lattice study of the entanglement entropy in yang-mills theory the structure of this work is the...

Universita degli Studi di Torino

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica Teorica

Tesi di laurea magistrale

Lattice study of the entanglemententropy in Yang-Mills theory

Candidato:Emanuele Maunero

Relatore:Marco Panero

Anno Accademico 2017–2018

Contents

Introduction ix

1 Quantum entanglement 1

1.1 EPR Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Bell Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Conformal Field Theory 5

2.1 Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Conformal Group In 2 Dimensions . . . . . . . . . . . . . . . 7

2.2 Conformal Quantum Field Theory . . . . . . . . . . . . . . . . . . . 9

2.2.1 Operator Product Expansion . . . . . . . . . . . . . . . . . . 11

2.3 Conformal Field Theory in 2 Dimensions . . . . . . . . . . . . . . . . 12

2.3.1 Stress Energy Tensor . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Virasoro Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Hilbert space of the conformal theory . . . . . . . . . . . . . . . . . 19

2.4.1 Null vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Ward Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Lattice Gauge Theories 25

3.1 Introduction To Lattice Gauge Theories . . . . . . . . . . . . . . . . 25

3.2 Lattice Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 Wilson Action . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 The Partition Function . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Setting the Scale . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Montecarlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Heath-Bath and Overrelaxation Algorithms . . . . . . . . . . 34

3.3.2 Errors In Lattice Results . . . . . . . . . . . . . . . . . . . . 36

4 Entanglement Entropy 37

4.1 Entanglement Entropy in Quantum Field Theories . . . . . . . . . . 37

4.2 Entanglement Entropy in Conformal Field Theories . . . . . . . . . . 40

4.2.1 Renyi Entropy and The Replica Trick . . . . . . . . . . . . . 40

4.2.2 Entanglement Entropy in a 2D Conformal Field Theory . . . 41

4.2.3 Entanglement Entropy in a massive Quantum Field Theory . 43

4.2.4 Entanglement Entropy of a Conformal Field Theory in d+1Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iii

iv CONTENTS

5 Lattice Simulations 495.0.1 Lattice Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Measurement Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.1 The Divergent Term . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 The Non-divergent Term . . . . . . . . . . . . . . . . . . . . . 53

Conclusions 55References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Abstract

The quantum entanglement of states of systems with many degrees of freedom isa very useful, model-independent, physical property, characterizing the structureof the ground state of quantum fields. It can be used to classify different phases offield theories, which are associated with different patterns of entanglement. In thiswork, we carry out a high-precision non-perturbative study of the behaviour of theentanglement entropy in SU(2) Yang-Mills theory in 3+1 dimensions. We regularizethe theory on a Euclidean hypercubic lattice of spacing a, which automaticallyprovides a gauge-invariant momentum cut-off O(a−1). We focus on the entanglementbetween a 3-dimensional slab of thickness l and its complement, and investigatequantitatively the quadratically ultra-violet divergent, l-independent, contributionto the entanglement entropy, as well as the non-divergent terms, scaling like l−2.

v

Acknowledgments

First of all i would like to thank my supervisor, Marco Panero, whose help hasbeen decisive for the completion fo this work. He has been able to guide me throughall the workflow with always inspired advises.

A special thanks goes to my family, who supported me in mi life choices withoutever doubting me.

All the code I have written for this work could have taken much more time if ithad not been for my brother’s help, who has had the patient to endure my outburstswhen things did not work.

I can not forget all my friends, the old ones and the new ones. I could not bethe person I am without all of them.

vii

Introduction

In recent years, it has become increasingly clear that the entanglement structureencoded in a many-body wave function provides important insight to the structureof the quantum state under consideration [33]. The simplest illustrative example isthe notion of topological entanglement entropy in (2 + 1)-dimensional topologicalfield theories. These are systems with no dynamical degrees of freedom, whichnevertheless exhibit an interesting phase structure. As discussed in [21] and [22],the entanglement entropy provides a useful oder parameter for characterizing thedistinct phases in such systems. Moreover, for quantum fields on the lattice in thevicinity of a quantum phase transition, the ground state is a strongly entangledsuperposition of the states of all elementary degrees of freedom, and different phasescan be characterized by different patterns of entanglement [27].

While the application of quantum entanglement to distinguish phases of many-body dynamics is fascinating in its own right, a remarkable connection linkinggravitational dynamics and entanglement, which emerged in the context of holography[29], provides further reason to delve deeper into the subject. This connection comesfrom the similarities between the entanglement entropy and the entropy of blackholes. As we will see in chapter 4 the entanglement entropy is defined as the vonNeumann entropy for the reduced density matrix ρA. This matrix describes thestate of quantum fields as seen by an observer who can only perform measurementswithin a subsystem A of a total system A ∪ Ac. The region Ac is inaccessible forthis observer, as if it was separated from A by a sort of event horizon. Moreover,the entanglement entropy in 4D shows an area law SA ∝ |∂A|ΛD−2

UV , where |∂A|is the entangling surface between A and Ac. Thus the entanglement entropy is insome sense similar to the entropy of black holes [17]: as was shown by Bekensteinand Hawking, the entropy of a black hole is proportional to the area of its horizon[4], [3].

The entanglement entropy is an intriguing subject, which plays an importantrole both in studies of quantum phase transitions and gravitational dynamics. Inthe present work we are investigate the entanglement entropy in Yang-Mills theoriesin flat four-dimensional spacetime.

The structure of this work is the following. In chapter 1 we give a brief overviewon entanglement is and why it is an important aspect of quantum mechanics. Inchapter 2 we introduce basic notions about conformal field theory and then inchapter 3 we briefly explain how to build a lattice simulation for a gauge theory. Inchapters 4 we give an overview on how to compute the entanglement entropy for a 4dimensional field theory. Finally, in chapter 5 we present the original results of thiswork, obtained via lattice simulations on SU(2) Yang-Mills theory.

ix

Chapter 1

Quantum entanglement

Quantum mechanics distinguishes itself from classical physics via the presenceof entanglement. Classically, one imagines situations where components of a singlesystem may be separated into non-interacting parts, which can be separately ex-amined, and then put back together to reconstruct the information about the fullsystem. This intuition fails in quantum mechanics, since the separate pieces, whilstnon-interacting, could nevertheless be entangled. This quintessential feature of thequantum world has been a source of great theoretical interest over the past decades.The initial debate started with the Einstein-Podolsky-Rosen (EPR) paper [13].

1.1 EPR Paper

When we talk about a macroscopic object we assume that its physical propertiesare independent of measurements, that is measurements merely reveal such physicalproperties. The case of quantum mechanics is remarkably different, as we knowthat quantum systems do not possess physical properties that are independent ofobservation. In fact such physical properties arise as a consequence of measurementupon the system following a mechanism called collapse of the wave function.

In the ’EPR paper’, co-authored with Nathan Rosen and Boris Podolsky, Einsteinproposed a Gedankenexperiment which, assuming locality and realism, demonstratedthat quantum mechanics is not a complete theory of nature. As we know, this is notcorrect. The EPR argument was crucially based on what they called ’element ofreality’ which, they believed, must be present in any complete physical theory.

A sufficient condition for a physical property to be an element of reality is thatit can be possible to predict with precision the value of that property before themeasurement. As we said before, however, physical properties of a quantum systemcan be detected only through measurements.

Consider a pair of entangled quantum bits (qubits) belonging to observers Aliceand Bob

1

2 CHAPTER 1. QUANTUM ENTANGLEMENT

|ψ〉 =|01〉 − |10〉√

2(1.1)

that is known as spin singlet. Suppose Alice and Bob are sufficiently far fromeach other (so that the measurements they perform are causally separated). Aliceperforms a measurement of spin along the s~x axis, so she measures the observable

~v · ~σ = σ1x1 + σ2x2 + σ3x3 (1.2)

Suppose Alice obtains +1 from the measurement: then we know with absolutecertainty that Bob will measure −1 by the same measurement. In fact we candemonstrate that whatever axis we choose to perform the measurement the resultabout the two qubits will be opposite.

Suppose |a〉 and |b〉 are the eigenstates of ~x ·~σ, then there exist complex numbersα, β, γ, δ such that

|0〉 = α |a〉+ β |b〉|1〉 = γ |a〉+ δ |b〉 . (1.3)

Substituting we obtain

|01〉 − |10〉√2

= (αδ − βγ)|ab〉 − |ba〉√

2. (1.4)

The quantity (αδ − βγ) is the determinant of the unitary matrix

(α βγ δ

), hence it

is equal to a phase factor eiθ with θ ∈ R. Therefore

|01〉 − |10〉√2

=|ab〉 − |ba〉√

2. (1.5)

As a result, if a measurement of ~x·~σ on the first qubit returns a result of ±1, it impliesa result of ∓1 on the second qubit. Therefore, since Alice, after a measurement, canpredict the value of the measurement performed by Bob before the measurement itself,this property must be an element of reality according to the EPR paper statement.As we know quantum mechanics allow us to calculate only the probability with whichwe can obtain a certain value from a measurement. So, following this reasoning,EPR aimed to demonstrate that quantum mechanics is not a complete theory ofnature.

1.2 Bell Inequality

About 30 years after the EPR paper publication, an experiment was proposed,that invalidates the theory proposed by Einstein, Podolsky, and Rosen. The key ofthis retraction is in the Bell inequality.

1.2. BELL INEQUALITY 3

The Bell inequality can be achieved with a thought and, most importantly,classical experiment. So initially we do not use any notion of quantum mechanics.We have to introduce a third element named Charlie. Charlie has the importanttask of preparing two particles, it is not important how, as long as he can repeat theexperimental procedure whenever he wants.

Once the two particles are prepared, one is given to Alice, the other to Bob. OnceAlice obtains her particle she can perform two kind of measurements among whichshe chooses randomly every time. These are measurements of physical propertiesthat we call PQ and PR from which Alice obtains the values Q and R respectively.Both Q and R can have the value +1 or −1. We suppose PQ and PR to be objectiveproperties of the particle that can be merely revealed by measurement. In the sameway Bob does with properties PS and PT .

The experimental timing must be such that Alice and Bob perform the measure-ment simultaneously (or in a casually disconnected manner) so that measurementsperformed by one cannot influence that of the other. Indeed, physical influencepropagates at most at the speed of light.

We are going to do some algebra with the quantity

QS +RS +RT −QT = (Q+R)S + (R−Q)T. (1.6)

Since Q and R are equal to ±1, it follows that either (Q+R)S = 0 or (Q−R)T = 0.In both cases it is easy to see that

QS +RS +RT −QT = ±2. (1.7)

Now we suppose p(q, r, s, t) to be the probability that the system is in the state

Q = qR = rS = sT = t (1.8)

before the measurement. Denoting the expectation value of an observable as 〈·〉, wehave

〈QS +RS +RT −QT 〉 =∑qrst

p(q, r, s, t)(qr + rs+ rt− qt)

≤∑qrst

p(q, r, s, t) · 2 = 2(1.9)

〈QS +RS +RT −QT 〉 = 〈QS〉+ 〈RS〉+ 〈RT 〉 − 〈QT 〉. (1.10)

Comparing equation (1.9) and (1.10) we obtain the Bell inequality

〈QS〉+ 〈RS〉+ 〈RT 〉 − 〈QT 〉 ≤ 2 (1.11)

Coming back to quantum mechanics, we return to our system of two entangledqubits (1.1). As in the classical experiment, suppose that Alice and Bob can performthe following measurements

Q = Z1 S =−Z2 −X2√

2

R = X1 T =Z2 −X2√

2

(1.12)

4 CHAPTER 1. QUANTUM ENTANGLEMENT

defined as follows

Z = σ3 =

(1 00 −1

)X = σ1 =

(0 11 0

)(1.13)

so that

Z |0〉 = |0〉X |0〉 = |1〉Z |1〉 = − |1〉X |1〉 = |0〉 (1.14)

With some algebra we find

〈QS〉 =1√2

〈RS〉 =1√2

〈RT 〉 =1√2

〈QT 〉 = − 1√2

(1.15)

hence

〈QS〉+ 〈RS〉+ 〈RT 〉 − 〈QT 〉 = 2√

2. (1.16)

This result violates the Bell inequality (1.11). Experiments with quantum particleshave been done to test whether Nature obeys or violates Bell’s inequality, and it hasbeen found that Nature violates it.

This implies that one or more of the assumptions underlying Bell’s inequalitymust be incorrect. The key assumptions underlying the proof of (1.11) are:

1. The assumption that the physical properties PQ, PR, PS and PT have definitevalues that exists independent of observation. This is known as the assumptionof realism.

2. The assumption that measurements performed by Alice do not influence theresults of Bob’s measurements. This is known as the assumption of locality

The assumption of realism is inherently rejected by quantum mechanics, so Bell’sinequality together with experimental evidence implies that quantum mechanics isnot a local theory.

This shows that quantum entanglement is an important property of quantumsystems: the fact that entangled states can violate Bell’s inequality implies thatentanglement is a distinctive feature of quantum systems, which has no counterpartin the classical world.

Further details about this topic are discussed in ref. [26].

Chapter 2

Conformal Field Theory

2.1 Conformal Group

A conformal field theory (CFT) is a physical theory that is invariant underconformal transformations, namely all those transformations which preserve anglesdefined in the following way

g′µν =∂xα

∂x′µ∂xβ

∂x′ν= Ω(x)gαβ (2.1)

where gµν is the metric and Ω(x) is a positive function called scale factor.If we consider an infinitesimal transformation x′µ = xµ− εµ we can see that (2.1)

becomes

gµν + ∂µεν + ∂νεµ = Ω(x)gµν . (2.2)

We set Ω(x)− 1 = Ω(x) and we find

∂µεν + ∂νεµ = Ω(x)→ Ω(x) =2

d∂ · ε, (2.3)

where d is the spacetime dimension (gµνgµν = d), so we can rewrite (2.2) in thefollowing way

∂µεν + ∂νεµ =2

dηµν∂ · ε. (2.4)

From this point we consider only the diagonal metric ηµν = diag(−1, 1, 1, 1).Now we apply the d’Alembert operator to both sides of (2.4) in order to obtain

the following differential equation

(ηµν + (d− 2)∂µ∂ν)∂ · ε = 0 (2.5)

This differential equation teaches us that ∂ · ε must be linear in x and so εµ must beat most quadratic in x:

εµ = aµ + bµνxν + cµνρxνxρ. (2.6)

5

6 CHAPTER 2. CONFORMAL FIELD THEORY

Of course (2.6) is a solution of (2.5), now we have to test if it verifies (2.4). Alsowe have to consider d ≥ .3 because d = 2 is a particular case.

• aµ

aµ satisfies Eq. (2.4) because it is a constant term; it induces a coordinatetransformation which is a translation: x′µ = xµ + aµ.

• bµνxν

By inserting this term in (2.4) we obtain

bµν + bνµ =2

dηµν(ηαβbαβ) (2.7)

so we understand that bµν can be expressed as a combination of a symmetric andan anti-symmetric term as follow

bµν = αηµν + βmµν (2.8)

where mµν = −mνµ and ηµν is the symmetric term because the elements out ofdiagonal do not satisfy (2.7).

Let us concentrate on the symmetric term first.

αηµν + αηµν =2

dηµν(ηαβαηαβ)→ 2αηµν =

2

dηµναd (2.9)

so αηµν is a solution of (2.4) and corresponds to a dilation x′µ = xµ + αxµ

Now consider the anti-symmetric term

mµν +mνµ =2

dηµν(ηαβmαβ)→ 0 = 0 (2.10)

so βmµν satisfies Eq. (2.4) and corresponds to a rotation x′µ = xµ +mµνxν .

• cµνρxνxρ

By inserting in Eq. (2.4) we find

∂µεν + ∂νεν = cνµρxρ + cνρµx

ρ + cµνρxρ + cµρνx

ρ

∂ · ε = cµµρxρ + cµρµx

ρ (2.11)

so that (2.4) becomes

cνµρxρ + cνρµx

ρ + cµνρxρ + cµνρx

ρ =2

dηµν(cααρx

ρ + cαραxρ) (2.12)

after contracting both sides with the metric ηµν we find

cµµρxρ + cµρµx

ρ + cµµρxρ + cµρµx

ρ = 2(cµµρ + cµρµ)xρ. (2.13)

Therefore this term is a solution of (2.4) but to understand what kind oftransformation it is, some further steps are needed. By deriving Eq. (2.4) withrespect to xρ:

2.1. CONFORMAL GROUP 7

name transformation generator

translation x′µ = xµ + aµ Pµ = −i∂µdilation x′µ = αxµ D = −ixµ∂µrotation x′µ = Lµνxν Lµν = i(xµ∂ν − xν∂µ)

SCT x′µ = xµ−(x·x)bµ

1−2(b·x)+(b·b)(x·x) Kµ = −i(2xµxν∂ν − (x · x)∂µ)

Table 2.1: Conformal transformations and their generators.

∂ρ[∂µεν + ∂νεµ] =2

dηµν∂ρ∂ · ε (2.14)

Permuting the indices, we obtain

∂ρ∂µεν + ∂ρ∂νεµ =2

dηµν∂ρ∂ · ε

∂ν∂ρεµ + ∂µ∂ρεν =2

dηρµ∂ν∂ · ε

∂µ∂νερ + ∂ν∂µερ =2

dηνρ∂µ∂ · ε.

(2.15)

We subtract the second and third equations to the first, so that

∂µ∂νερ =1

d[−ηµν∂ρ + ηρµ∂ν + ηνρ∂µ]∂ · ε. (2.16)

Now we substitute εµ = cµνρxνxρ in (2.16) so that

cρµν + cρνµ =1

d[−ηµνcααρ − ηµνcαρα + ηρµc

ααν + ηρµc

ανα + ηνρc

ααµ + ηνρc

αµα]. (2.17)

By this equation one can see that the symmetry cρµν = cρνµ is satisfied and oneobtains

cρµν = [−ηµνbρ + ηρµbν + ηνρbµ] (2.18)

where bµ = 1dcααµ. Thanks to this result we can find that the quadratic term

corresponds to the transformation x′µ = xµ + x2bµ − 2(b · x)xµ which is called aspecial conformal transformation (SCT).

Thus, we have found all coordinate transformations that form the conformalgroup: they are listed in Table 2.1.

2.1.1 Conformal Group In 2 Dimensions

In two spacetime dimensions the conformal transformations have a peculiarbehaviour: indeed, if we consider Eq. (2.5), we clearly see that in d = 2 the εν are


eigenfunctions of the d’Alembert operator and so they are analytic functions. Ind = 2 Eq. (2.4) becomes ∂µεν + ∂νεµ = ηµν∂ · ε. We set µ = 1, 2 and so we obtain 4equations

∂1ε2 + ∂2ε1 = 0

∂2ε1 + ∂1ε2 = 0

∂1ε1 + ∂1ε1 = ∂1ε1 + ∂2ε

2

∂2ε2 + ∂2ε2 = (∂1ε1 + ∂2ε

2)(−1)p.

(2.19)

With Euclidean signature (p = 0) the last two equations can be identified,whereas with Minkowskian signature (p = 1) this cannot be done. We choose p = 0because in statistical mechanics it does not give any problems. So we find

∂1ε2 = −∂2ε1

∂1ε1 = ∂2ε2(2.20)

that are the Cauchy-Riemann conditions for the analytic functions.Now we introduce the complex variables defined as

z = x1 + ix2 ε(x) = ε1(x1, x2) + iε2(x1, x2)

z = x1 − ix2 ε(z) = ε1(x1, x2)− iε2(x1, x2)(2.21)

We perform a Laurent series expansion around z = 0 on the ε(z) functions and weobtain

ε(z) =∑n

anzn

ε(z) =∑n

anzn

(2.22)

by introducing the generators of the infinitesimal transformation

ln = −zn+1∂z

ln = −zn+1∂z(2.23)

one can find

ε(z) = −∑n

anln−1z

ε(z) = −∑n

anln−1z(2.24)

At the end we can build the generators algebra, that is

[ln, lm] = (m− n)lm+n [ln, lm] = (m− n)ln+m [ln, lm] = 0 (2.25)

2.2. CONFORMAL QUANTUM FIELD THEORY 9

name transformation Jacobian

translation x′µ = xµ + aµ |∂x′∂x | = 1

rotation x′µ = Rνµxν |∂x′∂x | = 1

dilation x′µ = λxµ |∂x′∂x | = λd

SCT x′µ =xµ+bµx2

1+2b·x+b2x2 |∂x′∂x | = 1(1+2b·x+b2x2)d

Table 2.2: Conformal transformations and their Jacobians.

This result teaches us that we have two copies of the same algebra and this allowsus to consider z and z as distinct variables although z = z∗. This algebra is knownas Witt Algebra and it is the classical version of the Virasoro Algebra.

The problem of this algebra is that it is not possible to build up a group starting

from it: for example, the inverse of the transformation z′ = a2z2 is z =

√z′

a2, which

is not differentiable. However it is possible to find a group structure for a finitesubalgebra built upon the space s2 = C

⋃∞, that is the Riemann sphere. Thisalgebra is built starting from the most general transformation of the complex planein itself:

z → az + b

cz + d. (2.26)

All possible values that a, b, c and d can take are given by the group SL(2,C)/Z2,the Small Conformal Group. This group has 6 generators l−1, l0, l1 and l−1, l0, l1 andit is isomorphic to SO(3), in fact:

l1, l−1 → translation (2.27)

l0 + l0 → dilation (2.28)

l0 − l0 → rotation (2.29)

l1, l−1 → SCT (2.30)

2.2 Conformal Quantum Field Theory

Before introducing the fundamental aspects of a conformal quantum field theoryit is useful to show what are the Jacobians of the coordinate transformations listedin table 2.2

Now we have to build up a quantum field theory that is conformally invariant.

1. First of all, there must exist a ground state |0〉 that is invariant under conformaltransformations;

2. There exists a set of fields Ai that compose the Hilbert space;


3. There exists a subset of fields φi ∈ Ai that compose the base of the Hilbertspace. All of the fields Ai can be expressed as a linear combination of φi

4. At the end we request that, under a conformal transformation, φi transformsas

φi(x)→ φ′i =

∣∣∣∣∂x′∂x

∣∣∣∣∆id

φi(x′) (2.31)

where ∆i is the scale dimension of the field ad d is the spacetime dimension.

The fields φi defined as above are called quasi-primary fields.The conformal symmetry of the system imposes some constraints on the correla-

tors:

a) For the one-point function 〈φ〉:– Translation invariance:

〈φ〉 = 〈φ′〉 = |∂x′

∂x|∆d 〈φ(x+ a)〉 = 〈φ(x+ a)〉 (2.32)

Translation invariance implies 〈φ〉 = const.

– Dilation invariance:

〈φ〉 = 〈φ′〉 = |∂x′

∂x|∆d 〈φ(λx) = λd〈φ〉 (2.33)

the only way to verify the invariance 〈φ〉 = λd〈φ〉 is to impose 〈φ〉 = 0. So fora conformal field theory the one-point correlator is

〈φ〉 = 0 (2.34)

b) For the two-point correlation function 〈φ1(x1)φ2(x2)〉– Translation invariance:

〈φ1(x1)φ2(x2)〉 = |∂x′1

∂x1|

∆1d |∂x

′2

∂x2|

∆2d 〈φ1(x1 + a)φ2(x2 + a)〉 (2.35)

if we choose to call 〈φ1(x1)φ2(x2) = G(x1, x2) the request of invariance undertranslations reduces to G(x1, x2) = G(x1 + a, x2 + a), so we have to imposeG(x1, x2) = G(x1 − x2).

– Rotation invariance:

it is easy to realize that the invariance is achieved by imposing G(x1 − x2) =G(|x1 − x2|).– Dilation invariance:

〈φ1(x1)φ2(x2)〉 = λ∆1+∆2〈φ1(λx1)φ2(λx2)〉 (2.36)

2.2. CONFORMAL QUANTUM FIELD THEORY 11

so that G(|x1 − x2|) = λ∆1+∆2G(λ|x1 − x2|). In order to verify the equationwe have to impose that the 2-point Green function has the following form:

G(|x1 − x2|) =c

|x1 − x2|∆1+∆2(2.37)

where c is a constant term.

– Special conformal invariance:

〈φ1(x1)φ2(x2)〉 = 1(1+2bx1+b2x2

1)∆1

1(1+2bx1+b2x2

2)∆2〈φ1(x′1)φ2(x′2)〉

〈φ1(x′1)φ2(x′2)〉 = c|x′1−x′2|∆1+∆2

(2.38)

The only way we have to satisfy the equivalence is to impose ∆1 = ∆2 so thatthe 2 point correlator is non-zero only when the two fields have the same scaledimension and it is:

〈φ(x1)φ(x2)〉 =c

|x1 − x2|2∆φ(2.39)

c) For the tree-point correlator 〈φ1(x1)φ2(x2)φ3(x3)〉In order to obtain invariance under translation, rotation and dilation we haveto impose

〈φ1(x1)φ2(x2)φ3(x3)〉 =∑abc

cabc

ra12rb23r

c13

(2.40)

where a+ b+ c = ∆1 + ∆2 + ∆3. Then the request of invariance under specialconformal transformation imposes that

〈φ1(x1)φ2(x2)φ3(x3)〉 =c123

r∆1+∆2−∆312 r∆2+∆3−∆1

23 r∆1+∆3−∆213

(2.41)

where c123 is a constant term called structure constant.

2.2.1 Operator Product Expansion

Correlators with more than three points are complex and it seems not possibleto find out an exact form. To overcome this problem Wilson introduced the operatorproduct expansion that allows us to take a many point correlator, which we do notknow how to evaluate, and reduce it to a two or three point correlator, which weknow the exact form. The operator product expansion works as follow: take forexample a two point function


A(x)B(x) ∼ limx→y

∑i

ci(x− y)Oi(x) (2.42)

where ci(x− y) diverges at x = y.This operation can be used to evaluate the structure constants for all the three-

point correlators:

A(x)B(y)F (z) ∼ limx→y

∑i

ci(x− y)O(x)F (z) (2.43)

If we evaluate the expectation value of these quantities we have on the left handside a 3 point function, whose we know the exact dependence from (x, y, z), thenon the right hand side we have a 2 point function, which we know how to evaluate.The operator product expansion is also useful for the 2 point function because onecan write

φi(x)φj(y) ∼∑k

cijkφk(x)

|x− y|∆i+∆j−∆k(2.44)

2.3 Conformal Field Theory in 2 Dimensions

First of all we have to complexify a Euclidean R2 space by introducing thevariables z = x0 + ix1 and z = x0− ix1. As we have previously seen we can considerthe two complex variables as independent.

Now we define a primary field with conformal dimension (h, h) as the field thattransforms under a conformal transformation z → f(z) in the following way:

φ(z, z)→ φ′(z, z) = (∂f

∂z)h(

∂f

∂z)hφ(f(z), f(z)) (2.45)

where h+h = ∆ is the link between the scale dimension and the conformal dimension.Now we have to quantize the theory. First of all we identify x0 with the time

direction and x1 with the spatial direction. To remove infrared divergences due tolarge-momentum modes, we compactify the spatial direction x1 on a circle of radiusR = 1. So changing x1 → x1 + 2π we obtain a theory defined on an infinitely longcylinder. Then we introduce the complex variables:

ξ = x0 + ix1

ξ = x0 − ix1

(2.46)

in this way we can map the cylinder described by (x0, x1) in the z-plane C using thefollowing coordinate change

z = eξ

z = eξ(2.47)

2.3. CONFORMAL FIELD THEORY IN 2 DIMENSIONS 13

these are conformal transformations so one can equivalently use (x0, x1) or (z, z).Because of this transformation, the time variable becomes the radius and, as aconsequence, equal-time commutators will be traded for equal-radius commutators.

We can note that in this quantization

x0 → x0 + a = z → eaz, (2.48)

so a time translation translates into a a dilation, and

x1 → x1 + b = z → eibz, (2.49)

so a translation in space translates into a rotation. From quantum mechanics, weknow that the generator of time translations is the Hamiltonian, that in this casecorresponds to the dilations’ operator

H ∝ l0 + l0 (2.50)

2.3.1 Stress Energy Tensor

To define the stress-energy tensor we have to recall Noether’s theorem, whichstates that there exists a conserved current associated with each parameter of anexact symmetry. The generic conserved current can be expressed as:

jµ = Tµνεν (2.51)

where Tµν is the stress-energy tensor and εν is an infinitesimal symmetry transfor-mation.

Associated to the conserved current there is a conserved charge Q =∫ddxj0,

which in quantum field theories is the generator of symmetry transformations ofoperators (A)

δεA = [Q,A]ε (2.52)

This commutator must be evaluated at equal times, but, as we said before, in thequantization we are currently using this is equivalent to compute the commutator atequal radii.

From Eq. (2.51) one can obtain useful information about the stress-energy tensor,which follows from the conservation of the current:

∂µjµ = 0→ ∂µTµνεν = (∂µY

µν)εν + (∂µεν)Tµν (2.53)

∂µTµν = 0 because of the conservation of the tensor [28], so

∂µTµνεν = Tµν(∂µεν). (2.54)

Now we use the symmetry of the stress-energy tensor to obtain


∂µTµνεν =

1

2Tµν(∂µεν + ∂νεµ) (2.55)

at the end we recall (2.4) and we find

1

dTµνηµν∂ · ε = 0. (2.56)

As the latter equation must hold for all ε, we find

Tµνηµν = 0→ TrT = 0 (2.57)

This result is important because it is the condition for the scale invariance of asystem.

Coming back to two dimensions, we take

z1 = z = x0 + ix1 z2 = z = x0 − ix1

x0 =1

2(z + z) x1 =

1

−2i(z − z)

(2.58)

so we can write Tαβ = ∂xi

∂zα∂xj

∂zβTij and find the following components of the stress

energy tensor:

Tzz =1

4(T00 − 2iT01 − T11) Tzz =

1

2(T00 + 2iT10 − T11)

Tzz =1

4(T00 + T11)

(2.59)

In this notation the constraints on the trace of the tensor result in

Tzz = Tzz = 0. (2.60)

So the non-zero components of the stress-energy tensor are:

Tzz = T (z) (2.61)

that is a pure anti-holomorphic function and

Tzz = T (z) (2.62)

that is a pure holomorphic function. This can only happen in two dimensions becausewe have to impose infinitely many conditions to have an holomorphic field and in a2-dimensional conformal field theory we have infinitely many conserved currents.

Now we can extend the conserved charge to the complex coordinates (z, z) usingthe stress-energy tensor

Q =1

2πi[

∮ΩdzT (z) +

∮ΩdzT (z)] (2.63)

where Ω stands for a closed path at fixed radius travelled counterclockwise. Wecan now compute the infinitesimal transformation of a field φ with respect to thesymmetry


zw

=−

z

w

z

w

Figure 2.1: Cartoon of a radially ordered integral. Image taken from [18]

δε,εφ(w,w) =1

2πi

∮Ωdz[T (z)ε(z), φ(w,w)] +

1

2πi

∮Ωdz[T (z)ε(z), φ(w,w]. (2.64)

In quantum field theory we have to respect the time ordering of the fields’ productin order to preserve the causality principle. Although in the quantization we arecurrently using the time ordering results in radial ordering:

R[A(z)B(w)] = A(z)B(w)θ(|z| − |w|) +B(w)A(z)θ(|w| − |z|) (2.65)

the radial ordering allows us to avoid singularities throughout the integration’s pathin w = z (w = z). Consider now only the integral in z∮

ΩR[T (z)ε(z), φ(w,w)] =

∮Ωdzε(z)T (z)φ(w,w)θ(|z| − |w|)

−∮

Ωdzε(z)φ(w,w)T (z)θ(|w| − |z|)

(2.66)

graphically the expression shows up to be as is shown in figure 2.1. So we can write:∮ΩdzR[T (z)ε(z), φ(w,w)] =

∮Ωw

dzR[T (z)ε(z)φ(w,w)] (2.67)

where Ωw stands for a circulation around w. The same procedure is applied to z soat the end we have

δεεφ(w,w) =1

2πi

∮Ωw

dzε(z)R[T (z)φ(w,w)]+

+1

2πi

∮Ωw

dzε(z)R[T (z)φ(w,w)]

(2.68)

Now consider the following infinitesimal transformation:

f(w) = w + ε(w) f(w) = w + ε(w)

(∂f

∂w)h = (1 +

∂ε

∂w)h ∼ 1 + h

∂ε

∂w(∂ε

∂w)h ∼ 1 + h

∂ε

∂w

(2.69)

by the definition of primary fields (2.45) we have

δε,εφ(w, (w)) = (∂f

∂w)h(

∂f

∂w)hφ(f(w), f(w)). (2.70)


If we consider how the coordinates transformation affects a field:

φ(w + ε(w), f(w)) = φ(w, f(w)) + ε(w)∂

∂wφ(w, f(w)) (2.71)

we find that

δεεφ(w,w) = [(h∂ε

∂w+ ε

∂

∂w) + (h

∂ε

∂w+ ε

∂

∂w)]φ(w,w). (2.72)

We have to match this expression with (??), and to do that we can impose thatthe left- and right-hand sides of the equations are equal, because we can treat (z, w)and (z, w) as independent variables. To match the two equations it must be truethat:

R[T (z)φ(w,w)] =1

z − w∂

∂wφ(w,w) +

h

(z − w)2φ(w,w)

R[T (w)φ(w,w)] =1

z − w∂

∂wφ(w,w) +

h

(z − w)2φ(w,w) :

(2.73)

We can add any number of positive (z −w) power but their contribution in the loopintegral is zero.

We have just found and explicit example of an operator product expansion,furthermore this is also a way to identify a primary field, in fact if we find, throughthe operator product expansion, poles of higher than two, then the field is not aprimary field but a secondary one. Moreover if we recall equation (??) we can seethat the stress-energy tensor has scale dimension ∆ = 2.

Now if we calculate the stress-energy tensor’s operator product expansion wefind:

T (z)T (w) =1

z − w∂

∂wT (w) +

2

(z − w)2T (z) (2.74)

from (??) we expect that 〈T (z)T (w)〉 =∑i ci

(z−w)4 but it is also true that 〈T (z)〉 = 0 so

in order to have consistency among the two formulas we have to write:

T (z)T (w) =1

z − w∂

∂wT (w) +

2

(z − w)2T (w) +

c

2(z − w)4(2.75)

in the same way we have

T (z)T (z) =1

z − w∂

∂wT (w) +

2

(z − w)2T (w) +

c

2(z − w)4(2.76)

All in all the stress-energy tensor is not a primary field unless c = c = 0. c and c areconstants such that c− c = 0, while T (z)T (w) = 0.

At the end under a conformal transformation f(z) the energy-momentum tensortransforms as

T (z)→ T ′(z) = (∂f

∂z)2T (f(z)) +

c

12S(f, z) (2.77)

where S(f, z) is the Schwartzian derivative


S(fez) =

[∂f

∂z

∂3f

∂z3− 3

2(∂f

∂z)2

](∂f

∂z)−2 (2.78)

We can demonstrate the tensor’s transformation formula, therefore consider aninfinitesimal transformation f(z) = z + ε(z) so that

δεT (z) =1

2πi

∮Ωz

dwT (w)T (z)ε(w) (2.79)

now we use the operator product expansion

δεT (z) =1

2πi

∮Ωz

dwε(w)[1

w − z∂T (z)

∂z+

2

(w − z)2T (z) +

c

2(w − z)4] =

= limw→z

[ε(w)∂T (z)

∂z] + 2 lim

w→z[∂

∂wε(w)T (z)] +

c

2 · 3!limw→z

[∂3

∂w3ε(w)] =

= [ε(z)∂

∂z+ 2

∂ε(z)

∂z]T (z) +

c

12

∂3ε(z)

∂z3

(2.80)

With such an infinitesimal transformation the Schwartzian derivative becomesS(fez) ∼ ∂3ε

∂z3 , so that (2.77) becomes:

T ′(z) = (1 +∂ε

∂z)2T (z + ε(z)) +

c

12

∂3ε

∂z3=

= (1 + 2∂ε

∂z)(T (z) + ε(z)

∂T

∂z) +

c

12

∂3ε

∂z3

(2.81)

therefore

T ′(z)− T (z) = [2∂ε

∂z+ ε

∂

∂z]T (z) +

c

12

∂3ε

∂z3= δεT (z). (2.82)

So the two expressions are consistent with each other.

2.3.2 Virasoro Algebra

We can use the Laurent expansion on the stress energy tensor and find

T (z) =∑n∈Z

z−n−2Ln

T (z) =∑n∈Z

z−n−2Ln.(2.83)

Now take a transformation ε(z) = −εnzn+1; the conserved charge becomes:


Qn =

∮dz

2πiT (z)(−εnzn+1) = −εn

∑m

∮dz

2πiLmz

−m−2+n+1 =

= −εn∑m

∮dz

2πiLmz

−m+n−1 = −εn∑m

Lmδmn = −εnLn,(2.84)

therefore the terms of the Laurent series in which we expand the stress-energy tensorare the generators of the infinitesimal conformal group’s transformations, so theyhave to obey the conformal algebra.

We can write the Ln as functions of T (z), so:

Ln =

∮dz

2πizn+1T (z) Ln

∮dz

2πizn+1T (z), (2.85)

in this way we can compute the correlator

[Ln, Lm] =

∮dw

2πi

∮dz

2πizn+1wm+1[T (z), T (w)] =

=

∮dw

2πi

∮dz

2πizn+1wm+1T (z)T (w)−

∮dz

2πizn+1wm+1T (w)T (z) =

=

∮dw

2πi

∮Ωw

dz

2πizn+1wm+1R[T (z)T (w)] =

=

∮dw

2pii

∮Ωw

dz

2πizn+1[

1

z − w∂T (z)

∂z+

2

(z − w)2T (z) +

c

2(z − w)4]wm+1 =

=

∮dw

2πilimz→w

zn+1[∂T (z)

∂z+ 2

∂T (z)

∂w+

c

12

∂3

∂w3]wm+1 =∮

dw

2πiwn+1[

∂T (w)

∂w+ 2T (w)

∂

∂w+

c

12

∂3

∂w3]wm+1 =

=

∮dw

2πi[wn+m+2∂T (w)

∂w+ 2(m+ 1)wn+m+1T (w)+

+c

12m(m+ 1)(m− 1)wn+m−1] =

=

∮dw

2πi[−(n+m+ 2)wn+m+1T (w) + 2(m+ 1)wn+m+1T (w)+

+c

12m(m2 − 1)wn+m−1] =

=

∮dw

2πi[(m− n)wn+m+1T (w) +

c

12m(m2 − 1)wn+m−1].

(2.86)

At the end we find that the commutator is:

[Ln, Lm] = (m− n)Ln+m +c

12(m2 − 1)mδm,−n. (2.87)

The second term is known as central extension of the algebra (or conformal anomaly)and because of this the constant c is known as the central charge of the conformaltheory. The generators L0, L1, L−1 do not exhibit conformal anomaly and are theonly ones that generate invertible transformations.

2.4. HILBERT SPACE OF THE CONFORMAL THEORY 19

2.4 Hilbert space of the conformal theory

In a quantum field theory we define the initial state as an asymptotic state,thanks to the adiabatic hypothesis:

|Ain〉 = limz,z→0

A(z, z) |0〉 , (2.88)

now we transform the coordinates as w = 1z and w = 1

z . So, following (2.45), thefield changes as:

A(w,w) = (− 1

z2)h(− 1

z2 )hA(1

z,

1

z), (2.89)

therefore we can define the asymptotic final state as:

〈Aout| = limw,w→0

〈0| A(w,w) = limz,z→0

〈A| (1

z,

1

z)

1

z2h

1

z2h. (2.90)

At this point we can define the Hermitian conjugate of the field as

A†(z, z) = A(1

z,

1

z)

1

z2h

1

z2h, (2.91)

so that

〈Aout| = limz,z→0

〈0|A†(z, z). (2.92)

In the same way we can define the Hermitian conjugate of the stress-energy tensor:

T †(z) =∑m

L†

zm+2 =1

z4T (1

z) =

∑m

Lmz−m−2+4 =

∑m

Lmz−m+2 , (2.93)

so that

L†m = L−m (2.94)

This equivalence says that the Hamiltonian is exact.

Now we want that T (z) |0〉 = 0 because |0〉 is the void, so

limz→0

T (z) |0〉 = limz→0

∑m

Lmzm+2

|0〉 ⇐⇒ m ≥ .− 1, (2.95)

therefore we can see that

∀m ≥ .− 1 Lm |0〉 = 0 and 〈0|L†m = 0

∀m ≤ .1 〈0|Lm = 0(2.96)

and because of that we understand that 〈0| and |0〉 are both annihilated byL−1, L0, L1, so the void is invariant under the small conformal group SL(2,C).

Define now the initial state of a primary field:


limz,z→0

φ(z, z) |0〉 =∣∣h, h⟩ = |h〉 , (2.97)

form now on we will take h = h and with abuse of notation we will refer to |h〉 asprimary field.

Now we compute [Ln, φ]

[Ln, φ(0)] = limz,z→0

[Ln, φ(z, z)] = limz,z→0

∮dw

2πiwn+2[T (w), φ(z, z)] =

= limz,z→0

∮dw

2πiwn+1[

1

w − z∂φ

∂z+

h

(w − z)2φ]

= limz,z→0

[zn+1∂φ

∂z+ (n+ 1)hznφ]

(2.98)

so we find:

[Ln, φ(0)] = 0 ∀n > 0

[L0, φ(0)] = hφ(0)(2.99)

With the result we can calculate:

a) Lnφ(0) |0〉

Lnφ(0) |0〉 = [Ln, φ(0)] |0〉 − φ(0)Ln |0〉 = 0 (2.100)

with n > 0. We can see that

Ln |h〉 = 0 ∀n > 0, (2.101)

and because of this Ln acts as an annihilator of primary fields.

b) L0φ(0) |0〉

L0φ(0) |0〉 = [L0, φ(0)] |0〉 − φ(0)L0 |0〉 = hφ(0) |0〉 (2.102)

we can see that

L0 |h〉 = h |h〉 , (2.103)

and as L0 is the dilations’ generator, it naturally returns the scale dimension.Applying the Hamiltonian to Ln |h〉:

L0L−n |h〉 = [L0, L−n] |h〉+ L−nL0 |h〉 = nL−n |h〉+ hL−n |h〉= (n+ h)L−n |h〉 .

(2.104)

By this result we understand that also L−n |h〉 are eigenstates of the Hamiltonianbut with energy moved by n respect to that of the fundamental state. For thisreason we call L−n |h〉 secondary fields.

2.4. HILBERT SPACE OF THE CONFORMAL THEORY 21

At this point we affirm that all the primary fields and all the secondary fieldsgenerated by the primary ones compose the Hilbert space. The Hilbert space buildin this way has the following structure:

|h〉L−1 |h〉

L−2 |h〉 , (L1)2 |h〉L−3 |h〉 , L−2L−1 |h〉 , L−1L−2 |h〉 , (L−1)3 |h〉

...

(2.105)

We have to investigate if L2L−1 |h〉 and L−1L−2 |h〉 are independent states or not,so :

L−2L−1 = [L−2, L−1] + L−1L−2 = L−3 + L−1L−2, (2.106)

therefore at the third level of the conformal family we have 3 independent states.If we want an irreducible representation the states’s number at the n-th level willbe given by the not-ordered partition of n. For example if n = 4 we will haveL−4 |h〉 , (L−1)4 |h〉 , (L−2)2 |h〉 , L−2(L−1)2 |h〉 and L−3L1 |h〉. At the n-th level thenumber of independent states is given by the Dedekind function

η =1∏∞

n=1(1− qn). (2.107)

2.4.1 Null vectors

If we compute the norm of the state L−n |h〉 we find:

〈h|L†−nL−n |h〉 = 〈h|LnL−n |h〉 = 〈h|L−nLn |h〉+ 〈h| [Ln, L−n] |h〉= 0 + 〈h| 2L0 |h〉+

c

12(n2 − 1)n |h〉 = 2nh+

c

12n(n2 − 1).

(2.108)

From this result we can bring some useful considerations:

a) the result must be true ∀n, so if we want states with positive norm and save theunitary of the theory, it must be h ≥ .0.

b) ∀h, c∃n/ c12n(n2 − 1) 2nh, so if we want to preserve the unitary of the theory

it must be c ≥ .0.

c) If h = 0 then L−1 |h〉 has norm equal to zero.

Every state, such as L−1 |h〉, that have norm equal to zero is called null vector.The peculiarity of null vectors is that all the secondary fields generated startingfrom them are also null vectors.


If we want to understand for which value of h we have a null vector at the secondlevel of the conformal family, we have to calculate the determinant of the norm’smatrix that have the following form:(

〈h|L2L−2 |h〉〈h| (L1)2L−2 |h〉〈h|L2(L−1)2 |h〉〈h| (L1)2(L−1)2 |h〉

). (2.109)

With some math we find:

〈h|L2L−2 |h〉 = 4h+c

2〈h| (L1)2L−2 |h〉 = 6h

〈h|L2(L−1)2 |h〉 = 6h

〈h| (L1)2(L−1)2 |h〉 = 4h(2h+ 1),

(2.110)

so (2.109) becomes: (4h+ c

2 6h6h 4h(2h+ 1)

)(2.111)

Now we have to compute ∣∣∣∣4h+ c2 6h

6h 4h(2h+ 1)

∣∣∣∣ = 0 (2.112)

and we find:

h[(16h+ 2c)(2h+ 1)− 36h] = 0 (2.113)

First of all we see that for h = 0 we have a null vector at the second level of theconformal family fo any value of c, then we have:

(16h+ 2c)(2h+ 1)− 36h = 0→ h =(5− c)±

√(5− c)2 − 16c

16. (2.114)

Therefore by fixing c we find which are the primary fields that give a null vectorat the second level of the Hilbert space. For example if c = 1

2 , that is the centralcharge of the Ising model, we have a null vector at the secondary level for h = 1

2 andh = 1

16 , that are respectively the spin and the temperature fields of the Ising model.

2.5 Ward Identity

The Ward identity comes from the study of the behaviour of the N-points corre-lation function under infinitesimal symmetry transformations. So let us compute:

〈∮

dz

2πiε(z)T (z)φ1(w1, w1) . . . φN (wN , wN )〉, (2.115)

2.5. WARD IDENTITY 23

we have N singular points, and thanks to the radial ordered product we can deformthe integration’s path and integrate only on loops around the singularities. So theresult will be the sum of all these integrals:

N∑i=1

〈φ1(w1, w1)· · ·∮

dz

2πiε(z)R[T (z)φi(wi, wi)] . . . φN (wN , wN )〉 = (2.116)

=

N∑i=1

〈φ1(w1, w1)· · ·∮

dz

2πi[

hi(z − wi)2

φi(wi, wi) +1

z − wi∂φi(wi, wi)

∂wi] . . . φN (wN , wN )〉.

Because of the final result must not depend on ε we can write the Ward identity forthe conformal field theory as

〈T (z)φ1(w1, w1) . . . φN (wN , wN )〉 = (2.117)

=N∑i=1

(hi

(z − wi)2) +

1

z − wi∂

∂wi〈φ1(w1, w1) . . . φi(wi, wi) . . . φN (wN , wN )〉

that is how a N-points function transforms under a conformal transformation.

Chapter 3

Lattice Gauge Theories

3.1 Introduction To Lattice Gauge Theories

The modern approach to quantum field theories is based on Feynman’s pathintegral formulation. Given that an exact evaluation of Feynman’s path integralsis generally not possible for the field theories of the Standard Model, one usuallyexpands them in a perturbative series, in powers of some “small” parameter. Thismethod can not be used in the case of phenomena governed by a large couplingconstant (like the coupling of the strong interaction at low energies). To overcomethis difficulty, and to provide a non-perturbative formulation of gauge theories,Wilson [35] suggested to formulate the theory on a discrete lattice of points inEuclidean spacetime. Such a proposal has some important advantages:

• the inverse of the lattice spacing acts as an ultraviolet cut-off

• if the number of sites is kept finite all the infrared divergences are removed

• all quantum averages are given by mathematically well defined expressions

• quantum field theory in (d+1) spacetime dimensions regularised on a latticebecomes equivalent to an equilibrium statistical mechanics model in (d+1)space dimensions.

3.2 Lattice Discretization

We want to find an effective lattice regularisation for quantum field theories,including, in particular, non-Abelian gauge theories with SU(N) gauge group.

First of all we have to find an action that in the continuum limit (or for latticespacing going to zero) reduces exactly to the Yang-Mills action. Moreover the

25

26 CHAPTER 3. LATTICE GAUGE THEORIES

lattice must be Euclidean, so in the continuum limit we have to find the EuclideanYang-Mills action, that is

SY.M. =1

4

∫d4xF iµνF

µνi (3.1)

where F iµν = ∂µAiν − ∂µAiν − gf ijkA

jµAkν , and i is an index that runs over the number

of generators. fijk are the structure constants of the algebra of the generators of thegauge group, given by [τi, τj ] = ifijkτ

k. Finally, g is the coupling constant.We use a four-dimensional Euclidean lattice with one Euclidean-time coordinate

and three spatial coordinates. Let us denote with n = (n0, n1, n2, n3) the sites of thelattice and with µ = (0, 1, 2, 3) the unit vector in the four directions. we denote witha the lattice spacing. With this notation we denote with nµ the link starting fromand pointing in the positive µ direction (i.e. the link joining the two sites n andn+µ). Similarly nµν denotes the plaquette joining the four sites n, n+µ, n+µ+ ν,and n+ ν.

The standard way of define a quantum field theory on a lattice is to definethe scalars on the sites, the vectors on the links and all the two index tensorson the plaquettes. Thus the field Aµ must be defined on the links of the lattice.However Wilson made the choice, which ensures both gauge invariance and a smoothcontinuum limit, by putting on the link the variable Uµ(n) that is an element of thegroup from which the algebra is derived and is defined as

Uµ(n) = expcAµ(n) (3.2)

where c is a constant proportional to the lattice spacing. Uµ(n) can be thought ofas a parallel transporter in color space, that takes the color reference frame definedon the site n and rotates it into the one defined on n+ µ. So, for the variable Uµ(n)must hold U−µ(n+ µ) = U−1

µ (n).Now consider a gauge transform V (n) that affects Uµ(n) in the following way:

Uµ(n)→ V (n)Uµ(n)V −1(n+ µ). (3.3)

So the simplest way to construct a gauge invariant observable is to choose a closedpath Γ on the lattice and then construct

W = Tr∏nµ∈Γ

Uµ(n) (3.4)

where the product is assumed to be ordered along the path Γ. This observable isusually called Wilson loop.

3.2.1 Wilson Action

The action defined on the lattice will be built from the Uµ(n) and must begauge invariant. As we already stated, the trace of a path-ordered product of Uµ(n)

3.2. LATTICE DISCRETIZATION 27

x x

x x + ν + µ + ν

+ µ

^ ^

^

^

Figure 3.1: A figurative example of the loop in (3.5)

variables along a loop is a gauge invariant observable, so we construct the simplestloop, that is (see Fig. 3.1):

Uµν(n) = Uµ(n)Uν(n+ µ)U−µ(n+ µ+ ν)U−ν(n+ ν). (3.5)

The Wilson action is

SW = − β

2N

∑n,µ,ν

Re[TrUµν(n)] (3.6)

where

• β = 2Ng2

• ∑n,µ,ν is the sum over all the plaquettes travelled both clockwise and counter-clockwise

• 1/2 avoids the double counting of the plaquettes

• 1/N is a normalization factor for the trace of the plaquettes.

At the beginning we asked that in the continuum limit the Wilson action mustreturn the Euclidean Yang-Mills action. Then we can define

Uµ(n) = eiBµ(n) (3.7)

with


Bµ(n) = agτiAiµ(n), (3.8)

a being the lattice spacing and g the coupling constant. Therefore

Uµν(n) = eiBµ(n)eiBν(n+µ)eiB−µ(n+µ+ν)eiB−ν(n+ν) (3.9)

.We may expand the fields Bµ the first order in a as follows:

B−µ(n+ µ+ ν) ≡ −Bµ(n+ ν) ≈ −Bµ(n)− a∇νBµ(n)

Bν = Bν(n) + a∇µBν(n)

B−ν(n+ ν) = −Bν(n)

(3.10)

where we have denoted with

∇µBν(n) =Bν(n+ µ)−Bν(n)

a(3.11)

the finite difference on the lattice, whose continuum limit is a partial derivative (∂µ).Therefore we obtain

Uµν =eiBν(n) expiBν(n) + ia∇µBν(n)·· exp−iBµ(n)− ia∇νBµ(n)e−iBν(n).

(3.12)

Now we use the Baker-Campbell-Hausdorff formula

exey = exp

x+ y +

i

2[x, y] + . . .

, (3.13)

where the ellipsis denotes terms proportional to higher powers of x or y.Keeping in mind that Bµ ∼ a and that in the expansion we keep only terms up

to o(a2). We obtain

Uµν(n) = expia(∇µBν(n)−∇νBµ(n))− [Bµ(n), Bν(n)]. (3.14)

Let us define

ia(∇µBν(n)−∇νBµ(n))− [Bµ(n), Bν(n)] = ia2gFµν . (3.15)

If we substitute (3.8) we obtain

ia2gFµν = a(∇µagτiAiµ(n)−∇νagτiAiν(n))− a2g2[Aiµ(n), Ajν(n)]τiτj (3.16)

and so

Fµν = (∇µAiν −∇νaiµ)τi + ig[Aiµ(n), Ajν(n)]τiτj , (3.17)

that in the continuum limit (∇ν → ∂ν) is exactly the field-strength tensor for theSU(N) Yang-Mills theory.

Because the lattice spacing a is a small parameter we can expand the exponentialof (3.15) as follows


SW = − β

2N

∑n,µ,ν

Re[TrUµν(n)]

= − β

2N

∑n,µ,ν

Re[Tr

1 + ia2gFµν −

1

2a4g2FµνF

µν

]

=β

2N

∑n,µ,ν

Re[Tr

1 + ia2gF iµντi −

1

2a4g2F iµνF

jµντiτj

].

(3.18)

The normalization of the generators is such that Trτ iτ j

= 1

2δij . So neglecting theconstant coming from Tr1 we have

SW =βa4g2

8N

∑n,µ,ν

F iµνFiµν . (3.19)

To have the same expression as for the Yang-Mills action we have to impose that inthe continuum limit

∑n

→∫d4x

a4(3.20)

and

β =2N

g2. (3.21)

3.2.2 The Partition Function

Our real interest is not in the gauge action but in the partition function Z andin expectation values of physical quantities. In the continuum formulation of thetheory, to constructing Z one must address the problem of the integration measureand the related problem of gauge fixing. One of the most interesting advantages oflattice regularization is that the integrals involved in the construction of Z are linkby link ordinary integrals. So we have a natural choice for the integration measureand it is the Haar measure. The Haar measure dUµ(n) does not break the gaugeinvariance and makes the gauge fixing unnecessary.

Thus, for the partition function we have

Z =

∫ ∏n,µ

dUµν(n)e−SW (3.22)

and for a generic expectation value of an observable O we have

〈O〉 =1

Z

∫ ∏n,µ

dUµν(n)O(Uµν(n))e−SW (3.23)


3.2.3 Setting the Scale

We have just seen how the partition function and the expectation value of anobservable are defined for a lattice regularised Yang-Mills theory. Thus we can seeeq. (3.22) and eq. (3.23) as the equations that define the lattice regularisation inthe Feynman path-integral formalism. In this way the functional integrals of thecontinuum theory are replaced by discretely many, ordinary group integrals. This letus use the computational tools of statistical systems, both analytical and numericalones, to investigate the lattice counterpart of the continuum Yang-Mills theory.

Up to this point equations (3.22) and (3.23) describe nothing more than astatistical-mechanics system of SU(N) matrices defined on the oriented lattice links,which has little to do with the continuum gauge theory. However the continuum limitcan be achieved by sending the lattice spacing a to zero, keeping some dimensionfulphysical quantity fixed to a specific value.

To reach the continuum limit from eq. (3.19) we had to impose β = 2Ng2 and

this is the only parameter on which the dynamics of the lattice Yang-Mills theorydepends. Even the lattice spacing a can be considered as a function of β. This isparticularly important, because all dimensionful quantities in the lattice theory aremost naturally expressed in the appropriate power of the lattice spacing a.

Thus in order to study the continuum limit of the lattice theory, one shouldfirst compute the expectation value of some dimensionful physical quantity in theappropriate units of lattice spacing, at fixed β. Then, assuming that such physicalquantity has a well defined physical value the physical value of a can be deduced.This procedure is known as setting the scale.

An appropriate observable to carry out this procedure should satisfy some request:

• the observable should be as insensitive as possible to lattice discretizationartefacts. Since these artefacts are maximally manifest at length scales equalto the lattice spacing a, an appropriate observable should be one characteristicof length scales much higher than the lattice spacing

• the observable should be on for which the comparison between lattice resultsand physical vale should be as strong as possible, due to avoid any sort ofuncertainties

• the observable should be simple to compute to high precision

One of the most popular ways to set the scale has ben based on one of thecharacterising non-perturbative low-energy property of the Yang-Mills theory: thefact that the theory is confining, or in other words the fact that the physical spectrumonly contains colour singlets, and characterised by an asymptotically linear potentialV between a quark and antiquark pair. Phenomenologically that the quark andantiquark in a meson are tied together by a linear rising potential [12]. The simplestway such a behaviour is to assume that the infrared regime of QCD is described byan effective string, which joins together quark and antiquark.

On the lattice the simplest way to reproduce quark and antiquark pair is tostudy the mean value of a large rectangular Wilson loop of sizes R× T , which we


T

R

Figure 3.2: Wilson loop of size R× T

denote with 〈W (R, T )〉 (see fig. 3.2). Let us also assume that T is a segment inthe time direction and R a segment in any other spatial direction. The physicalinterpretation of 〈W (R, T )〉 is that it represent the creation of a quark antiquarkpair at t = t0, which immediately separate to a distance R from each other, keepingtheir positions fo a time T and then instantaneously annihilate at time t = t0 + T .According to this description we expect for large T :

〈W (R, T )〉 ∼ e−TV (R). (3.24)

The quark and antiquark pair is considered in the fundamental representation ofthe gauge group, thus, the absence of dynamical matter fields in a representation thatcould screen fundamental colour charge forbids string breaking. So at sufficientlylarge distances R between the charges the potential become linear in R. At finitedistance other terms included in V (R) become relevant: a constant term, whichsimply plays the role of a sort of renormalisation term, a 1

r term (Luscher term [24]),and other subheading terms [2]

V (R) ' σR+ V0 −π

12R+ . . . , (3.25)

where σ is the string tension. Phenomenologically studying models of quarkoniumstates observed in nature it was possible to suggest that the physical value for thestring tension in QCD would be (450MeV )2, and one can take this value to be validalso in the pure Yang-Mills theory.


Form eq. (3.24) we can derive the lattice evaluate of the inter quark potentialV (R) at a given β:

aV (R

a) = − lim

T→∞a

Tln(〈W (

R

a,T

a)〉), (3.26)

then one can fit

aV = (σa)2(R

a) + aV0 −

π

12Ra, (3.27)

to obtain σa2 and consequently derive the physical value of a at that β.

At small β one can compute the expectation values by expanding the exponentialin eq. (3.23) in a power series in β, and using the properties of the Hair measure tocalculate exactly the series coefficients. At large β one can compute the expectationvalues in eq. (3.23) by expanding the lattice theory perturbatively around the freelimit. At intermediate β one can compute the lattice expectation values of physicalobservables numerically.

A way of setting the scale is explained in [10].

3.3 Montecarlo Method

Each of the multiple integrals in (3.22) and (3.23) is done over the whole groupmanifold. If the integrals are approximated by sums over a sufficiently dense set ofpoints or if the gauge group is discrete the integrals reduce to sums over a finitenumber of terms. We shall say that each of these terms specifies a configuration Cof the system. The fact that quantum averages are expressed by sums over finitesets of configurations leads one to think that they could be calculable by numericalmethods. This is indeed the case, but not through a straightforward summationprocedure, in fact even for the simplest gauge model (for a modest 44 lattice) thenumber of configurations is such a large number that a direct numerical sum isbeyond the possibilities of the most powerful computer. On the other hand, as isthe case for thermodynamical systems, only a subset of configurations effectivelycontributes to the quantum averages. If a sufficient number of configurations isselected among the relevant ones with a probability distribution proportional to themeasure

∏n,µ dUµν(n) exp−S, the exact quantum averages may be approximated

by averages taken over this sample of configurations.

This numerical method goes under the name of Monte Carlo integration. Thealgorithm works as follows. An initial configuration C1 is stored into the memoryof the computer (i.e. the values of all gauge dynamical variables Uµ(n). From C1

the computer generates a new configuration C2, which replaces C1 in the memory,by a stochastic procedure. Actually, C2 is generated deterministically from C1, butthrough an algorithm that involves pseudo-random numbers. Thus, only a transitionprobability matrix P (C → C ′ is defined for the passage from a configuration to the

3.3. MONTECARLO METHOD 33

next. From C2 the computer generates C3, and so on. The stochastic process isdesigned in such a way the probability of encountering any definite configurationC at the kth step in the sequence converges, as k grows larger, to a distributionproportional to the correct measure.

If we assume that after n0 steps the probability distribution has come closeenough to the correct one, we approximate the exact quantum expectation valuesdefine in (3.22) and (3.23) with

〈O〉 =1

n

n∑k=n0+1

O(Ck) (3.28)

where O(C) stands for the value of the observable O in the configuration C.The Boltzman distribution P (C) ∝ exp−S(C) must be an eigenvector of the

stochastic matrix P (C → C ′). This is guaranteed if P obeys detailed balanceconditions:

P (C → C ′)P (C ′ → C)

=e−S(C′)

e−S(C). (3.29)

The normalization of the stochastic transition matrices∑

C′ P (C → C ′) = 1 andthe detailed balance conditions imply that∑

C

e−S(C)P (C → C ′) =∑C

e−S(C′)P (C ′ → C) = e−S(C′). (3.30)

The detailed balance conditions is not a necessary requirement for convergenceto the Boltzman distribution, but it is satisfied in the most commonly implementalgorithms. However, it does not fix the algorithm completely; the simplest algorithmthat is used to gnerate the configurations is the one that was proposed by Metropoliset al. [25], known as Metropolis procedure. It works as follows.

The transition matrix P (C → C ′) is determined in two steps. First a newcandidate configuration C ′ is selected starting from C according to some probabilitydistribution P0(C → C ′), that satisfies the equality P0(C → C ′) = P0(C ′ → C).Then the variation in action

∆S = S(C ′)− S(C) (3.31)

caused by the change of configuration is calculated. Now a random number r isselected with uniform probability distribution between 0 and 1, and

• if r ≤ e−∆S , then the change is accepted and the new configuration will be C ′

• if r > e−∆S , then the change is rejected and the new configuration will be Cagain.

Clearly, if the passage from C to C ′ lowers the action the change is always accepted.Vice versa, if the change of configuration increases the action the new configurationis accepted only with conditional probability e−∆S . This occasional acceptance ofchanges which increase the action simulates the effect of quantum fluctuations. It isstraightforward to verify that the matrix P (C → C ′) defined by the above procedure


satisfies the detailed balance condition. Indeed, assuming for example S(C ′) < S(C),we have

P (C → C ′) = P0(C → C ′)e−∆S

P (C ′ → C) = P0(C ′ → C).(3.32)

Therefore

P (C → C ′)P (C ′ → C)

=P0(C → C ′)P0(C ′ → C)

e−∆S =e−S(C′)

e−S(C). (3.33)

In practice, the passage from a configuration to the next is obtained by changing,or, in technical language, updating, just one of the dynamical variables, or at mosta limited set of variables. In this way the variation in action is kept small and oneavoids large changes in the action, which lead to a rejection of the move. Furtherdetail on the Montecarlo method can be found in [30].

3.3.1 Heath-Bath and Overrelaxation Algorithms

Monte Carlo calculations are indispensable in evaluating lattice expectationvalues and the Metropolis algorithm described previously is used to generate thechain of configurations essential to compute the integrals in eq. (3.22) and eq. (3.23).However the Metropolis algorithm is not the most efficient at generating such chainof configurations, thus we use a mix of eath-bath [11] [20] and overrelaxation [1][5] algorithms. these algorithms generate the chain of configurations, distributedaccording to the desired probability distribution, in less computer time than theMetropolis one.

The way they achieve such an improvement respect to the Metropolis algorithmis by generating a new value for the variable to be update that is more independentfrom the original value than the Metropolis algorithm does. However, even thoughthe heath-bath and overrelaxation algorithms works in a quite similar way, thereis a substantial difference between the two. The overrelaxation algorithm is notergodic, because it is a micro canonical algorithm. In other words, this means thatthe overrelaxation is an Hamiltonian or Euclidean action preserving algorithm. Thus,in order to preserve the ergodicity the overrelaxation algorithm needs to be mixedwith an ergodic algorithm, like the heath-bath one.

We briefly describe the implementation of these algorithm in the case of a S(2)lattice Yang-Mills theory. The generalisation to S(N) (with N ≥ 3) was proposed byCabibbo and Marinari in [7] and consists in applying these updates to a sequenceof, at least N − 1, S(2) subgroups of each S(N) matrix.

The heath-bath algorithm works as follow. Consider the update of a givenUµ(x) matrix (a link variable). Its contribution to the Wilson action is given bythe real part of the trace of the plaquettes in which Uµ(x) enters (eq. (3.6)). Forthe purpose of explaining the heath-bath algorithm we can write the plaquettes as

3.3. MONTECARLO METHOD 35

Uµ(x)S†µ(x), where Sµ(x) is the sum of horseshoe-shaped parallel transporters fromx to x+ aµ. A linear combination of SU(2) matrices is still a SU(2) matrix, so thecontribution to the Wilson action of the link under exam is proportional to the traceof Uµ(x)s†µ(x), where sµ(x) = ksµ(x) and sµ(x) ∈ SU(2). Thus the statistical weightof the configuration in given by

dUµ(x) exp

kβ

2TrUµ(x)s†µ(x)

. (3.34)

Now, if we set Vµ(x) = Uµ(x)s†µ(x), the invariance of the Haar measure underright group translations transform the problem of generating a new variable Uµ(x),distributed according to eq. (3.34), to generating an S(2) matrix V distributedaccording to

dVµ(x) exp

βk

2TrV

. (3.35)

Because of S(2) can be treated as an element of S3, we can rewrite the matrix V as:

V = v0 + iσ · v, (3.36)

where v20 + viv

i = 1. Because of TrV = 2v0, eq. (3.35) can be written as

1

2π2dv0

√1− v2

0 expkβv0dcosθdφ. (3.37)

In this equation v0 and cosθ take values in [−1, 1], while φ ∈ [0, 2π). Moreover wehave used a spherical coordinates parametrization in which

• v1 =√

1− v20sinθcosφ

• v2 =√

1− v20sinθsinφ

• v3 =√

1− v20cosθ

At this point the new value for Uµ(x) is built as Uµ(x) = Vµ(x)sµ(x).

The overrelaxation algorithm has the same base of the heath-bath algorithm, butworks as follows. It is based on the observation that if the contribution to the Wilsonaction from a given link variable Uµ(x) is proportional to the trace of Uµ(x)s†µ(x),then the product

sµ(x)Uµ(x)s†µ(x) (3.38)

gives the same contribution to the action. Moreover this contribute is as far aspossible from the original Uµ(x).


3.3.2 Errors In Lattice Results

The Monte Carlo method consists in extrapolating from the lattice simulationthe same quantity over and over for each iteration of upgrading, At the end of therun we have a large bunch of measures, whose mean value will be the expectationvalue we are interested in. At this point we have to face the problem of evaluatingwhich error we have on such mean value.

based on the previous description of the Monte Carlo method one may thinkthat the best way of treating errors is using the Gaussian distribution:

σx =

√∑Ni=1(xi − x)2

N(N − 1), (3.39)

where σx is the error on the mean value, x is the mean value, xi is the single measureand N is the total number of measurements. The problem with this estimationof the error on x is that the Gaussian distribution treats decorrelated measures.However the path-bath and overrelaxation algorithms we implemented are basedon generating a new configuration weighted with the previous one. This proceduregenerate correlated data and this means they are not independent one from another.

Thus we use the jackknife approach (the delete-d one) [32], This method ofestimating errors is based on the fact that measures are less and less related themore upgrade are done between measurements. The jackknife method works asfollows.

Assume to have a set of measurements X = (x1, . . . , xN ). Then we bin our datainto J subset of M = N

J consecutive measurements. To be more specific, after bingowe have J bins we call Yi (i = 1, . . . , J) that contains the measurements as follows:

Yi = (x1+(i−1)M , . . . , x1+iM ). (3.40)

Now we build J more bins we call Ki (i = 1, . . . , J). Each Ki contains the originalset X, to which we subtract the measures in Yi: Ki = X − Yi. Then we averagemeasurements in each Ki bin and call ki the mean value. Finally we can evaluatethe error on the mean value x using:

σx =

√√√√J − 1

J

J∑i=1

(ki − x)2. (3.41)

In this work we use J = 10, because an interval of M = N10 measurements is a

reasonable waiting time after which we can assume measurements are not correlated.

Chapter 4

Entanglement Entropy

4.1 Entanglement Entropy in Quantum Field Theories

In this section we discuss the entanglement entropy in a quantum field theory.For simplicity, we consider the regularization of the theory on a spacetime lattice.We assume that at each site we have a finite-dimensional Hilbert space Hα with αindexing the sites. Then, a pure quantum state of the system is an element of thetensor product Hilbert space:

|ψ〉 ∈ ⊗αHα. (4.1)

We want to understand how a subset of the lattice degrees of freedom are entangledwith the rest in a given state of the above kind. We divide the lattice sites into twosets by drawing a boundary across the lattice. We will label the region within theboundary as A and the complementary region as B. We call the artificial boundaryas the entangling surface ∂A. With this kind of spatial decomposition we have aparticular bipartitioning of the Hilbert space:

⊗α Hα = HA ⊗HB. (4.2)

Now we can define the reduced density matrix ρA as:

ρA = TrB|ψ〉〈ψ| (4.3)

which formalizes the idea of capturing the state of the degrees of freedom in Aassuming complete ignorance of what happens in B.

We are interested in quantifying the amount of entanglement that exists in |ψ〉,partitioned as described above. This can be obtained from the von Neumann entropyof the reduced density matrix, which is often referred to as the entanglement entropy:

SA = −TrρA ln ρA (4.4)

In continuum quantum field theories, we find that given a wavefunctional Ψ[φ(x)]for the instantaneous state of the system, we can mimic the previous construction todefine ρA and its associated entanglement measure. φ(x) indicates the collection of

37

38 CHAPTER 4. ENTANGLEMENT ENTROPY

fields that characterize the system and x is a set of spatial coordinates that describesthe spatial location on a time-slice.

It is clear that the construction of ρA involves ignoring the part of wavefunctionalthat corresponds to the spatial region B. The process of tracing over the degreesof freedom of the complementary region B consists in integrating over all fieldsconfigurations in that domain. Once we have the reduced density matrix, we cancompute SA as defined in Eq. (4.4).

Now assume we have a d-dimensional relativistic quantum field theory on someLorentzian spacetime L. We choose a Cauchy slice Σd−1 that is a space-like slice,defining in the quantum field theory a moment of simultaneity. On Σd−1 we havea state of the system given by the wavefunctional Ψ[ψ(x)]. Then we pick somecodimensional region A on the Cauchy slice, which allows for a spatial bipartitioningof the form A ∪B = Σd−1. The boundary of the region is, as before, the entanglingsurface ∂A. As before, we decompose the Hilbert space of the quantum field theoryinto HA ⊗HB. Then the reduced density matrix ρA = TrHBρΣ is defined. Hencewe can give a quantitative measure of the entanglement between regions A and Bby computing the von Neumann entropy.

Since Σ is a Cauchy slice, the future (and past) evolution of the initial data onit allows us to reconstruct the state of the quantum field theory on the entiretyof L. In fact, the past and future domain of dependence of Σ, that we denoteas D±[Σ], together make the background of the spacetime on which the quantumfield theory is defined [34]. In the same way, the domain of dependence of A,D[A] = D+[A]∪D−[A], is the region in which the reduced density matrix ρA evolvesonce we know the Hamiltonian acting on the reduced system A.

Now the domains of dependence of A and B do not make up the full space-timeby themselves (D[A] ∪D[B] 6= L), but we have to consider also the regions whichcan be influenced by the entangling surface ∂A. Denoting with J±(p) the causalfuture and past of a point p, we have to account for the regions J±[∂A], that arenot contained in D[A] or D[B]. As a result, the full spacetime L decomposes intofour casually-defined regions: the domain of dependence of the region A and itscomplement, and the causal future and past of the entangling surface [19]:

L = D[A] ∪D[B] ∪ J+[∂A] ∪ J−[∂A]. (4.5)

The above decomposition depicted in Fig. ?? is useful to formulate someconstrains on the entanglement entropy, that follows from the relativistic causality.If we unitarily evolve the reduced density matrix ρA with transformations thatare supported only by HA or HB, the eigenvalues of ρA remain unaffected. Thesetransformations could include perturbations of the Hamiltonian and local unitarytransformation supported in the domain of dependence D[A] or D[B]. Now considera deformation of the region A into another region A′ on a different Cauchy sliceΣ′. The state ρΣ′ on the new slice is related by unitary transformation to the stateρΣ. Such transformation can be constructed from operators localized in A and sodoes not affect the entanglement spectrum of ρA. We of course can make similararguments for the complementary region B.

Now if we fix the state at t→ −∞, and consider perturbations of the Hamiltoniansupported in some region τδH , then by virtue of causality, we can only affect the statein the causal future of this region. In other words, in any region of the spacetime

4.1. ENTANGLEMENT ENTROPY IN QUANTUM FIELD THEORIES 39

AD+[A]

D[A]

D[A] = D+[A] [ D[A]

D[A]D[Ac]

J+[@A]

J[@A]

Figure 4.1: On the left we have an illustration of the causal domains of A. On theright we have a depiction of the decomposition indicated in (4.5)

which does not intersect J+[τδH ], our changes have no effect. By reversing the timeordering, if we fix the state at t→ +∞, we can only affect the state in the causalpast of the region τδH .

We can summarize this discussion through the following properties of ρA:

• the entanglement spectrum of ρA depends only on the domain D[A] and noton the particular choice of Cauchy slice Σ

• fixing the state in the far future or in the far past, the entanglement spectrumof ρA is insensitive to any local deformations of the Hamiltonian in D[A] orD[B].

The entanglement entropy enjoys other useful properties which are summarizedbelow. Their derivation may be found in [26].

• If the ground state wave function is pure, the entanglement entropy of thesubsystem A is equal to that of its complement B:

SA = SB. (4.6)

SA is no longer equal to SB at finite temperature because the system is in amixed state.

• For any subsystem A, B and C, that do not intersect the following inequalitiesholds:

SA+B+C + SB ≤ SA+B + SB+C

SA + SC ≤ SA+B + SB+C(4.7)

These inequalities describe the property that is called the strong subadditivityof the entanglement entropy [23]

• If B is empty, the inequality (4.7) yields the subadditivity

SA+C ≤ SA + SC , (4.8)

Similarly, setting C be empty leads to the triangle inequality

|SA − SB| ≤ SA+B (4.9)


4.2 Entanglement Entropy in Conformal Field Theories

4.2.1 Renyi Entropy and The Replica Trick

We are interested in evaluating the entanglement entropy for a conformal fieldtheory and to do so, in this section we follow the discussion of Ryu and Takayanagiin [31]. In order to compute the entanglement entropy we introduce the Renyientropy:

Sn = − ∂

∂nln(Trρn), (4.10)

that is a one parameter generalization of the von Neumann entropy. In the limit ofn→ 1 the Renyi entropy reduces to the entanglement entropy:

SA = − limn→1

∂

∂nln(TrA ρ

nA). (4.11)

So first of all one has to compute TrA ρnA and to do this the so called “replica trick”

can be used.

For illustration purposes, we consider a 2D dimensional system and we assumeA to be the single interval x ∈ [u, v] at tE = 0 in the flat Euclidean coordinates(tE , x) ∈ R. The ground state wave function Ψ can be found by path-integratingfrom tE = −∞ to tE = 0 in the Euclidean formalism

Ψ(φ0(x)) =

∫ tE=0

tE=−∞Dφe−S(φ), (4.12)

where φ(tE , x) denotes the field which define the 2D conformal field theory. The totaldensity matrix ρ is given by two copies of the wave function: ρφ0φ′0

= Ψ(φ0)Ψ(φ′0).

the complex conjugate one Ψ can be obtained by path-integrating from tE = ∞to tE = 0. to obtain the reduced density matrix ρA, we need to integrate φ0 on Bassuming φ0(x) = φ′0(x) when x ∈ B:

[ρA]φ+φ− = (Z1)−1

∫ tE=∞

tE=−∞Dφe−S(φ)

∏x∈A

δ(φ(+0, x)− φ+(x))δ(φ(−0, x)− φ−(x)),

(4.13)

where Z1 is the vacuum partition function on R2 and we introduce (Z1)−1 in orderto normalize ρA such that TrA ρA = 1. This computation is sketched in Fig. 4.2(a).

To find the TrA ρnA we can prepare n copies of (4.13)

[ρA]φ1+φ1− [ρA]φ2+φ2− . . . [ρA]φn+φn− , (4.14)

and take the trace successively. In the path-integral formalism this is realized bygluing φi±(x) as φi+(x) = φ(i+1)+(x) (i = 1, 2, . . . , n− 1) and φn−(x) = φ1+(x),

4.2. ENTANGLEMENT ENTROPY IN CONFORMAL FIELD THEORIES 41

tE

+∞

0

u v

−∞x

x

tE

A BB

φ+

φ−

(a) (b)

φ1

φ2

φ3

Figure 4.2: (a) The path integral representation of the reduced density matrix. (b)The n-sheeted Riemann surface Rn. (we take n = 3 for simplicity.)

and integrating φi+(x). In this way TrA ρnA is given in terms of the path-integral on

a n-sheeted Riemann surface Rn (see Fig 4.2 (b)):

TrA ρnA = (Z1)−n

∫(tE ,x)∈Rn

Dφe−S(φ) ≡ ZnZn1

. (4.15)

To evaluate the path-integral on Rn, it is useful to introduce replica fields. Letus first take n disconnected sheets. The field on each sheet is denoted by φk(tE , x)(k = 1, 2, . . . , n). In order to obtain a conformal field theory on the flat complex planeC which is equivalent to the present one on Rn, we impose the twisted boundaryconditions:

φk(e2πi(w − u)) = φk+1(w − u) , φk(e

2πi(w − v)) = φk−1(w − v), (4.16)

where we employed the complex coordinate w = x+ itE . Equivalently we can regard

the boundary conditions (4.16) as the insertion of two twist operators φ+(k)n and

φ−(k)n at w = u and w = v for each (k-th) sheet. Thus we find

TrA ρnA =

n−1∏k=0

〈φ+(k)n (u)φ−(k)

n (v)〉. (4.17)

4.2.2 Entanglement Entropy in a 2D Conformal Field Theory

We have shown that the ratio of partition functions in (4.15) is the same as thecorrelation function arising from the insertion of primary scaling operators φ+

n and


φ−n . Moreover this correlation function is computable from the Ward identity ofconformal field theory.

First of all we use the conformal mapping w → ζ = w−uw−v to map the branch

points to (0,∞), then, by the mapping ζ → z = ζ1n = (w−uw−v )

1n , we map the whole

of the n-sheeted Riemann surface to the z-plane C. We know the transformationof the stress-energy tensor under coordinate transformations from (??), and using〈T (z)〉C = 0 we find:

〈T (w)〉Rn =c

24

(1− 1

n2

)(v − u)2

(w − u)2(w − v)2. (4.18)

This has to be compared with the form of the correlation of T with two primaryoperators φ+

n and φ−n , which is given by the Ward identity (2.117). We assume thatφ+n and φ−n have the same scaling dimension h = h = c

24(1− 1n2 ), thus we find

〈T (w)φ+n (u)φ−n (v)〉C =

[h

(w − u)2+

h

(w − v)2+

1

w − v∂

∂u+

1

w − v∂

∂v

]〈φ+n (u)φ−n (v)〉C.

(4.19)

The fields φ±n are normalized so that

〈φ+n (u)φ−n (v)〉C = |v − u|−2h|v − u|−2h, (4.20)

so we can compute the Ward identity:

〈T (w)φ+n (u)φ−n (v)〉C =

=

[h

(w − u)2+

h

(w − v)2+

2h

(w − u)(v − u)− 2h

(w − v)(v − u)

]〈φ+n (u)φ−n (v)〉C

=h(v − u)2

(w − u)2(w − v)2〈φ+n (u)φ−n (v)〉C (4.21)

In writing the equation above we are assuming that w is a complex coordinate on asingle sheet C, which is now decoupled from the others. We have therefore shownthat

〈T (w)〉Rn ≡∫DφT (w)e−S(Rn)∫Dφe−S(Rn)

=〈T (w)φ+

n (u)φ−n (v)〉C〈φ+n (u)φ−n (v)〉C

. (4.22)

Since the Ward identity determines all the properties under conformal transfor-mations, we conclude that, under a scale transformation, TrA ρ

nA behaves like the

n-th power of the two-point correlation function of a primary field operator φn withh = c

24(1− 1n2 ):

ZnZn1

= TrA ρnA =

n−1∏k=0

〈φ+(k)n (u)φ−(k)

n (v)〉C = (〈φ+n (u)φ+

n (v)〉C)n. (4.23)

Hence


TrA ρnA = |v − u|−4nh = |v − u|

nc6

(1− 1

n2

)(4.24)

Therefore using (4.11) we find the entanglement entropy for a single interval in a2D dimensional conformal field theory, according to

SA =c

3ln

(l

a

), (4.25)

where l = v − u is the length of the interval and 1/a is an ultraviolet cut-off.

4.2.3 Entanglement Entropy in a massive Quantum Field Theory

As we said before, we are interested in evaluating the entanglement entropy ina conformal field theory. The reason why we are going to compute the value ofthe entanglement entropy in a massive quantum field theory will be clear later. Inthis section we consider an infinite non-critical model in 1+1 dimensions, in thescaling limit where the lattice spacing a→ 0 with the correlation length ξ (inversemass) fixed. This corresponds to a massive relativistic quantum field theory. Weconsider the case when the subset A is the negative real axis, so that the appropriateRiemann surface has branch points of order n at 0 and infinity. However for the noncritical case, the branch point at infinity is unimportant: we should arrive at thesame expression by considering a finite system whose length L is much greater thanξ.

Let us consider the expectation value of the stress tensor Tµν of a massiveEuclidean quantum field theory on such a Riemann surface. In complex coordinates,there are three non-zero components: T ≡ Tzz, T ≡ Tzz and the trace Θ = 4Tzz =4Tzz. These are related by the conservation equations:

∂zT +1

4∂zΘ = 0

∂zT +1

4∂zΘ = 0.

(4.26)

Consider the expectation values of these components. In the single-sheetedgeometry 〈T 〉 and 〈T 〉 both vanish, but 〈Θ〉 is constant and non vanishing: itmeasures the explicit breaking of the scale invariance in the non-critical system. Inthe n-sheeted geometry, however, they all acquire a non trivial spatial dependence.By rotational invariance about the origin, they have the form:

〈T (z, z)〉 =Fn(zz)

z2

〈Θ(z, z)〉 − 〈Θ〉1 =Gn(zz)

zz

〈T (z, z)〉 =Fn(zz)

z2 .

(4.27)


From the conservation conditions (4.26) we have

(zz)(F ′n +1

4G′n) =

1

4Gn. (4.28)

Indeed:

(zz)F ′n = (zz)∂

∂(zz)Fn(zz) = (zz)

[1

2z

∂

∂z+

1

2z

∂

∂z

]Fn(zz)

= (zz)1

2z

∂

∂zz2〈T (z, z)〉+ (zz)

1

2z

∂

∂zz2〈T (z, z)〉

=zz2

2

∂

∂z〈T (z, z)〉+

z2z

2

∂

∂z〈T (z, z)〉.

(4.29)

For the other derivative we have

zz

4

∂

∂(zz)Gn(zz) =

zz

4[

1

2z

∂

∂z+

1

2z

∂

∂z](zz)(〈Θ(z, z)〉 − 〈Θ〉1)

=z

8

∂

∂z(zz)(〈Θ(z, z)〉 − 〈Θ〉1) +

z

8

∂

∂z(zz)(〈Θ(z, z)〉 − 〈Θ〉1)

=zz

8(〈Θ(z, z)〉 − 〈Θ〉1) +

z2z

8

∂

∂z〈Θ(z, z)〉+

zz

8(〈Θ(z, z)〉 − 〈Θ〉1)+ (4.30)

+zz2

8

∂

∂z〈Θ(z, z)〉

=1

4Gn(zz) +

z2z

8

∂

∂z〈Θ(z, z)〉+

zz2

8

∂

∂z〈Θ(z, z)〉.

Therefore

(zz)(F ′n +1

4G′n) =

1

4Gn(zz) +

zz2

2

∂

∂z〈T (z, z)〉+

z2z

2

∂

∂z〈T (z, z)〉+

+z2z

8

∂

∂z〈Θ(z, z)〉+

zz2

8

∂

∂z〈Θ(z, z)〉 (4.31)

=1

4Gn(zz)

We expect that Fn and Gn both approach zero exponentially fast for |z| ξ,while in the opposite limit, on distance scales |z| ξ they approach the conformalfield theory values Fn → c

24(1− n−1) and Gn → 0. This means that if we define afunction

Cn(R2) ≡(Fn(R2) +

1

4Gn(R2)

), (4.32)

then we have

R2 ∂

∂R2C(R2) =

1

4Gn(R2). (4.33)

Using the boundary conditions we can evaluate the integral


∫ ∞0

Gn(R2)

R2d(R2) =

∫ ∞0

∂

∂(R2)Gn(R2) = 4Cn(R2)|∞0

= −4c

24

(1− 1

n2

)= − c

6

(1− 1

n2

).

(4.34)

Equivalently we can see the integral above in a different manner:∫(〈Θ〉n − 〈Θ〉1)d(R2) = −nc

6(1− 1

n2), (4.35)

where the integral is over the whole of the n-sheeted surface. Now this integralmeasures the response of the free energy − lnZ to a scale transformation. Since themass m is the only dimensional parameter of the renormalized theory, the left handside of (4.35) equal to:

−m ∂

∂m[ln(Zn)− n ln(Z1)], (4.36)

which give

ZnZn1

= Kn(ma)c6

(n−n−1), (4.37)

where kn is a constant and we have inserted a power of a to make the resultdimensionless.

Using (4.11) we find (according to [8])

SA ∼c

3ln(

ξ

a). (4.38)

4.2.4 Entanglement Entropy of a Conformal Field Theory in d+1Dimensions

Now we would like to move on to the computation of the entanglement entropyin higher dimensional conformal field theory (d+ 1 ≥ 3).

As in the 2D case explained in subsection 4.2.1, we assume the conformal fieldtheory to be defined on the d + 1 dimensional manifold R × N . We define thesubsystem A as the submanifold on N at the fixed time t = t0 ∈ R. The strategyof calculating the entanglement entropy SA is the same as in the 2D case. Firstfind the reduced trace TrA ρ

nA and then plug this in (4.11) to obtain SA. We can

compute TrA ρnA from the partition function Zn on the n-sheeted d+ 1 dimensional

manifold Mn as in the 2D case:

TrA ρnA =

ZnZn1

. (4.39)

The n-sheeted manifold Mn can be constructed as follows. First we remove theinfinitely thin d dimensional slice A from M1 = R×N . Then we call the boundary


L

L

L

l

l

(a) (b)

A

B BA

Figure 4.3: The straight belt in d = 3

of such a space Aup and Adown. Next we prepare n copies of such a manifold. Theirboundaries are denoted by Aiup and Aidown (i = 1, 2, . . . , n). Now we glue Aiup with

Ai+1down for every i. As we take the trace of ρnA, Ai=nup is glued with A1

down. In the endthis procedure leads to a manifold Mn with conical singularities where all n cutsmeet.

It is not straightforward to calculate Zn for an arbitrary choice of A even in freefield theories. This is so because the conformal structure is not as strong as in the2D conformal field theory case. Therefore for our purpose we restrict our argumentsto a specific form of A. We also assume N ≡ Rd for simplicity. We choose A to be astraight belt of width l:

A = xi|x1 ∈ [− l2,l

2], x2,3,...,n ∈ [−∞,∞] (4.40)

as shown in figure 4.3. Although the lengths in the directions of x1, x2, . . . , xn areinfinite, we can regularize the theory in the infrared by assuming the system tohave length L in each of these directions. The boundary in this case is given by thestraight surface ∂A = Rd−1.

Thus we consider the entanglement entropy of d+ 1 dimensional conformal fieldtheory on R1,d defined by massless free fields. This can be regarded as the infinitevolume limit L→∞ of the conformal field theory on M = R1,1 × T d−1, where thevolume of the torus is Ld−1.

Because this theory is free, we can perform the dimensional reduction on T d−1

and obtain infinitely many two dimensional free massive theories, whose masses aregiven by

m2 =

d∑i=2

k2i =

(2π

L

)2 d∑i=2

n2i , (4.41)


where ki = 2πniL are the quantized momenta such that ni ∈ Z in the torus directions.

We reduce to the case where the subsystem A is the straight belt because in thisway the computation of the entanglement entropy SA is reduced to the computationof SA in massive 2D quantum field theories.

As we found previously we know the formulas for the entanglement entropy bothin the massless limit (l ξ) (4.25) and the massive limit (l ξ) (4.38). Thereforewe can estimate SA by replacing the sums of infinitely many modes ni with theintegral of ki in the L→∞ limit:

SA =

ξ≤l∑k2,...,kd

c

3ln

(ξ

a

)+

ξ≥l∑k1,...,kd

c

3ln

(l

a

)

=

(L

2π

)d−1 c

3

[∫ a−1

l−1

dd−1k ln

(ξ

a

)+

∫ l−1

0dd−1k ln

(l

a

)]

=c

3(d− 1)2d−1πd−1

2 Γ(d+1

2

) (Ld−1

ad−1− Ld−1

ld−1

).

(4.42)

If we set d = 3 we find the entanglement entropy of conformal field theory in 4spacetime dimensions [31]:

SA =c

24

(L2

a2− L2

l2

), (4.43)

where c is the two dimensional central charge and L2 is the boundary surface ∂A.

Chapter 5

Lattice Simulations

In the previous section we have discussed the formula for the entanglemententropy of a field theory in four spacetime dimensions. To further clarify themeaning of the two terms in (4.43), we can write the entanglement entropy in thefollowing way:

1

|∂A|S(l) =1

|∂A|SUV +1

|∂ASf (l) = kΛD−2UV − kL2−D, (5.1)

where ΛUV is the ultraviolet cut-off scale, which, in our case, is proportional to theinverse of the lattice spacing. In D=4, Eq. (5.1) becomes:

1

|∂A|S(l) =k

a2− k

l2. (5.2)

For the purposes of lattice simulations, it is useful to represent the result obtainedcombining (4.11) and (4.15) in terms of free energy:

S[A] = limT→0

( limn→1

∂

∂nF [A,n, T ]− F (T )), (5.3)

where n is the number of cuts, F (T ) = − ln(Z(T )) is the free energy at temperatureT and F [A,n, T ] = − ln(Z[A,n, t]) is the free energy obtained by integrating overall fields with the boundary conditions explained in the following subsection.

5.0.1 Lattice Setup

The method we use to compute the entanglement entropy relies on the replicatrick which involves integration of fields all over an n-sheeted Riemann surface. Toreproduce this method on the lattice we have to impose the following boundaryconditions.

First of all, because we want to investigate the entanglement entropy in 4D, weuse a four dimensional lattice with one time-like coordinate (t) and three space-likecoordinates (x, y, z). The subsystem A is a slab of thickness l in the x direction,

49

50 CHAPTER 5. LATTICE SIMULATIONS

Figure 5.1: Topology of the space for n = 2 cuts.

which extends maximally in all other directions. To reproduce the n-sheeted Riemannsurface we impose the following boundary conditions for fields in the lattice:

• if the spatial coordinates lie within A, fields are periodic in time direction withperiod nt

• if the spatial coordinates lies within B, fields are periodic in the time directionwith period nt

n

• fields are periodic with period nx, ny, nz respectively in the x, y, z direction

ni indicate the number of sites in the respective direction. Such topology is depictedin Fig. 5.1 for n = 2 cuts.

Thus F [A,n, T ] is the free energy computed on the lattice with n cuts, whileF (T ) is the free energy computed on the lattice without cuts. In four dimensionalspace-time these cuts are the three volumes B at the time slice t = nt

k (k = 0, 1, . . . , n)which have a common boundary ∂A = ∂B. At this boundary the branching pointsof the cuts are situated.

At the end, to perform the simulations via Monte Carlo method, we decided torandomly choose the sites to upgrade at each iteration. Moreover we cut the linksthat cross A and B because of the ambiguity in the subsystem they belong to.

5.1. MEASUREMENT PROCEDURE 51

5.1 Measurement Procedure

To compute the entanglement entropy we start from the formula (5.3), so firstof all we have to find a way to estimate the derivative over n at n = 1. Followingwhat Buividovich and Polikarpov [6] done, we use the simplest way to estimate thederivative:

limn→1

∂

∂nF [A,n, T ] = F [A, 2, T ]− F [T ]. (5.4)

This finite difference of free energies well estimates the derivative over n for latticewith 2 cuts, while for lattice with a number of cuts n ≥ 3 a better estimation shouldbe used. Thus the formula for the entanglement entropy becomes:

S[A] = limT→0

(F [A, 2, T ]− 2F (T )). (5.5)

5.1.1 The Divergent Term

In the following we concentrate on testing the divergence of the first quadraticallydivergent term of (5.2). we have a formula for the entanglement entropy in terms offree energies, but free energies can not be directly ,measured in lattice simulation,thus we use the average plaquette action, which is linked to the free energy by theformula: ∑

p

〈β(1− 1

2Trgp) =

∂

∂lnβF (β). (5.6)

We can use the average plaquette action to study the divergent term because thederivative over β in (5.6) can only affect logarithmic dependence on the UV cutoffscale (which is too weak to be observed in lattice simulations) and can not affectthe power-like dependence on the lattice spacing the entanglement entropy has.This follows from the fact that, at small coupling, the lattice spacing a = Λ−1

UV

depends on β as a(β) ∼ exp−γβ, where the coefficient γ can be obtained fromthe Gell-Mann–Low β-function [9].

From the computation of the entanglement entropy via the replica trick, weknow that on the branching points of the cuts there are conical singularities, thuswe expect that the divergent part of the entanglement entropy depends on thosesingularities. To test this we studied the action density in the (t, x) plane, wherethe cuts are located. The action density for a given lattice site in (t, x) is obtainedby averaging the action of all plaquettes to which this site belongs.

In Fig. 5.2 is shown the plot of the action density of those sites near the branchingpoints as function of the lattice spacing a.We have used lattices with sizes form


Figure 5.2: Plot of the action density for those sites near the branching points asfunction of the action density.

18× 123 to 24× 123 with different values of l, n = 2 cuts and lattice spacing froma = 0.096 to a = 0.16. According to (5.3) the action density for lattices withoutcuts is subtracted to the action density for lattices with cuts. To evaluate the pointin the plot we take 50k configurations for every simulation. To calculate the latticespacing a from β we use the following formula [10]:

ln(σa2) =

3∑j=0

aj(β − β0)j , (5.7)

with

• β0 = 2.4

• a0 = −2.09

• a1 = −6.82

• a2 = −1.90

• a3 = 9.96

• √σ = 450MeV

The data in Fig. 5.2 are fitted with the straight line a+bx with a = 0.409±0.009and b = −0.07± 0.06. The fact that the data do not depend on the lattice spacingimplies that the action density for those sites near the branching points includesthe quadratically divergent term ∼ a−2|∂A|. This follows from the fact that thebranching points covers the hypersurface of total area |∂A| = 2nynza

2. Since the

5.1. MEASUREMENT PROCEDURE 53

divergence in the total average action comes from the vicinity of branching points, itis clear that the divergent term in the entanglement entropy does not depend on l.

5.1.2 The Non-divergent Term

To test the l-dependence of the second non-divergent term of (5.2) we have tofind a way to avoid the divergence. Thus if we consider the derivative over l ofthe entanglement entropy 1

|∂A|∂∂lS[A], we can estimate the derivative with finite

difference of free energies:

1

|∂A|∂

∂lSA(l +

a

2) ∝ F [l + a, 2, T ]− F [l, 2, T ]

a|∂A| . (5.8)

In this way, because of the divergent term does not depend on l, in the numeratorof (5.8) the divergence naturally cancels out.

Moreover the derivative over l of the entanglement entropy can be computedwithout need of extracting free energies from the lattice. In fact, we can use amethod proposed in [16] and [14], which involves the integration over a certainparameter α ∈ [0, 1] of differences of actions to obtain the finite difference in (5.8).

Assume that we have two different actions S1[φ] and S2[φ] for some set of fieldsφ defined on the lattice of some fixed size. We would like to calculate the differenceF2 − F1 of free energies F1 = −ln(Z1) and F2 = −ln(Z2), where the partitionfunctions are calculated by integrating over all fields φ with the weights exp−S1[φ]and exp−S2[φ], respectively. Now we define the interpolating partition functionZ(α) (α ∈ [0, 1]), so that Z(0) = Z1 and Z(1) = Z2:

Z(α) =

∫Dφ exp−(1− α)S1(φ)− αS2(φ). (5.9)

Thus the difference F2 − F1 can be represented in the following way:

F2 − F1 = −∫ 1

0dα

∂

∂αln(Z(α)) =

∫ 1

0dα〈S2[φ]− S1[φ]〉α, (5.10)

where the average 〈S2[φ] − S1[φ]〉α is defined by integrating over all fields withthe weight Z−1(α) exp−(1− α)S1[φ]− αS2[φ]. To integrate over α we use theextended Simpson’s rule [15], and we take 11 values of α between α = 0 andalpha = 1 at intervals ∆α = 0.1.

In table 5.1 are shown the data point we used to estimate the derivative ∂∂lS[A].

All the data are extracted from lattices with sizes of 24 × 123. The number ofconfiguration depicted indicates the number of configuration taken for every step ofα.

In Fig. 5.3 the results are shown, fitted with the line ax−b with a = 0.00007 andb = 3. Unfortunately we don’t have enough data to obtain a decent fit. Howeverin a qualitative manner we can see that the initial data we have follows an l−3

dependence, as we expect.


β la−1 # conf.

2.50 2 100k2.45 2 100k2.30 3 100k2.30 4 100k2.40 3 100k

Table 5.1: Points used to evaluate the derivative of the entanglement entropy withrespect to l.

Figure 5.3: l-dependence of the derivative of the entanglement entropy ∂∂lS[A].

Conclusions

The aim of this work is to show how to calculate the entanglement entropy fora 4-dimensional quantum field theory and then find an effective way to evaluate itusing lattice simulations. In chapter 5 we have presented how we have set up thelattice in order to achieve the theoretical results.

The set of lattice data obtained in this work are compatible with the analyticalexpectations and with lattice previous results obtained by Buividovich and Polikarpovin a work of the same kind [6]. We tested both the area law of the entanglemententropy, which depends on the quadratically divergent term shown in (5.2) and thel-dependence of the non-leading term.

The results we acquired via lattice simulations show that the lattice setup weimplemented can be used to perform large-scale simulations with the goal of a moredetailed study of the entanglement entropy. This goal extends well beyond the scopeof the present thesis work, but the strategy to successfully achieve it is well defined:

• by acquiring a very large sample of numerical data, one can reach sufficientprecision for the estimate of the l-dependent term;

• the simulations can then be repeated on lattices of increasingly large volume,possibly much larger than 24a × (12a)3, in order to reliably estimate finite-volume corrections, and to properly extrapolate them to the infinite-volumelimit;

• a generalization of this study to a setup with more than 2 cuts may revealif the results depend strongly on the number of cuts (and, thus, give moreconfidence on the connection between the quantity evaluated here and theactual entanglement entropy).

Once this program will be fully carried out, it will be possible to proceed andperform a first-principle non-perturbative study of the entanglement entropy innon-Abelian, non-supersymmetric gauge theories. This may lead, for example, to aclassification of the phase transitions in terms of the entanglement entropy.

55

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