λ−si lnλi

= −∞∑i=1

lnλi = − ln( ∞∏i=1

λi

)= − ln detO.

Compendiously,

ln detO = Tr lnO (4.6)

= − dζO(z)dz

∣∣∣∣z=0

(4.7)

is achieved, as long as the operator zeta-function ζO(z) is well defined. Moreover, it isuseful to define a heat kernel

Kt(A) := Tr e−tA, (4.8)

so that a suitable t ∈ R insures that the trace is well defined. The kernel defines a zetafunction

ζKt(A)(z) = 1Γ(z)

∞∫0

dttz−1Kt(A), (4.9)

because

ζ ′Kt(A)(0) = − ln detA = − ln∏

λi = −∑

lnλi.

And with (4.7), it can be determined that

Tr lnA = −ζ ′A(0) (4.10)

= −∞∫0

dtt−1Kt(A), (4.11)

where from now on, the operator zeta function of a heat kernel will be identified byζA(z) := ζKt(A)(z).As there are more operators involved in the following sections, the factorization proper-

ties of heat kernels will be useful. For two operators O1 and O2 the heat kernel factorizesdue to the Baker-Campbell-Hausdorff formula

Kt(O1 +O2) = Tr e−t(O1+O2) = Tr(e−tO1e−tO2et2[O1,O2]/2+...

).

40

4.2. Transforming the determinants into integrals

Here, the last factor depends on convoluted commutators of the operators, so that forcommutating operators

Kt(O1 +O2) = Tr(e−tO1e−tO2

)= Tr

(e−tO1

)Tr(e−tO2

)= Kt(O1)Kt(O2) (4.12)

the properties of the trace Tr(O1 ⊗ O2) = Tr(O1) Tr(O2) in the last step are used. Inthis case, the exponential function provides the direct sum, which is the sum of theeigenvalues of the operators O1 and O2.


From (4.1), the evaluation of the fraction of the determinants needs to be calculated. Inthe last section, it could be made obvious, that it is common for the evaluation determi-nants of operators containing an unbounded spectrum to use zeta function-regularizingtechniques for expressions like


∣∣∣∣∣ (4.13)

instead of using the expression in (4.1). For equation (4.13), there exists a representationin integrals, which will be developed in the following theorem.

THEOREM 4.1: TRANSFORMING THE DETERMINANTS IN INTEGRALS

The determinants in (4.1) can be transformed in integrals according to


∣∣∣∣∣ = − ddz ζ(z)

∣∣∣∣z=0

(4.14)

with

ζ(z) = ζ1(z) + ζ2(z) + ζ3(z),

ζ1(z) = 1Γ(z)

∞∫0

dttz−1L1L24πt

[Kt(Q)− 1

],

ζ2(z) = 1Γ(z)

∞∫0

dttz−1 [Kt(∆)− 1] ,

ζ3(z) = 1Γ(z)

∞∫0

dttz−1[Kt(∆)− L1L2

4πt

] [Kt(Q)− 1

],

where ζ1(z) and ζ2(z) are defined for <(z) > 1 and analytically continued to z = 0 andζ3(z) is valid for all z and the difference of the heat kernels Kt(M) = Kt(M)−Kt(M0).

41


The proof of this theorem will take several transformations, which involve studying thezero-modes of the operator M and M0 as well as investigating the asymptotic behaviourof the involved heat kernels. At first, the zero-mode will be removed and the factorizationproperty will be used.

LEMMA 4.2: FACTORIZATION OF HEAT KERNELS

Introducing a regularizing mass µ2 > 0 the expression (4.2) can be rewritten in

ln det′MdetM0

= − limµ→0

∞∫0

dtt−1Kt(M + µ2)− lnµ2

, (4.15)

= − ddz

∣∣∣∣z=0

1Γ(z)

∞∫0

dttz−1Kt(M + µ2) (4.16)

with where Kt(M + µ2) is defined as

Kt(M + µ2) = e−tµ2Kt(∆) [Kt(Q)−Kt(Q0)] . (4.17)

Proof of lemma 4.2. In order to handle the zero mode, a regularizing mass µ2 is intro-duced which is chosen to be arbitrary large to dissolve the zero mode of M . The shiftin the spectrum of M and M0 with µ2 has been carried out in order to ensure thatthe spectrum is always positive. In order to calculate the left hand side of (4.14) thedeterminants are transformed into a trace according to (4.15) and the zero mode of Mis split off

ln det′MdetM0

= Tr′ ln M

M0(4.18)

= limµ→0

[Tr ln

(M + µ2

M0 + µ2

)+ lnµ2

](4.19)

and following (4.11) the heat kernel can be identified

ln det′MdetM0

= − limµ→0

∞∫0


, (4.20)

where Kt(M + µ2) is defined as the difference of the heat kernels

Kt(M + µ2) : = Kt(M + µ2)−Kt(M0 + µ2) (4.21)

= e−tµ2Kt(∆)Kt(Q), (4.22)

where the fact that heat kernels factorize (4.12) was used, because all commutators arezero

Kt(M + µ2) = e−tµ2Kt(∆)Kt(Q). (4.23)

42


In [Hop93] a proof is given, that expression (4.15) exists for t → ∞ and µ → 0. Theasymptotic behaviour of this expression for t → 0 will be investigated later on. Toachieve this, the heat kernel of the operator Q will be calculated at first.

LEMMA 4.3: HEAT KERNEL FOR THE OPERATOR QUsing the spectrum and the spectral densities of Q the limit T → ∞ the heat kernelKt(Q) is given by

Kt(Q) = erf(m0√t) + e−

34m

20t erf

(m02√t

),

where erf(x) denotes the Gaussian error function.

Proof of lemma 4.3. First, the spectrum of Q is evaluated. Following [Hop93] and[MF53] the spectrum of Q and Q0 can be evaluated, which is given by the eigenmodefor translation

ε1 = 0 (4.24)

and the discrete eigenmode

ε2 = 34m

2, (4.25)

corresponding to vibrational modes. The continuous part is given by

εk = m2 + k2, (4.26)

with the spectral density

g0(p) = 12π

T − 3mp2 + 1

2m2(

p2 + 14m

2)

(p2 +m2)

+O(T−1).

The spectrum of Q0 can be obtained by solving the wave equation

d2

dz2ψn(z) = (εn +m2)ψn(z)

which gives

εn =(2πTn

)2+m2, n ∈ Z.

43


For large T , the spectrum goes over to a quasi-continuous spectrum

εp = p2 +m2 (4.27)

with the following spectral density

g0(p) = T

2π .

The difference of the spectra is given by

g0(p)− g0(p) = 12π

−3mp2 + 1

2m2(

p2 + 14m

2)

(p2 +m2)

+O(T−1), (4.28)

where the lower orders can be neglected in the limit T →∞.Using the eigenvalues ε1 and ε2 from (4.24) and (4.25) as well as spectral densities of

Q and Q0 from (4.28), the heat kernel Kt(Q) in the limit T →∞ reads

Kt(Q) = 1 + e−3m2/4 +∞∫−∞

dp [g0(p)− g0(p)] e−t(p2+m2)

T→∞= 1 + e−34m

2t +∞∫−∞

dpg(p)e−t(p2+m2), (4.29)

where the spectral densities (4.28) can be used in the limit T →∞

g(p) = −m2π

2p2 +m2

0+ 1p2 + m2

04

. (4.30)

These integrals (as shown in the appendix C) are given by

∞∫−∞

dxe−t(x2+a2)

x2 + a2 = π

a

[1− erf(a

√t)].

This simplifies the heat kernel to

Kt(Q) = 1 + e−34m

2t − 2m2π

π

m(1− erf(m

√t))− e−

34m

2t m

2π2πm

[1− erf(m2

√t)]

= erf(m√t) + e−

34m

2t erf(m

2√t

). (4.31)

44


REMARK 4.4. The asymptotic behaviour in the area x ≈ 0 of the error function are givenas

erf(x) = 2x√π− 2x3

3√π

+O(x5),

so that (4.31) conducts

Kt(Q) = 3m√π

√t− 3m3

3√πt3/2 +O(t5/2), (4.32)

where the series expansion of the exponential function

exp(−x) = 1− x−O(x2)

is used as well.

In the limit t → 0, the asymptotic behaviour of the heat kernels are given by theWeyl law [Wey11]. A physical reasoning of this law can be found in the article ’Canone hear the shape of a drum?’ by Marc Kac [Kac66]. Heat kernels can be treated likethe integral kernel of the heat equation, therefore, physically, the limit t→ 0 representsthe initial state. Imagining, that the initial state of a heat source can be described bya Dirac δ-function, the boundaries of the system play no role in the limit t → 0. Thisis similar to a diffusion equation in statistical physics, where these thoughts can be aswell applied.

LEMMA 4.5: ASYMPTOTIC BEHAVIOUR OF THE HEAT KERNEL

In the limit t→ 0 the asymptotic expression of the heat kernels are given by

Kt(M + µ2) = L1L2√4πt

(a1 + a0t) +O(t3/2),

with

a1 = 3m2π , a0 = −3m

2π

(12m

20 + µ2

).

Proof of lemma 4.5. In Remark 4.4, the heat kernel of Kt(Q) was observed to behavelike

Kt(Q) t→0∼ t1/2.

The limit for the e−tµ2 part is due to the exponential function

e−tµ2 = 1− tµ2 −O(t2) (4.33)

45


so that

e−tµ2 t→0∼ 1.

This leads to the fact that the heat kernel of the Laplacian is viewed asymptotically.Here, the Weyl formula applies [ANPS09]

Kt(∆) = |Ω|4πt + α+O(t), (4.34)

where Ω = L1L2 is the area of integration and α is a constant not needed. Consequently,

Kt(∆) t→0∼ t−1.

Therefore, combing the above asymptotic behaviour for t→ 0 leads to

K(M + µ2)t

t→0∼ t−3/2,

implying the integral diverges at t = 0. To handle this divergency, the asymptoticbehaviour is examined later on in detail.Combining the asymptotic expression of the heat kernel of Q (4.32) and the series

expansion of e−tµ2 (4.33)

e−µ2tKt(Q) =[3m√

π

√t− 3m√

πt3/2

(12m

2 + µ2)]

+O(t5/2)

with (4.34), the term

Kt(M + µ2) = L1L2√4πt

[3m2π −

3m2π t

(12m

2 + µ2)]

+O(t3/2)

is found. For the integrand of the zeta function with

a1 = 3m2π , a0 = −3m

2π

(12m

20 + µ2

)is

tz−1Kt(M + µ2) = L1L2√4π

tz−1/2(a1

1t

+ a0 + a−1t+O(t2)).

Sequentially, the determined expression for the asymptotic expansion of the heat ker-nels will be used to split the integrands diverging parts from the non diverging parts inthe limits t→ 0 and µ2 → 0

46


LEMMA 4.6: SEPARATING THE DIVERGING PARTS

Using the asymptotic expansion derived in Lemma 4.5 the searched expression can betransformed into

ln det′MdetM0

= − ddz

∣∣∣∣z=0

1Γ(z)

∞∫0

dttz−1[Kt(M)− 1

], (4.35)

where Kt(M) is the difference of the heat kernels of Kt(M) and Kt(M0).

Proof of lemma 4.6. The asymptotical observations in Lemma 4.5 mean, that the diver-gencies, which come from µ → 0, can be isolated. The terms diverging for t → 0 areseparated with a1 := a1

L1L2√4π from (4.35) reads

ζM+µ2(z) = 1Γ(z)

∞∫0

dttz−1Kt(M + µ2) (4.36)

= 1Γ(z)

∞∫0

dttz−1Kt(M + µ2)− e−tµ2

−Θ(1− t)[a1t−1/2 − e−tµ2]

+ 1Γ(z) a1

1∫0

dttz−3/2

︸︷︷︸=:I1(z)

+ 1Γ(z)

∞∫1

dttz−1e−tµ2

︸︷︷︸=:I2(z)

, (4.37)

where Θ denotes the step function with Θ(x) = 1 for x > 0 and Θ(x) = 0 for x ≤ 0.The separation has been executed between the t ∈ [0, 1] interval and the t ∈ [1,∞]interval. In the t ∈ [0, 1] interval the asymptotic behaviour of Lemma 4.5 has been used.Combining the two intervals (4.36) is regained. Incidentally, the derivative

ddz

∣∣∣∣z=0

1Γ(z)I(z) = lim

z=0

( 1Γ′(z)I(z) + 1

Γ(z)I(z))

= limz=0

I(z)

with an integral I(z) will often be used, where the identities of the gamma function wereused according to appendix B.1.The last integral now will be evaluated. Since it converges for z = 0, the derivative

can be taken

ddz

∣∣∣∣z=0

I2(z) = ddz

∣∣∣∣z=0

1Γ(z)

∞∫1

dttz−1e−tµ2 =∞∫1

dtt−1e−tµ2

= −γ − lnµ2 +O(µ2). (4.38)

47


The last step results from the exponential integral Ei(x) and it’s series expansion (seee.g. [BMMS13])

Ei(x) = −∞∫−x

dte−t

t≈ γ + ln x+O(x),

where γ = 0.5772 . . . is the Euler-Mascheroni constant. The second integral in (4.37)will be evaluated for z > 1/2

I1(z) = 1Γ(z) a1

1∫0

dttz−3/2 = 1Γ(z) a1

22z − 1 (4.39)

The derivative at z = 0 is with abuse of the condition <(z) > 1/2 (for details see Remark4.7)

ddz

∣∣∣∣z=0

22z − 1 = −4

so that

ddz

∣∣∣∣z=0

I1(z) = ddz

∣∣∣∣z=0

1Γ(z) a1

1∫0

dttz−3/2 = −4a1. (4.40)

The first integral of (4.37) is renamed

ζµ(z) = 1Γ(z)

∞∫0

dttz−1Kt(M + µ2)− e−tµ2 −Θ(1− t)

[a1t−1/2 − e−tµ2]

,

where the index shows the value of the regularising mass µ2. Furthermore, it defines anew zeta function ζ ′µ(z). For t > 1, the above integral gives, due to the properties of thestep function,

ζ ′µ(z) := 1Γ(z)

∞∫0

dttz−1[Kt(M + µ2)− e−tµ2]

. (4.41)

This means, that including the values for t ≤ 1 from the step function

ζ ′µ(z) = ζµ(z) + 1Γ(z)

1∫0

dttz−1[a1t−1/2 − e−tµ2]

and, because the diverging parts have been split off, the limit µ = 0 can be taken here

ζ ′0(z) = ζ0(z) + 1Γ(z)

1∫0

dttz−1[a1t−1/2 − 1

]

48


where again for <(z) > 1/2

1Γ(z)

1∫0

dttz−1[a1t−1/2 − 1

]= 1

Γ(z)

( 2a12z − 1 −

1z

)

= 1Γ(z)

2a12z − 1 −

1Γ(z + 1) ,

using zΓ(z) = Γ(z + 1). Now taking the derivative, the first part equals (4.39), giving

ddz

∣∣∣∣z=0

( 1Γ(z)

2a12z − 1 −

1Γ(z + 1)

)= −4a1 − γ. (4.42)

(see appendix B.1). This concludes

ddz

∣∣∣∣z=0

ζ ′0(z) = ddz

∣∣∣∣z=0

ζ0(z)− 4a1 − γ. (4.43)

Remembering (4.15),

ln det′MdetM0

=− limµ→0

∞∫0


(4.36)= − lim

µ→0

[ ddz

∣∣∣∣z=0

ζM+µ2(z)− lnµ2]

(4.37)(4.38)= − ddz

∣∣∣∣z=0

ζ0(z) + 2a1 + γ + limµ→0

(lnµ2 − lnµ2

)(4.43)= − d

dz

∣∣∣∣z=0

ζ ′0(z)

can be found. Conclusively, this means, with the definition of (4.41), in the limit µ→ 0

ln det′MdetM0

= − ddz

∣∣∣∣z=0

1Γ(z)

∞∫0

dttz−1[Kt(M)− 1

], (4.44)

which concludes Lemma 4.6.

REMARK 4.7. The integrals in (4.40) and (4.42) were not defined for z = 0, thus thesolution

ddz

∣∣∣∣z=0

1∫0

dttz−3/2 = 4

is not valid for <(z) < 1/2 and requires a study of their analytic continuation to <(z) <1/2 or a dimensional renormalisation. But this is not necessary, because in the finalconclusion both of these integrals indicated with value 4 cancel out.

49


Now a simple expression of the heat kernel is found that can be used to separate thez-direction from the other directions. The following proof will complete Theorem 4.1.

Proof of theorem 4.1. With Kt(M) = Kt(Q)Kt(∆) the z-direction can be split off faraway from the other spatial directions

ζ ′(z) = ζ1(z) + ζ2(z) + ζ3(z) (4.45)

where

ζ1(z) = 1Γ(z)

∞∫0

dttz−1L1L2(4πt)

[Kt(Q)− 1

], (4.46)

ζ2(z) = 1Γ(z)

∞∫0

dttz−1 [Kt(∆)− 1] , (4.47)

ζ3(z) = 1Γ(z)

∞∫0

dttz−1[Kt(∆)−

(L1L24πt

)] [Kt(Q)− 1

], (4.48)

where in the symmetric case L × L the heat kernel of the Laplacian Kt(∆) can beidentified with the Jacobi θ-function

θ(z) =∑n∈Z

e−πn2z = z−1/2θ(1/z).

The left hand side of (4.45), the zeta function (4.44), can be achieved by summing thethree above integrals.

4.3. Calculating the interface tension

In the following, the previously obtained integrals will be evaluated. It can be seen, thatthe structure of the integrals ζ1 and ζ3 is not changing by reconsidering the geometryof the system. Thus, the solution of these integrals can be taken from the previousliterature.

Contribution of ζ1(z)From (4.29) with A := L1L2 the integration can be done

ζ1(z) = 1Γ(z)

∞∫0

dttz−1L1L2(4πt)−1[Kt(Q)− 1

]

= L1L24πΓ(z)

∞∫0

dttz−2e−3m2t/4 + L1L24πΓ(z)

∞∫0

dttz−2∞∫−∞

dpg(p)e−t(m2+p2)

50


This integral is solved in [Mün89] in four dimensions and in [Mün90] in three dimen-sions for the case L = L1 = L2. Another approach involving residue calculus and theproperties of the complex logarithm will be shown in appendix D. This provides thesolution

ddz

∣∣∣∣z=0

ζ1(z) = −L1L24π m2

(3 + 3

4 ln 3)

(4.49)

With L1 = L2 = L, it can be observed that the geometrical dependence does not changethe solution, which was obtained before.

Contribution of ζ2(z)The contribution of ζ2(z) is given by

ζ2(z) = 1Γ(z)

∞∫0

dttz−1 [Kt(∆)− 1] . (4.50)

The heat kernel Kt(∆), which is by knowing the eigenvalues of the Laplacian

Kt(∆) = Tr e−t∆, (4.51)=∑n1,n2

e−λn1,n2 t,

where the eigenvalues are given by

λn1,n2 = (2π)2[n2

1L2

1+ n2

2L2

2

].

In (4.50), the zero-mode of (4.51) for n1 = n2 = 0 is avoided by the subtraction of 1, soζ2 reads

ζ2(z) = 1Γ(z)

∑′

n1,n2

∞∫0

dttz−1e−λn1,n2 t

= 1Γ(z)

∑′

n1,n2

λ−zn1,n2

∞∫0

dttz−1e−t,

where the integral is exactly the gamma function that cancels out. Therefore,

ζ2(z) =∑′

n1,n2

λ−zn1,n2

= (2π)−2z∑′

n1,n2

[n2

1L2

1+ n2

2L2

2

]−z(4.52)

51


needs to be calculated. In the case of L1 = L2 = L, this function is called Epstein-Zeta function and has been calculated before. In the diploma thesis [Hop93], an explicitcalculation can be found. In [Eli95], another approach can be found. The result in thisliterature is

dζ2dz

∣∣∣∣z=0

= − ln L2

4π + 2 ln[√

2Γ(3/4)Γ(1/4)

]. (4.53)

≈ − ln L2

4π − 1.47634 . . . . (4.54)

The result implies, that there will be a similar first term ∼ lnL1L2 and a term, which isdepending on the geometric dependence of the problem, which must result in the secondterm of (4.53) for L1 = L2. Thus for L1 6= L2 equation (4.52) needs another calculationwhich will be done in the next chapter, including this geometric dependence.

Contribution of ζ3(z)The ζ3(z) contribution in [Mün90] is given by

ddz

∣∣∣∣z=0

ζ3(z) = 2

√√3Amπ

e−√

3mA/2 + faster decreasing terms.

In this calculation the properties of the Jacobi θ function were used, which do not applyin the present case, because there is no fixed L in both spacial directions. Nevertheless,it can be observed, that for large L, therefore for large L ∼

√A =

√L1L2 this term

vanishes. This observation will not be proved in detail here. These calculations of ζ1(z)and ζ3(z) leave the ζ2(z) contribution to be calculated.

4.3.1. Dimensional renormalization

The interface tension shall now be evaluated in dependence of physical quantities. Theseare given by the system area L1L2, the inverse mass m−1

R as the correlation length, andthe renormalized coupling gR. In the proceeding, the broken phase is observed in firstloop order. The relevant renormalization was derived in one loop order in [Mün90],explicit calculations are given for instance in [GKM96] and in the diploma thesis of M.Köpf [Köp08]. Setting the extra dimension d = 4 − ε in L =

√L1L2, all equations in

these works stay invariant. Therefore, the calculations will not be given in detail here.The renormalized coupling is given by

g = gR

(1 + 7

4uR8π

),

the renormalized mass is given by

m2 = m2R

(1− 3

8uR8π

)

52


with the dimensionless coupling

uR = gRmR

.

The aim is to give the dimensionless coupling uR in first order. In this one loop scheme,no orders above O(uR) fall into place. The energy of a kink

H0 = 2Am3

g

shall be expressed by the renormalized values. Therefore, a series expansion up to thefirst order is used for the mass

m = mR

√1− 3

8uR8π = 1− 3

128πuR +O(u2R) (4.55)

⇒ m3 = m3R

(1− 3

128πuR +O(u2R))3

= m3R

(1− 9

128πuR +O(u2R))

and the coupling constant

1g

= 1gR

(1− 7

32πuR +O(u2R)).

Now the dimensionless constant becomes

m3

g= m3

R

gR

(1− 9

128πuR +O(u2R))(

1− 732πuR +O(u2

R))

= m3R

gR

(1− 37

128πuR +O(u2R)). (4.56)

4.3.2. Geometric independence of the interface tension

The contributions of to the energy splitting of ζ1(z) and ζ3(z) were obtained before. Thecontribution of ζ2(z) contained a term dependent on the system size and another termindependent of the system size for L1 = L2 = L. Assuming, for the setup L1 6= L2, thiswill lead to a part depending on the system size and another term depending on the aspectratio L1/L2 of the system. The contribution of ζ2(z) has as well been seen only in theprefactor. Thus, in this section, it will be assumed that again only the prefactor changesbecause the analytic calculation of ζ2(z) in [Mün90] was similar. The derivative of thesum over the eigenvalues λn1,n2 will as well give a logarithmic contribution, dependingon the geometric dependence of the system which will expressed by the aspect ratioτ = iL1/L2. Thus, the prefactor is C(τ) in [Mün90]

∆E = C(τ) exp(−σ(A)A), (4.57)

53


setting A :=√L1L2. In conclusion, the exponent of the energy splitting (4.1)

∆E = 2e−H0

√H02π

∣∣∣∣∣ det′MdetM0

∣∣∣∣∣−1/2

can be calculated now depending only on ζ1(z). Here, H0 = 2Am3/g and

ln∣∣∣∣∣ det′MdetM0

∣∣∣∣∣ = − ddz ζ1(z)

∣∣∣∣z=0− d

dz ζ2(z)∣∣∣∣z=0− d

dz ζ3(z)∣∣∣∣z=0

.

It has been observed that ζ3(z) contains no relevant contribution to the interface tension.The following results were obtained for

ζ ′1(0) = − A

4πm2(

3 + 34 ln 3

)with A = L1L2. and in [Mün90]

ζ ′2(0) = − ln A

4π −B(τ),

where B(τ) is still to be calculated, because of the geometric dependence of this factor.Therefore, the determinants become∣∣∣∣∣ det′M

detM0

∣∣∣∣∣−1/2

= exp[− A

8πm2(

3 + 34 ln 3

)− 1

2 ln A

4π −B(τ)

2 + . . .

]

= e−B(τ)/2√A4π

exp[− A

8πm2(

3 + 34 ln 3

)+ . . .

].

This means, the energy splitting is given by

∆E = 2√

2Am3

2πge−B(τ)/2√

A4π

exp[−2Am3

g− A

8πm2(

3 + 34 ln 3

)+ . . .

]

= 4√

2m3

g

e−B(τ)/2√

2exp

[−2Am3

g− A

8πm2(

3 + 34 ln 3

)+ . . .

]= C(τ) exp(σ∞A),

where

σ∞ = limA→∞

1Aσ(A) = lim

A→∞

(2m3

g+ m2

8π

(3 + 3

4 ln 3)

+ . . .

)

= 2m3

g+ m2

8π

(3 + 3

4 ln 3).

54


In a last step, this interface tension is expressed by the physical variables achieved bydimensional renormalization. It follows with (4.56) and (4.55)

σ∞ = 2m3

g+ m2

8π

(3 + 3

4 ln 3)

= 2m3R

gR

(1− 37

128πuR +O(u2R))

+m2R

(1− 3

128πuR)

8π(

3 + 34 ln 3

)= 2m

2R

uR− 37

64πm2R +m2

R

18π

(3 + 3

4 ln 3)−m2

RuR3

128π1

8π

(3 + 3

4 ln 3)

= 2m2R

uR

(1− uR

4π

(1332 + 3

16 ln 3)

+O(u2R)).

This is precisely the result obtained in [Mün90]. Nevertheless, the precise calculation forthe case L1 = L2 = L in [Hop93] indicates, that the interface tension will not changeunder other geometries. But the prefactor C can contain a dramatic change for theenergy splitting, thus for the correlation length of the system. With these calculationsit should be clear, that for the case with geometric dependence on a L1 × L2 geometry,the prefactor must follow two restrictions:

• It must provide the result (4.53) for L1 = L2, which means for the aspect ratioτ = i. This indicates, that the theory can still be evaluated in the scheme of zetaregularization techniques.

• A certain modular invariance must be given, as the system is invariant under thechange of L1 ↔ L2, which means, that the prefactor must be the same for τ → 1/τ .

These considerations will lead to the formulation of the theory for the geometric depen-dence of the Laplace operator under the modular invariance τ → 1/τ .

55


56

5THE DETERMINANT OF THE LAPLACE OPERATOR

In the last chapter, the integral ζ2, which is the determinant of the two dimensionalLaplacian, was identified to play the key role for the geometric dependence. In theliterature, there are some approaches to calculate the determinant of the Laplacian thatwill be used here. The first approach is due to D. B. Ray and I. M. Singer [RS73] and inphysics C. Itzykson and J.-B. Zuber [IZ86]. Mathematically, this approach is even older,going back to the Kronecker limit formula [Sie61]. Thus, this calculation will need a deepknowledge of special functions, many properties are given in the appendix B. Then, thecalculation is some kind of straight forward and easy to understand. By this connectionit this calculation will be subsumed under the title Kronecker limit formula.The second approach uses less constructions over special functions, but a many prop-

erties of complex integrals and complex analysis, mostly governed by the Cauchy residuetheorem. The key idea is due to [Pol86] a Sommerfeld-Watson transformation. Therefore,this section is named after this transformation.The last approach on the determinant differs completely from the other two. In this

approach the structural analogy to a Bosonic string will be used in order to evaluatethe upcoming expressions and the Laplacian will be identified with an infinite set ofharmonic oscillators coming from the massless Klein-Gordon equation. This approachis motivated by J. Baez [Bae98a,Bae98b].Calculating the determinant of the Laplacian on an infinite plane, the plane is equiva-

lent to a sphere, which is a Riemanian surface. In the context of critical phenomena, thesimplest non-spherical case of a Riemanian surface is a torus, which is identical to a planewith periodic boundary conditions in two directions. The two directions are separatedin a holomorphic and anti-holomorphic sector. The interactions between these sectorsare revealed by modular transformations [DFMS97]. The torus can be imagined as theplane with periodic boundaries, where one direction is transformed by an exponentialtransformation into a circle, which leads to a cylinder. In the other direction, the peri-odic boundary conditions also apply, so that the endings of the cylinder are connectedleaving the form of a torus (see figure 5.1). This transformation is a conformal mapping,

57

THE DETERMINANT OF THE LAPLACE OPERATOR

Figure 5.1.: Transforming a plane with periodic boundary conditions into a torus.

leaving angles and local distances preserved but not distances at all. Thus, the torus canbe defined by two independent lattice vectors on a plane. On the complex plane, thesevectors are represented by two complex numbers ω1 and ω2, where the area of the torusis A = =(ω2ω1). The modular ratio therefore is τ = ω2/ω1. Under the imposed periodicconditions in x1 and x2, direction the functional is invariant with respect to translationsby the periods ω1 and ω2.

5.1. Kronecker limit formula

In this system, the function is restricted on asymmetric domains L1×L2 periodic bound-aries. This calculation follows the work of C. Itzykson and J.-B. Zuber [IZ86] and ispresented more detailed in the textbook [DFMS97]. Historically this method follows anapproach in mathematics outlined by D. B. Ray and I. M. Singer ’Analytic Torsion ontwo dimensional manifold’ using the Selberg trace [Sel56] which serves as an analogy tothe Poisson summation formula used in [Mün89]. The eigenfunctions of the Laplacian∆ for a system with periodic boundaries are given by the equation

∆ϕ(x1, x2) = λn1,n2ϕ(x1, x2)

where the eigenvalues λn1,n2 are given by

λn1,n2 = (2π)2[n2

1L2

1+ n2

2L2

2

]. (5.1)

By introducing the values

ω1 = L1, ω2 = iL2

with ω1 = ω1, ω2 = −ω2

a lattice Λ can be defined on a torus T = C/Λ

Λ = ω1n+ ω2m|n,m ∈ Z .

58


The dual lattice Λ∗ is then given by the generators

k1 = −iω2A, k2 = i

ω1A

with k1 = k1, k2 = −k2,

where A = Imω2ω1 is the area A = L1L2, which remains invariant under the modulartransformation. The eigenvalues take the form

(2π)2[(

n1L2A

)2+(n2L1A

)2]

= (2π)2[(

in1ω2A

)2+(n2ω1A

)2]

which can be rewritten in terms of the dual lattice(2πA

)2 [−n2

1ω22 + n2

2ω21

]=(2πA

)2|n1ω2 − n2ω1|2 . (5.2)

Keeping in mind that ω2 ∈ iR, resulting in ω2 = −ω2, now the eigenvalues (5.1), become

λn1,n2 = (2π)2 |n1k1 + n2k2|2 = (2π/A)2 |n1ω2 − n2ω1|2 .

The modular ratio is given by τ = ω2/ω1 = iL1/L2, which can be transformed in thefollowing way

τ−1 = ω1/ω2 = −k2/k1 = τk.

Introducing the modular ratio τ = ω2/ω1, the sum over the eigenvalues (4.52) can beexpressed by

−12 ln det′∆ =

∑lnλ−1/2

n1,n2 = 12G′(0) (5.3)

with the Eisenstein series

G(s) =∣∣∣∣ A

2πω1

∣∣∣∣2s ∑′

m,n

1|m+ nτ |2s

. (5.4)

Now the equivalence of the right hand side of (5.3) to (4.52) can be proven

12G′(s) =

∣∣∣∣ A

2πω1

∣∣∣∣2s log(

A

2πω1

) ∑′

m,n

1|m+ nτ |2s

−∣∣∣∣ A

2πω1

∣∣∣∣2s ∑′

m,n

log |m+ nτ ||m+ nτ |2s

and12G′(0) =

∑′

m,n

log A

2πω1|m+ nτ |,

so that

exp(12G′(0)) =

∏m,n

A2πω1

|m+ nτ |.

59


Conclusively, the Eisenstein series given in (5.4) can be used to evaluate the sum. Thefunction G(s) is analytic for Re s > 1 but can be continued [DFMS97]. Defining

G(s) =(

4π2

τ22

)−2s ∑′

n,m

[(m+ nτ)]−2s

=(

4π2

τ22

)−2s2ζ(2s) +

∑′

n

(∑m

[(m+ nτ)]−2s)

, (5.5)

where ζ(s) is the Riemann ζ-function with τ = τ1 + iτ2. Due to the representations ofthe cotangent [BMMS13], the first step is achieved by

π cot(πz) = 1z

+∞∑d=1

( 1z − d

+ 1z + d

)

π cot(πz) = πi− 2πi∞∑m=0

qm(z), q(z) = e2πiz.

Differentiating both representations (k − 1) times with respect to z gives

(−1)k−1Γ(k)∞∑d=1

( 1z + d

)k= −(2πi)k

∞∑m=1

mkqm

⇒∞∑d=1

( 1z + d

)k= (2πi)k

Γ(k)

∞∑m=1

mk−1qm,

where the fact can be used, that k is even. Therefore, the lattice consists for any cz + d∑′

c,d

( 1cz + d

)k=∑′

d

1dk

+∑′

c

(∑d

1(cz + d)k

)

This has been used in (5.5). The sum over m is periodic in the real part of nτ withperiod 1. With τ = τ1 + iτ2 the following can prove,

f(nτ1) =∞∑

m=−∞

1|m+ n(τ1 + iτ2)|2s

= . . .+ 1| − 1 + n(τ1 + iτ2)|2s + 1

|n(τ1 + iτ2)|2s + 1|1 + n(τ1 + iτ2)|2s + . . .

=∞∑

m−1=∞

1|m+ (nτ1 + 1) + iτ2|2s

=∞∑

m=∞

1|m+ (nτ1 + 1) + iτ2|2s

= f(nτ1 + 1),

because the sum goes over all m ∈ Z. A function f(x) periodic in [0, 1] can now beexpressed by a Fourier expansion

f(x) =∑n∈Z

cne2πinx, cn =1∫

0

dxf(x)e−2πinx.

60


As it was proven, the function f(nτ1) above is a periodic in [0, 1], so the Fourier expansionreads

f(nτ1) =∑p∈Z

e2πipnτ1

1∫0

d(nτ1)∑m

1|m+ n(τ1 + iτ2)|2s e−2πipnτ1

=∑p∈Z

e2πipnτ1

1∫0

dy∑m

1|m+ y + inτ2|2s

e−2πipy

and

∑m

1|m+ nτ |2s

=∑l

e2iπlnτ1

1∫0

dye−2iπly∑m

1|m+ y + inτ2|2s

=∑l

∞∫−∞

dye2iπl(nτ1−y) 1|m+ y + inτ2|2s

= 1Γ(s)

∑l

∞∫−∞

dy∞∫0

dtts−1e2iπl(nτ1−y)−t(y2+n2τ22 )

where the last transformation is due to the representation of the gamma function

1zk

= 1Γ(k)

∞∫0

dttk−1e−zt,

so that

1|m+ y + inτ2|2s

= 1Γ(s)

∞∫0

dtts−1e−t(y2+n2τ22 ).

Evaluating the integral with respect to y leads to

∑m

1|m+ nτ |2s

= π1/2

Γ(s)∑l

∞∫0

dtts−3/2e−(tn2τ22 +π2l2/t−2iπlnτ1)

=√π

Γ(s− 1/2)Γ(s) |nτ2|1−2s

+√π

Γ(s)∑′

l

e2iπlnτ1

∣∣∣∣ πlnτ2

∣∣∣∣s−1/2 ∞∫0

dtts−3/2e−π|ln|τ2(t+1/t), (5.6)

61


because for l = 0∞∫0

dtts−3/2e−tn2τ22 =

∞∫0

dtn2τ2

2

(t

n2τ22

)s−3/2e−t

=( 1n2τ2

2

)s−1/2 ∞∫0

dtts−3/2e−t

= |nτ2|1−2sΓ(s− 1/2)

with

Γ(z) =∞∫0

dttz−1e−t.

For the second term in (5.6), the substitution t→ |πl/(nτ2)|t is used as succeeding

∑′

l

∞∫0

dtts−3/2e−(tn2τ22 +π2l2/t−2iπlnτ1)

=∑′

l

e2iπlnτ1

∞∫0


∣∣∣∣ dt ∣∣∣∣ πlnτ2t

∣∣∣∣s−3/2e−πnlτ2(t+1/t)

=∑′

l

e2iπlnτ1


∣∣∣∣s−3/2 ∞∫0

dt |t|s−1/2 e−πnlτ2(t+1/t).

Consequently, the term results in

G(s) =∣∣∣∣∣4π2

τ22

∣∣∣∣∣−2s

2ζ(2s)

+√π

Γ(s− 1/2)Γ(s)

∑′

n

|nτ2|1−2s

+√π

Γ(s)∑′

n

∑′

l

e2iπlnτ1


∣∣∣∣s−1/2 ∞∫0

dtts−3/2e−π|ln|τ2(t+1/t). (5.7)

Summing the second term over n, subsequent functional relation (reflection formula B.3)can be used

π−s/2Γ(s/2)ζ(s) = π1/2−s/2Γ(1/2− s/2)ζ(1− s)

with the definition of the zeta function√π

Γ(s− 1/2)Γ(s)

∑′

n

|nτ2|1−2s = 2√π

Γ(s− 1/2)Γ(s) |τ2|1−2sζ(2s− 1)

= 2Γ(s)π

s−1/2|τ2|1−2sΓ(1− s)ζ(2− 2s).

62


In order to find the value of the derivative G′(s) at s = 0, the function G(s) is expandedto first order around s = 0. For the first term in (5.7), this leads to

2ζ(2s) = 2ζ(0) + 2sζ ′(0) +O(s2)= −1− 2s ln(2π) +O(s2)

with ζ(0) = −1/2 from (B.22) and 2ζ ′(0) = − ln(2π). The value of the second term in(5.6) becomes

Γ(1− s)ζ(2− 2s)Γ(s) = (1 + sΓ′(1) +O(s2))

(π2

6 + 2sζ ′(2) +O(s2))(s+O(s2))

= π2

6 s+O(s2),

where the value ζ(2) = π2/6 (B.23), the value of the derivative (B.24) and the seriesexpansion 1/Γ(s) = s + s2 + O(s3) is used. The third term can as well be expandedaround s = 0. The integral (see for instance [Sch81])

∞∫0

dtt−3/2e−α(t+1/t) =√π

αe−2α

together with the series expansion gives in the last term of (5.7)

1Γ(s)e2iπlnτ1

∞∫0

dtts−3/2e−π|ln|τ2(t+1/t)

= se2iπlnτ1

√π

π|ln|τ2e−2π|ln|τ2 +O(s2).

These calculations lead to

G(s) = −1− 2s ln∣∣∣∣ A

2πω1

∣∣∣∣− 2s ln 2π + π

3 sτ2

+ s∑′

n

∑′

l

1|l|

e2iπlnτ1−2π|ln|τ2 +O(s2).

With the definition

q(τ) = e2πiτ , η(τ) = q1/24(τ)∞∏n=1

(1− qn(τ))

63


of the Dedekind eta function η(τ) and because of |A/ω1| =√Aτ2, the derivative of G(s)

close to s = 0 reads

G′(s) = − ln τ2 + π

3 τ2 + η(τ)η(τ)

G′(0) = −2 ln(√

Aτ2|η(τ)|2)

= −2 ln(√

L1L2

√L1L2|η(τ)|2

)

= − ln L1L24π − 2 ln

(√

4π√L1L2|η(τ)|2

).

For L1 = L2 = L

G′(0) = − ln L2

4π − 2 ln(√

4π|η(i)|2)

= − ln L2

4π − 2 ln(√

4π|η(i)|2)

≈ − ln L2

4π − 1.47634 . . .

is found.

5.2. Sommerfeld-Watson transformation

A second way of calculating the determinant follows J. Polchinski [Pol86]. In this cal-culation, the Sommerfeld-Watson transformation plays a key role. The aim of thistransformation is to convert a slowly converging series into an contour integral usingthe Cauchy residue theorem. The sum as a sum over the residues corresponds to theintegrand’s poles and the integral can be evaluated.The problem of calculating infinite series over complex poles arises in the early 20th

century, when wireless communication began to grow. A. Sommerfeld calculated thepropagation of electro-magnetic surface-waves and transformed the contour of an inte-gral over infinite poles into a contour excluding only two branch cuts [Som09]. The otherrelated work is due to G. N. Watson on a similar problem, calculating by transforminga sum into a complex integral. This is called Watson transformation. The first usage ofthis transformation goes back to G. N. Watson [Wat18] who calculated the diffractionof electric fields in presence of the earth. A. Sommerfeld combined both methods. Thena different theory of diffraction, using integrals, where he transformed the integral overinfinite poles into an integral using only one branch cut, was created. The present formu-lation goes back to A. Sommerfeld applying the Watson transform in order to calculateinfinite series coming from spherical functions [Som47] arising from the calculation ofelectro magnetic for the above mentioned problem. It has later been used in order to cal-culate scattering amplitudes [Reg59]. These calculations became important in the high

64


energy sector of strong interactions, then the Regge theory was succeeded by quantumchromo dynamics.At first, the present form of ζ2 needs to be rewritten. The calculation begins with the

left side of (5.2) by transforming it to(2πA

)2 [−n2

1ω22 + n2

2ω21

]=(2πA

)2(n2ω1 − n1ω2)(n2ω1 + n1ω2)

=(2πAω1

)2(n2 − n1τ)(n2 + n1τ)

=(2πAω1

)2(n2 − n1τ)(n2 − n1τ)

= τ2A

4π2

τ2(n2 − n1τ)(n2 − n1τ),

where in the third step the fact is used that τ = τ1 + iτ2 is purely imaginary. The laststep is due to comparison reasons with Polchinski. The ζ2 function (4.52) can now beexpressed as

ζ2(s) =(τ2A

)−s ∑′

n1,n2

[4π2

τ22

(n2 − n1τ)(n2 − n1τ)]−s

ddz ζ2

∣∣∣∣z=0

= ln τ2A−G′(0), (5.8)

where the fact was used, that the sum gives1

∞∑′

n1=−∞

∞∑′

n2=−∞1 = 2ζ(0) = −1

to the derivative of the prefactor. The function G(s) is defined as

G(s) = −∑′

n1,n2

[4π2

τ22

(n2 − n1τ)(n2 − n1τ)]−s

= −∑′

n1,n2

(

4π2

τ22

)−s ∑n1,n2

[(n2 − n1τ)(n2 − n1τ) + µ2 − µ2

]−s= − lim

µ2→0

(

4π2

τ22

)−s ∑n1,n2

[(n2 − n1τ)(n2 − n1τ) + µ2

]−s−(

4π2µ2

τ22

)−s .The sum is prevented from diverging because the regularizing mass µ2 is taken intocalculation. The calculations are similar to the ζ-function regularization, meaning here,

1there is a small error in [Pol86]

65


that the derivative of G(s) at the value s = 0 is searched.The derivative of G(s) is givenby

ddsG(s)

∣∣∣∣s=0

= − lims→0µ2→0

dds

(

4π2

τ22

)−s ∑n1,n2

[(n2 − n1τ)(n2 − n1τ) + µ2

]−s−(

4π2µ2

τ22

)−s .The simplest part can be evaluated directly, as

lims→0

dds

(4π2µ2

τ22

)−s= lim

s→0

ddse

−s ln(

4π2µ2

τ22

)

= − lims→0

ln(

4π2µ2

τ22

)(4π2µ2

τ22

)−s= 2 ln τ2 − 2 ln 2πµ. (5.9)

The modular ratio was chosen to be a complex number, so splitting τ = τ1 + iτ2 thesummand gets

(n2 − n1τ)(n2 − n1τ) + µ2 = n22 + n2

1(τ1 + iτ2)(τ1 − iτ2)− 2n1n2τ1 + µ2

= (n2 − n1τ1)2 + n21τ

22 + µ2

= (n2 − n1τ1)2 +Q2(n1, µ),

where

Q2(n1, µ) = n21τ

22 + µ2. (5.10)

The sum becomes∑n1,n2

[(n2 − n1τ1)2 +Q2(n1, µ)

]−s=

∞∑n2=−∞

∞∑n1=−∞

[(n2 − n1τ1)2 +Q2(n1, µ)

]−s︸︷︷︸

=f(n2)

,

where the n2 sum can be evaluated explicitly using the Sommerfeld-Watson transforma-tion. Then

∞∑n=−∞

(−1)ng(n) = 12i

∮C

dz g(z)sin πz .

The contour C encloses all poles on the real axis, that are all zeros of sin(πz), which arethe numbers z = n, n ∈ Z. For non-alternating series f(n), this can be converted intointo an integral using g(n) = e−iπnf(n). This choice forces a changing of sign, because

e−iπz

2i sin πz < −1, for =(z) > 0,

e−iπz

2i sin πz > 0, for =(z) < 0.

66


<z

=z

× × ×××××

×z+

Figure 5.2.: Contour C closing at z → ±∞.

Therefore, introducing the substitution

− e−iπz

2i sin πz −12 , for =(z) > 0,

e−iπz

2i sin πz + 12 , for =(z) < 0,

can be used to change the integral into an integral using both contours C+ : z,∞+iε <z < −∞+ iε and C− : z,−∞− iε < z <∞− iε, (see Figure 5.2)

∞∑n2=−∞

f(n2) = −∫C+

dzf(z)(

e−iπz

2i sin πz + 12

)+∫C−

dzf(z)(

e−iπz

2i sin πz + 12

)

=(∫

C−−∫C+

)dz2 f(z)︸︷︷︸

=:I1

+∫C−

dzf(z) e−iπz

2i sin πz −∫C+

dzf(z) e−iπz

2i sin πz︸︷︷︸=:I2

.

In the succeeding, both integrals will be evaluated and their derivative will be taken.

Evaluating the I1 integral

For the I1 integral there is

I1(s) = 12

∞∑n1=−∞

(∫C−−∫C+

)dz[(z − n1τ1)2 +Q2(n1, µ)

]−s,

which by a shift of z → z + n1τ1 does not change on account of the infinite integrationlimits. For this reason, the new function is symmetric f(z) = f(−z) and under asubstitution in C+ with z → −z, it can be lined out that

I1(s) =∞∑

n1=−∞

∞∫−∞

dz[z2 +Q2(n1, µ)

]−s, (5.11)

67


<z

=z

Cr

CR−CR+

L

C1C2

×z+

Figure 5.3.: Contour Γ+ for integration.

where the limit ε → 0 has been taken. This integral can be evaluated using the Eulerbeta function, what is done in the appendix B.2. Using (5.10), the result is

lims→0

ddsI1(s) = −

∑n1

2πQ(n1, µ)

= −2πµ− 2πτ2∑′

n1

n1, (5.12)

where, in the last step, the n1 = 0 contribution was split up from the sum, and the limitµ2 → 0 has been taken. Using the properties of the Riemann zeta function at ζ(−1),(see appendix B.3) the contribution is

lims→0

ddsI1(s) = −2πµ− 2πτ22ζ(−1)

= −2πµ+ πτ23 . (5.13)

Evaluating the I2 integral

For the I2 integral there is

I2(s) =∞∑

n1=−∞

∫C−

dz e−iπz

2i sin πz[(z − n1τ1)2 +Q2(n1, µ)

]−s−

∞∑n1=−∞

∫C+

dz e−iπz

2i sin πz[(z − n1τ1)2 +Q2(n1, µ)

]−s,

where the contour C+ is given in Figure 5.2. This integral converges for s = 0, so firstthe derivative can be taken. The derive of the integral over the contour C+ is given by

lims→0

dds

−∞+iε∫∞+iε

dz[z2 + µ2

]−s e−iπz

2i sin πz = −−∞+iε∫∞+iε

dz e−iπz

2i sin πz ln[z2 + µ2

].

68


Furthermore, there is a branch point for the logarithm, when the argument is zero orless. This will be used to deform the contour C+ to a contour Γ+, which is given inFigure 5.3. From Cauchy’s residue theorem, it is known that for the new contour∮

Γ+= 0

because there are no singularities inside the contour Γ+, but there is a branch point atν+ = n1τ1 + iQ(n1, µ) = iz+ outside of the contour. While evaluating the integrals, thelimits R → ∞ and r → 0 will be taken. The part L of Γ+ running from ∞ + iε to−∞+ iε is given by

−∫L

=∫CR1

+∫CR2

+∫C1

+∫C2

+∫Cr.

It can be deduced that (substituting z = z+ + reiφ, φ ∈ (0, 2π))∣∣∣∣∫Cr

∣∣∣∣ ≤ 2πr →r→0

0,

as well as for the other contributions (substituting z = Reiθi , θ1 ∈ (π, 3π/2), θ2 ∈(3π/2, 2π)), it consequently ends in∣∣∣∣∣

∫CR+

+∫CR−

∣∣∣∣∣ →R→∞ 0,

due to [AGW96]. This means ∫L

= −∫C1−∫C2, (5.14)

where the contour of C1 runs from iz+ to i∞ and for C2 from i∞ to iz+ with z+ > 0:

C1 : y ∈ iR : iz+ ≤ y < i∞ and C2 : y ∈ iR : i∞ < y ≤ iz+

With a change of variables y = iz and dz = −idy for r → 0 the integral using contourC1 is given by ∫

C1= i

∞∫z+

dy e−πy

2i sin(−πiy) ln[−y2 + µ2

]

= i∞∫z+

dy e−πy

2 sinh(πy) ln[−y2 + µ2

]and the integral using C2 is given by∫

C2= i

z+∫∞

dy e−πy

2i sin(−πiy) ln[−y2 + µ2

]

= −i∞∫z+

dy e−πy

2 sinh(πy) ln[−y2 + µ2

].

69


To such a degree, the integrals should as well cancel out. This is not the case, as thereis a gap of 2δ over the branch cut of the logarithm. The complex logarithm will givea contribution of 2πi because the value is different when the branch cut is approachedfrom the left or from the right side by the contour. Now, the logarithm can be studiedexplicitly. Rewriting

ln(−y2 + µ2) = ln [(iy + µ)(iy − µ)]= ln(iy + µ) + ln(iy − µ),

while iy → i(y + z+) = iy − µ this is for the C1 part

ln(−y2 + µ2) = ln(iy − µ+ µ) + ln(iy − 2µ)= ln(iy + δ) + ln(iy − 2µ)

=δ→0+

ln y − π

2 i + ln(iy − 2µ)

as a result of the complex logarithm and because the branch cut is approached from theright side. For the C2 part, the same substitution gives

ln(−y2 + µ2) = ln(iy − µ+ µ) + ln(iy − 2µ)= ln(iy − δ) + ln(iy − 2µ)

=δ→0−

ln y + 3π2 i + ln(iy − 2µ)

so that

ln(−y2 + µ2)− ln(−y2 + µ2) = −π2 i− 3π2 i = −2πi.

Consequently, using the proposed substitution iy → i(y+z+) = iy−µ, the integrals overthe contours C1 and C2 are∫

C1+∫C2

= −i∞∫0

dy e−π(y+z+)

2 sinh [π(y + z+)] (−2πi)

= −2π∞∫0

dy e−π(y+z+)

2 sinh [π(y + z+)]

= ln[1− e−2πz+

].

Finally, the integration over the L part leads to∫L

= − ln[1− e−2πz+

],

as the sum of integrals over the contours C1 and C2 is the negative integral over L(5.14). For the C− contour, the same calculation applies using a contour Γ− in thenegative complex plane. The contribution will be∫

C−= − ln

[1− e−2πz−

],

70


ergo the derivative of I2 leads with z+ = −in1τ1+√n2

1τ22 + µ2 to the following expression:

lims→0µ2→0

ddsI2(s) = −2 lim

µ2→0

∑n1

ln[1− e−2πz+

]

= −2 limµ2→0

ln[1− e−2πµ

]− 2

∞∑n1=−∞n1 6=0

ln[1− e−2π(−in1τ1+n1τ2)

]

= −2 limµ2→0

ln[1− e−2πµ

]− 4

∞∑n1=1

ln[1− e−2πin1τ

], (5.15)

where again the the zero mode n1 = 0 was taken out, so that in the second term, thelimit µ2 → 0 could be taken. The first expression can be expanded because the limitµ2 → 0 will be taken later on

−2 ln[1− e−2πµ

]= −2 ln 2πµ+ 2πµ+O(µ2). (5.16)

Conclusively,

lims→0

ddsG(s)| = 2 ln τ2 − 2 ln(2πm)︸︷︷︸

(5.9)

+ 2πµ︸︷︷︸(5.12)

− πτ23︸︷︷︸

(5.13)

+ 2 ln(2πµ)− 2πµ︸︷︷︸(5.16)

+ 4∞∑

n1=1ln[1− e2πin1τ

]︸︷︷︸

(5.15)

,

so that the the zero mode cancels out and limit µ2 → 0 can be taken. The expression

−πτ23 + 4

∞∑n1=1

ln(1− e2πin1τ2

)= 2 ln |η(τ)|2

is just connected with the Dedekind eta function defined by

η(τ) = q1/24∞∏k=1

(1− qk

), q = e2πiτ ,

whose properties will be investigated later on. Combining the calculations

G′(0) = 2 ln |η(τ)|2 + 2 ln τ2 = 2 ln τ2|η(τ)|2

and adding the prefactor (5.8),

ζ ′2(0) = ln τ2A−G′(0)

= − lnA+ ln τ2 − 2 ln τ2|η(τ)|2

= − lnA− 2 ln√τ2|η(τ)|2

71


is obtained. This can be transformed in

ddz ζ2(z)

∣∣∣∣z=0

= − ln A

4π − 2 ln(√

4πτ2|η(τ)|2)

= − lnA− 2 ln(√

τ2|η(τ)|2)

= −2 ln(√

Aτ2|η(τ)|2).

5.3. Massless Klein-Gordon field

In a new setup, a more ’physical’ approach will be examined. This goes back to ablog entry of John Baez [Bae98a, Bae98b]. The main idea is to use the analogy, thatthe investigated field is analogous to field of a massless field theory, the massless Klein-Gordon field. As it has been seen before, the eigenvalues of the Laplacian are givenby

λn1,n2 = (2π)2[n2

1L2

1+ n2

2L2

2

], ni ∈ Z.

The determinant is ’ill-defined’ by the product

det′∆ =∏n1,n2

λn1,n2 ,

which can be redefined via zeta function regularizing

ln det′∆ = − limz→0

ddz ζ∆(z),

where the spectral zeta function is

ζ∆(z) =∑′

λ−zn1,n2 .

The partition function is given by a path integral, where the Euclidean action S is usedinstead of the Hamiltonian from the chapters before. Especially for a Euclidean freescalar field ϕ on L1 × L2, the action functional is defined as [PS95]

S = 12

∫L1×L2

d2xϕ(x)∆ϕ(x).

The partition function is given by the path integral

Z =∫Dϕe−S = (det′∆)−1/2.

For a scalar field with mass µ the Laplacian ∆ can be replaced by

∆→ ∆µ = ∆ + µ2,

72


which is needed later on.In statistical physics, the partition function can as well be written in terms of the

trace, as it has been used in the introductory chapters. In this section, the value L2 isused as the Euclidean time. The value L1 then represents one space dimension. Thepartition function is defined as

Z = Tr e−HL2 , (5.17)

where H is the Hamiltonian of the scalar field. The scalar field represents a free masslessKlein-Gordon field, where the Hamiltonian is given by a collection of harmonic oscillators

H =∑k

ω(k)[a†(k)a(k) + 1

2

].

Here a† and a are the common creation and annihilation operators. The circular fre-quency is given by

ω(k) =√k2 + µ2 µ=0= |k|, k = 2π

L1n, n ∈ Z. (5.18)

To perform the trace, the energies for the massless case are needed, which are given by

E(ln) =∑n∈Z

2πL1|n|(ln + 1

2

).

This calculation is as well known from the quantum mechanic harmonic oscillator. Num-ber l represents the different energy levels, which are possible for the collection of nharmonic oscillators given by the Hamiltonian H. The Klein-Gordon field used here isgiven by all frequency modes. This means, that the sum over n is over all numbers in Z.With the energies given, the trace can be evaluated, as the energies are the eigenvaluesof the Hamiltonian. The partition function now reads

Z =∏n∈Z

∞∑l=0

e−L22πL1

(ln+1/2)|n|

=∏n∈Z

e−πL2L1|n|

1− e−2πL2L1|n|

=∏n∈Z

q1/2|n|(1− q|n|)−1

which is owing to the representation of the geometric series, q is given by q(τ) = e2πiτ =e−2πL2/L1 . Omitting the zero mode for n = 0 in the partition function, a new partition

73


function is given by

Z ′ =∏

n∈Z\0q1/2|n|(1− q|n|)−1

=∞∏n=1

qn(1− qn)−2

= q∑∞

n=1 n

[ ∞∏n=1

(1− qn)]−2

. (5.19)

Now, the surprising result of the zeta function regularization (see appendix B.3) is neededin order to handle the infinite sum in the exponent of the prefactor

1 + 2 + 3 + . . . =∞∑n=1

n = ζ(−1) = − 112 .

This leads to

Z ′ = q−1/12[ ∞∏n=1

(1− qn)]−2

= |η(τ)|−2

with the Dedekind eta function

η(τ) = q1/24[ ∞∏n=1

(1− qn)].

From the path integral (5.17), it is known that

det′∆ ∼ |η(τ)|4,det′∆ = C|η(τ)|4,

where a constant C is needed as a consequence of the contribution of the zero mode. Inthe last steps, the divergency for n = 0 was excluded. To handle this factor in aboveequation, the case of a massive Klein Gordon field has to be analyzed

det ∆µ = det′∆λ0,0,

where the zero mode is given by the constant solution λ0,0 = µ2, so that

det′∆ = limµ→0

det ∆µ

µ2 .

In this case, the circle frequency (5.18) is given by

ω(k) =√k2 + µ2 = 2π

L1

√n2 + L2

14π2µ

2 =: 2πL1n′,

74


so that for n = 0, the value of n′ is n′ = mL1/(2π) and by a similar calculation as in(5.19), the partition function reads

Z =∏n∈Z

e−πL2L1n′(

1− e−2πL2L1n′)−1

= e−µL2/2(1− eµL2

)−1· Z ′

= Z ′

eµL2/2 − e−µL2/2

= Z ′

2 sinh(µL2/2)

Ensuing, the limit µ→ 0 can be taken

C det′∆ = limµ→0

det′∆(2 sinh(µL2/2))2

µ2 = L22 det′∆

because of

limx→0

4 sinh2(αx/2)x2 = α2.

Finally, this gives

det′∆ = L22|η(τ)|4

or

ln det′∆ = 2 lnL2|η(τ)|2.

This solution only handles the case that L2 was chosen as the Euclidean time before.Switching L1 ↔ L2 gives the solution

ln det′∆ = 2 lnL1|η(τ ′)|2,

where now τ ′ = 1/τ from before. It is found out, that

ln det′∆ = 2 ln(L2|η(τ)|2)

= 2 ln(√

L1L2

√L2L1|η(τ)|2

)

= 2 ln(√

L1L2=(τ)|η(τ)|2)

which ensures that the modular invariance τ → 1/τ is correct. Finally identifyingA = L1L2, it results in

ddz ζ∆(z) = − ln det′∆ = −2 ln

(√A=(τ)|η(τ)|2

)

75


A more elegant but less intuitive way to gain the correct zero mode contribution wouldbe to calculate the zero mode in the way it has been done before by the method ofcollective coordinates. A partition function here can be given by

Z =∫Dϕe−

12 (ϕ,∆ϕ)

=∫

dc0

∫ ∏n>0

dcne−12∑

n>0 λnc2n

=√A

∫ ∏n>0

dcne−12∑

n>0 λnc2n

=√A∏n>0

λ−1/2n ,

where the collective coordinate c0 was used as before.

5.4. Result

As the results in the above calculations gave the Dedekind eta function, the proper-ties of this function will be studied in detail before giving the result for the geometricdependence of the factor C(τ).

5.4.1. Properties of the Dedekind Eta function

The resulting determinant of the Laplacian was always given through the Dedekind Etafunction, which is defined by

η(τ) = q1/24∞∏k=1

(1− qk

), q = e2πiτ . (5.20)

A plot of the eta function is shown in Figure 5.4. To compare the results to other resultsin literature, the value for τ = i is needed. This can be derived from a special value ofthe Euler q-series Φ(q), which was given by Ramanujan in his ’Lost Notebook’, [AB05]

Φ(e−2π) = eπ/12Γ(1/4)2π3/4 , (5.21)

which is related to the Dedekind eta function by

Φ(q) = q−1/24η(τ), q = e2πiτ

so that

η(i) = e−π/12Φ(e−2π) = Γ(1/4)2π3/4 . (5.22)

This result is equivalently gained by using the Selberg-trace formula. But the resultin [Sel56] is achieved by calculating the Laplacian on a symmetric setup, therefore, this

76

5.4. Result

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

|η(iτ

)|

τ

|η(iτ)|

Figure 5.4.: Plot of the Dedekind eta function performed with numerical data fromMathematica.

result is gained with a similar calculation as before, setting L1 = L2 = L (see forinstance [Eli95]).Another property is the symmetry, which was indicated in the following scheme

η(−1/τ) =√−iτη(τ), (5.23)

implying√τ |η(τ)|2 =

√1/τ |η(1/τ)|2, (5.24)

what is just the modular invariance imposed in the introduction τ → 1/τ . This can alsobe observed in the plot of the right hand side of equation (5.24) in Figure 5.5.

5.4.2. Geometric dependence of critical fluctuations

After describing the properties of the Dedekind eta function, the result for the correlationlength can now be given. As this result differs only in the prefactor C(τ) from the presentresults, this factor is evaluated now. The result is

C(τ) =√

2√π=(τ)|η(τ)|2

. (5.25)

At first, this result can be compared to the results of the symmetric system. For L1 = L2the ratio is τ = i, so that

C = C(i) =√

2π

4π3/2Γ2(1/4) = 1.351956 . . . , (5.26)

77


which is the same result obtained analytically in [Mün90]. Furthermore, the solutionprovides the modular invariance τ → 1/τ , as this result is not changed qualitatively thebehaviour gained from the Dedekind eta function. The interface tension will not chang,because there is no factor τ included in the exponential that contributes to the interfacetension in (4.1). Therefore the only change with geometric dependence is included inC(τ). A comparison of the static factor C = C(i) with the geometric depending factorC(τ) is given in Figure 5.6. It can be seen, that there is a dramatic change in this factor.This means, the correlation length increases with the deviation of aspect ratio from thevalue 1.

78

5.4. Result

0

0.1

0.2

0.3

0.4

0.5

0.6

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

√τ|η

(iτ)|2

τ

√τ |η(iτ)|2

Figure 5.5.: Plot of√τ |η(τ)|2 with triangles denoting the values at integer values and

their inverse.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.5 1 1.5 2 2.5 3 3.5 4iτ

C(τ)/C(i)

Figure 5.6.: The factor C(τ) depending on the aspect ratio of L1 and L2 compared to theconstant factor C (blue line).

79


80

6CONCLUSION AND OUTLOOK

As proposed in the introduction, field theoretical methods have presented a fruitfuluniversal method for analysing interface fluctuations in three dimensions. This thesisextended the examination of interface fluctuations on new structures with a genuineattention on the geometric dependences. As underlying geometrical structure, a cuboidwas investigated and the suggested solution has been calculated by different methods.

In 2, it could be shown that the derivative raised for fluctuations, is responsible forthe geometric dependence of the correlation length in the Ginzburg-Landau theory. Re-ferring to [Mün89], field theoretical methods, reaching from path integral formalism overthe methods of collective coordinates and the instanton pseudo-particles, have been pro-lific to develop this theory in chapter 3. The reasons using these methods were gatheredin detail in the last part of the first chapter, where the correlation length was identifiedby the energy spitting of a massless particle in field theories. In chapter 4, it was shownby zeta regularization and heat kernel methods that only the transversal modes dependon the geometrical setup of the system. Therefore, for new geometries only these modes,corresponding to the two-dimensional Laplacian, need to be recalculated. Especially thezeta regularization techniques have shown universal applications in these fields, sincethey are applied in almost every calculation in chapter 4 and 5.In chapter 5, the geometry was focussed on a rectangular setup. Before calculating the

determinant explicitly, the geometrical structure with the imposed boundary conditionswas transformed into a torus by considerations from conformal field theory. Then, threedifferent methods were used. The first method has shown to be quite direct, gettingalong without a deep understanding in complex analysis. The second method, due tothe Sommerfeld-Watson transformation, turned out to be rather complicated but alsoquite elegant – it can be reduced on a few steps, when the reader is used to complexanalysis. Whereas the existing literature on this method is presented fairly abbreviated,this thesis made a contrast and exposed this calculation most precisely. Both calculationsdelivered an expression involving the Dedekind eta function insuring that the solutionis invariant under the transformation L1 → L2 as proposed in the introduction. In a

81

CONCLUSION AND OUTLOOK

third method, the calculation was redone and the analogy between the Laplacian and amassless Klein-Gordon field was used in order to write the partition function, obtainedfrom statistical mechanical considerations, in terms of harmonic oscillators. This methodhighlighted to be a very intuitive approach for physicists, delivering the same solution.However, this calculation is not as elegant as the two others, as the solution needs to bemodified by an argument in order to cover the proposed invariance. It could be shownthen, that for a strict solution, the method of collective coordinates needs to be appliedagain.As there is a deeper connection to conformal field theory in this rectangular setup

with periodic boundary conditions, another method, involving Virasoro algebra couldbe possible.A further calculation on a similar problem could be to impose other geometries as

basal area. As the calculation of the normal mode is invariant under the imposed ge-ometric structure, only the geometry of the Laplacian needs to be recalculated. Forinstance, there are solutions for the circular Laplacian [Wei87], eventually there is anapproach solving elliptic problems, thus where it should also be a factor depending onthe geometrical structure.In the future, it should be researched, if there is a proper limit law for the Dedekind

eta function, which would make a reduction on two dimensions possible, where thecorrelation length can be compared to a direct two dimensional calculation, e.g. in[PF83].Because of the structure of the problem, which was motivated by the Ising model

in this thesis, there should also be an approach solving the geometric dependence ofthe correlation length numerically by a Monte-Carlo calculation in order to test thepresented solution, as presented in the work of other papers [Mün89,Mün90]. It shows,that a quantitative experimental analysis of the interface fluctuations is difficult to reach,but it should be possible to test the geometrical dependence qualitatively in experiments.A similar calculation applies for an effective string theory in quantum chromo dy-

namics. In the confining regime, the quark anti-quark pair qq is described by WilsonLoops [LW02] where the partition function can be written as

Z ∼ e−V (R)T ,

where V (R) is the potential and T is the time-line of the world sheet. As confining regimeof quantum chromo dynamics contains in general two phases: the strong coupling phaseand the rough phase. The two are separated by the so-called roughening transition whichis the point in which the strong coupling expansion in the Wilson loop ceases to converge.These two phases are related to two different behaviours of the quantum fluctuations.In the strong coupling phase, these fluctuations are massive, while in the rough phasethey become massless. This fact can be used in order to describe this transition bythe massless theory in two dimensions. Calculating this transition by setting the time-line direction L2 and R = L1 ends up in the same calculations as before, leading to theDedekind eta function. Therefore, this so called Lüscher term [LSW80] can be calculatedthis way. As there are too many expressions that have to be defined to reach the effective

82

string theory, this has been omitted in this work, but it is an interesting fact, applyingthese calculations in other fields of physics.

83

CONCLUSION AND OUTLOOK

84

AFORM AND ENERGY OF THE KINK-PROFILE

A.1. Form of the kink

The variation of the Hamiltonian near the classical solution provides

δH

δφ

∣∣∣∣φ=φ0

= 12(−2∇2φ0) + g

4!4φ30 −

m2

4 2φ0 = 0, (A.1)

or to be more general

−∇2φ0(x) + V ′(φ(x)) = 0,

where V ′(φ) means the derivative of the potential V with respect to the function φ. Asit was proposed, the boundary conditions in z-direction were defined anti-periodic, sothat

φ = limz→±∞

φ0(z) = ±v.

In this direction, the variation of the Hamiltonian (A.1) provides

− ∂2

∂z2φ0(z) + g

6φ30(z)− m2

2 φ0(z) = 0. (A.2)

This equation maintains for all z ∈ [−∞,∞]. Therefore, the constant values of φ0 in thelimits cancel out the second derivative, so that the vacuum expectation is

g

6v3 = m2

2 v ⇒ v =√

3m2

g(A.3)

besides the trivial solution v = 0. In equation (A.2), the z-direction was observed andcan be rewritten more generally performing a saddle point evaluation

∂2

∂z2φ0 = V ′(φ0) (A.4)

85

FORM AND ENERGY OF THE KINK-PROFILE

Now a solution of (A.4) is searched, which gives a minimum of the Euler-Lagrangeequation. It is a non-linear differential equation of second order that can be solved byusing the substitution

∂

∂z

(φ′(z)

)2 = 2 ∂∂zV ′(φ). (A.5)

From integration, it follows

z∫a

dz = ±φ∫

0

dφ√2V [φ]

= ±φ∫

0

dφ√g12(φ2 − v2)

⇔ z − a∓√g

12arctanh

(φv

)v

.

Therefore, the solution

φ0(z) = ∓v tanh[m

2 (z − a)]

(A.6)

is obtained with (A.3). It shows, that from (A.5), two solutions follow, where the signcan be chosen.

A.2. Energy of the kink

The energy integral is given by the spatial integration over the Hamiltonian density

H[φ0] = 12(∇φ0)2 + V [φ0].

The first derivative of the kink solution (A.6) is

∂

∂zφ0 = mv

2 sech2[m

2 (z − a)]. (A.7)

Substituting this into the z-component of the energy integral ,∫dz 1

2

(∂

∂zφ0

)2= 1

2

∞∫−∞

du v2m2

4 sech4(m

2 u)

= m2

8

2 tanh(m2 u) [

sech2 (m2 u)

+ 2]

3m

∞−∞

= mv2

3

86

A.2. Energy of the kink

is obtained and with (A.3), it follows∫dz 1

2

(∂

∂zφ0

)2= m3

g.

The other spatial directions with the dilations L1 and L2 with periodic boundary condi-tions then lead to ∫

d3x12 (∇φ0)2 = L1L2

m3

g.

The same result follows for the potential V (φ0)∫d3xV [φ0] =

∫d3x

g

4!φ40 −

m2

4 φ20 + 3

8m4

g

= L1L2

∫dug

4!v4 tanh4(u)− m2

4 v2 tanh2(u) + 38m4

g

= L1L2

[g

4!v4u− 8

3m tanh(u) + 23m tanh(u) sech2(u)

− m2

4 v2(u− tanh(u)) + 38m4

gu

]∞−∞

.

Now v can be substituted with (A.3) and the linear terms, each individually leadingto divergence, cancel out, whereas the tanh(x) as an odd function cancels out too bysymmetric limits∫

d3xV (φ0) = L1L2

[g

249m4

g2 z − m2

43m2

gz + 3

8m4

gz

]∞−∞

+ L1L2m3

g

= L1L2

[ 924 −

34 + 3

8

m4

gz

]∞−∞

= L1L2m3

g

Therefore, the solution of the energy integral

H [φ0(x)] = 2L1L2m3

g

is the energy of a kink.

87

FORM AND ENERGY OF THE KINK-PROFILE

88

BPROPERTIES OF SPECIAL FUNCTIONS

B.1. The Gamma function

A generalization of the factorial n!, n ∈ N for real or complex numbers leads to therepresentation

Γ(n) = (n− 1)!,

where the gamma function is defined as

Γ(z) =∞∫0

dttz−1e−t

for all z > 0. For the factorial the following equation

(n+ 1)! = (n+ 1)n!

is valid for all n ∈ N. This property can be transferred to the gamma function usingintegration by parts and the L’Hôpital rule

Γ(z + 1) = zΓ(z) ⇒ Γ(z) = (z − 1)Γ(z − 1). (B.1)

This recursive property can be used to define the gamma function on all real numbersz, except for nonpositive integers. A specific value of the Gamma function is

Γ(1/2) =√π, (B.2)

which results by a substitution t→√t in the above integral giving a Gaussian integral.

With this value, all values of half integer numbers can be achieved by (B.1).Another important result is the reflection formula for 0 < z < 1 due to L. Euler

[AAR99] that reads

Γ(1− z)Γ(z) = π

sin(πz) , (B.3)

89

PROPERTIES OF SPECIAL FUNCTIONS

which can be used to calculate the products

Γ(1/2)Γ(1/2) = π (B.4)Γ(−1/2)Γ(1/2) = Γ(−1/2 + 1− 1)Γ(1/2) = −2Γ(1/2)Γ(1/2) = −2π, (B.5)

where (B.1) was used in the second relation. The following relations are obtained fromthe result Γ(1) = 1 and (B.1) or obtained from the derivatives of the gamma functiongiven in [AAR99] and [BMMS13]

1Γ(0) = 0 (B.6)

Γ′(1) = −γ (B.7)Γ′(0)→∞ (B.8)

Γ′(1/2) = −√π(γ + ln 4), (B.9)

limz→0

ddz

1Γ(z) = 1 (B.10)

limz→0

ddz

1Γ(1− z) = γ, (B.11)

where γ is the Euler-Mascheroni constant.

B.2. The Euler beta function

The beta function is closely related to the gamma function. It is defined by

B(x, y) =∞∫0

tx−1

(1 + t)x+y dt

for <(x) > 0 and <(y) > 0. By the following representation

B(x, y) = Γ(x)Γ(y)Γ(x+ y) ,

the analytic continuation of the gamma function applies. The beta function is used inthis thesis to evaluate expressions like

limz→0

ddz I(z) = lim

z→0

ddz

∞∫−∞

dx(a2 + x2)−z (B.12)

= 2 limz→0

ddz

∞∫0

dx(a2 + x2)−z (B.13)

90

B.3. The Riemann zeta function

for any a and by the substitution t = x2/a2, dt = 2x/a2dx, the integral becomes

I(z) =∞∫0

dt a2√t(a2 + a2t)z

= a1−2z∞∫0

dt t1/2

(1 + t)z

= a1−2zB(1/2, z − 1/2).

And the derivative with respect to z is

I ′(z) = −2 ln(a)a1−2zB(1/2, z − 1/2) + a1−2zB′(1/2, z − 1/2), (B.14)

where in the limit z → 0, the first term vanishes according to B(1/2, z − 1/2) → 0.Because of (B.6) the derivative of the beta function is given by the derivatives of thegamma function which give here

limz→0

B′(1/2, z − 1/2) = limz→0

Γ(1/2)Γ′(z − 1/2)Γ(z)− Γ(z − 1/2)Γ′(z)Γ2(z)

= −√π limz→0

−2√πΓ′(z)

Γ2(z) .

This expression gives

limz→0

Γ′(z)Γ2(z) = − lim

z→0

ddz

1Γ(z) = −1

from (B.10). Conclusively for (B.12)

limz→0

ddz I(z) = −2πa. (B.15)

Instead of the limit z → 0, in another calculation the limit z → 1 is needed. Beginningwith (B.14), it follows

limz→1

ddz I(z) = −2 ln(a)a−1π + a−1(−π ln 4), (B.16)

where in the first term, the value of the gamma function at 1/2 was used and in thesecond term, the values of the derivative of the gamma function (B.7) and (B.9) wereused.

B.3. The Riemann zeta function

The Riemann zeta function gives a connection between series and integrals. The famousseries

ζ(z) =∞∑n=1

n−z, (B.17)

91

PROPERTIES OF SPECIAL FUNCTIONS

which converges for <(z) > 1, can also be defined as the integral

ζ(z) = 1Γ(z)

∞∫0

dt tz−1

et − 1 . (B.18)

The zeta function can be continued analytically by the so called reflection formula (usingthe reflection property of the gamma function)

π−s/2Γ(s/2)ζ(s) = π1/2−s/2Γ(1/2− s/2)ζ(1− s) (B.19)

ζ(s) = 2sπs−1 sin(πs

2

)Γ(1− s)ζ(1− s) (B.20)

so that the zeta function is defined for any z ∈ C\1. The reflection formula providesthe surprising solution

ζ(−1) =∑n

n = − 112 . (B.21)

The value at s = 0 is the sum

ζ(0) =∑n

1 = −12 . (B.22)

Another important value is

ζ(2) =∑n

n−2 = π2

6 . (B.23)

The derivative of the zeta function provides [AAR99]

ζ ′(0) = −12 ln(2π). (B.24)

92

CTHE ERROR FUNCTION INTEGRAL

The integral

∞∫−∞

f(x)dx =∞∫−∞

dxe−t(x2+a2)

x2 + a2 (C.1)

for a > 0 needs to be solved. For this the following expression can be achieved

∞∫−∞

dxe−t(x2+a2)

x2 + a2 =∞∫−∞

dx

1x2 + a2 −

t∫0

dαe−α(x2+a2)

, (C.2)

because the function f(x) can be rewritten by the following integral

t∫0

dαe−α(x2+a2) = −e−t(x2+a2)

x2 + a2 + 1x2 + a2 ,

using the derivative. This has been used in (C.2) to replace η(x) in (C.1).The integral over the first term in (C.2) is easy to perform by using the arctan. The

result is for a > 0∞∫−∞

dx 1x2 + a2 = π

a. (C.3)

Succeeding, the second term has to be solved. From the theorem of Fubini follows

∞∫−∞

dxt∫

0

dαe−α(x2+a2) =t∫

0

dα∞∫−∞

dxe−α(x2+a2).

93

THE ERROR FUNCTION INTEGRAL

By substituting z =√αx, this integral separates into two parts which can be solved

independentlyt∫

0

dαe−a2α

√α

∞∫−∞

dze−z2.

The inner integral here is the Gaussian integral. The solution can be obtained by goingover to polar coordinates

∞∫−∞

dze−z2 =√π. (C.4)

Finally, the integral

I =√π

t∫0

dαe−a2α

√α

(C.5)

needs to be solved. Here, the lower boundary of the integral is zero, so that the improperintegral has to be solved, taking the limit

limβ→0

t∫β

dαe−a2α

√α.

By substituting u = a√α, the limit β → 0 can be taken

limβ→0

a√t∫

a√β

2√α

adue−u2

√α

= limβ→0

2a

a√t∫

a√β

due−u2

= limβ→0

2a

a√t∫

−∞

due−u2 −a√β∫

−∞

due−u2

= 2a

a√t∫

−∞

due−u2 −√π

a.

The Gaussian integral is symmetric and its value is√π. We can use this property to

rewrite the integral above because the upper boundary is a√t > 0. Thus, the integration

from −∞ to 0 can be taken out and the integral can be rewritten in terms of the errorfunction erf(x)

a√t∫

−∞

due−u2 =√π

2 +a√t∫

0

due−u2

=√π

2 +√π

2 erf(a√t).

94

Finally, the integral (C.5) reads

I = π

a+ π

aerf(a

√t)− π

a= π

aerf(a

√t),

which can together with the solution of the first part (C.3) be used in (C.2) to solve theproblem

∞∫−∞

dxe−t(x2+a2)

x2 + a2 = π

a− π

aerf(a

√t)

= π

a

(1− erf(a

√t)).

95

THE ERROR FUNCTION INTEGRAL

96

DCALCULATION OF ζ1

The following integral is going to be evaluated

ζ1(z) = 1Γ(z)

∞∫0

dttz−1L1L2(4πt)−1[Kt(Q)− 1

]

= L1L24πΓ(z)

∞∫0

dttz−2e−3m2t/4 + L1L24πΓ(z)

∞∫0

dttz−2∞∫−∞

dpg(p)e−t(m2+p2), (D.1)

where the derivative with respect to z is going to be taken in the limit z → 0. Substitutingx = 3/4m2t and y = (m2 + p2)t in (D.1) leads to the following integral, replacingL1L2/4π =: C by a constant C

ζ1(z) = C

Γ(z)

∞∫0

dx( 4

3m2

)z−1xz−2e−x +

∞∫−∞

dpg(p)∞∫0

dy yz−2e−y

(m2 + p2)z−1

.The first integral here is a representation of the Γ function and gives Γ(z − 1). Thesecond integral evaluated with respect to y gives the same contribution Γ(z − 1). Thenthe integral reads

ζ1(z) = C

Γ(z − 1)Γ(z)

( 43m2

)z−1+ Γ(z − 1)

Γ(z)

∞∫−∞

dpg(p)(m2 + p2)1−z

The first contribution is

F0(z) = 1z − 1

(3m2

4

)1−z

⇒ F ′0(0) = −3m2

4 + 34 ln 3

4 + 32 lnm

97

CALCULATION OF ζ1

This leaves the integral

I = Γ(z − 1)Γ(z)

∞∫−∞

dpg(p)(m2 + p2)1−z (D.2)

to be evaluated. The spectral density (4.30)

g(p) = −m2π

(2

p2 +m2 + 1p2 + m2

4

)

can be transformed into

g(p) = −m2π

3p2 +m2 +

3m2

4(p2 +m2)2 +

(3m2

4

)2

(p2 +m2)2(p2 + m2

4 )

.The following three integrals are gained for (D.2) using this spectral density

F1(z) = − 1z − 1

3m2π

∞∫−∞

dp(p2 +m2)−z

= − 1z − 1

32πm

2−2z∞∫−∞

dp(p2 + 1)−z

F2(z) = − 1z − 1

3m3

8π

∞∫−∞

dp(p2 +m2)−1−z

= − 1z − 1

38πm

2−2z∞∫−∞

dp(p2 + 1)−1−z

F3(z) = − 1z − 1

m

2π

(3m2

4

)2 ∞∫−∞

dp 1(p2 +m2)1+z(p2 + m2

4 )

= − 1z − 1

m1−2z

2π

(34

)2 ∞∫−∞

dp 1(p2 + 1)1+z(p2 + 1

4),

Here, all transformations can be achieved by a substitution of p→ pm. The solution forF1 can be found be the beta function

F1(z) = − 1z − 1

3m2−2z

2π B(1/2, z − 1/2)

⇒ F ′1(0) = 32πm

2(−2π) = −3m2,

98

where the derivative of the beta function and B(1/2,−1/2) = 0 has been used. Thesecond integral F2 is found in the same manner

F ′2(z) = −(− 1

(z2 − 1)3

8πm2−2z − 2

z − 13

8πm2−2z lnm

) ∞∫−∞

dp(p2 + 1)−1−z

− 1z − 1

38πm

2−2z ddz

∞∫−∞

dp(p2 + 1)−1−z

⇒ F ′2(0) = m2( 3

8π −3

4π lnm) ∞∫−∞

dp(p2 + 1)−1

− limz→0

1z − 1

38πm

2−2z ddz

∞∫−∞

dp(p2 + 1)−1−z

The first integral gives ∫dp(p2 + 1)−1 = B(1/2, 1/2) = π,

by the transformation of the integral into a beta function (see B.2). The second integralcan solved with beta function (s = z + 1) as well

dds

∞∫−∞

dp(p2 + 1)−s = B′(1/2, s− 1/2)→ −2π ln 2

and the expression becomes

−34m

2 ln 2.

Therefore, in the limit z → 0 the expression is

F ′2(0) = 3m2

4

(12 − lnm

)− 3

4m2 ln 2.

For F3 the following applies

F ′3(z) = − 9m32π

(− 1

(z − 1)2m−2z + 2

z − 1m−2z lnm

) ∞∫−∞

dp 1(p2 + 1)1+z(p2 + 1

4)

− 9m32π

1z − 1m

−2z ddz

∞∫−∞

dp 1(p2 + 1)1+z(p2 + 1

4)

⇒ F ′3(0) = 9m32π (1− 2 lnm)

∞∫−∞

dp 1(p2 + 1)(p2 + 1

4)

+ limz→0

9m32π

ddz

∞∫−∞

dp 1(p2 + 1)1+z(p2 + 1

4).

99

CALCULATION OF ζ1

<z

=z

CR

R−R

× i

Figure D.1.: Contour ΓR in order to integrate the integral (D.5).

As the second integral converges, the following expression is achieved

limz→0

ddz

∞∫−∞

dp 1(1 + p2)1+z(p2 + 1

4)= −

∞∫−∞

dp ln(p2 + 1)(1 + p2)(p2 + 1

4). (D.3)

For both integrals in (D.3), a fraction decomposition can be applied, leaving for the firstintegral

43

∞∫−∞

dp[

1p2 + 1

4− 1p2 + 1

]= 4

3π,

because of the arcus tangens again. For the second integral, the following is achieved

43

∞∫−∞

dp[

ln(p2 + 1)p2 + 1/4 −

ln(p2 + 1)p2 + 1

], (D.4)

which can be evaluated in the complex plane (see figure D.1) considering the integral∮ΓR

dz ln(z + i)z2 + 1 . (D.5)

This integral can be evaluated due to the residue theorem because of the the residue atz = i ∮

ΓRdz ln(z + i)

z2 + 1 = Resz=i( ln(z + i)z2 + 1

)= 2πi ln(2i)

2i = π

(ln 2 + iπ

2

), (D.6)

where the last expression follows because of the complex logarithm. The integralR∫−R

dx ln(x+ i)x2 + 1 =

R∫0

dx ln(−x+ i)x2 + 1 +

R∫0

dx ln(x+ i)x2 + 1

=R∫

0

dx ln(x2 + 1) + iπx2 + 1

100

then follows by equaling the real parts, because the integral over the contour CR goesto zero as R→∞. The solution (D.6) the limit R→∞ is given by

π

(ln 2 + iπ

2

)=∞∫0

dx ln(x2 + 1) + iπx2 + 1 .

Therefore, equaling the real parts, the solution of the second term in (D.4) is∫dp ln(p2 + 1)

p2 + 1 = 2π ln 2,

while for the first term the same procedure applies, resulting in∫dp ln(p2 + 1)

p2 + 14

= 2π ln 94 .

And together the expression (D.4) results in

8π3

(ln 9

4 − ln 2)

= 8π3 ln 9

8 ,

giving

F ′3(0) = 3m2

8 (1− 2 lnm)− 3m2

4 ln 98 .

Conclusively, the contribution of ζ1(z) is given by

ζ ′1(0) = C

(− 3m2

4 + 3m2

4 ln 34 + 3m2

2 lnm

− 3m2

+ 3m2

8 − 3m2

4 lnm− 3m2

4 ln 2

+ 3m2

8 − 3m2

4 lnm− 3m2

4 ln 98

)= Cm2

(−3 + 3

4 (2 lnm− lnm− lnm) + 34

(ln 3

4 − ln 2− ln 98

))= −Cm2

(3 + 3

4 ln 3).

101

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106

Danksagung

Zum Ende dieser Masterarbeit möchte ich folgenden Personen meinen Dank aussprechen:

• Herrn Prof. Dr. Gernot Münster für die Ausgabe des interessanten Themas, sowiedie fachliche Unterstützung und richtungweisenden Ratschläge während der Bear-beitung.

• Herrn Dr. Jochen Heitger für die Bereitschaft, die Zweitkorrektur zu übernehmen.

• Meinen Eltern und meinem Bruder für die Unterstützung während des gesamtenStudiums, meinem Vater insbesondere für die Bereitschaft, diese Arbeit Korrekturzu lesen.

• Ninja Schmiedgen für die Hilfestellung in vielfachen organisatorischen Fragen unddas Korrekturlesen dieser Arbeit.

• Meinen Büronachbarn Walter Tewes und Felix Tabbert, sowie Evgenij Travkin undBenedikt Seitzer für die gute Atmosphäre, interessante Diskussionen und hilfreicheMotivationsschübe.

• Dem AStA der Universität Münster für die spannende und interessante gemein-same Zeit, sowie für die Drucklegung dieser Arbeit.

• Dem Falken Harras für die schlussendliche Motivation, endlich fliegen zu wollen,sowie seiner Pflegemutter für viele hilfreiche Tipps.

• Frau Prof. Dr. Cornelia Denz und Herrn Prof. Dr. Michael Klasen für dieUnterstützung bei der Einwerbung eines Stipendiums für die nachfolgende Zeit.

107

Erklärung zur Masterarbeit

Hiermit versichere ich, dass die vorliegende Arbeit über “Geometric dependenceof interface fluctuations” selbstständig verfasst worden ist, dass keine anderenQuellen und Hilfsmittel als die angegebenen benutzt worden sind und dass die Stellender Arbeit, die anderen Werken – auch elektronischen Medien – dem Wortlaut oder Sinnnach entnommen wurden, auf jeden Fall unter Angabe der Quelle als Entlehnung ken-ntlich gemacht worden sind.

(Datum, Unterschrift)

108

geometricdependenceofinterfaceﬂuctuations€¦ · contents 1. introduction 1 2. theory of phase...

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