Emergent spacetime from simple field theories

Emergent spacetime from simple field theories

Julius Engelsöy

Julius Engelsöy Emergent spacetime fr om simple field theories

Doctoral Thesis in Physics at Stockholm University, Sweden 2021

Department of Physics

ISBN 978-91-7911-474-9

Emergent spacetime from simple field theories

Julius Engelsöy

Academic dissertation for the Degree of Doctor of Philosophy in Physics at Stockholm University to be publicly defended on Tuesday 15 June 2021 at 13.15 online via Zoom, public link is available at the department website.

Abstract

Thermalization is an elusive phenomenon in quantum mechanics. Since according to the AdS/CFT correspondence, a black hole in the bulk spacetime is dual to a thermal state in the boundary CFT, thermalization in the CFT is dual to black hole formation in AdS. Thus, understanding quantum thermalization is likely a key component in understanding the information paradox—the contradiction between QFT and general relativity occurring when a black hole seemingly erases the information of whatever went into creating it.

In this thesis we investigate different aspects of quantum thermalization in simple field theories that have conjectured holographic duals, or more specifically, free large N singlet models. Since such theories are free and hence integrable, thermalization is more subtle but nevertheless appears in different forms. We investigate the late-time quench dynamics of a version of the O(N) vector model with probes such as the effective density matrix and the spectral density function.

We refine an operator thermalization hypothesis and explore its consequences in different spacetime dimensions in general large N singlet models. We also discuss the prospect of using thermal mixing in the boundary theory as a diagnostic of strong gravity in the bulk.

Keywords: AdS/CFT, thermalization, quench, large N singlet models, emergent spacetime.

Stockholm 2021

http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-192633

ISBN 978-91-7911-474-9 ISBN 978-91-7911-475-6

Department of Physics

Stockholm University, 106 91 Stockholm

EMERGENT SPACETIME FROM SIMPLE FIELD THEORIES

Julius Engelsöy

Emergent spacetime from simple field theories

Julius Engelsöy

©Julius Engelsöy, Stockholm University 2021 ISBN print 978-91-7911-474-9

ISBN PDF 978-91-7911-475-6

Printed in Sweden by Universitetsservice US-AB, Stockholm 2021

Abstract

Thermalization is an elusive phenomenon in quantum mechanics. Since according to the AdS/CFT correspondence, a black hole in the bulk spacetime is dual to a thermal state in the boundary CFT, thermalization in the CFT is dual to black hole formation in AdS. Thus, understanding quantum thermalization is likely a key component in understanding the information paradox—the contradiction between QFT and general rel- ativity occurring when a black hole seemingly erases the information of whatever went into creating it.

In this thesis we investigate different aspects of quantum thermal- ization in simple field theories that have conjectured holographic duals, or more specifically, free large N singlet models. Since such theories are free and hence integrable, thermalization is more subtle but nevertheless appears in different forms. We investigate the late-time quench dynamics of a version of the O(N ) vector model with probes such as the effective density matrix and the spectral density function. We refine an opera- tor thermalization hypothesis and explore its consequences in different spacetime dimensions in general large N singlet models. We also dis- cuss the prospect of using thermal mixing in the boundary theory as a diagnostic of strong gravity in the bulk.

iii

Svensk sammanfattning

Termalisering ¨ ar ett g¨ ackande fenomen inom kvantmekaniken. Eftersom AdS/CFT-korrespondensen implicerar att ett svart h˚ al i bulkrumtiden ¨ ar dualt med ett termiskt tillst˚ and i rand-CFT:n, ¨ ar termalisering i CFT:n dualt med bildandet av ett svart h˚ al i AdS. S˚ aledes ¨ ar f¨ orst˚ aelsen av kvanttermalisering sannolikt en nyckelkomponent i uppl¨ osningen av in- formationsparadoxen – mots¨ agelsen mellan QFT och allm¨ an relativitet- steori som uppst˚ ar d˚ a ett svart h˚ al till synes raderar informationen om vad som bildade det.

I denna avhandling unders¨oker vi olika aspekter av kvanttermaliser- ing i enkla f¨altteorier som har f¨ormodade holografiska dualer, n¨armare best¨amt fria stora-N -singlettmodeller. Eftersom s˚ adana teorier ¨ar fria och d¨arf¨or integrerbara ¨ar termalisering mer subtilt men uppst˚ ar ¨and˚ a i olika former. Vi unders¨oker sentida quench-dynamik i en version av O(N )-vektormodellen med prober s˚ asom den effektiva densitetsmatrisen och spektraldensitetsfunktionen. Vi f¨orfinar en operatortermaliseringsf¨or- modan och utforskar dess konsekvenser i olika rumtidsdimensioner i allm¨anna stora N -singlettmodeller. Vi diskuterar ocks˚ a m¨ojligheten att anv¨anda termisk mixning i randteorin f¨or att diagnosticera stark gravi- tation i bulken.

v

Contents

Abstract iii

Svensk sammanfattning v

Preface xi

I Emergent spacetime from simple field theories 1

1 Introduction 3

2 Quantum statistical mechanics 11

2.1 Density matrices and ensembles 11

2.2 Thermal ensembles 14

2.3 Quantum entanglement 16

2.4 Eigenstate thermalization hypothesis 21

3 Linear response theory 23

4 Quantum quenches 33

5 Holography 39

5.1 QFT/gravity duality 39

5.2 Evanescent modes 42

6 Large N singlet models 45

7 Operator thermalization hypothesis 55

8 Thermal behavior from a quench 61

vii

viii Contents

9 Operator thermalization in d > 2 67

10 Thermal mixing 75

11 Conclusion and outlook 83

A Computation of the spectral density function 87

Bibliography 91

II INCLUDED PAPERS 97

I Quenched coupling, entangled equilibria, and corre- lated composite operators: a tale of two O(N ) models 99 II Operator thermalisation in d > 2: Huygens or resur-

gence 127

List of accompanying papers

The following papers are included in this thesis, and are referred to by the Roman numbers assigned below.

I S. Banerjee, J. Engels¨ oy, J. Larana-Aragon, B. Sundborg, L. Thor- lacius, and N. Wintergerst, “Quenched coupling, entangled equilib- ria, and correlated composite operators: a tale of two O(N ) mod- els,” JHEP 08 (2019) 139, arXiv:1903.12242v2 [hep-th]

II J. Engels¨oy, J. Larana-Aragon, B. Sundborg, and N. Wintergerst,

“Operator thermalisation in d > 2: Huygens or resurgence,” JHEP 09 (2020) 103, arXiv:2007.00589v1 [hep-th]

The following paper is not included in the thesis, and is cited as an ordinary reference in the main text.

[45] J. Engels¨oy, T. G. Mertens, and H. Verlinde, “An investigation of AdS

backreaction and holography,” JHEP 07 (2016) 139, arXiv:1606.03438v3 [hep-th]

The chronological order of the papers is: I, II.

Reprints were made with permission from the publishers.

ix

Preface

This thesis is the result of some of my research carried out at the Cos- mology, Particle Astrophysics and Strings (CoPS) group at Stockholm University from September 2016 to April 2020. The thesis consists of two parts: Part I is devoted to background theory and a discussion of the results of the papers on which the thesis is based. Part II consists of the appended papers.

Contributions to the papers

Paper I I performed or contributed to all calculations involving the spectral density function as well as the basic setup with the diagonalization of the action and Bogolyubov transfor- mations. I joined and contributed in discussions about all results of the paper.

Paper II I performed or contributed to all calculations involving the response function. I joined and contributed in discussions about all results, all conceptual issues and the interpreta- tion of results.

Material from the Licentiate thesis

All chapters involving background theory and discussions related to Pa- per I are based on material from my Licentiate thesis “Quenched coupling and thermal behavior in the O(N ) vector model,” 2019 (unpublished).

Some chapters were modified so as to harmonize with the chapters per-

xi

xii Preface

taining to Paper II.

Acknowledgements

First and foremost, I would like to express my gratitude to my supervisor Bo Sundborg without the support and guidance of whom this thesis would not have been produced. Always willing to make time to discuss any issue, Bo has taught me a great deal about physics over the years and provided great support to me in my role as a junior researcher. I am also very grateful to my co-supervisor L´ arus Thorlacius who through thoughtful discussions always tend to expand my physics horizons. I want to thank my PhD student colleague, Jorge Lara˜ na Arag´ on, for joining me in this physics journey as well as on many physical journeys (including one to visit L´ arus in Iceland). We have had many fruitful discussions that have led to a deeper understanding of what we are actually up to.

My gratitude goes to Nico Wintergerst and Souvik Banerjee who both have generously, patiently, and pedagogically basically given me private lessons on topics that are fundamental in my research as well as answered any and all of my technical questions.

I also wish to extend my gratitude to David Marsh and Fawad Has- san for invaluable feedback on my thesis and to Francesco Torsello for tirelessly proof-reading it and coming up with great suggestions.

I am grateful to Francesco Torsello and Emil Blomquist for our re- warding discussions on general physics topics as well as for working out with me in the gym. Thanks also to my former office neighbors An- ders Lundkvist and Mikica Kocic for helping me on several occasions with their insight when I have been stuck in some calculation or just on general conceptual issues.

Julius Engels¨oy

Stockholm, March 2021

Part I

Emergent spacetime from simple field theories

1

Chapter 1

Introduction

The past couple of centuries have seen a vast development in the field of physics. Thermodynamics and statistical mechanics were developed during the 1800’s by the likes of Boltzmann, Lord Kelvin, Gibbs, and Maxwell. The conundrum of the diverging black body spectrum encoun- tered in these fields led Planck to develop the notion of quanta in 1900 whereby quantum mechanics was born (see [1]). Between the publica- tion of his discovery of special relativity in 1905 (see [2]), and the year 1915, Albert Einstein developed his theory of general relativity (GR) describing gravity as a curving and rippling of spacetime (see e.g., [3]).

In the year 1915 Hilbert formulated GR as an action later known as the Einstein–Hilbert action [4].

Shortly after the publication of Einstein’s first paper on GR, the astronomer Karl Schwarzschild found an analytical solution to the Ein- stein field equations describing the effect on spacetime of a spherically symmetric, static, massive body surrounded by vacuum (see [5]). Later, it was understood that for a sufficiently dense body a sort of (coordi- nate) singularity would form on what would be named the event horizon.

Schwarzschild had found the metric of a black hole, an object so massive that not even light can escape its gravitational pull.

However, decades later, around the year 1970 it was argued by Beken- stein and showed by Hawking that black holes have entropy [6, 7]

S

= A

4 , (1.1)

3

4 1. Introduction

where A is the horizon area and all quantities here and throughout are measured in Planck units. This relation—that the amount of information needed to describe certain systems is proportional to their area and not their volume—later gave rise to what is known as the holographic principle.

Furthermore, Hawking showed that black holes in fact do emit ra- diation. This so-called Hawking radiation turns out to have a thermal

spectrum with the temperature [7]

T

= 1

8πM , (1.2)

where M is the mass of the (Schwarzschild) black hole. These discoveries gave rise to the concept of black hole thermodynamics.

The fact that black holes radiate means that they eventually evap- orate. Shortly before the inception of black hole thermodynamics it was shown that the only distinguishing external features of a black hole are its mass, angular momentum, and electric charge (see e.g., [8])—something later named the “no-hair theorem.” The implication of this is that any other information thrown into a black hole gets lost as the black hole evaporates into “informationless,” thermal radiation. This violates the postulate of unitarity in quantum mechanics and became known as the information paradox—an incompatibility between GR and QFT.

In the 1960’s and early 1970’s the Standard Model (SM) of parti- cle physics describing the quantum origin of electromagnetism and the strong and weak nuclear forces was completed. The model is based on quantum field theory (QFT) which in turn was developed from quan- tum mechanics in the early half of the twentieth century. In the late 1940’s it was realized that the experimentally measurable parameters in QFT are the energy-dependent “renormalized” versions of the “bare”

ones showing up in the action of the theory. Not all classical theories are renormalizable in this sense after the usual prescription of second quan- tization and annoyingly, it turns out that GR is not renormalizable—the best one can do is to formulate a low-energy effective quantum field the-

1Throughout this thesis the word “thermal” refers to whether the state of a system can be described by a thermodynamic equilibrium ensemble of states such as the canonical ensemble. Quantum mechanically this means a mixed state. These concepts will be expanded upon in Chapter 2.

1. Introduction 5

ory that breaks down at higher energies. The running of the renormalized couplings in QFT as a function of energy was phrased in the language of renormalization group (RG) flow by Kenneth Wilson around the year 1970 (see [9]).

A possible resolution to the conflict between QFT and GR was dis- covered as a mere side-effect of phenomenological studies of hadron colli- sions in the 1960’s. A model was proposed which described intermediate states as quantized oscillations of a string. However, the spectrum invari- ably seemed to include a massless spin-2 particle. In the years following its discovery, this so-called string theory was reinterpreted as a candidate theory of quantum gravity where the spin-2 particle is interpreted as the quantum of the gravitational field—the graviton.

Since the development of string theory, a few consistency checks have been made comparing its predictions to those of more accepted theories such as GR and black hole thermodynamics. Most notably, string theory leads to (the slightly modified) Einstein field equations. Furthermore, it was shown by Strominger and Vafa that the formula (1.1) can be reproduced by microstate counting in string theory for special kinds of black holes [10].

Since its inception, string theory has needed to deal with some ad- versities such as it only being physically consistent in a particular (high) number of spacetime dimensions. This has the consequence that in order to obtain from string theory a theory that describes our Universe one needs to compactify all but four spacetime dimensions. The problem is that there are virtually endless ways to do this, and there is no real clue as to which way leads to a correct description of our Universe.

These problems and more have at times negatively influenced the

field of string theory. However, in 1995 Polchinski [11] published his

theory on so-called D-branes—dynamical membranes on which the end-

points of strings are attached—which sparked a surge in the field. Two

years later, in 1997, Juan Maldacena [12] published a seminal paper

partly based on Polchinski’s findings that conjectured a duality between

a type of string theory in Anti-De Sitter space (AdS)—a specific solution

to the Einstein field equations with a negative cosmological constant re-

sulting in a timelike boundary—and a type of QFT in a non-gravitational

spacetime with so-called conformal symmetry (CFT). What is fascinat-

6 1. Introduction

ing with this duality is that the two dual spacetimes—AdS and the non-gravitational spacetime containing the CFT—differ in dimension- ality; the CFT can be viewed as living on the boundary of the AdS spacetime and hence this duality is seen as a manifestation of the holo- graphic principle. The term “AdS/CFT correspondence” is sometimes used interchangeably with “holography.”

There are differing views as to how powerful holography is. Some proponents argue that there is in fact a full gravity/QFT duality in which any gravitational system can be described by a dual QFT in one less dimension. The most conservative definition of the word only per- tains to the particular type of top–down AdS/CFT correspondence that Maldacena conjectured.

Holography allows understanding one side of the duality by studying the other. Hence, in order to understand quantum gravity one can study the dual QFT using normal field theory methods and then use a “dic- tionary” to translate the results over to the gravity side. The existing dictionary developed in the years following Maldacena’s paper relates fields and partition functions on the two sides of the AdS/CFT duality (see e.g., [13, 14]). Hence, correlators on the gravity side can be related to correlators on the CFT side.

Since the tension between GR and QFT is very much pronounced in the context of black holes they provide an interesting subject to be studied in holography. According to the dictionary, a black hole in an asymptotically AdS spacetime is dual to a thermal state in the CFT.

Thus, thermalization at high energies in the CFT should be dual to black hole formation.

As will be expanded upon in Chapter 5, the types of CFTs that are dual to semiclassical Einstein gravity are strongly coupled and thus not susceptible to perturbation theory, which makes them difficult to study.

Hence, one approach to feasibly study emergent spacetime

would be to identify simpler theories that nevertheless share some key features with these strongly coupled CFTs and use them as toy models that might teach us something about universal properties of quantum gravity and black holes. This approach is the underlying principle of this thesis and

2Emergent spacetime in this context means a gravitational theory that “emerges”

from an underlying field theory without gravity.

1. Introduction 7

the appended papers.

There are several different aspects of simple field theories that could be studied in principle. In this thesis, we will focus on the ones studied in the attached papers and provide the relevant background in the coming chapters. One might crudely divide the papers into the two categories of “state thermalization” (Paper I) and “operator thermalization” (Pa- per II). They represent two different perspectives on thermalization in the context of simple field theories and we will here introduce the basic concepts underlying them.

Since a black hole in principle could be formed as a pure state, the evolution from a pure state to (what looks like) a thermal state is of great interest. One way of dealing with the problem is to argue that even pure states look thermal. This line of reasoning was codified in the so-called eigenstate thermalization hypothesis (ETH) first coined by Srednicki in 1994 (see [15]). ETH roughly claims that for non-integrable

systems, observable expectation values taken in energy eigenstates agree with thermal expectation values, i.e., the state “looks” thermal in a sense even though it is pure.

However, even in free (integrable) theories it is possible to study pure states that look thermal and if they have a dual gravitational interpre- tation one might even be able to obtain new insight into the information paradox. In order to do that we need to be able to prepare a state that looks thermal. A convenient way to dynamically prepare such a candi- date state is to impose a so-called quantum quench on a system of choice.

A quench is when a state is prepared with one Hamiltonian and then at later times time-evolved with another Hamiltonian. The transition from the first Hamiltonian to the second can be instantaneous or continuous.

A well-known pure state that under certain conditions looks thermal is the so-called thermofield double (TFD) state,

|T F Di = 1

√ Z X

n

e

|ni

|ni

, (1.3)

3An integrable (quantum) system has an infinite number of mutually commuting conserved charges and is thus highly constrained.

8 1. Introduction

where Z is the partition function,

Z = X

n

e

, (1.4)

β is the inverse temperature 1/T , |ni is an energy eigenstate, E

is its associated energy level, and the subscripts 1 and 2 refer to what is spe- cial in this construction, namely that we have two identical (uncoupled) copies of the same theory making the Hilbert space H = H

⊗ H

. We will show in Section 2.1 that an expectation value in this state of an operator acting on one of the theories is equal to a thermal expectation value. Hence, we have “purified” the thermal state by creating a highly entangled, pure state.

Interestingly, the TFD in certain CFTs has been conjectured to be dual to an eternal (two-sided) black hole [16] and this proposal and its generalizations have been given the name “ER = EPR” by Malda- cena and Susskind [17]. ER stands for Einstein–Rosen bridge, meaning a two-sided, eternal black hole also known as a wormhole. EPR in turn, stands for Einstein–Podolsky–Rosen entanglement also known as quan- tum entanglement. The expression “ER = EPR” seeks to emphasize that entanglement is crucial in black hole physics.

Some of the considerations discussed throughout this chapter lead us to consider a special type of quench in our quest to find how close we can get to a real thermal state, namely a mixing quench whereby two initially coupled theories are suddenly decoupled. This is the subject of Paper I. Starting in the ground state of the coupled theory, this will create a highly entangled state in the post-quench theory. We can then use different probes to investigate the extent to which this state looks thermal and how its behavior deviates from conventional thermality.

The type of theory we consider is the so-called free (massive) O(N )

vector model which we will describe in Chapter 6. The reason for this

choice of theory is twofold. First, it is a particularly simple theory which

allows us to do explicit calculations and obtain closed-form results. Sec-

ond, it (or rather its massless relative) does have a proposed holographic

dual, namely the so-called Vasliev higher spin theory which, even though

it is not semiclassical Einstein gravity, might still provide some insight

into the dual gravitational interpretation of such a quench.

1. Introduction 9

More explicitly, the action of the free O(N ) vector model is

S = 1 2

Z

d

x ∂

ϕ

∂

ϕ

, (1.5)

where d is the spacetime dimension, {ϕ

} are scalar fields, and the su- perscript a is the O(N ) vector index. Here and throughout the thesis, repeated indices imply summation unless stated otherwise. As is im- plied by the sign of the above expression, the metric signature here and throughout the thesis is (+ −− . . .) unless otherwise is stated. Our aim is now to couple this theory to an identical one with the simplest coupling imaginable, namely a bilinear coupling. However, doing so would lead to a tachyonic instability. To regulate this we introduce masses which are sufficiently big compared to the coupling.

The full action we will consider is hence S = 1

2 Z

d

x



∂

ϕ

∂

ϕ

+ ∂

ϕ

∂

ϕ

− m

ϕ

ϕ

− m

ϕ

ϕ

− 2h

(t)ϕ

ϕ



, (1.6)

where the subscripts L and R denote the different copies of the theory and h(t) is the time-dependent coupling that we will turn off at some point.

A different perspective on thermalization in simple field theories is given by studying which operators allow a system to quickly relax back to equilibrium. Fundamental fields in free theories perturb a system in a way that does not let it relax back to thermal equilibrium in a short amount of time but it turns out that composite operators of the type that would be holographically dual to particle fields in the bulk theory do.

In Paper II we study this phenomenon and refine a conjecture proposed in [18] that gives a condition for which operators thermalize—the oper- ator thermalization hypothesis (OTH)—as well as investigate operator thermalization in different spacetime dimensions.

We study OTH in the context of so-called large N singlet models,

a class of theories to which the O(N ) vector model mentioned above

belongs. They provide a fertile testing ground since—even though they

are simple theories—they have conjectured holographic duals and con-

10 1. Introduction

tain rich physics including a phase transition indicating the existence of black hole-like states at high temperatures. These properties are de- scribed in Chapter 6.

The rest of the thesis is devoted to giving an account of the back- ground theory needed in order to discuss the appended papers. In Chap- ter 2 we go over the fundamentals of quantum statistical mechanics in- cluding density matrices and entanglement. In Chapter 3 we derive re- sults from linear response theory that are important for discussing the results of Paper I. In Chapter 4 we describe the very basics of quantum quenches. In Chapter 5 we expand on the main ideas behind holography and describe the origin of so-called evanescent modes which will play a role in the holographic interpretation of the quench we are considering.

In Chapter 6 we go over the basics of the class of simple field theories

considered in this thesis including the O(N ) vector model. In Chapter 7

we describe the operator thermalization hypothesis (OTH). In Chapter 8

we discuss the results of our investigation of the quench described above

as presented in Paper I. In Chapter 9 we discuss our modification of OTH

and our investigation of operator thermalization in different spacetime

dimensions as presented in Paper II. In Chapter 10 we outline some work

in preparation related to thermal mixing. Lastly, in Chapter 11 we sum-

marize the work presented in thesis and provide an outlook for further

research.

Chapter 2

Quantum statistical mechanics

The classical microscopic description of thermodynamics relies on statis- tical uncertainty. However, for microscopic or inherently quantum sys- tems a correct description requires quantum mechanics as a fundamen- tal ingredient. Quantum mechanics introduces a new kind of uncertainty that is not the usual classical ignorance of something we could know hypothetically. In order to deal with this inherent quantum uncertainty one needs to express statistical mechanics in a quantum formalism, the basics of which we will review below.

2.1 Density matrices and ensembles

A quantum expectation value of the quantum operator A is given by hψ| A |ψi where |ψi is the state. If we do not know the state of the system under consideration or if we are considering a statistical ensemble of states the expectation value is given by

hAi = X

i

p

hψ

| A |ψ

i , (2.1)

where p

is the probability of the system being in the (normalized but otherwise arbitrary) state |ψ

i. In general, such an expectation value can

11

12 2. Quantum statistical mechanics

be computed using a so-called density matrix, ρ = X

i

p

|ψ

i hψ

| , (2.2)

by taking the trace defined as

hAi = Tr(ρA) ≡ X

n

hn| ρA |ni , (2.3)

where the states {|ni} form an arbitrary, complete, orthonormal set of states. It shares all the relevant properties with the perhaps more familiar matrix trace such as cyclicity and basis-independence as can easily be checked. We note that because of these properties and the unitarity of time-evolution operators it is immaterial which quantum picture we choose to work in.

If the density matrix describes a pure state, say, |ψi, it follows that we can write

ρ = |ψi hψ| , (2.4)

whereby we see that for a pure state, there is only one p

= 1 as defined in (2.1). A state that is not pure is called “mixed” and is described by an ensemble of states characterized by the probablilities {p

}.

A general density matrix ρ has three basic properties [19], namely 1. ρ is Hermitian, i.e., ρ = ρ

,

2. ρ has unit trace, i.e., Tr ρ = 1, and

3. ρ is positive semidefinite, i.e., hφ| ρ |φi ≥ 0 for any vector |φi ∈ H.

All of the above properties follow—in one way or another—from the fact that the {p

} form a probability distribution.

The density matrix has another property which is important in the study of thermality, namely

Tr ρ

= 1 ⇐⇒ ρ describes a pure state . (2.5)

2. Quantum statistical mechanics 13

The reason is that, in general,

Tr ρ

= X

n

hn| X

i

p

|ψ

i hψ

|

! 

 X

j

p

|ψ

i hψ

|



 |ni

= X

i,j

p

p

hψ

| X

n

|ni hn|

!

|ψ

i hψ

|ψ

i

= X

i,j

p

p

| hψ

|ψ

i |

≤ X

i,j

p

p

hψ

|ψ

i hψ

|ψ

i

= X

i

p

! 

 X

j

p



 = 1 , (2.6)

where we used the Cauchy–Schwarz inequality. Equality is achieved if and only if |ψ

i = |ψ

i for every i, j. Since by construction, |ψ

i 6= |ψ

i for all i 6= j, equality is attained only if the density matrix describes a single, pure state.

Since ρ is Hermitian its eigenvectors {|φ

i} are orthogonal and its eigenvalues {P

} are real. Writing ρ in the spectral representation

ρ = X

i

P

|φ

i hφ

| , (2.7)

and imposing the properties 2 and 3 shows that the {P

} form a proba- bility distribution, just like the original {p

}. For a pure state |ψi we triv- ially have only one eigenvector |φi = |ψi of ρ and only one P

= p

= 1.

In the Schr¨odinger picture, states evolve as

|ψ(t)i = U(t) |ψ(0)i , (2.8)

where

U (t) ≡ T

 exp



−i Z

0

dt

H(t

)



, (2.9)

and where T { · } stands for time-ordering.

Hence the density matrix evolves as

ρ(t) = U (t)ρ

U

(t) , (2.10)

where ρ

describes states at t = 0. Starting from the Schr¨odinger equa-

14 2. Quantum statistical mechanics

tion it can be shown that this time-evolution is also described by dρ(t)

dt = −i[H, ρ] . (2.11)

Thus, if the density matrix is only a function of H such as (2.17) it is constant over time in the Schr¨ odinger picture. In the Heisenberg picture, the density matrix does not evolve in time but the expectation value of an operator A is of course independent of the picture since

hA(t)i = Tr(ρ(t)A) = Tr U(t)ρ

U

(t)A

= Tr ρ

U

(t)AU (t)

= Tr(ρ

A(t)) , (2.12) where A(t) is now in the Heisenberg picture.

The von Neumann entropy S of a quantum system is defined as

S = − Tr(ρ log ρ) , (2.13)

and can in general be expressed as S = − X

i

P

log P

, (2.14)

where {P

} are the eigenvalues of ρ and hence the same as in (2.7). Recall that for a pure state we have only one P

= 1 and hence

S = 0 ⇐⇒ ρ describes a pure state . (2.15) Furthermore, since 0 ≤ P

≤ 1 we have S ≥ 0.

2.2 Thermal ensembles

In order to assess thermality we need to be able to make measurements on a system and compare the results with what we would expect from a thermal system. In practice, one way of doing this is to compute an observable expectation value in the particular state the system is in and compare the result to an expectation value taken in a thermal ensemble.

There are several ways to obtain a “thermal-looking” system in this

2. Quantum statistical mechanics 15

sense—two of which we will discuss in the subsequent two sections. In the current section, we introduce two different thermal ensembles required for discussions in later chapters, and their mutual relation.

Arguably the most fundamental thermal ensemble is the so-called canonical ensemble, which describes the state of thermal equilibrium at constant temperature. An expectation value in the canonical ensemble can be written

hAi

= 1 Z

X

n

e

hn| A |ni , (2.16)

where {|ni} are the energy eigenstates associated with the energy eigen- values {E

} and β = 1/T is the inverse temperature. We can compare with the general expression (2.1) to identify

ρ

= 1 Z

X

n

e

|ni hn| = 1 Z

X

n

e

|ni hn|

= e

Z

X

n

|ni hn| = e

Z , (2.17)

where Z is the canonical ensemble partition function given by the sum of so called Boltzmann factors,

Z = X

n

e

= X

n

e

hn|ni

= X

n

hn| e

|ni = X

n

hn| e

|ni = Tr e

. (2.18)

Thus, (2.16) can be expressed as

hAi

= Tr e

A

Tr e

. (2.19)

Another ensemble that is frequently used in equilibrium statistical mechanics is the microcanonical ensemble, which describes a system con- tained in a small energy range around some fixed energy E

. Expectation values in this ensemble are given by

hAi

≡ 1 N

X

En∈[Ei,Ei+∆E]

hE

| A |E

i , (2.20)

16 2. Quantum statistical mechanics

where ∆E is the thickness of the energy range, which is arbitrary but usually taken to be small. Since finite systems have a discrete energy spectrum, the sum is over a finite number of states N = N (E

) which is just the number of energy eigenstates corresponding to energy eigenval- ues in the range [E

, E

+ ∆E]. Again comparing to (2.1) yields

ρ

= 1 N

X

En∈[Ei,Ei+∆E]

|E

i hE

| . (2.21)

One of the fundamental implications of statistical mechanics is that for most systems, expectation values taken in the canonical ensemble are equivalent to those taken in the microcanonical ensemble. The reason for this is that in the thermodynamic limit, the canonical ensemble average can be approximated as

hAi

= 1 Z

X

n

N (n∆E)e

hAi

. (2.22)

For most systems, the number of states N (E) grows as a very high power of the energy [20] and thus conspires with the Boltzmann factors e

to give a sharply peaked distribution that is centered around E = E

given that we identify

β = ∂

∂E log N (E)    

i

. (2.23)

Thus, the term hAi

is picked out from the sum in (2.22). Saying that a state or system is “thermal” implies that expectation values of observables agree with expectation values taken in the canonical or mi- crocanonical ensembles. The process (in time) of approaching such an agreement is called thermalization.

2.3 Quantum entanglement

The above formalism of density matrices and entropy is very useful in

the study of quantum entanglement. As mentioned in Chapter 1, entan-

glement is related to thermodynamics in the context of black holes and

purification of thermal states. To begin the discussion of entanglement

2. Quantum statistical mechanics 17

let us consider a product Hilbert space

H = H

⊗ H

. (2.24)

A system with the above binary Hilbert space decomposition is called a bipartite system. A general pure state in this Hilbert space can be expressed as

|ψi = X

i,j

c

|φ

i

|φ

i

, (2.25)

where c

are complex, normalized phase factors, {|φ

i

} form an or- thonormal basis in H

, {|φ

i

} form an orthonormal basis in H

, and we implicitly assume a tensor product ( ⊗) in between the kets. Now, if there exist sets of numbers {c

} and {c

} such that

c

= c

c

, for all i, j , (2.26) then we can write

X

i,j

c

|φ

i

|φ

i

= X

i

c

|φ

i

! X

i

c

|φ

i

!

, (2.27)

and we say that the (pure) state is “separable” or a “product state.” By contrast, a (pure) state is said to be “entangled” if there do not exist numbers that fulfill (2.26).

Now, in order to compute an expectation value of an operator A that only acts on the subspace H

we follow the usual procedure defined in (2.3) to obtain,

hAi = Tr(ρA) = X

i,j

hφ

|

A B

hφ

| ρA |φ

i

|φ

i

= X

i

hφ

|

A



 X

j

hφ

|

B

ρ |φ

i



 A |φ

i

≡ X

i

hφ

|

A

ρ

A |φ

i

≡ Tr

(ρ

A) , (2.28)

18 2. Quantum statistical mechanics

where we have defined the reduced density matrix ρ

≡ Tr

ρ ≡ X

j

hφ

|

B

ρ |φ

i

, (2.29)

which has all the properties of a density matrix mentioned above, and the partial trace Tr

( · ) with respect to H

. As we can see, all the relevant information about the state from the perspective of an observer with access only to H

is encoded in ρ

. Only tracing with respect to H

is called “tracing out B” or taking the partial trace.

Now, a general density matrix ρ describing a pure state in the full Hilbert space can be expressed, using (2.4) and (2.25), as

ρ = X

i,j,k,l

c

(c

)

|φ

i

|φ

i

hφ

|

hφ

| . (2.30)

Taking the partial trace with respect to H

yields ρ

= Tr

ρ

= X

n

hφ

|

B



 X

i,j,k,l

c

(c

)

|φ

i

|φ

i

hφ

|

hφ

|



 |φ

i

= X

n,i,k

c

(c

)

|φ

i

hφ

| , (2.31)

and hence,

ρ

= X

n,i,k

c

(c

)

|φ

i

hφ

|

× X

a,c,d

c

(c

)

|φ

i

hφ

|

= X

n,a,d, i,k

c

(c

)

c

(c

)

|φ

i

hφ

| . (2.32)

2. Quantum statistical mechanics 19

Taking the trace yields

Tr

ρ

= X

l

hφ

|

A



 

 X

n,a,d, i,k

c

(c

)

c

(c

)

|φ

i

hφ

|



 

 |φ

i

= X

n,a,l,k

c

(c

)

c

(c

)

≤ X

n,a,l,k

c

(c

)

c

(c

)

= 1 , (2.33)

where we used

X

i,j

c

(c

)

= 1 , (2.34)

due to the normalization of (2.25), and where equality is attained if and only if the rows labeled by i of the matrix c

(and columns labeled by j) are proportional to one another. This in turn can only happen if and only if the matrix has rank one which is both a necessary and sufficient condition for the matrix being the outer product of two vectors, say, c

and c

, and thus the condition (2.26) is fulfilled. Hence, we conclude

Tr

ρ

= 1 ⇐⇒ ρ describes a separable state , (2.35) or in other words, only a ρ describing an entangled state will produce a reduced density matrix describing a mixed state. This means that ρ

contains information about the degree of entanglement between H

and H

. A measure of entanglement is the so-called entanglement entropy which is just the von Neumann entropy of ρ

,

S

= − Tr

ρ

log ρ

. (2.36) Using (2.15) one can show that

S

> 0 ⇐⇒ ρ describes an entangled state , (2.37)

As mentioned in Chapter 1, a very special, entangled, pure state is

20 2. Quantum statistical mechanics

the TFD state

|T F Di = 1

√ Z X

n

e

|ni

|ni

, (2.38)

where the states {|ni

} are energy eigenstates in the subspace H

. The associated density matrix is

ρ

= |T F Di hT F D| = 1 Z

X

n,m

e

|ni

|ni

hm|

hm| , (2.39) and we find that an expectation value of an operator acting only on the subspace H

in this state is

Tr(ρ

A) = hT F D| A |T F Di

= 1 Z

X

n,m

e

hm|ni

hm| A |ni

= 1 Z

X

n,m

e

δ

hm| A |ni

= 1 Z

X

n

e

hn| A |ni

, (2.40)

which is equal to the canonical ensemble expectation value (2.16). The reduced density matrix ρ

is hence given by

ρ

= 1 Z

X

n

e

|ni

hn| , (2.41)

which can also be seen from ρ

= Tr

(ρ

) = X

k

2

hk|T F Di hT F D|ki

= 1 Z

X

n,m,k

e

|ni

hk|ni

hm|

hm|ki

= 1 Z

X

n,m,k

e

|ni

δ

hm| δ

= 1 Z

X

k

e

|ki

hk| . (2.42)

2. Quantum statistical mechanics 21

Thus, to an observer with access only to H

, the pure state |T F Di will

“look” thermal. The construction of |T F Di is called “purifying” the canonical ensemble. This can in fact be done for any mixed state by introducing an auxiliary Hilbert space and creating an entangled state whose reduced density matrix will correspond to the original mixed state [19].

We thus see how entanglement can make a pure state look thermal to an observer with access to only part of the full Hilbert space. Al- though the above review pertains to quantum mechanics and not QFT, it will be sufficient in order to discuss the results of Paper I since we are considering a free field theory where modes experience an approximate thermalization separately as will be discussed in Chapter 4.

2.4 Eigenstate thermalization hypothesis

For completeness we will also briefly discuss the eigenstate thermaliza- tion hypothesis (ETH) and how its underpinning ideas are related to the approach to thermality taken in Paper I. Although Srednicki [15]

was first to coin the term “eigenstate thermalization,” he did it in the context of discussing the behavior of energy eigenfunctions of certain quantum systems. For the purposes of this brief introduction to ETH however, the language of Deutsch [21] is more illuminating and therefore we will follow the review [22]. The statement of the hypothesis is that for a large class of operators O in non-integrable systems, the expectation values taken in an energy eigenstate E

take the form [22]

hE

| O |E

i = hOi

+ ∆

, (2.43) where h · i

is the microcanonical ensemble average at energy E

defined by (2.20). ∆

has zero mean averaged over i and a variance of order hO

i

×e

where S(E

) is the microcanononical entropy log N at energy E

.

The physical meaning of the hypothesis is that pure states of a given

energy in isolated quantum systems with many degrees of freedom will

appear thermal in the sense that observable expectation values are ex-

tremely close to the microcanonical ensemble average. The motivation

22 2. Quantum statistical mechanics

for this hypothesis can be illustrated by considering an integrable sys- tem (which will not thermalize) with Hamiltonian H

and perturbing it with the small random matrix Hamiltonian H

. It turns out that the energy eigenstates {|E

i} of the total Hamiltonian H

= H

+ H

are a superposition of a large number of energy eigenstates {|E

i

} of the integrable H

sharply peaked around the energy E

[22], i.e.,

|E

i = X

j

c

|E

i

, (2.44)

where the {c

} are stochastic and sharply peaked at i = j. Using the statistics of the {c

} one can show that [22]

hE

| O |E

i = X

k

hc

i

hE

| O |E

i

, (2.45)

where h · i

is the averaging over the random matrices H

. Since the {c

} are sharply peaked at i ≈ j, (2.45) gives the microcanonical ensemble.

We thus see that an externally imposed disorder in an originally inte- grable system can give rise to thermal behavior even for pure states. A similar superposition of many energy eigenstates can be achieved by a quench that takes an integrable system into another integrable system.

In Paper I we employ this latter approach and use different probes to

elicit how thermal the state can appear.

Chapter 3

Linear response theory

An important diagnostic tool for any system is the so-called response function and its relative, the spectral density function. The latter turns out to play an important role in the connection between field theory and gravitational geometry as will be discussed in Section 5.2. In this chapter we will develop the tools required in order to discuss the results of the appended papers. We choose to derive the needed results in quantum mechanics since it turns out that they can be directly extended to a QFT setting. Hence, one might think of arguments (of operators) otherwise containing the spatial position vector x to be suppressed in the following discussion.

Now, consider any theory to whose original Hamiltonian H we add a perturbation

H

(t) = φ

(t) O

, (3.1) where {φ

(t) } are called (time-dependent) sources of the operators {O

} and where summation of repeated indices is implicitly understood here and throughout this thesis. The question then arises, “What is the effect on the expectation value of an operator O

by this addition?” For suffi- ciently small perturbations we expect the shift of the expectation value to be linear in the source φ(t) implying that we can write

hO(t)i|

− hO(t)i|

≡ δhO

(t) i = Z

−∞

dt

χ

(t, t

)φ

(t

) , (3.2)

23

24 3. Linear response theory

where we have implicitly defined the response function χ

(t, t

). Since the LHS clearly is state-dependent we conclude that χ

is also state- dependent. By the linearity of quantum mechanics we further conclude that this state can be either pure or mixed. The form of equation (3.2) makes us identify χ

(t, t

) as a Green’s function. We will see below how in fact, in QFT, it can be identified with the usual retarded Green’s function. We also note that since we are dealing with expectation values, the quantum picture we choose to work in is immaterial.

Assuming that our original system is invariant under time-translations we can write

χ

(t, t

) = χ

(t − t

) . (3.3) This assumption also makes it convenient to work in frequency space.

Fourier transforming (3.2) now yields

δ hO

(ω) i = Z

dt e

δ hO

(t) i = Z

dt e

Z

dt

χ

(t − t

)φ

(t

)

= Z

dt e

Z

dt

χ

(t − t

)e

e

φ

(t

)

= Z

dt

e

φ

(t

) Z

dt e

χ

(t − t

)

= Z

dt

e

φ

(t

) Z

du e

χ

(u)

= χ

(ω)φ

(ω) , (3.4)

where it is understood that even though the functions retain their orig- inal name, the argument distinguishes the original function from its Fourier transform. We note that although (3.2) is non-local in the sense that it involves an integral over all time, the relation is completely local in frequency space. Technically, this is just a consequence of the convo- lution theorem and physically it just means that a harmonic source will generate a response of the same frequency.

We will be interested in the case of a source with one kind of operator

and its effect on the expectation value of the same operator and hence we

will drop the labels i, j in the following discussion. We first note that with

the Hamiltonian and O Hermitian, φ(t) must be real and, furthermore,

3. Linear response theory 25

χ(t) must be real. Using conventional notation, we can define

χ(ω) ≡ χ

(ω) + iχ

(ω) , (3.5) where χ

(ω) = Re χ(ω) and χ

(ω) = Im χ(ω). Since the primes are not to be confused with derivatives the convention is somewhat unfortunate.

Now, χ

(ω) can also be written

χ

(ω) = − i

2 (χ(ω) − χ

(ω))

= − i 2

Z

dt e

χ(t) − e

χ(t)

= − i 2

Z

dt e

(χ(t) − χ(−t)) , (3.6)

and we see how the presence of an imaginary part of the response func- tion is associated with breaking of time-reversal symmetry. Time-reversal symmetry breaking is in turn associated with dissipation and hence, χ

(ω) is called the dissipative or absorptive part of the response func- tion. We also note that χ

(ω) is an odd function of ω and it turns out that χ

(ω) is an even function.

Causality implies that χ should vanish for t

> t in (3.2) since a quantity at time t should not depend on future events. This means that χ(t) only has support for t ≥ 0. When t < 0 we can close the contour of the Fourier representation of χ(t),

χ(t) = Z

−∞

dω

2π e

χ(ω) , (3.7)

in the upper half plane. Since the result should vanish we conclude, due to the residue theorem, that χ(ω) does not have poles in the upper half plane.

Now, let us derive the so-called Kubo formula. The derivation is the most straightforward in the interaction picture where states evolve with the time evolution operator

U (t, t

) ≡ T

 exp



−i Z

t0

dt

H

(t

)



, (3.8)