Studying Lee-Yang zeros using graph theory and complex dynamics

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Studying Lee-Yang zeros using graph theory and

complex dynamics

Jos van Willigen

June 28, 2020

Bachelor thesis Mathematics and Physics & Astronomy

Supervisors: prof. dr. Han Peters, prof. dr. Jean-S´ebastien Caux, Wouter Buijsman MSc

Institute of Physics

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

The first of the two distinguished Lee-Yang theorems states that a system exhibits no phase transitions, in case a neighbourhood of the positive real line in the complex ξ-plane exists on which the partition function Z 6= 0, for ξ = exp(−2βH). The second states that for the ferromagnetic Ising model all complex zeros of the partition function lie on the unit circle. We apply methods from graph theory and complex dynamics and find a sufficient condition for the existence of a zero-free neighbourhood of the positive real line in the complex ξ-plane for bounded degree graphs. Besides, we discuss a pair of experiments that measure the locations of the complex Lee-Yang zeros in the lab.

Title: Studying Lee-Yang zeros using graph theory and complex dynamics Author: Jos van Willigen, josvwilligen@gmail.com, 11616350

Supervisors: prof. dr. Han Peters, prof. dr. Jean-S´ebastien Caux, Wouter Buijsman MSc Second graders: dr. Hessel Bouke Posthuma, dr. Vladimir Gritsev

End date: June 28, 2020

Cover: Zeros of the partition function on the unit circle in the complex ξ-plane, for four differently-sized systems. The gap indicated with the arrows remains as N → ∞. Institute of Physics

University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1 Introduction

In the field of statistical physics, systems with a large number of interacting particles are studied. Due to this large amount of particles and interactions, exact computation of properties of these systems is very hard to perform. Therefore, the field of study uses statistical methods and probability theory to approximate the macroscopic behaviour of these large systems.

A prime example of macroscopic behaviour studied by statistical physics is that of a phase transition. A phase transition is a sudden change in the properties of a system as a result of a little change in the system’s environment. A phase transition can only occur when the system has a very large number of interacting particles.

To successfully study a many-particle system, typically a certain model will be ap-plied, which usually regards it in a more ideal setting where only the largest effects and influences are considered. Of course, the model must approximate the system well, if one wants the calculations to predict the behaviour adequately.

One such model, which is often used to determine properties of magnets, is the Ising model. This is the most famous model that exhibits phase transitions. In this paper, we will investigate when a magnet described by the Ising model is free from phase transitions. Because of the challenge that lies in trying to make direct computations regarding these large systems, it is worth investigating whether an easier method exists to determine when a system undergoes a phase transition.

Fortunately, in 1952, Lee and Yang discovered the Lee-Yang theorems [1, 2] which provide us with such a method. They proved that the zeros of the partition function of a system, a function ubiquitous in statistical physics, give insight in the occurrence of phase transitions as a result of changes in the external magnetic field H. More specifically, for ξ := exp(−2H/kBT ), when a neighbourhood of the positive real axis in

the complex ξ-plane is free of zeros, no phase transitions occur as a result of variations in ξ. The existence of such a neighbourhood depends on the structure of the system and the temperature T at which the system is being kept.

The main goal of this paper is to investigate for which temperatures and structures such a zero-free region of the positive real axis exists, meaning no phase transitions can occur. To achieve this goal, we will use a combination of methods from the mathematical fields of graph theory and complex dynamics.

We will now discuss the general structure of this paper.

In Section2.1 the reader will be given the necessary introduction into the mathemat-ical field of complex analysis, which one will need for a better understanding of this paper. Furthermore, in Section 2.2, the Ising model and the partition function will be formally introduced and discussed. With this introduction in both mathematics and physics presented in Chapter 2, the remainder of this paper should be accessible for

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both physics and mathematics students, without prior knowledge of the other field. An exception is made for Chapter 4, where additional physical knowledge is required for a full understanding of the chapter.

Thereafter, in Chapter 3 the aforementioned Lee-Yang theorems will be discussed, Theorems 3.1.1 and 3.3.2 in this paper. We will go in-depth into the correspondence between the Ising model and a model for monatomic gases, for which Theorem3.1.1was originally proved. This correspondence implies that the theorem can be translated into the Ising model setting, which results in Theorem 3.3.1.

Afterwards, in Chapter 4, we will discuss a pair of experiments that measure the complex zeros of the partition function. We find this achievement to be quite stunning, for the complex zeros were originally deemed purely theoretical. Since any real external magnetic field H produces values of ξ bound to the positive real axis, one might be susceptible to thinking that other values of ξ are inherently meaningless. These two experiments prove the contrary.

After Chapter 4, we will pursue the paper’s main goal in Chapters 5 and 6. In Chapter 5, a new function R is introduced. This function R is proved to be useful in the analysis of the zeros of the partition function Z, as discussed in Lemma 5.1.2, and is easier to calculate than Z, as discussed in Lemma 5.1.3 using graph-theoretical methods. This last property is only true for certain structures, however. Fortunately, we can assume without loss of generality that the system has such structure, as proved in Theorem5.2.1.

The mathematical core of this paper is concluded by Chapter 6, where methods from complex dynamics are used to find a sufficient condition for when R(ξ) stays away from −1 as the graph grows, for ξ in a certain neighbourhood of R+. This is done

by first defining a function fξ,b that corresponds to increasing the system for a given

temperature and value of ξ. In Section 6.1, we prove that an interval for fξ,b exists

that is mapped within itself, for certain temperatures and ξ-values close enough to the positive real axis. With this invariant interval in hand, we prove the existence of an invariant cone in Section6.2. Finally, in Section6.3, we combine the previous results to prove Theorem6.3.2, which gives sufficient conditions for the temperature and structure of a non-trivial system such that a neighbourhood of R+ exists on which Z is not zero.

Hence, Theorem 6.3.2completes the main goal of this paper.

We end this paper with Chapter 7. In Section7.2we discuss applications in, perhaps surprisingly, computer science. Finally, in Section 7.3 we pose interesting questions in this field of study that we wish to see answered in the foreseeable future.

Thanks go to prof. dr. Han Peters, prof. dr. Jean-S´ebastien Caux and Wouter Bui-jsman MSc for their guidance and fruitful discussions during this project. Furthermore, thanks go to dr. Hessel Bouke Posthuma and dr. Vladimir Gritsev for being the second graders of this paper.

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2 Preliminaries

We will commence this paper by discussing the knowledge one needs to read and under-stand the paper. Both a mathematical and physical underunder-standing of certain topics are necessary, which we will discuss in that order.

2.1 Mathematical Background

Before we are ready to understand the physical context of this project, we need to lay the mathematical foundations. These foundations lie in the field known as function theory, or complex analysis. The statements I will make are widely discussed in literature such as [3].

2.1.1 Power Series

We start by investigating a power series.

Definition 2.1.1. A power series centred at z0 ∈ C is a series of the form ∞

X

n=0

cn(z − z0)n,

in which z0 and all cn are complex numbers, and z is a complex variable. The complex

number z0 is called the centre of the power series.

We are interested in whether this series converges for a given value of z. It is clear that it converges at its centre z = z0, but for other values the following theorem by Abel

is very helpful.

Theorem 2.1.2 (Abel). If a power series converges for any value of z0, such that z 6= z0,

then for every value of z satisfying

|z − z₀| < |z0− z₀| the series converges absolutely.

Assume that there is at least one z0 6= z0 for which the power series around z0

con-verges. Let us denote by S the set of points for which the power series concon-verges. Then, we define

R := sup

z0_∈S

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Corollary 2.1.3. The power series centred at z converges absolutely at every point of the open disk

{z ∈ C; |z − z0| < R}, (2.1)

and it diverges at every point outside the closure of this disk.

Because of this property of the disk (2.1), it will be called the disk of convergence of the power series centred at z0, with R its radius of convergence.

2.1.2 Analytic Functions

Analytic functions are closely related to power series, and are defined as follows: Definition 2.1.4. Let O ⊂ C be an open set and let f : O → C. Then, f is analytic on O if for every a ∈ O there exists some ra> 0 such that f can be written as power series

centred at a on B(a, ra).

We say f is analytic at z if z is contained in an open set on which f is analytic. Example. Consider ez : C → C and a ∈ C. Then we can write

ez= ea+z−a= eaez−a = ∞ X n=0 ea n!(z − a) n_, using ez−a =P∞

n=0(z − a)n/n!. This series converges for all a, so ez is analytic on C.

Analytic functions turn out to be exactly the functions that are complex differentiable (or holomorphic). To make this more precise, we need one more definition.

Definition 2.1.5. Let O ⊂ C be an open set and let f : O → C. Then, f is complex-differentiable at a when

lim

z→a

f (z) − f (a) z − a

exists, and its limit is called the derivative of f at a, which is written as f0(a). The function f is called holomorphic on O if f is complex-differentiable at every a ∈ O.

We now state the following theorem: Theorem 2.1.6. If the power series P∞

n=0cn(z − a)n converges for every z in B(a, r),

then its corresponding analytic function defined on B(a, r), equal to

f (z) =

∞

X

n=0

cn(z − a)n,

is holomorphic. Furthermore, on B(a, r), it holds that

f0(z) =

∞

X

n=0

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In the following theorems, we see properties that portray really well how ‘nice’ holo-morphic functions are.

Theorem 2.1.7. Let U ⊂ C and f : U → C holomorphic. If f0(z0) 6= 0 for some z0 ∈ U ,

then f preserves angles between directed curves through z0, as well as orientation. We

call f conformal at z0.

Theorem 2.1.8 (Schwartz). Let D ⊂ C be the open unit disk and let f : D → D be holomorphic such that f (0) = 0 and |f (z)| ≤ 1 on D. Then, for every z ∈ D, |f (z)| ≤ |z| and |f0(0)| ≤ 1. Furthermore, if |f (z)| = |z| for any z 6= 0 or if |f0(0)| = 1, then f is a rotation, meaning that f (z) = az for some a ∈ C with |a| = 1.

We can use the latter theorem to make a statement about fixed points of holomorphic functions.

Definition 2.1.9. Let f a complex function and p ∈ C such that f (p) = p, then p is called a fixed point of f . Moreover, if there exists an open neighbourhood U of p such that f◦n(z) → p uniformly as n → ∞ for all z ∈ U , we call p attracting. For an attracting fixed point p, we call the open set of points that converge to p as n → ∞ the basin of attraction.

Corollary 2.1.10. Let f : C → C be holomorphic and p a fixed point of f . Then p is attracting if and only if |f0(p)| < 1.

Proof of Corollary 2.1.10. Assume |f (p)0| < 1. Then there exists some bounded neigh-bourhood U = B(p, r) of p and some c < 1 such that

f (p) − f (z) p − z < c if z ∈ U. We find that |f (z) − p| < c|z − p|, which implies that

|f◦n(z) − p| < cn|z − p| < cnsup

z∈U

kz − pk, which converges to 0 uniformly.

Conversely, assume that p is attracting and let U the neighbourhood as discussed in Definition2.1.9. Many authors, such as [4], assume p = 0. We will follow that practice because it presents a more clear proof. Note that p can be taken equal to 0 without loss of generality, by translating C such that p is mapped to 0 before applying f and translating 0 to back to p afterwards.

Find B(0, r) ⊂ U for some r ≤ 1 and define ˜f (z) := r−1f◦n(rz) for n large enough such that |f◦n(z)| < |z| for all z ∈ B(0, r). Now we see that ˜f (0) = 0 and that for all

z ∈ D ˜ f (z) = r−1f◦n(rz)= r−1|f◦n(rz)|<∗ r−1|rz| < 1

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holds, where ∗ holds because rz ∈ B(0, r). We know ˜f cannot be a rotation (then f would be as well), so ˜ f0(0)

< 1 must hold by Theorem2.1.8. Since

˜ f0(z) _z=0 = (f◦n)0(z) _z=0 = n−1 Y k=0 f0(f◦k(z)) z=0 = (f0(0))n, we see that | ˜f (0)0| < 1 =⇒ |f0(0)|n< 1, so |f0(0)| < 1.

Another useful theorem when considering analytic functions is the Implicit Function Theorem as stated below. The proof can be found in [5, pp. 34-35].

Theorem 2.1.11 (Implicit Function Theorem). Let U ⊂ Cn+m and f = (f1, . . . , fm) : U → Cm

holomorphic. If (z0, w0) ∈ U is a point where f (z0, w0) = 0 and

det   ∂fµ ∂zν 1≤µ≤m n+1≤ν≤n+m  6= 0,

then there exist an open neighbourhood V × W of (z0, w0) and a unique holomorphic map

g : V → W such that g(z) = w and f (z, g(z)) = 0 for all z ∈ V .

2.2 Physical Background

This project investigates the zeros of the partition function. The relevance of the parti-tion funcparti-tion from the physical point of view will be discussed in this secparti-tion, whereas its zeros specifically will not be discussed until Chapter3.

This section is written in a writing style that is very common in physics, whereas the previous section was written in a mathematical style. Expect Chapters 3 and 4 to be in the same writing style as this section, and Chapters 5 and 6 to again use the more mathematical way of writing. This stresses the different backgrounds of this topic.

We will restrict ourselves to the Ising model in this project;1 a certain model of reality. First, we start with a short overview of what will be discussed in this section. In that way, we have a better understanding of what we are studying.

Intuitively, the Ising model consists of many spins situated on the vertices of a graph. These spins can be either positively or negatively oriented and can change sign over time. The different spins will interact with each other, influencing neighbouring spins to change orientation, in exchange for energy. Together, the spins will form a configuration describing how they are each aligned. In thermal equilibrium, the partition function will grant us a tool to determine the probability to find the system in a certain configuration

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at a fixed time, given the total energy of the system. How the system changes will not be discussed, only the probability to find it in a certain state when we pause time.

Several factors will influence the partition function, such as the temperature of the sys-tem, how strongly spins interact with each other and whether an external field is applied to the system. Ultimately, we are interested in the zeros of the partition function. The zeros provide us with information about when the system undergoes a phase transition, as explained in Chapter 3. A phase transition is what we call a big, sudden change in the macroscopic behaviour of a system when only a small change in the parameters is applied. For the Ising model, an example is a magnet which loses its magnetic properties at a certain temperature.

In Section 2.2.1 the interaction energy and the external field will be elaborated on. Thereafter, in Section 2.2.2, a mathematical tool will be introduced, which will help us find the partition function and its relevance in Section 2.2.3.

2.2.1 The Ising Model

The Ising model, introduced by Wilhelm Lenz and first studied by his student Ernst Ising, consists of a lattice of spin variables σα which can be either 1 or -1. Any two of

these spins have an interaction energy

−E(α, α0)σασα0.

If σαand σα0have the same sign, this interaction energy is equal to −E(α, α0). Otherwise,

the interaction energy is equal to E(α, α0).

We call a material ferromagnetic if E(α, α0) ≥ 0 for all distinct α, α0. In that case, the interaction energy is lower when two variables are the same. When E(α, α0) ≤ 0 for all distinct α, α0, we call a material antiferromagnetic. In that case, the interaction energy is lower when two variables are different. If the interaction energy between to variables is zero, we call those two variables non-interacting.

Aside from this interaction energy between different spin variables, a spin may interact with an external field. This field is denoted by H and has an interaction energy with the spin variable σα equal to

−Hσ_α.

Again, it depends on the value of H and σα whether this energy is positive or negative.

Note that the external field H, in this case, is homogeneous; every spin variable expe-riences the same external field, but its interaction energy depends on the value of the spin variable.

A schematic representation of a system which can be described by the Ising model is shown in Figure2.1.

2.2.2 Statistical Mechanics

For this subsection, we follow the explanation given in [6, pp. 7-18]. Statistical mechanics is discussed in multiple other sources as well, such as [7], but [6] has a more mathematical

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Figure 2.1: A schematic representation of the two-dimensional Ising model on a square grid. The arrows indicate whether the spin variable is +1 or -1.

point of view and does not require knowledge of thermal physics, which will be more suitable for our intended audience.

In the case of a large number of variables, we need to introduce the theory of statistical mechanics to study the behaviour of the system. We will make two assumptions on the system which will make our upcoming work more bearable. First, we will only take care of systems with discrete energies, and we also assume there exists some maximal value ∆E such that all differences between energy levels are a multiple of ∆E. Fortunately, this value of ∆E will not remain in the expressions we derive.

If we only consider the one sample of an Ising model we are interested in, with a finite amount of particles, we have too few particles to effectively use statistical methods. Therefore, we must consider the Ising model sample which has captured our interest in a ‘bath’ of other Ising models. We will consider N copies of the original Ising model, each connected by infinitely weak interaction energies. These interactions will allow for an exchange of energy, without themselves contributing to the total energy of the system. We are interested in the limit N → ∞. This bath allows the system to freely exchange energy with the environment, and thus to reach thermal equilibrium.

In other words, we copy our graph a several times and create a bigger graph in which all these copies are sub-graphs. All sub-graphs are connected with these very small interaction energies, which are small enough to not contribute to the total energy even if N → ∞. All sub-graphs have the same structure, but the copied variables do not have to take the same value for each sub-graph. Using the same structure for the sub-graphs in the environment as the original system has a mathematical advantage since we can treat the environment and the system of interest similarly. We study this system using a postulate in statistical mechanics:

Postulate 2.2.1. If we fix the total energy Etot of a system in thermal equilibrium, then all configurations corresponding to the total energy are equally probable.

Thus, we focus on the set of configurations which all have the same total energy Etot. These configurations are often called the microstates.

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We can mathematically formulate this postulate. Let σ(n)represent the set of variables for the n-th Ising model, and let E(n) σ(n) be the corresponding total energy of that configuration in that Ising model. If we only know that the ensemble has energy Etot, the probability that the N Ising models have configurations σ(1), σ(2), . . . , σ(N ) is given by

P

σ(1), σ(2), . . . , σ(N )|Etot=

( ₁

Ω(Etot₎, Etot :=

PN

n=1E(n) σ(n) = Etot

0, otherwise, (2.2)

where Ω(Etot) is the number of configurations corresponding to a total energy equal to Etot_{. The quantity Ω(E}tot_{) can be expressed as}

Ω(Etot) =X σ(1) X σ(2) · · ·X σ(N ) δEtot_,Etot.

Using a Kronecker delta once more, we can rewrite Equation (2.2) as P

σ(1), σ(2), . . . , σ(N )|Etot= δEtot,Etot

Ω(Etot₎. (2.3)

2.2.3 Partition Function

We have now extended our original one Ising model with N − 1 additional systems. We are, however, not necessarily interested in these additional Ising models. We would like to know the probability P σ(1)|Etot

that the first Ising model (which we have chosen to be the original we are interested in) has configuration σ(1) given the ensemble has energy Etot. For a system in thermal equilibrium, this probability is computed, modifying Equation (2.3), by P σ(1)|Etot= P σ(2)· · · P σ(N )δ_Etot_,Etot Ω(Etot₎ . (2.4)

For small N , we can no further simplify this for the general case. However, we are interested in the case where N is large, or rather the limit N → ∞. That corresponds to having one Ising model connected to a ‘bath’ with infinite degrees of freedom. That bath acts as the environment for the structure we are interested in. A collection of Ising models with a probability given by Equation (2.4) is called a canonical ensemble.

The following results are taken from [6]. The calculations and derivations are left out, as they can be found in [6, pp. 10-15]. The most important intermediate results will be mentioned here.

Using the definitions

ζ := iϑ

∆E and Z(ζ) := X

σ

e−ζE(σ), (2.5) the quantity Ω(Etot) can be rewritten as

Ω Etot = ∆E 2πi

Z iπ/∆E

−iπ/∆E

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Similarly, the numerator of the fraction in Equation (2.4) can be written as X σ(2) · · ·X σ(N ) δEtot_,Etot = ∆E 2πi Z iπ/∆E −iπ/∆E exp ζ h Etot− E(1)σ(1) i + (N − 1) log Z(ζ) dζ .

Now, to find an approximation of these integrals, the method of steepest descent is used. Because Z is a finite sum of analytical functions, it is analytical itself as well. This allows for the integration contour to be deformed in the complex plane. Instead of integrating from −iπ/∆E to iπ/∆E in a straight line, we now integrate via −iπ/∆E + β and iπ/∆E + β in straight lines. Here, β is chosen to satisfy

¯

E := Etot/N = − ∂

∂βlog Z(β)

which turns out to be well defined; exactly one β satisfies this condition.

When β is determined from this equation, Z(β) is called the partition function. From physical considerations, it follows that the variable β can be shown to equal (kBT )−1,

for kB the constant of Boltzmann and T the temperature.

Using the method of steepest descents, it follows that as N → ∞, P σ|Etot = exp(−βE (σ))

Z(β) (1 + O(N

−1_{)) →} exp(−βE (σ))

Z(β) ,

where σ is the configuration of the original Ising model.

This is where the relevance of Z becomes clear. The partition function Z can be of great use when determining properties of the system. See the following example. Example. Given the partition function Z, the expected energy of the original copy of the Ising model given the total energy Etot _{is given by}

E E |Etot = − ∂

∂βlog Z(β). This follows from

− ∂ ∂β log Z(β) = −_∂β∂ Z(β) Z(β) = X σ E(σ)e −βE(σ) Z(β) = X σ

E(σ)P σ|Etot = E E|Etot_.

By definition, the partition function as we have seen until now is a finite sum of exponentials, and, therefore, analytical. However, ultimately this paper will consider only systems with infinite size. To clarify, in this chapter we considered N → ∞ copies of the same system in order to use statistical physics, but the original system did not have to have infinite size. From this moment onward, however, we will assume that the system of interest has infinite size as well. When we have a system of infinite size, the partition function is described by an infinite sum of analytical functions, which is no longer necessarily analytical itself.

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Definition 2.2.2. Let the partition function Z, which may take more variables, and one of its variables x be given. If log(Z) is non-analytic as a function of x at some x0,

we say that the corresponding system has a phase transition in x at x0.

Examples include iconic examples as the melting of ice, the boiling of water, sublima-tion of a solid, but also examples lesser known by the general public such as a magnet losing its magnetisation (see Section 2.2.4).

For this paper, we are interested when these phase transitions do (or do not) occur. In the next chapter, we will discuss useful theorems linking complex zeros of the partition function to phase transitions. In the chapters after that, we will investigate these zeros using graph theory and complex dynamics.

We will now also define the partition function more formally than previously done in Equation (2.5).

Definition 2.2.3 (Physical partition function). Given a graph G = (V, E), we can consider this as a system described using the Ising model where V represent the set of vertices v, each assigned a value σ(v) = ±1. We consider two variables u, v to be interacting precisely when {u, v} ∈ E, with interaction energy J . The system can also be affected by an external field H, which acts uniformly on the system.

We define the corresponding partition function ZG(J, β, H) as

ZG(J, β, H) := X σ exp(−βE (σ)) =X σ exp  β X {u,v}∈E J σ(u)σ(v) + βX v∈V Hσ(v)  .

Please note that the sum over σ is shorthand for the sum over all functions (or configu-rations) σ : V → {−1, 1}.

2.2.4 Example

In this section, we will discuss an example of a system where we can use the partition function to determine phase transitions:

Example. Given a ferromagnet of infinite size with no external field H. At which tem-perature T does the magnet lose its magnetic properties?

Even though the remainder of this project discusses zeros of the partition function, we will not use this method yet. This problem is one of the ‘easier’ two-dimensional problems which can be solved with the partition function. Nevertheless, we will have to make some assumptions and will, therefore, not solve it exactly. An exact solution is possible, but well beyond the scope of this introductory section.

We will make a total of three restrictions to this model. Firstly, we demand variables which are not nearest neighbours to be non-interacting, i.e. when α and α0 are not nearest neighbours, then E(α, α0) = 0. Two variables are considered nearest neighbours when the distance between their locations is one unit. In Figure 2.1, such variables are directly connected with a line.

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Secondly, we assume the structure to be regular. So, every variable has the same number of nearest neighbours and all nearest neighbours have the same distance. The uniform distance implies that all interaction energies are equal. Although many text-books, such as [7], assume the variables to be situated on a square grid, we will not make that assumption. Thirdly, we will use an approximation which will be introduced when we need it, as we are not quite ready for it now.

These two restrictions greatly increase our ability to make calculations on this model. For example, in two-dimensions, exact calculations for all temperatures can only be performed when only nearest-neighbour interactions are considered. However, in one dimension, this restriction is not necessary. An exact computation with only these two assumptions is performed in [8]. Instead of presenting this method, we will use one approximation that will reduce the number of computations we have to perform to find an approximate solution.

Note that, mathematically, these two restrictions can be captured in one restriction. We assume the graph to be regular (every vertex has the same degree, or number of neighbours) and all interaction energies E to be equal when two vertices are neighbours in the graph. The updated formulation of the example now reads:

Example. Given a ferromagnet represented by an infinite d-regular graph with interaction energy E precisely between neighbours, with no external field H. At which temperature T does the magnet lose its magnetic properties?

Solution. This solution is similar as discussed in [7, pp. 344-345] and [9, pp. 97-99]. We focus on one of the vertices viin our graph with neighbours {u1, . . . , ud}. If σv = 1, then

the total energy of the interaction energies the vertex vi has with its neighbours is

E(1) = −1 2E d X k=1 σ(uk) = − 1 2Ed¯σ,

where ¯σ is the mean of the spins σ(u1), . . . , σ(ud). Similarly, for σ(v) = −1 we find

E(−1) = 1 2E d X k=1 σ(uk) = 1 2Ed¯σ.

We can use the partition function as if vi is the only variable spin. Then we can define a

local partition function Zi, which only takes into account the configurations σ(vi) = ±1.

Then we find that Zi is equal to

Zi =

X

σ(vi)

e−12βE(σ(vi)).

If we fill in our results, we find that Zi= e

1

2βEd¯σ+ e− 1

2βEd¯σ = 2 cosh βEd¯σ

2

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(a) 1₂βEd > 1 (b) 1₂βEd < 1

Figure 2.2: The orange line portrays tanh(βEd¯σ), whereas the blue line represents ¯σ. In both figures, these functions are plotted for a representative value of βEd on ¯

σ ∈ [−1, 1].

This implies that the average expected value of the spin of vi is given by

E(σ(vi)) =   X σ(vi) σ(vi)P(σ(vi))  = 1 Zi e12βEd¯σ− e−βEd¯σ = tanh βEd¯σ 2 .

Now is the time for our third and final approximation. We approximate that E(σ(vi)) =

¯

σ. In other words, we assume that the expected value of the spin of vi is equal to

the mean of the spin of its neighbours. This approximation is called the mean field approximation. The physical motivation of this approximation is that no spin variable behaves differently than its surrounding. This is true, under the assumption that the orientation of vi does not affect its neighbours, which is true for d → ∞.

With this approximation, we have found the following self-consistency equation: ¯

σ = tanh βEd¯σ 2

.

This equation has, depending on βEd either 3 or 1 solution, see Figure 2.2. We were interested in for what value of β this system shows a phase transition. When βEd = 2, the system shows a pitchfork bifurcation. This bifurcation indicates the phase transition, which corresponds to demagnetisation.

Why the phase transition is demagnetisation, and not some other phenomenon, can be seen by the following observation. When βEd > 2 there were two stable configurations for the magnet: most of the spins up, or most of the spins down. When βEd → 2, these stable configurations converge to ¯σ = 0, implying no net spin orientation. When βEd < 2, we find that the system only has a stable configuration with no magnetic behaviour.

Using that β is given by (T kB)−1, we find that the temperature at which we have

demagnetisation is given by

Tc=

Ed 2kB

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This is when the phase transition happens. Because the material is assumed to be a ferromagnet, this value is positive.

We have found a phase transition in T at Tc= _2kEd_B, which means that log ZG(E, β, 0)

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3 Lee-Yang Theorems

Having concluded the review of both the mathematical and physical prior knowledge required for this project, we will now discuss the Lee-Yang theorems. The first of these theorems will provide us with a tool to determine when phase transitions cannot occur, by making use of zeros of the partition function.

We will state Theorem3.1.1, which originally applies to a model of monatomic gases, before discussing how it applies to the model we are interested in: the Ising model. We will discuss the similarities between the monatomic gas and the Ising model in Section3.2 and state the theorem for the Ising model in Section 3.3, together with the famous Theorem3.3.2.

3.1 Monatomic gases

The first theorem discusses the behaviour of the partition function of a series of graphs {ZV} as the number of vertices |V |, from here on denoted as the volume V , goes to

infinity.

The model to which the theorem is originally applied in [1] is a monatomic gas. This gas has an interaction energy between two atoms as a function of their relative distance. Furthermore, the interaction energy has the following properties:

1. The atoms have a finite impenetrable core of diameter a, so that E(r) = ∞ if r ≤ a.

2. The interaction has a finite range b, so that E(r) = 0 if r ≥ b. 3. E(r) 6= −∞ for all r.

When considering gases, the partition function as we have seen in Definition 2.2.3 is not correct. Instead, we need the grand canonical partition function ZV, of a gas with

volume V , equal to Z_V = M X N =0 ZNyN,

where M is the maximum number of atoms that could fit in V , ZN is the partition

function of Definition 2.2.3and

y = 2πm/βh2

3

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is the fugacity of the gas.1 We now state the first theorem of Lee and Yang.

Theorem 3.1.1. If in the complex y-plane a region R containing a segment of the positive real axis is free of roots for all V , then in this region as V → ∞ all the quantities

1 V log ZV, ∂ ∂ log y 1 V log ZV, ∂ ∂ log y 2 1 V log ZV, . . . approach limits which are analytic with respect to y. Furthermore, the operations

∂ ∂ log y

and limV →∞ commute in R.

Remember that we defined a phase transition as non-analytic behaviour of log Z. We conclude by Theorem 3.1.1 that, in the case that we can find a neighbourhood of R+

which contains no zeros for any V , there cannot be a phase transition in y at any positive value of y. With other words, for all positive y, the logarithm log Z of the partition function is an analytic function of y. Furthermore, in the case that there are several pieces of the positive real line which allow for the existence of such neighbourhood, these pieces individually allow only one phase. See Figure3.1for an example.

Figure 3.1: This is an example to clarify the implications of Theorem3.1.1. The coloured lines represent the positive real line and the black curve the set of zeros of the partition function. By the theorem, there are no phase transitions on each of the coloured lines, only possibly at the transition from one colour to another.

3.2 Comparison Lattice Gas and Ising Model

In the previous section, we have discussed two theorems which apply to monatomic gases. We are, however, primarily interested in the phase transitions of the Ising model. In this section, we will discuss how these two models compare, which was originally discussed in [2]. We will see that they are mathematically equivalent. Afterwards, we will be able to apply the theorems to the Ising model, as well in Section3.3.

In the monatomic lattice gas, the energy of the gas is determined by a configuration as well. This configuration describes whether locations on the lattice are occupied or vacant, whereas the configurations of the Ising model describe the orientation of the

1_{In this expression of y, m represents the mass of an atom, h is Planck’s constant and µ is the chemical}

potential. We will not go into depths about the meaning of the chemical potential or the fugacity, as they will only be mentioned in this chapter and are not very relevant for the remainder of this paper.

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spin at every lattice point. It is possible to translate these configurations between the models. We will say a vacant lattice point in the monatomic gas corresponds to a spin variable of +1 and an occupied point with a spin of −1.

We now see that the volume of the lattice gas corresponds to the total number of spins and that the number of atoms in the gas corresponds to the number of downward oriented spins.

For our following discussion of the Ising model, we will use some new notation, as defined in Table3.1.

Symbol Definition

N The total number of spins.

[↑] The number of spins oriented upwards.

[↓] The number of spins oriented downwards.

[↑↑] The number of nearest neighbouring spins that are parallel and upward. [↓↓] The number of nearest neighbouring spins that are parallel and downward. [↑↓] The number of nearest neighbouring spins that are antiparallel.

Table 3.1: Notation used in this section as defined for the Ising model.

Remark. Suppose the Ising model is defined on a d-regular lattice. Then the identities below are true.

N = [↑] + [↓] d[↑] = [↑↓] + 2[↑↑] d[↓] = [↑↓] + 2[↓↓]

dN = 2([↑↑] + [↓↓] + [↑↓])

Using this remark, the total energy of a configuration σ is given by E(σ) = H([↓] − [↑]) + E([↑↓] − [↑↑] − [↓↓]) = H([↓] − [↑]) + E 2[↑↓] −dN 2 . Since −EdN/2 is independent of σ, we can redefine the total energy E (σ) such that E(σ) = H([↓] − [↑]) + 2E[↑↓]. We find the partition function of the Ising model Z_I is equal to

ZI = ZI(E, β, H) =

X

σ

e−β(H([↓]−[↑])+2E[↑↓]).

It will prove useful to introduce the notion of free energy per lattice point, which is defined as F = −kBT N−1log(ZI) in [7, p. 247], so that we may write the partition

function as

ZI = exp(−F N β). (3.1)

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We now consider a lattice gas on the same lattice as we used for the Ising model. Then, the volume per atom, or specific volume v, corresponds to

N [↓] = 2N N + [↓] − [↑] = 2 1 − I.

Furthermore, we wish these two systems to correspond and no two atoms to occupy the same lattice site, so we consider the following potential energy u between two atoms:

  

 

u = +∞ if the two atoms occupy the same lattice point, u = −4E if the two atoms are nearest neighbours, and u = 0 otherwise.

Since [↓↓] corresponds to the number of nearest neighbours of the atomic gas by the correspondence, the total energy of the gas UG is given by UG = −4[↓↓]E. Therefore,

the grand partition function of this gas is given by ZG=

X

σ

y[↓]e4[↓↓]Eβ.

Similar to Equation (3.1), this may be written as ZG= exp(pN β), where p is the pressure

of the gas.

If we choose the fugacity of the gas y to equal exp(−2β(H + dE)), we find Z_G=X σ e−β(2H[↓↓]+2d[↓]E−4[↓↓]E)=X σ e−β(2H[↓]+2E[↑↓])=X σ e−β(E(σ)+N H), (3.2)

which equals exp(−N β(F + H)) using Equation (3.1). From the last line of equations, we find that p corresponds to −F − H, since exp(pN β) = exp((−F − H)N β).

We have found the corresponding quantities as shown in Table 3.2. Using the table, we can derive the thermodynamic properties of one system from the other. This shows the two problems are equivalent.

Ising model Lattice gas

Number of spins Volume

Number of spins downward Number of atoms 2/(1 − M ) Specific volume v

−F − H Pressure p

Table 3.2: Corresponding quantities between Ising model and lattice gas.

3.3 Application of Theorems to Ising Model

We will now use the correspondence between the lattice gas and the Ising model discussed in the previous section to formulate Theorems3.1.1and3.3.2in the setting of the Ising model as well.

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First, note that only considering a lattice gas in the previous section could be done with no loss of generality, as all monatomic gases can be considered the limit of a lattice gas as the lattice constant becomes arbitrarily small.

Second, we have seen that the grand canonical partition function of the monatomic gas and the canonical partition function of the Ising model are proportional.

We will now write y = exp(−2β(H + dE)) as x · ξ, where ξ = exp(−2βH) is indepen-dent of the structure of the graph, and x does not depend on H but does depend on the structure of the graph. For a given temperature and structure, x can be considered as a constant.

This implies that Theorem 3.1.1, when applied to the Ising model, reads as follows. Theorem 3.3.1. If in the complex ξ-plane a region R containing a segment of the positive real axis is free of roots for all N , then in this region as N → ∞ all the quantities

1 N log ZN, ∂ ∂ log ξ 1 N log ZN, ∂ ∂ log ξ 2 1 N log ZN, . . .

approach limits which are analytic with respect to ξ. Furthermore, the operations_{∂ log ξ}∂ and limN →∞ commute in R.

Furthermore, it holds that for the Ising model with N spins

ZN = e−F N β ∗= eHN β N X n=0 X σ [↓]=n x[↓]e4[↓↓]Eβξ[↓] = eHN β N X n=0 Pnξn,

where Pn is a positive real number that is independent of ξ. Note that ∗ follows from

Equation (3.2). We define the polynomial P (ξ) =PN

n=0Pnξn.

If we now set some very general restrictions for the interaction energy, one can prove the famous Lee-Yang theorem [2], stated below.

Theorem 3.3.2 (Lee-Yang). If the interaction E between two gas atoms is such that (

E = +∞, if the two atoms occupy the same node in the lattice E ≤ 0, otherwise,

then all the roots of the polynomial P lie on a circle centred on the origin in the complex y-plane.

This theorem states that the zeros of the partition function are not scattered around the complex plane as we could have thought a priori. The locations of the zeros are very orderly: they are confined to the unit circle. Moreover, as N → ∞, all zeros must remain on the unit circle. Therefore, one of the results of Theorems 3.1.1 and 3.3.2 is that a monatomic gas with an interaction energy as described in the latter can have only one phase transition, if it has any at all. The same holds for a system described by a ferromagnetic Ising model: there can only be a phase transition when |ξ| = 1.

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-1.0 -0.5 0.5 -1.0 -0.5 0.5 1.0 (a) -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 (b) -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 (c)

Figure 3.2: In (a) to (c), the blue curve represents the zeros of the partition function, which all lie on the unit circle by Theorem3.3.2. In (a) there exists a region containing the positive real line that contains no zeros, hence there is no phase transition. In (b), the zeros get arbitrarily close to the real line, but the red point at ξ = 1 is not a zero. In (c) all points of the unit circle are a zero for the partition function. In (b) and (c), Theorem 3.3.1does not give any information.

Since any physical external field H and β are real, we find that all physical values of ξ are real and even positive. Therefore, for ferromagnetic Ising models, there can only be a phase transition when ξ = 1.

It is worth noting that all values of ξ that are a zero of the partition function are unphysical,2 as physical values of ξ are bound to the positive real axis where Z > 0 because all Pnare positive. However, even though the zeros of the partition function are

unphysical values of ξ, Theorem3.3.1states that they can be useful in the determination of the physical phenomenon of phase transitions.

See Figure 3.2 for three different scenarios that can occur when analysing the ze-ros of the partition function for the Ising model, according to Theorem 3.3.2, and the conclusions that can be drawn using Theorem 3.3.1.

In Chapters 5 and 6, we will investigate a condition for the existence of a neighbour-hood of the positive real line, as described in Theorem 3.3.1, using a method which combines graph theory with complex dynamics. However, we will first investigate two experiments that measure or observe the Lee-Yang zeros in the complex plane in Chap-ter4.

2

We use the word ‘unphysical’ to make clear that no real external field can produce these values of ξ. In Section4.2we will discuss an experiment that measures these zeros.

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4 Experimental Determination of

Complex Zeros

In this chapter, we will discuss several experiments that try to measure the Lee-Yang zeros or their distribution in the complex plane. These experiments try to give a more physical meaning to the complex zeros of the partition function, apart from their theo-retical application in Theorem 3.3.1.

Both experiments require a lot of prior knowledge of either statistical physics or quan-tum mechanics. Therefore, we will not go into too much detail when discussing the experiments but will mention the key aspects of each method. This chapter aims to give insight into how the complex zeros of the partition function can be interpreted or observed experimentally in a laboratory, although they may appear unphysical a priori.

4.1 Density of zeros from magnetisation

We will commence with the experiment that was first conducted, by discussing the experiment described in [10]. This experiment aims to relate the density function of Lee-Yang zeros on the unit circle, g(ϑ), to the magnetisation of ferromagnets which can be described by the Ising model. The density function g gets its name from the property that g(ϑ) dϑ gives the fraction of zeros between ϑ and ϑ+dϑ on the unit circle in the case N → ∞ in Theorem 3.3.2. Recall that as N → ∞ the partition function has infinitely many zeros on the unit circle (counting multiplicity).

We start with the relation for the Ising model from [2]

I(z) = 1 − 4z Z ∞

0

g(ϑ)(z − cos ϑ) z2_{− 2z cos ϑ + 1}dϑ ,

where the normalised magnetisation I(z) = m/ms is a measurable quantity, with m

the magnetic moment and ms its saturation value. The normalised magnetisation I is

defined such that I = 1 when all spins from Figure2.1point upwards, and I = −1 when they all point downwards.1 For all other configurations, I ∈ (−1, 1). Using intermediate steps from [11], we can express g(ϑ) in terms of I as

g(ϑ) = 1

2πr→1lim−Re{I(r exp(iϑ))}. (4.1)

1

In Section3.2, we wrote an M for the normalised magnetisation but here we use the notation from [10].

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Using experimental data of m(H), the researches determined the values of I(H) for several values of the external magnetic field H. By combining this result with Equa-tion (4.1), the researchers have determined g(ϑ).

In order to do this, they have used the ansatz formula for the normalised magnetisation I(z) of the form

f (z) = 1 + n1z − (1 + n1)z

2

1 + d1z + d2z2

, (4.2)

where n1, d1 and d2 are parameters. Substituting I(z) 7→ f (z) in Equation (4.1) yields

the relation g(ϑ) = 1 2π 1 + (d1− d2)n1− d2+ (1 − d1+ d2)n1cos ϑ − (1 − d2+ n1) cos(2ϑ) 1 + d2 1+ d22+ 2d1(1 + d2) cos ϑ + 2d2cos(2ϑ) . (4.3) After fitting of Equation (4.2) with the experimental data for I(H), the values obtained by fitting for the parameters n1, d1 and d2 are substituted in Equation (4.3) to obtain

an approximate formula for g(ϑ). In Figure 4.1, several approximations made in [10] of g(ϑ) are portrayed. Please note that ϑ ∈ [0, π] in the figure, but this information is sufficient, since g(−ϑ) = g(ϑ).

Figure 4.1: This figure is taken from [10]. Here, the experimentally determined Lee-Yang density function g is shown for T = 49, . . . , 53, 99 K, respectively. The data for curves 5 and 6 are scaled by factors 1/2 and 1/4, respectively. In this figure, the zeros stay away from the positive real axis, hence the systems exhibit no phase transitions. However, it suggests that g(0) > 0 at temperatures below those measured in this experiment. In that case, phase transitions could occur.

The validity of the experiment is determined by comparing the results of two calcula-tions of the magnetic specific heat c(H), one based on the data of m(H) and the other using g(ϑ). The two values are found to be within a 6% error margin of each other for T = 49, . . . , 53 K, implying that the experimentally determined function g(ϑ) is a good

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approximation of the theoretical g(ϑ) (which should yield the exact value of the mag-netic specific heat) for these temperatures. The correspondence between the different values for c(H) is not present for T = 99 K. The researches note that this may be due to the high temperature violating one of the necessary conditions for approximating this specific system well with the Ising model.

4.2 Correspondence between Lee-Yang zeros and critical

times in decoherence

We will now examine the experiment first theoretically described in [12] and experimen-tally performed in [13]. This experiment is the first to directly observe the complex zeros of the partition function, which were before then considered unobservable because of their complex nature. Please note that [13] was published in 2012, making this very recent research. We will deviate our notation from [12, 13] to better accommodate notation elsewhere in this paper.

In this experiment, a so-called probe spin is coupled to a ferromagnetic Ising model system, such that it has equal interaction with all spins of the Ising model. The probe is a quantum mechanical particle that can be either in the up or down state, similar to the spins in the Ising model, but can also be in a superposition of these states. In the latter case, only the measurement of the probe spin will determine the direction of its spin. The interaction between the probe spin and the Ising model is equal to HI = −λσzPv∈V σ(v). Here, λ is a constant and σz is the Pauli z-matrix2 acting on

the probe spin. The factor B :=P

v∈V σ(v) acts as a random field variable for the probe

spin.

The Ising model has the corresponding partition function with J ≥ 0 in Defini-tion 2.2.3, which can be written in the form of a polynomial when the system is finite, as we have seen in Section 3.3.

The probe spin is initially in a superposition state of |Ψ(t = 0)i = √1

2(|↑i + |↓i), and

the Ising model is in thermal equilibrium.3 Fluctuations of the random field cause a random phase such that

|Ψ(t)i = √1 2

|↑i + eiλtB|↓i describes the superposition state of the probe spin at time t.

Now we introduce the coherence L of the quantum probe spin, which corresponds to the spin polarisation in the x − y plane. More formally, it is defined as the expectation value of σx+ iσy. Importantly, the coherence L is a measurable quantity. At time t, the

2_{Very bluntly put, this matrix represents the measurement that decides whether the quantum probe spin}

is up or down. A more elaborate explanation of the quantum mechanics involved in this experiment can be found in [14].

3

The researchers note that the time evolution of the probe spin can be considered an equilibrium-state phenomenon as well, because the coupling with the Ising model can be made arbitrarily small.

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Figure 4.2: This figure is taken from [13]. In (a) to (c), the crosses indicate the measured probe spin coherence L(t) for (a) a laboratory temperature of T = 300 K, (b) a simulated temperature Teff = 15 J/8 and (c) a simulated temperature

Teff = 9 J/40. The solid lines are the numerically calculated values of L(t).

In (d) to (f), the red crosses indicate the Lee-Yang zeros measured from the zeros of L(t) corresponding to (a) to (c). The theoretical predictions of the Lee-Yang zeros are shown as blue circles for comparison.

coherence L is shown to equal the expectation value of the phase factor, given by L(t) = EeiλtB=∗ Z(J, β, H + 2iλt/β)

Z(J, β, H) ,

where ∗ is a non-trivial result discussed in [12]. If we were to ignore the normalisation factor Z(J, β, H), we would find that L(t) is equivalent to the partition function with a complex magnetic field equal to H0 := H + iλt/β. For the ferromagnetic Ising model the Lee-Yang zeros all lie on the unit circle, by Theorem 3.3.2, meaning they occur exclusively at imaginary values for the external field H0. Hence, the actual external magnetic field H is set to zero in this experiment.

Therefore, the probe spin coherence L vanishes when the time t is such that ξ0 := exp(−4iλt) reaches a Lee-Yang zero. By measuring the times t at which the coherence L vanishes, the researches determined which corresponding values for the complex magnetic field constitute a Lee-Yang zero. The results from [13] are shown in Figure4.2.

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It must be noted that the Ising model used for the experiment has only nine particles and one probe spin, which is far from N → ∞. To increase the number of particles N in the Ising model, each of the newly added particles will have to be connected to the probe spin. The measurement itself will not be more difficult to perform, as only the coherence L of the probe spin has to be measured. The researches mention this as an advantage of their setup, as compared to other methods.

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5 Graph Theory

As motivated by Theorem3.3.1, we need to establish whether there is a region of R+that

is free of zeros of the partition function as the size of the graph becomes infinitely large. So, for a given graph G of infinite size and bounded degree and a sequence of subgraphs of G that approximate G increasingly well as their sizes increase, we wonder whether there is a region of R+ that is free of roots of Z for all sub-graphs in the sequence.

Therefore, it would prove useful if the partition function of a graph could be defined inductively from its sub-graphs. However, Z has no such property.

In this section, we will introduce a function R that has a similar property. For a tree, the function R can be determined inductively from subgraphs, which is proven in Lemma 5.1.3. Moreover, we will argue that the study of trees is sufficient, as for every graph of bounded degree there exists a tree with the same bounded degree that has the same function R. This is proven in Theorem 5.2.1, which is the main goal of this chapter. The usefulness of R follows from Lemma 5.1.2, which relates the zeros of the partition function with zeros of R + 1.

In Chapter 6 we will use these results and use methods from the field of complex dynamics to observe the behaviour of the partition function when we take the limit |V | → ∞, so that Theorem 3.3.1may be applied.

The statements made in the following two chapters follow the results of [15].

5.1 Bounded Degree Trees

The following definitions allow us to transfer from a physical into a more mathematical setting.

Definition 5.1.1. 1. Let G = (V, E) be a graph and let X ⊆ V . We call any map τ : X → {0, 1} a boundary condition on X.

2. Let τ be a boundary condition on X ⊆ V . We say U ⊆ V is compatible with τ when both (τ (x) = 1 =⇒ x ∈ U ) and (τ (x) = 0 =⇒ x /∈ U ) hold for each x ∈ X. If U is compatible with τ , we write U ∼ τ .

3. We define ZG,τ(ξ, b) := X U ⊆V U ∼τ Y u∈U ξu ! · b|δ(U )|. (5.1)

When X = ∅, this is equivalent to the partition function in Definition 2.2.3, up to a factor, where ξ = exp(−2βH), as before, and b := exp(−2J β). The boundary

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condition determines, for the vertices in X ⊆ V , whether they must or must not be included in the subsets U ⊆ V .

4. Given a boundary condition τ of X and a vertex v of G. We construct boundary conditions τv,0, τv,1 on X ∪ {v} where v is set to 0 and 1, respectively. All elements

of x ∈ X are mapped to τ (x) by these boundary conditions. It is possible that v is given two different values by one of these boundary conditions, in which case no subset U ⊆ V is compatible with that boundary condition.

5. We finally define RG,τ,v :=    ∞ if ZG,τv,0 = 0, Z_G,τv,1(ξ,b) Z_G,τv,0(ξ,b) otherwise, (5.2) which is called the ratio of the partition function.

The following lemma relates the zeros of the partition function with the values for which R = −1.

Lemma 5.1.2. If ZG,τv,0 6= 0, then:

RG,τ,v 6= −1 ⇐⇒ ZG,τ 6= 0. (5.3)

Proof. We observe that RG,τ,v 6= −1 ⇐⇒ ZG,τv,1 ZG,τv,0 6= −1 ⇐⇒ ZG,τv,1+ ZG,τv,0 ZG,τv,0 6= 0 ⇐⇒ ZG,τ ZG,τv,0 6= 0, which is true precisely when ZG,τ 6= 0.

Using this lemma, finding a zero-free region of R+in the complex ξ-plane is equivalent

to finding a region of R+ where R(ξ) 6= −1, as long as ZG,τv,0(ξ) 6= 0.

For bounded degree trees, we have the following lemma. This lemma will prove useful in Sections 5.2 and 6.3. The lemma provides us with a way to inductively determine RG,τ,v for a tree by considering the trees that remain after a vertex is removed.

Lemma 5.1.3. Let G = (V, E) be a tree with boundary condition τ on X ⊆ V . Let u1, . . . , ud be the neigbours of v in G, and let G1, . . . , Gd be the components of G − v

containing u1, . . . , ud respectively. We just write τ for the restriction of τ to X ∩ V (Gi)

for each i. For i = 1, . . . , d let τi,0 and τi,1 denote the respective boundary conditions

obtained from τ on (X ∪ {u}i) ∩ V (Gi) where ui is set to 0 and 1 respectively. If for

each i, not both ZGi,τi,0 and ZGi,τi,0 are zero, then

RG,τ,v = ξv d Y i=1 RGi,τ,ui+ b bRGi,τ,ui+ 1 .

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Proof. Close inspection of Equation (5.1) shows we can write ZG,τv,1 = ξv d Y i=1 ZGi,τi,1+ bZGi,τi,0 and ZG,τv,0 = d Y i=1 bZGi,τi,1+ ZGi,τi,0

in the case G is a tree. Therefore,

RG,τ,v = ξv d Y i=1 ZGi,τi,1+ bZGi,τi,0 bZGi,τi,1+ ZGi,τi,0 .

Now, fix i ∈ {1, . . . , d}. If ZG,τv,0 6= 0, we can divide the numerator and denominator

by it, so that ZGi,τi,1+ bZGi,τi,0 bZGi,τi,1+ ZGi,τi,0 = RGi,τ,ui + b bRGi,τ,ui+ 1 . If, on the other hand, ZG,τv,0 = 0, we have RGi,τ,ui = ∞ and

ZGi,τi,1+ bZGi,τi,0 bZGi,τi,1+ ZGi,τi,0 = 1 b = RGi,τ,ui+ b bRGi,τ,ui+ 1 . This concludes the proof.

5.2 General Bounded Degree Graphs

In Lemma 5.1.3, we have found a way to inductively determine the ratio RG,τ,v of a

tree with subtrees connected by the vertex v, using the ratios of these subtrees with v removed. In Chapter 6 we will see how we can use this result to determine when the zeros of the partition function stay away from the positive real line.

We will first expand the scope of Lemma 5.1.3 so that it can be applied to general bounded degree graphs. We do this using an algorithm which constructs a tree of self-avoiding walks, which corresponds to the general graph. We observe that this tree has the same ratio of the partition function R as the general graph. Because of this, we can study a general bounded degree graph by studying its tree of self-avoiding walks. Construction. Given a graph G and a self-avoiding walk tree TSAW = TSAW(G, v) of

G, we want to construct a boundary condition τG on TSAW. We do this the following

way:

1. For each vertex u of G, assign an ordering of its edges. A leaf in TSAWcorresponding

to a walk closing a cycle in G is assigned a 0 if the edge closing the cycle is higher in the ordering than the edge starting the cycle. It is assigned a 1 otherwise. 2. If σ is a boundary condition on a subset of the leaves of G, we extend τG by

assigning to any vertex of TSAW corresponding to the end of a path in a leaf the

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Furthermore, for (ξu)u∈V complex numbers associated with the vertices of G, we

assign every vertex of TSAW one such value. For w ∈ TSAW, let u be the last vertex of

the corresponding walk. Then ξw is set to be equal to ξu.

Now we state and prove the following theorem.

Theorem 5.2.1. Let G = (V, E) be a connected graph of maximum degree at most d + 1, and with vertex v ∈ V . Let X ⊂ V \ {v} be a collection of leaves of G, and let σ : X → {0, 1} be a boundary condition on X. Set ξu = ξ for all u /∈ X, and choose

ξu 6= 0 arbitrarily for u ∈ X. Then

R_T_SAW_(G,v),τ_G_,v= RG,σ,v, (5.4)

where both sides are considered as rational functions in ξ.

Proof. We use induction on the number of free vertices n of G. A vertex is called free if it is not in the domain of the boundary condition.

If n = 1, then v must be that free vertex. By assumption, all other vertices are leaves. Thus, G is a tree and equals its tree of self-avoiding walks. Hence, the statement is immediate. In Figure5.1, an example of such a graph and its tree of self-avoiding walks is shown.

v

a b c

v

va vb vc

Figure 5.1: On the left a tree is shown, with on the right its tree of self-avoiding walks starting in v. It is obvious that the graphs are identical.

We now consider a graph G for which n ≥ 2, and we assume that Eq. (5.4) holds for all graphs with less than n free vertices. We denote the neighbours of v by u1, . . . , um.

We now want to split the vertex v in m pieces. We do this rigorously the following way: we construct a graph ˆG by replacing v in G with v1, . . . , vm, such that each vi only has

ui as its neighbour. To the vertex v was assigned an external field parameter of ξ, by

assumption. To the vertices v1, . . . , vm we assign a field parameter of ξ

1

m, choosing the

same holomorphic branch for all vi. An example of this procedure is shown in Fig.5.2.

We also extend the boundary condition σ in m + 1 different ways. For each i ∈ {0, . . . , m}, we define an extension σ_i of σ to X ∪ {v1, . . . , vn} defined by σi(vj) = 1

when j ≤ i and 0 when j > i. We observe that σ0 maps all vi to 0 and σm assigns all vi

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v u1 u2 u3 w1 w2 w3 v1 v2 v3 u1 u2 u3 w1 w2 w3

Figure 5.2: On the left a graph G is shown with a vertex v, all nodes of this graph are coloured blue. On the right is shown a graph ˆG constructed by splitting the vertex v as described in the text. This graph has red nodes.

Because we have chosen our parameter ξm1 , it follows that

Z_G,σˆ ₀ = ZG,σv,0 and Z_G,σˆ _m = ZG,σv,1. Therefore, RG,σ,v = ZG,σv,1 ZG,σv,0 = ZG,σˆ m Z_G,σˆ ₀ = m Y i=1 Z_G,σˆ _i Z_G,σˆ _i−1 . (5.5)

If we define ˆσi as the restriction of σi to X ∪ {v1, . . . , vi−1, vi+1, . . . , vm}, by freeing vi,

it follows from Equation (5.2) that Z_G,σˆ _i

Z_G,σˆ _i−1

= R_G,ˆˆ_σ_i_,v_i. (5.6)

Now, we have almost done all the preparations so that we can successfully use the induction hypothesis. However, we need to do one more thing.

We write ˆGi for the connected component of ˆG that contains the vertex vi. Observe

that ˆGi has at most as many free vertices with boundary condition ˆσi as G has with its

boundary condition σ. That is, because ˆG has as many free vertices under ˆσi as G has

under σ, and ˆGi only contains a subset of the vertices of ˆG. Take notice of Fig. 5.3for

an example of such a ˆGi.

Since ˆGi has at most as many free vertices as G, one of which is vi, the graph ˆGi− vi

will have less free vertices. It is here that we can apply our induction hypothesis to conclude that

R_Gˆ_i−vi,ˆσi,ui = RTSAW( ˆGi−vi,ui),ˆτi,ui,

where ˆτi is the boundary condition induced by ˆσi as described before.

We can write R_Gˆ_i_,ˆ_σ_i_,v_i = ξ 1 m R_Gˆ_i−vi,ˆσi,ui+ b bR_Gˆ_i−vi,ˆσi,ui+ 1 = ξm1 R_T SAW( ˆGi−vi,ui),ˆτi,ui+ b bR_T SAW( ˆGi−vi,ui),ˆτi,ui+ 1 , (5.7)

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v1 v2 u1 u2 w1 u1 u1w1 u1w1u2 u1w1u2u1 u1w1u2v2 u1u2 u1u2v2 u1u2w1 u1u2w1u1

Figure 5.3: Here, ˆG1 = ˆG2 is shown for the graph in Fig. 5.2on the left. On the right,

one can see the graph TSAW( ˆG1− v1, u1).

making a few easy steps first,1 and applying our induction hypothesis second. If we con-sider Eq. (5.5) and apply Eq. (5.6), then Eq. (5.7) and finally the result of Lemma5.1.3, we find RG,σ,v = m Y i=1 Z_G,σˆ _i Z_G,σˆ _i−1 Eq. (5.5) = m Y i=1 R_Gˆ_i_,ˆ_σ_i_,v_i Eq. (5.6) = m Y i=1 ξm1 R_T SAW( ˆGi−vi,ui),ˆτi,ui+ b bR_T SAW( ˆGi−vi,ui),ˆτi,ui+ 1 Eq. (5.7) = ξ m Y i=1 R_T SAW( ˆGi−vi,ui),ˆτi,ui+ b bR_T SAW( ˆGi−vi,ui),ˆτi,ui+ 1 = R_T_SAW_(G,v),τ,v Lemma 5.1.3.

If we can verify that τ satisfies the conditions we set for τG, the proof is complete.

In order to do so, we make the following observations. The boundary condition τ is obtained from the boundary conditions ˆτi, which are induced by ˆσi, and thus satisfies

the following conditions:

1. walks ending in a leaf u ∈ X are assigned σ(u), 2. walks ending in a leaf u /∈ X are not assigned a value,

1_{These steps are the following, where we use that v}

i only has neighbour ui:

RGî,ˆσi,vi= Z_Gˆ_i_,ˆ_σ iv_{i ,1} ZGî,ˆσiivi,0 = ξm1 Z_Gˆ_i−vi,ˆσ_iu i ,1 + bZ_Gˆ_i−vi,ˆσ_iu i ,0 bZGî−vi,ˆσiu_{i ,1}+ ZGî−vi,ˆσiu_{i ,0} = ξm1 RGî−vi,ˆσi,ui+ b bRGî−vi,ˆσi,ui+ 1 .

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3. the boundary condition of a cycle depends on the ordering of the neighbours and 4. the boundary condition for a walk that merely ends in a cycle is determined in the

induction process, and is determined.

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6 Complex Dynamics

In the previous chapter, we have found a method to find the ratio of the partition function RG,τ,v of any bounded degree graph G by using its tree of self-avoiding walks

TSAW. In this chapter, we will use complex dynamics to determine when this ratio is not

equal to −1, which is useful because of Lemma 5.1.2.

However, we will first prove Theorem6.1.2, which concerns maximal circular intervals which are contracted under the dynamics. This will be done in Section 6.1, together with the necessary preparation. Thereafter, in Section 6.2, we will use Theorem 6.1.2 to deduce the existence of an invariant cone for fξ,b, which will have two important

consequences, Lemma 6.3.1and Theorem6.3.2, which will be discussed in Section 6.3. Similar to those in Chapter 5, the results of this chapter will closely follow [15]. However, especially in the proof provided in Section 6.1, we will deviate from [15], at certain times.

6.1 Invariant Interval

We will start with a formal definition of the function fξ,b, which shows resemblance to

the equation in Lemma 5.1.3when RGi,τ,ui = z for all i ∈ {1, . . . , d}. Applying fξ,b will

correspond to connecting d copies of the same tree G to a single vertex with external field ξ acting on it.

Definition 6.1.1. Given b, ξ ∈ C and d ∈ N, we define 1. g : C → C : z 7→ _bz+1z+b, and 2. fξ,b: C → C : z 7→ ξ · g(z)d= ξ z+b bz+1 d .

Below is the statement of the theorem we will use in the next section. We will only prove the first part.

Theorem 6.1.2. Let d ∈ N≥2 and let b ∈

d−1 d+1, 1

. There exists ϑb ∈ (0, π) such that

(a) for each ξ ∈ ∂D with Arg(ξ) ∈ (−ϑb, ϑb) there exists a closed circular interval

Ib ⊂ ∂D, with 1 as boundary point, which is forward invariant under fξ,b and does

not contain −1. In particular, the orbit of z = ξ under fξ,b avoids the point −1;

(b) The interval (−ϑb, ϑb) is maximal: The collection {ξ} ⊂ ∂D for which the orbit of

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Before we prove the first part of this theorem, we will state and prove some results which will be useful in the proof.

Lemma 6.1.3. Let d ∈ N and b > 0. Then |f_ξ,b0 (z)| is monotonically increasing in | Arg(z)|.

Proof. We compute |f_ξ,b0 (z)| and see that

f_ξ,b0 (z)= ξd z + b bz + 1 d−1 1 − b2 (zb + 1)2 = d 1 − b 2 (zb + 1)2 = |d − db 2_| |zb + 1|2.

Since |zb + 1| decreases monotonically in | Arg(z)|, this proves the claim. Definition 6.1.4. We call a set I ∈ ∂D a circular interval if I is of the form

I = {eiϕ|ϕ ∈ (a, b)},

which we may write as ‘the circular interval Arc(eia_{, e}ib_)’. _{Similarly, we may write}

{eiϕ_{|ϕ ∈ [a, b]} as Arc[e}ia_{, e}ib_].

Corollary 6.1.5. Let d ∈ N≥2 and let b ∈

d−1 d+1, 1

. Then the set J :=z ∈ ∂D

f_ξ,b0 (z)< 1

is a circular interval containing 1, excluding −1, and is independent of ξ. Proof. The point 1 lies in J , since

|f_ξ,b0 (1)| = dξ z + b zb + 1 d−1 1 − b2 (zb + 1)2 z=1 = d 1 − b 2 (b + 1)2 = d1 − b 1 + b ∗ < 1.

where ∗ holds because d < b+1_b−1 by assumption. The point −1 does not lie in J , since

f_ξ,b0 (−1) = d 1 − b 2 (1 − b)2 = d1 + b 1 − b > 1.

The fact that J is a circular interval now follows from Lemma 6.1.3. Since |f_ξ,b0 (z)| is independent of ξ, J is as well.

Note that every point z on the unit circle has a unique ξz associated with it such that

fξz,b(z) = z. Actually, ξz is given by z · (f1,b(z))

−1_{= z · f} 1,b(z).

Studying Lee-Yang zeros using graph theory and complex dynamics