Continue matrix product toestanden voor Gaussische velden

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Continuous matrix product states for

Gaussian fields

Bastiaan Aelbrecht

Faculty of Sciences

Department of Physics and Astronomy

Academic year: 2019-2020

Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Physics and Astronomy

Promotor: Prof. dr. Jutho Haegeman

Advisor: Quinten Mortier

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D A N K W O O R D

Altijd leuk zo’n dankwoord voor een thesis, zeker als je jezelf voorneemt dat je zeker niemand wilt vergeten, en dan plots beseft hoeveel mensen er een rol hebben gespeeld in het (succesvol) volbrengen van mijn nakende status van fysicus. De lezer is dus gewaarschuwd voor de compleet onverantwoorde langdradigheid aan bedankingen als gevolg van dat voornemen. Ik zou het kunnen houden bij de mensen die me dit jaar tot aan die eindstreep hebben geloodst, maar echt eerlijk is dat niet. Laten we beginnen bij vader en moeder Van Velthoven.

Zonder twijfel staan zij bovenaan. Zonder hen zou ik dit nooit aan het schrijven geweest zijn. Ik heb altijd kunnen doen wat ik wil, en ze hebben me daar ook altijd in gesteund. Dankjewel mama voor je nooit aflatend enthousiasme en vertrouwen in mij. Dankjewel papa om me dagelijks te verlichten met je vaste verzameling van 15 daddy jokes en er al die jaren voor mij geweest te zijn. Ik moet toegeven, die laatste weken voor de thesisdeadline moeten voor hen -hoe zal ik het zeggen - een hel geweest zijn. Iets met een andere planeet... Gelukkig hielden onze dagelijkse familiewandelingen me op tijd en stond terug met mijn beentjes op de aarde. Ook mijn liefste broer en zussen ben ik heel dankbaar voor de immer geanimeerde sfeer thuis. Het is ook niet meer dan gepast dat de ongelooflijke kookkunsten van Anke hier een speciaal plaatsje verdienen.

Toegegeven, een thesis in combinatie met een pandemie is nog zo slecht niet. Zeker als je dan een eigen plekje hebt thuis en helemaal in de watten wordt gelegd. Als ik het over watten heb, dan mag ik zeker ook Joos niet vergeten. Mijn ’tweede verblijf’ aan de kust was werkelijk een droompaleis. Met iedere dag heerlijk eten, kilo’s koffie op voorraad, nog meer daddy jokes, en bovendien asperges op maandag.

Echter, wie hier eigenlijk hoort te staan als we het over watten hebben, is mijn schat van een meter. Dit zou de examenperiode geweest zijn dat we iedere avond samen naar de Koningin Elizabethwedstrijd piano kijken, zoals we dat in mijn eerste jaar deden. In een ’normaal’ academiejaar was ik meer in Bellem dan ik thuis was. Dat was toch wel het tweede verblijf van alle tweede verblijven. Ik heb me daar altijd thuis gevoeld. Zeker als het ambiance was op zondag. Al mijn neefjes, nichtjes, nonkels en tantes (die ik ondertussen al maanden niet meer heb gezien) bij elkaar: geroep door elkaar aan tafel, veel te veel frietjes met kip in venkel en de beroemde ’poivronsaus’, zoals dat hoort bij de Millets.1

Ik mis dat. En wat mis ik mijn teerbeminde metekindje Aria, waarmee ik nog héél veel tijd in te halen heb deze zomer.2 Meter en opa Bonheiden en oom Jelle verdienen hier absoluut ook een plaatsje. Keer op keer drukten zij me op de borst hoe belangrijk het is om hard te werken. Meter en opa trots maken is altijd één van de grootste drijfveren geweest om door te blijven gaan. Tenslotte wil ik de familiebladzijde absoluut afronden met Tante Mieke, en de beste manier om dat te doen, is om een berichtje van haar enkele dagen voor de thesisdeadline te citeren:

VOORTDOEN EN GOED ETEN EN DRINKEN EN RUSTEN!

1 Shoutout naar nonkel Koenraad en onkie Stijn die dit weliswaar een fascinerend schouwpel vinden, maar toch opteren voor de barbecue.

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Die familiebladzijde mag gerust doorlopen, want de vriendengroep die ik in al die jaren rondom mij heb verzameld, voelt echt wel als een familie. Ik weet niet goed waar te beginnen, of wat de volgorde zou moeten zijn. Misschien is chronologisch geen slecht begin. In dat geval moet ik absoluut beginnen bij Brent, Joren, Jonas en Brian. Vier mannen waarvan ik weet dat ik ´altijd op hen kan rekenen, ongeacht hoelang we elkaar niet gehoord hebben. Dat zijn de besten. Vriendschappen die de tijd doorstaan. Merci, maatjes.

Heel veel van die vriendschappen zijn ontstaan op Home Astrid, die prachtige home aan de Sterre. Zonder twijfel zal ik de geweldige dynamiek daar het meest missen volgend jaar. De koffiepauzes met mijn lieve bovenbuurvrouw Elien, het hyperkinetisch dagelijks geklop aan mijn deur van spring in’t veld Hannelore, de muzkale sessies met Yelke, de geestverruimende avonden bij Bard Wogaerts, de memorabele fuiven, de solarium gang... Het is echt een geschenk om al je vrienden op kousenafstand van je te hebben.

Twee mede-Aristotelische blonde goden wil ik in het bijzonder bedanken. ”Ofniet, ofwat, ofwel?” en ”Mijn gras is zo slecht nog niet” zijn twee standaarden geworden in het Leonar-diaans denken. Hoeveel elitaire bijeenkomsten op zondagavond beginnen en eindigen met het sleutelwoord ’emergentie’? Heel veel van mijn motivatie om nieuwe dingen te leren, om kritisch en out of the box te denken, heb ik van jullie geleerd. Bedankt Jarige jolige jongeman, om samen met mij alles in vraag te stellen, om alles te laten vallen om me mijn verhaal te laten doen, en vooral: om me keer op keer te doen rollen van het lachen. Weet dat je me altijd mag Stueren. Seksueel savante snoeperkont Sander, in zijn contreien beter gekend als Barry Baardman, is heel wat minder barbaars dan zijn naam doet uitschijnen. Ik herinner me de dag dat ik hem leerde kennen op het gangfeestje van het vijfde verdiep nog als gisteren. Enthousiast, openhartig en een dromer. Dat is nooit veranderd. Je hebt me een rijker persoon gemaakt, op zovele vlakken. Blijf altijd wie je bent. Zak.

Als ik het over zakken heb, dan komt Lander op de proppen. Sinds die winterse namiddag vorig jaar in de Basiel, heb ik altijd het merkwaardige gevoel gehad dat er continu iemand een oogje in’t zeil hield. Iets dat ik heel hard appreci¨eer. Ook mede Rach-freak Lukas en zijn blijde wederhelft Bjorge wil ik graag bedanken, en niet enkel voor koffie en cocktails.3 Maar dat kan volgend jaar nog meer dan genoeg in onze wekelijkse sessies van jazz-ensemble ’Speldenstraat’.

Het is een vreemd soort, die fysici. De meest vreemdsoortigen groeperen zich dan nog eens in een studentenvereniging om gezamenlijk aan idealisme en alcoholisme te doen. Bij de VVN gegaan was ongetwijfeld de beste beslissing die ik kon maken tijdens mijn opleiding. Bijna dagelijks met gelijkgezinden bezig zijn met manieren om fysica ’sexy’ naar het publiek te brengen, heeft me naast een breder zicht op wetenschap en wetenschapscommunicatie, ook op persoonlijk vlak veel doen groeien. Die groep van ’gelijkgezinden’ is doorheen de jaren een hechte vriendengroep geworden. Iedereen, enorm bedankt voor al de veel te lange vergaderingen, goedkope pintjes, en Ewaldsfeer-referenties. En Fleur, Julian en Bolle: ik weet 100% zeker dat volgend jaar een topjaar gaat worden. Make your senior proud!

Een persoon die een speciaal plaatsje inneemt in mijn hart en deze thesis is Lotte. Wat een geschenk om een vriendin te hebben die je altijd kan bellen, no matter what, en je keer op keer goed doet voelen. Florencia neemt hier absoluut ook een plaatsje, zonder wiens canciones de la puta madre deze periode heel wat minder kleurrijk zou zijn geweest.

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Zo zijn we bijna beland bij het langverwachte einde, en het minstens even langverwachte begin van deze thesis. Mijn keuze om een thesis te doen bij de Quantum Group was niet zozeer een ongetemperde passie voor het kwantumveeldeeltjesprobleem.4

Neen, wat me aanvankelijk enorm aantrok tot de ’kwantumtempel’ op het tweede verdiep van de S9 was de levendigheid van die groep, de informele ’kousen’sfeer en de onmetelijkheid aan krijtborden. En Gertian natuurlijk. Gertian’s geestdrift is sinds mijn eerste jaar een enorme inspiratiebron geweest, hij liet me al snel inzien dat de theoretische fysica werkelijk het walhalla is.

Mijn topbegeleider Quinten wil ik graag bedanken voor zijn betrokkenheid bij mijn project, voor zijn uitgebreide antwoorden op al mijn vragen en voor zijn enthousiasme over mijn resultaten (een ’nice’ betekende echt veel voor mij). Moesten de vleermuizen geen roet in het eten gegooid hebben, had ik ongetwijfeld veel meer in die cMPS-bureau gezeten. In het bijzonder wil ik ook mijn promotor Jutho in de bloemetjes zetten, die me dit jaar met veel ijver ingewijd heeft in de wereld van continue tensornetwerken, en vooral: de wereld van het onderzoek. Ik wil hem bedanken voor al de tijd die hij heeft vrijgemaakt, voor zijn onvermoeibare trein aan suggesties en idee¨en, maar bovenal voor het vertrouwen dat hij in mij had.

Hoe kan het ook anders dan deze compleet onverantwoorde langdradigheid af te sluiten met mijn pracht van een vriendin Mathilde. We leerden elkaar kennen toen ik me in dit ’kwantumavontuur’ stortte en zij heeft het avontuur en mijn mentale toestand van op dichte voet kunnen volgen, en met enig succes kunnen stabiliseren. Niemand anders kan me zo tot rust brengen als haar. Nu ik Wally en Sjaapie gevonden heb, ´en deze thesis binnen is, ben ik een gelukkig man. Ik zie je graag.

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A B S T R A C T

This dissertation concerns the development of the formalism of Gaussian continuous tensor network states as a variational ansatz for quantum field theories, following the work of A. Tilloy.[1] This is preceded by a torough discussion of the class of tensor network states and

the required quantum-informatical concepts. This chapter is followed by the introduction of the continuous matrix product state, in which we re-derive some key results and extract the properties that will be essential in higher-dimensional generalizations. The model that we will test our variational class on, is a quasi-free non-relativistic boson model, that we obtain as a regularized version of the relativistic free boson. Finally, after deriving the key calculational rules for Gaussian continuous tensor network states and with that variationally optimizing the bosonic Hamiltonian, we aim to examine the entanglement structure of the state.

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C O N T E N T S

Acknowledgements 3

Abstract 7

1 t e n s o r n e t w o r k s tat e s 11

1.1 Strongly correlated quantum systems 12

1.1.1 Optical lattices 12

1.1.2 The variational method 13

1.2 Basics of quantum information theory 14

1.2.1 Density matrix formulation 15

1.2.2 Quantum entanglement 18

1.2.3 Quantifying entanglement 19

1.2.4 Generalized measurements 20

1.2.5 Time development 22

1.3 Hilbert spaces: a convenient illusion 22

1.4 Tensor network states 24

1.5 Matrix product states 25

1.5.1 Completeness and naturalness of the MPS 26

1.5.2 Observables & MPS 27

1.5.3 Gauge freedom 29

1.6 Holographic quantum states 29

1.7 Projected entangled pair states 30

1.7.1 Parent Hamiltonians 32

2 c o n t i n u o u s m at r i x p r o d u c t s tat e s 33

2.1 Feynman’s objections 34

2.1.1 Sensitivity to high frequencies 34

2.1.2 Only Gaussian trial states 35

2.1.3 We still have to do a functional integral 35

2.1.4 Density matrices 35

2.2 Continuous matrix product states 36

2.2.1 cMPS as a generalization of field coherent states 39

2.2.2 Observables & cMPS 41

2.2.3 Gauge freedom 44

2.2.4 Uniform cMPS 44

2.2.5 Application to the Lieb-Liniger model 45

2.3 Holographic quantum field states 46

2.4 Curing the sensitivity to high frequencies 47

2.5 Path integral representation of the cMPS 49

3 b o s o n i c m o d e l s 55

3.1 Klein-Gordon theory in d+1 dimensions 55

3.2 Regularization of the Klein-Gordon Hamiltonian 56

3.2.1 Bogoliubov transformation 58

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c o n t e n t s 3.2.3 Two-point functions 60 3.2.4 High-momentum behaviour 60 3.2.5 Dispersion relation 61 3.3 Lieb-Liniger model 61 4 c o n t i n u o u s t e n s o r n e t w o r k s tat e s 63

4.1 Continuous tensor network states 63

4.1.1 Functional integral representation 63

4.1.2 Operator representation 65

4.2 Gaussian cTNS 66

4.3 Expectation values 68

4.3.1 Gaussian cTNS 69

4.4 Characterization of the D=1 gcTNS 70

4.4.1 Exact mapping to the free bosonic model in the small ν

µ limit 71

4.4.2 Variational optimization 71

4.5 Variational optimization for general D 74

4.5.1 Spectrum of K(p) 75 4.5.2 The algorithm 77 4.6 Entanglement in the gcTNS 81 4.6.1 D=1 82 4.6.2 General D 89 4.6.3 The algorithm 92 4.7 Outlook 93 a c o h e r e n t s tat e s 95

a.1 Coherent states in quantum mechanics 95

a.1.1 Fock states 96

a.1.2 Coherent states 96

a.1.3 Overlap between coherent states 98

a.2 Coherent states in bosonic quantum field theory 98

b g au s s i a n f u n c t i o na l i n t e g r a l s 99

b.1 Multidimensional Gaussian integrals 99

b.2 Functional derivation of Gaussians 100

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1

T E N S O R N E T W O R K S TAT E S

Exactly solvable systems in physics are very rare. As long as we’re not occupied with ’spherical cow’ textbook examples as the two-body problem, we have to resort to approximation methods. Actually that’s not entirely true. Many classical many-body systems are still ’numerically exactly’ solvable, for example we can simulate the solar system, or do a molecular dynamics simulation in a biophysical system. If we have N planets which mutually interact, the computation time scales as N2. This is quite challenging for galaxies, but actually an astronomer shouldn’t be complaining. In systems of many interacting quantum particles, even numerically exact solving (in this case it corresponds to exactly diagonalizing the many-body Hamiltonian) is impossible for all but the smallest systems. This is because in quantum mechanics we have to account for all possible superpositions of particle states. We will see in section1.3that the

number of involved basis states for a typical number of NAinteracting spins, is of the order 21023 ∼ 1080, which is much larger than the number of atoms in the observable universe![2]

Essentially there are two methods to cope with this exponential complexity of the many-body problem. The most common one is perturbation theory, which has application in all of physics, ranging from the discovery of Neptune to the splitting of spectral lines due to a magnetic field. In perturbation theory it is implicit that the interaction strengths are only weak and that the system behaves free in zeroth order. However for strongly interacting systems, which includes many systems in condensed matter physics as well as the quantum field theory of the strong interaction, this will not work. In those cases we have to resort to the variational method, in which we work with the full Hamiltonian, at the price of having a good insight of how the wave function actually looks like.

We discuss strongly interacting quantum many-body systems and the variational method in section1.1. Having insight about the form of the wave function of low-energy states (the states

we are interested in) is absolutely crucial, because we don’t want our variational optimization to sweep over all (exponentially many) possible states in Hilbert space. Luckily it turns out that the relevant states, i.e. ground states of Hamiltonians with local interactions, obey a so-called area law for the entanglement entropy. In other words, the entanglement is much lower than we would expect. We should therefore look for a class of variational ansatz states that exhibit this nice property. The tensor network states (TNS) of section1.4are exactly these natural states that

we’re looking for. Not only are they able to describe quantum correlations accurately (thereby going well beyond mean field theory), they are also a very efficient class, as they exhibit in general polynomial scaling in the number N of particles, instead of exponential. In section

1.5we go on with discussing the most famous TNS, the matrix product state (MPS), which is

able to describe all properties of (gapped) one-dimensional strongly interacting systems. We discuss explicitly its entanglement properties using the machinery of section1.2and see that

indeed these states obey the forementioned area law. We give attention to the ’calculus of MPS’, i.e. how to calculate expectation values and normalize these states, and the inherent gauge freedom that allows us to simplify the problem even more. These are in particular important

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t e n s o r n e t w o r k s tat e s

for generalizations to the continuum in chapters 2 and 4. We conclude this chapter with

alternative constructions of the MPS, that give us clear insights of why these states work so well. In section1.6we discuss the intrinsic holographic property of MPS, in that sense that they

are completely described by the dissipative dynamics of a zero-dimensional auxiliary system. This too will be essential in continuum generalizations. Finally in section 1.7we discuss how

the MPS can be generated by introducing virtual spins on the bonds, maximally entangling them and at last projecting them out. This will be the starting point of the two-dimensional generalization of MPS and gives us an insight about the concept of ’parent Hamiltonians’.

1.1 Strongly correlated quantum systems

Although many problems in condensed matter physics can be described efficiently within a single-particle picture, i.e. with mean field theory, in a certain class of systems, mapping the many-particle model to an effective single-particle model gives completely the wrong answers. This is the class of strongly correlated quantum systems. In those cases it is necessary to treat the full many-body problem, instead of treating the inter-particle interactions as a weak perturbation starting from a system of non-interacting particles.[3] Typical systems where the

interactions are not weak are magnetic systems, quantum Hall systems, (high-temperature) superconductors, one-dimensional systems1

and optical lattices.[4] The latter will be discussed

in subsection1.1.1. Theoretically such systems are often modelled by quantum spin systems,

that provide effective models for describing magnetism and quantum phase transitions. In these models, we put the physical system on a lattice, and position finite-dimensional Hilbert spaces (as is the Hilbert space of a spin degree of freedom) at each site. The huge surge of interest into these systems is again due to the development of optical lattice experiments. 1.1.1 Optical lattices

The development of optical lattices is accompanied by a tremendous advancement of ex-perimental techniques in ultracold atomic physics. This trend started with the creation of the Bose-Einstein condensate in a vapor of rubidium-87 atoms in 1995.[6] The study of cold

quantum gases has been one of the hottest - and coolest - areas of physics since the creation of the BEC. An optical lattice traps neutral atoms in deep potential wells, called trapping potentials, so that the atoms become very localised. The way this is done is by interfering two or more laser beams and thereby creating standing waves. For the atoms this works as a periodically oscillating potential, because the lasers will polarize the charge cloud of the atom, i.e. induce an electric dipole moment, which interacts with the electric field of the laser.[5] Suppose we

put a BEC on such an optical lattice. The bosons in the potential well will be coupled by tunneling, so that they are delocalized over the lattice. However, if the intensity of the standing

1 As this work mainly is considered with such one-dimensional quantum systems, we will sketch why in those cases weak interactions are often a very bad assumption. Indeed, whereas the properties of an interacting gas in three dimensions may remain essentially similar to the ground state properties of a free system, in one dimension this completely changes due to inevitable collisions between the particles. An excited particle will always cause interactions with neighbouring particles, so that excitations in one-dimensional systems are unavoidably collective phenomena. Apart from that there are in very few cases also exact analytical solutions available, for example by Bethe ansatz (section3.3), which allows the condensed matter physicist to benchmark his approximation method

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1.1 strongly correlated quantum systems

Figure 1: Arrays of one-dimensional quantum gases can be created by using two-dimensional lattice potentials. The interference pattern of the lasers ensures that there cannot be hopping between the arrays. These kind of systems are the systems cMPS (chapter2) would like to describe.[5]

wave laser is high enough, the atoms in the superfluid phase will undergo a phase transition to the Mott insulator phase.[7] This is the state where there is a fixed occupation number at

each lattice site, and the atoms cannot move freely anymore. This Mott insulating state was first realized in 2002 and was a major breakthrough in the study of cold gases.[8] Because this

resembles very much the aforementioned quantum spin systems, optical lattices allow us to build universal quantum simulators, i.e. to study (possibly computation-intractable) condensed matter Hamiltonians.[9] This will very likely gives us insight about e.g. the mechanism behind

high-temperature superconductivity. 1.1.2 The variational method

As we mentioned in the introduction, perturbation theory will certainly fail for these strongly correlated systems. Luckily there is another approximative way to extract the low-energy properties of such a system. The variational method has proven his power for solving quantum many-body systems2

time by time, ranging from quantum chemistry (Hartree-Fock theory and density functional theory) to the BCS wave function for superconductivity3

, and eventually, to quantum lattice systems. The idea behind it is very simple. You have some idea about how the ground state wave function would look like and you use this as a trial state|Ψito minimize the expectation value of the Hamiltonian

E[Ψ] = hΨ|Hˆ |Ψi

hΨ|Ψi , (1)

2 For quantum field theories it was not tought to be useful, due to Feynman’s criticism (section2.1). However, as the

title of this dissertation suggests, nowadays they too are beaten by the variational principle.

3 I wrote a popular (Dutch) article on the website of the UGhent physics society (VVN) about John Bardeen and the BCS theory:https://vvn.ugent.be/blog/613/.

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which is called the Rayleigh-Ritz quotient. Actually you don’t even have be to so clever, because the trivial observation

hΨ|_Hˆ |Ψi hΨ|Ψi =

∑

_n hΨ|_Hˆ |ni hn|Ψi hΨ|Ψi =

∑

_n En | hn|Ψi |2 hΨ|Ψi ≥E0

∑

_n | hn|Ψi |2 hΨ|Ψi = E0 (2) guarantees that for any |Ψi, E[Ψ] will always be greater than or equal to the exact ground state energy. The Rayleigh-Ritz method comes down to minimizing this quotient with respect to all the variational parameters ci in the trial state.4 In chapter 4 we will minimize this

quotient using a steepest descent method.5

In other words, we calculate the variational parameter-gradient of the Rayleigh-Ritz quotient

∇_cE[Ψ] = 1

hΨ|Ψi∇chΨ|Hˆ |Ψi − 1

| hΨ|Ψi |2hΨ|Hˆ |Ψi ∇chΨ|Ψi (4) and do a trial step in the direction of steepest descent of this gradient, until there is no such direction anymore (then the energy is in its minimum).6

We mentioned that we could use any state as a trial state. We could even use general states in the N-particle Hilbert space (a complete expansion with respect to the Hilbert space basis), but as we’ll see in section1.3this comes down to minimizing dN coefficents, where d is the

dimension of the one-particle Hilbert space. We clearly don’t want to run into this exponential wall again (this was entirely the point of introducing approximation methods beyond exact diagonalization), so as a first requirement the class of ansatz states should be efficient, in other words we would like a polynomial scaling with N in the number of parameters. Secondly, just as the best runner in the world is not per se the best person in the world, a good approximation for the ground state energy does not imply that other physical observables (like the correlation function) are well approximated. We thus have as a second requirement that the class of ansatz states is natural. As a third requirement we should have the possibility to eventually reach all states in Hilbert space by enlarging the ansatz states somehow: the class should be complete. The last requirement will be that the calculation of expectation values is not too hard. We will see in section1.4and beyond that tensor network states are the right way to go.

1.2 Basics of quantum information theory

[About entanglement] I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. - Erwin Schr ¨odinger [10]

4 The Hartree-Fock/BCS gap/Gross-Pitaevskii... equations are exactly obtained this way. 5 In many cases one however uses imaginary-time evolution,

|Ψ0i = lim

τ→∞

e− ˆHτ_|_Ψ_i

||e− ˆHτ_|_Ψ_{i ||}. (3)

On other words, we work with the imaginary evolution operator on our trial state, and let it evolve for infinite time. Again inserting a resolution of the identity, guarantees that the term E0[Ψ]with the lowest energy will be

least damped.

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1.2 basics of quantum information theory

The tensor network language is in essence the language of quantum information theory, which is based on one exceptional ’spooky’ property of quantum mechanics, namely the idea of entanglement.7

Therefore we will provide the reader with a dictionary of the fundamental concepts. When Einstein, Podolsky and Rosen tried to embarrass the quantum community in 1935by exposing the fundamental non-local character of quantum correlations, they actually gave birth to an inexhaustible source of dazzling non-classical features and new research areas.[11] Quantum entanglement is the phenomenon that two particles can be correlated with

each other, no matter how far apart. This means that if one carries out an experiment on an electron here, it will immediately affect the other electron, that can be on the other side of the Universe. For example if the total spin of a pair of entangled electrons is zero and we measure the spin of one electron on a certain axis, then the spin of the other electron, measured on the same axis, will immediately be opposite.

Quantum entanglement only became a real research topic after the work of John Bell, who showed that this phenomenon cannot be described by a classical ’hidden-variable’ theory if the so-called Bell inequalities for classical correlations are violated.[12] And that this was indeed

the case was first shown experimentally by Aspect’s experiments concerning correlations between polarizations of entangled photons.[13] Figure2is a nice visual example of a more

recent experiment with entangled photon phases. Nowadays the overwhelming majority of experiments are in favor of the reality of violation of the Bell inequalities and hence quantum non-locality.

The theory of quantum information concerns the use of this entanglement phenomenon as a source of information which can be used e.g. for the exponential speed-up in quantum computation.[14]

1.2.1 Density matrix formulation

The fundamental entity in quantum mechanics is the quantum state, which is represented as a ray8

in a Hilbert space and contains all possible information about the system considered.[16]

However in most cases we are not interested in the behaviour of the entire universe, but rather in a selected part of the whole.[17] Our observations will then be limited to a small part of a

much larger quantum system. We will see that in that case the state is not represented by a ray, but rather by a density operator (or ’matrix’ in a chosen basis) working on the Hilbert space.

Suppose |Ψi is the quantum state vector representing the universe, consisting of two subsystems i.e. |Ψi ∈ H = H₁⊗ H₂. We are interested in the value of an observable A of subsystem 1 only, this means we want to evaluate the expectation value of ˆA⊗_{1. This can be}ˆ written in terms of the spectral decomposition of ˆA

hΨ|Aˆ⊗1ˆ|Ψi = hΨ|

∑

n

anPˆn⊗1ˆ|Ψi, (5)

7 I wrote a popular (Dutch) article on the website of the UGhent physics society (VVN) about quantum entanglement:

https://vvn.ugent.be/blog/kwantumverstrengeling/.

8 A ray is a set of unnormalized vectors which represent the same state. Therefore it is in fact an equivalence class with respect to the relation|ψi ∼λ|ψi.

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Figure 2: In a recent paper by Moreau et al. a way to ’image’ the entanglement between two photons is demonstrated.[15] In the experiment a laser source fires photons at a liquid crystal (BBO)

which causes entanglement between the phases of some photons. Next, the beam is splitted (BS) into two equal arms. Some of the entangled photon pairs have thus parted ways. In one arm there is a phase filter (SLM2) which effectively ’measures’ a photon to one of four possibilities and hence causes the second photon to choose a definite state. Then, a super-sensitive camera capable of detecting single photons records the first photon, but only if the second photon triggers it (this is, if it has the opposite phase and has traveled the same amount of time). The researchers showed that correlations between some of these pairs of photons were stronger than what classical probability would predict.

H₁ H₂

H

Figure 3: Hilbert space of the bipartite quantum system. In general the states live in bothH₁andH2

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where ˆPnis the projecor onto the eigenspace of an.9 After this measurement (projection onto a certain eigenspace), our (normalized) state will be collapsed to

ˆ

Pn⊗1ˆ|Ψi q

h_Ψ|_Pˆ_n⊗₁ˆ|_Ψi

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with a probability (Born rule)

Prob(n) = hΨ|Pˆn⊗1ˆ|Ψi. (7)

Let’s introduce the density operator

ˆρ= |Ψi hΨ| (8)

and upon partial tracing over one subsystem the reduced density operator

ˆρ1= TrH2|Ψi hΨ|, (9)

which acts only onH₁. The expectation value (5) can than be rewritten in terms of this operator,

i.e.

h_Ψ|_Aˆ ⊗₁ˆ|_Ψi =

∑

n,m

ˆρ1 →Tr2 ˆ Pn⊗1ˆ|Ψi hΨ|Pˆn⊗1ˆ hΨ|_Pˆ_n⊗₁ˆ|Ψi = ˆ Pnˆρ1Pˆn Tr1Pˆnˆρ1 (11) So we see that the density operator language is a natural language when describing subsystems of larger quantum systems, and thus the language to describe entanglement of one subsystem with another.

Density operators also arise for another kind of preparation of a quantum system, namely in the case where we have a classical probability distribution, an ensemble, of quantum states. Suppose we have a source of photons which spits out photons in state|Ψ₁iwith probability p1 and in state|Ψ₂iwith probability p2=1−p1. Then the average value of an observable A is

p1hΨ1|Aˆ|Ψ1i +p2hΨ2|Aˆ|Ψ2i =Tr(ˆρ ˆA), (12) where

ˆρ= p1|Ψ1i hΨ1| +p2|Ψ2i hΨ2|. (13) This is different from a coherent superposition of the two photon states as there is no relative phase information and we just have an incoherent mixture of two photon states. An important ensemble in statistical mechanics is the thermal state,10

a mixture of states at a non-zero temperature. In this case the probabilities are Boltzmann distributed pn = e

−βEn

∑ne−βEn where β

denotes the inverse temperature and En the energy eigenvalues: ˆρ= e

−β ˆH

Tr(e−β ˆH₎. (14)

9 In the case of a non-degenerate a_nthis is simply|ni hn|.

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We conclude that we can understand the density operator as follows: the state of a subsystem is actually a statistical ensemble of states. Tracing out the degrees of freedom of one subsystem, will inevitably be accompanied with throwing away some of the information about the other -entangled - subsystem.11

We say the state is pure if (8) holds and mixed if (13) holds.

1.2.2 Quantum entanglement

The whole is more than the sum of its parts. - Jan Ryckebusch

Now what is quantum entanglement in the language of section1.2.1? We look again at the

density operator ˆρ∈ L(H₁⊗ H₂). Suppose that it decomposes as

ˆρ=

∑

pjˆρj₁⊗ ˆρ₂j, (15)

i.e. it has separate parts corresponding to subsystem 1 and 2. In this case doing a measurement on subsystem 1 won’t effect the state in subsystem 2, hence there is no entanglement. We call this a classical separable state. In any other case ˆρ is called an entangled state. The big difference in the quantum mechanical case is the lack of separability. One cannot measure subsystem 1 without disturbing subsystem 2. So we say a state is entangled if its reduced states are not pure, but mixed.

We would like to quantify the amount of entanglement present in a bipartite pure quantum state|Ψi ∈ H₁⊗ H₂. To that end we choose two orthonormal bases{|ii₁}and{|ji₂}forH₁ andH₂ respectively, in which we expand|_Ψi

|Ψi =

∑

i,j

Cij|ii1|ji2 (16)

and apply a singular value decomposition12

on the expansion coefficient Cij = hi|1hj|2|Ψi. This gives us |_Ψi =

∑

i,j,k U_ikΣ_kkV_jk∗|ii₁|ji₂ = r

∑

k=1 Σkk|ki1|ki2 (18)

because the unitary matrices U and V just rotate the original orthonormal bases to new orthonormal bases|ki₁and|ki₂. This is called the Schmidt decomposition of a bipartite quantum state. We call the rank of the matrix C, thus the number r of singular values, the Schmidt number. This Schmidt number is invariant under unitary evolutions that do not couple subsystem 1 and 2 and hence a first measure to evaluate the entanglement between the two partitions. Because the singular values are non-negative real numbers, we will denote them as√λk = Σkk. On the level of the reduced density matrices, (9) becomes

ˆρ1= r

∑

k=1

λk|ki1hk|1, (19)

11 Unless the density operator is of the form (15), but then there are no quantum correlations between the subsystems.

12 The singular value decomposition (SVD) is a generalization of the diagonalization of square normal matrices to arbitrary complex matrices. Given an m×n complex matrix C, it decomposes into

C=UΣV† (17)

where U and V are m×m and n×n unitary matrices respectively andΣ is an m×n rectangular diagonal matrix. The diagonal entriesΣii are called the singular values of C, which are all non-negative real numbers, and the

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so it corresponds to an eigendecomposition with clearly ˆρ2 having the same eigenvalues. Because (19) represents a probability distribution of states, we must have that

Tr(ˆρ1) =Tr(ˆρ2) = r

∑

k=1

λk =1. (20)

If one of these eigenvalues is one, all the other should thus be zero. In this case, the reduced density operators are pure and our original state is thus a separable product state |Ψi = |χi₁|φi₂. In all other cases, the state is entangled. It is clear that ’maximal entanglement’ will occur when all Schmidt values are equal to λk = 1_r. Indeed in that case the reduced density operators

ˆρ1= ˆρ2= 1

r1ˆ (21)

of either of the two subsytems contains no information: in whichever basis we choose to measure the subsystem, the r outcomes are equally likely.

1.2.3 Quantifying entanglement

The first crude entanglement measure is just the Schmidt number, the rank of our original wavefunction. However the ensemble interpretation of the mixed density operator also guides us to a fundamental information-theoretic concept. That is, the concept of entropy. The Shannon entropy of a probability distribution quantifies the amount of ’missing information’, of uncertainty. How bigger the entropy, how less we know about the state of the system. Therefore the Von Neumann entropy, defined as the Shannon entropy of the Schmidt values distribution function, is a suitable measure for the entanglement of a pure state

S({λ_k}) =:− r

∑

k=1

λkln λk = −Tr(ˆρ1ln ˆρ1) ≡S(ˆρ1) =S(ˆρ2) = −Tr(ˆρ2ln ˆρ2). (22)

This ’entanglement entropy’ quantifies the amount of ’mixing’ in a quantum system. Indeed, the more mixed the system, the larger the spread in the probability distribution and the higher the entropy. When the reduced density operator is pure (the bipartite state was a product state), the entropy is zero as there is no uncertainty with respect to which state the system is in. On the other hand, in the worst case scenario of a ’flat’ entanglement spectrum13

like (21),

the entropy will be maximal S(ˆρ1=

1

r1ˆ) =ln r → S(ˆρ1) ≤ln r. (23)

We can illustrate how this entropy is related to the spread of the Schmidt values by looking at a simple two-qubit system, e.g. a system of two spin-1₂ particles. We study the Bell state (the singlet state)

|Ψ−i = √1

2 |↑i1|↓i2− |↓i1|↑i2, (24)

13 The entanglement spectrum can be obtained by ordering the Schmidt values λ_kfrom big too small as a function of their index k. In much cases (see also later on when we’re talking about truncated SVD) we’re more interested in how the entanglement is organised in a state, rather than just one single number, namely the entanglement entropy. [18]

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Figure 4: Entanglement spectrum of the Bell state, the modified Bell state and the product state. The entropy is maximal when the Schmidt values are all the same, and reduces gradually when one Schmidt value starts to dominate.

the ’modified’ Bell state

|_Ψ˜−_{i =} _√1

3|↑i1|↓i2− r

2

3|↓i1|↑i2 (25)

and finally the product state (which equals of course|←i₁|←i₂)

|ΨPi = 1

2(|↑i1− |↓i1) ⊗ (|↑i2− |↓i2). (26)

After writing the density matrix of these states and tracing out one (doesn’t matter which) subsystem, we get respectively the eigenvalues (1₂,1₂), (2₃,1₃) and (1, 0). Figure4shows clearly

that the entanglement is maximal (S=ln 2) for the Bell state and 0 for the product state. The more ’localized’ the distribution of the Schmidt values, the smaller the uncertainty about the state and hence the entropy.

Of course there exist other measures for the entanglement,14

such as the R´enyi entropy and the logarithmic negativity, but for our purposes the Von Neumann entanglement will be sufficient.[19]

1.2.4 Generalized measurements

The measurement postulate in quantum mechanics states that we can expand our quantum state|_Ψiin any basis (countable or uncountable) we like, e.g.

|Ψi =

∑

n

cn|Ψni, (27)

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and if we then measure the system by projecting it, like in (6), it will indeed end up in one

of these basis states. If the system is initially in state |Ψithen the probability that we’ll get |_Ψ_niis given by|cn|2= Tr(ˆρ ˆPn), where ˆPn= |Ψni hΨn|and ˆρ= |Ψi hΨ|. This is called the Von Neumann projective measurement. However, in many cases the projective measurement is too restrictive to describe a general measurement, for example when the system of interest is part of a bigger system. Below we will describe the generalized measurement procedure, which will turn out to be a projective measurement in an extended space. [20]

Instead of a set of projectors ˆPn, we now have a set of ’measurement operators’ ˆEn, one for each outcome of the experiment, and require that the probability of obtaining n still obeys Born’s rule

Prob(n) =Tr(Eˆ_nˆρ). (28)

Of course we must have that all probabilities are real and greater than zero. In addition the probabilities should sum up to zero. This results in

∑

n ˆ En=1 (29) ˆ En≥0 ∀n (30) ˆ En=Eˆ†n ∀n. (31)

The same conditions hold for the projection operators ˆPn but now the ˆEnare not required to be projectors. We call these measurement operators positive operator-valued measures (POVM). We can write ˆEn= Mˆn†Mˆn, where ˆMn are called Kraus operators, to see that

Prob(n) = hΨ|_Mˆ†

nMˆn|Ψi. (32)

Hence the state after the measurement is

|Ψi → Mˆn|Ψi hΨ|Mˆ† nMˆn|Ψi (33) or ˆρ→ Mˆnˆρ ˆM † n Tr(Mˆnˆρ ˆM†n) , (34) the equivalent of (11).

The generalized measurement can be realized upon enlarging the original target systemAwith Hilbert spaceH_A with an auxiliary probe or ancilla systemB, with another Hilbert spaceH_B. The ancilla system can be seen as a purely virtual mathematical apparatus to describe quantum measurements, but as we’ll see later, it is often useful to interpret it as an extra quantum system that couples to the original system. Suppose we want to measure an observable A of system A. The measuring device can detect D eigenvalues of the operator ˆA, as the ancilla Hilbert space has a D-dimensional (orthonormal) basis{|ni |n=1, . . . , D}. These probe states will be in one-to-one correspondence with the possible outcomes of the experiment. Consider a fixed state|0iof systemB, and a unitary operator ˆU acting on the product state|Ψi |0i ∈ HA⊗ HB. This unitary operator will allow interaction, and thus entanglement, between the two systems according to ˆ U|Ψi |0i = D

∑

n=1 ˆ Mn|Ψni |ni. (35)

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Unitarity of ˆU requires that∑_nMˆ†_nMˆn=1. If we would do a projective measurement on thisˆ probe+target system of the form ˆ1A⊗ |ni hn|, i.e. forcing the probe system in one of its D states, we obtain state|niwith probability

Prob(n) = hΨ| h0|_Uˆ† ˆ 1A⊗ |ni hn| ˆ U|Ψi |0i (36) = hΨ|Mˆ_n†Mˆn|Ψi (37)

which has the same form as (32). Therefore this measurement prescription (unitary evolution

of the joint system, followed by a projective measurement on the joint system), describes a generalized measurement with POVM’s ˆEn = Mˆn†Mˆn. So in a general quantum measurement one obtains information about the system of interest by coupling it to an auxiliary degree of freedom (the probe) and then detecting some property of the probe.

1.2.5 Time development

In a closed quantum system, the evolution of the state vector is governed by the Schr ¨odinger equation

id

dt|Ψ(t)i = Hˆ(t) |Ψ(t)i, (38)

which has formally the solution

|_Ψ(t)i =Uˆ(t, 0) |Ψ(0)i (39)

with

ˆ

U(t, 0) = Te−iR0tdt0H(t0), (40)

a time-ordered exponential. It follows then immediately that the (pure) density operator corresponding to this state evolves as

ˆρ(0) → ˆρ(t) =Uˆ(t)ˆρ(0)Uˆ†(t). (41) The equation of motion for the density operator (time evolution for Heisenberg operators)

d

dtˆρ(t) = −i[Hˆ(t), ˆρ(t)] =L(ˆ t)ˆρ(t), (42) whereLis the Liouville superoperator,15

is called the Liouville-Von Neumann equation. For a time-independent Hamiltonian the solution to the Liouville-Von Neumann equation is clearly

ˆρ(t) =eLˆtˆρ(0). (43)

1.3 Hilbert spaces: a convenient illusion

Hilbert space is gratuitously big. - Carlton Caves [21]

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1.3 hilbert spaces: a convenient illusion

H

M

Figure 5: Full Hilbert spaceHand the subspaceMof area-law states. We want our variational manifold to cover this corner as good as possible. MPS (and in general TNS) with moderate bond dimension D are very good at doing this. On the other hand when we let D evolve from 1 (a product state) to O(eN)MPS will eventually cover the whole Hilbert space. This is the completeness property of MPS.

A

Ac

Figure 6: For a random state in Hilbert space, there would be entanglement (represented by the red lines) from all points inAto all points in its environmentAc_{. However, for the low-energy}

states of local Hamiltonians, there is almost only entanglement across the boundary of that region. In other words, the entanglement property will not grow as the volume ofA, but rather as the boundary A ∩ Ac _{between system and environment, which is in this case}

one-dimensional.

A generic state in an N-particle product spaceHof d-dimensional Hilbert spaces (N ’qudits’) is

|Ψi =

∑

s1,...,sN

Cs1...sN|s1, . . . , sNi. (44)

Suppose d=2 (a system of qubits), in that case this state is a superposition over all 2N states |01, 02, . . . , 0Ni, |11, 02, . . . , 0Ni,. . . , |11, 12, . . . , 1Ni. In other words, a generic state in Hilbert space requires dN _{coefficients to specify it. The Hamiltonian is even worse, as it contains} (dN)2 coefficients. This curse of dimensionality makes, as we mentioned in the introduction, exact diagonalisation, and hence the extraction of observables, completely unfeasible for more than a few particles. However, it turns out that this exponential wall is in fact a convenient illusion.[22] The states that we’re interested in are not at all generic. These interesting states

are the low-energy states of Hamiltonians with short-range interactions.16

The difference between a generic state and a ’naturally occurring’ state, is shown in figure

6. Naturally occurring states are only slightly entangled, as supposed to a typical state in

Hilbert space. This means that we expect a quick fall-off in the entanglement spectra of such states (i.e. more like the ’modified Bell state’ (25) than the maximally entangled Bell state

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(24)). Intuitively, this is clear: local Hamiltonians couple e.g. site i and site i+1, inducing

entanglement between the sites. Due to the monogamous nature of entanglement, there cannot be much entanglement left to couple to sites further away.[23]

That the entanglement entropy scales as the area instead of the volume has been proven for a number of instances, including for gapped one-dimensional systems with a unique ground state.[24][25] Choosing a variational class that is ’natural’, corresponds to using this neat

property to our advantage. This will greatly reduce the exponential complexity of quantum many-body systems. The class of tensor network states that we will now discuss, implicitly relies on such area laws.

1.4 Tensor network states

Maybe there is some way to surround sthe object, or the region where we want to calculate things, by a surface and describe what things are coming in across the surface. It tells us everything that’s going on outside. I’m talking about a new kind of idea but that’s the kind of stuff we shouldn’t talk about at a talk, that’s the kind of stuff you should actually do! -Richard Feynman [26]

What this quote of Feynman has to do with tensor network states (TNS), is the fact that TNS provide us with an efficient local descripton of the relevant many-body states, that can be obtained by modelling the way in which the entanglement is distributed. They will in other words exactly cover the area-law manifold of figure 5. Tensor networks are the ideal marriage

between quantum information theory and quantum many-body physics.

But first, what is a tensor network? Tensor networks were first introduced by Roger Penrose in the 70’s to visually depict multilinear functions of tensors.[27] A tensor network consists of

several vertices linked by edges. In the context of condensed matter physics, this is perfectly suitable for representing quantum lattice systems, with the lattice sites being the vertices and the interactions between them located on the edges. We will first introduce the graphical language, which is fairly simple. A tensor is represented by a solid shape, and tensor indices are lines attached to these shapes. The number of these ’legs’ thus equals the rank of the tensor and connecting legs equals contracting that index (similar to Einstein’s summation convention). The lowest-order tensors look as follows

c scalar c (45) v i vector vi (46) M i j matrix Mij (47) T i j k

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1.5 matrix product states

How they arise on the level of the many-body wavefunction Cs1...sN in (44), is that the original

rank N tensor (so a shape with N legs attached to it), factorizes as a number of smaller rank tensors which are connected by fully contracted legs. Accordingly, the introduction of a TN comes not for free: the indices that glue the different tensors together introduce some new degrees of freedom. We call these virtual or bond indices. It turns out that the virtual indices exactly represent the structure of the many-body entanglement. This will be more clear in the upcoming section.

1.5 Matrix product states

The most elemental tensor network state is the matrix product state (MPS), which is - upon truncation of the Schmidt number - an efficient ansatz for one-dimensional quantum spin systems.[2] A brief history. The matrix product state famously has its origins in Wilson’s

numerical renormalization group (NRG) to tackle the problem of a magnetic impurity in a metal.[28] Given the great amount of correlations in this system, this could not be tackled

with a perturbative approach. The NRG however had little succes for anything but quantum impurity problems, and failed to do well for general lattice problems like the Heisenberg and Hubbard model. Upon improving on the NRG, Steve White developed the density matrix renormalization group (DMRG) in 1992, which turned out to be equivalent to a variational optimization over the set of matrix product states.[29][30]

Although this thesis won’t perform explicit calculations on specific models with these lattice variational ansatzes, they expose a lot of the general features of tensor network states that will be present in the continuum too. In the MPS, the expansion coefficient Cs1...sN of (44) with

respect to the spin basis|sii⊗N gets factorized as a chain-like product of three-index tensors Asi

α,β(i), with one leg in the physical Hilbert space and two legs in an auxiliary Hilbert space

|A(1), . . . , A(N)i_MPS =

∑

s1,...,sN Tr[BAs1₍₁₎_As2₍₂₎_{. . . A}sN₍_N_{)] |}_s 1, . . . , sNi. (49) A(1) s1 A(2) s2 A(3) s3 . . . _A(N−1) sN−1 A(N) sN

Figure 7: MPS with periodic boundary conditions.

hωL| A(1) s1 A(2) s2 A(3) s3 . . . A(N−1) sN−1 A(N) sN |ωRi

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For each spin index si, Asi represents a Di−1×Di matrix, where the maximum D of the Di is called the bond dimension of the MPS. In many cases one restricts to square matrices Asi

with the same bond dimension. The amount of variational parameters is thus of the order N·d·D2 =(the number of sites)·(the local physical dimension, so the number of matrices per site)·(the number of matrix elements of Asi_{). Compare this linearity in N to the previous}

exponential scaling. The boundary term B is given by1D×D in the case of periodic boundary conditions (figure7), and B= |ω_Ri hω_L|in the the case of open boundary conditions (figure 8), where|ω_Riand|ω_Liare D-dimensional vectors.

1.5.1 Completeness and naturalness of the MPS

The class of matrix product states is complete, hence obeying our third requirement of subsection1.1.2.[31] This means that eventually every state in Hilbert space can be written as a

matrix product state (possibly with infinite-dimensional matrices). However, we saw above that physical states are not just any state; the relevant states have bounded entanglement. We will now quantify these statements and argue why matrix product states are such natural, economic representations of ground states of gapped, local Hamiltonians. Therefore we use the machinery from section1.2.

We consider a quantum state|Ψiof a one-dimensional spin chain which we will cut between site 1 and 2 as in (50). We then do a Schmidt decomposition (18) on this bipartition, and apply

a unitary transformationΓ(1)on the first ket to transfer to the spin basis

Cs1...sN = Γ(1) Σ R(2) , (50) in equations, |_Ψi =

∑

s1,...,sN Cs1...sN|s1, . . . , sNi =

∑

α1 Σα1|Lα1(1)i |Rα1(2)i =

∑

α1,s1 Σα1Γ s1 α1(1) |s1i |Rα1(2)i. (51) We can do this recipe N−1 times (6 cuts in the above picture) and obtain schematically

Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ , (52) in equations, |_Ψi =

∑

α1,s1α

∑

2,s2 · · ·

∑

αN,sN Γs1 α1(1)Σα1Γ s2 α1,α2(2)Σα2. . .Γ sN−1 αN−2,αN−1(N−1)ΣαN−1Γ sN αN−1(N) |s1, . . . , sNi (53) The summations over the αnare on the ’virtual level’, meaning that the expansion coefficient Cs1...sN is now a product of matrices. The matricesΓ(1)and Γ(N)are 1×D and D×1

rspec-tively, such that this is an open boundary MPS. On the other hand, we can promote these vectors to matrices and trace over the auxiliary indices to obtain a periodic boundary MPS. We

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1.5 matrix product states

have now shown that any one-dimensional spin chain can be written as a MPS. However, one might wonder what is the benefit of doing this (as it seems we have added more to the state instead of reducing it)? The formulation in terms of all these Schmidt decompositions immedi-ately allows us to use the entanglement properties of low energy states to our advantage. If indeed the entanglement property obeys an area law (in the one-dimensional case, it means that it is bounded by a constant), the entanglement spectrum (remember figure4) falls of very

quickly. For example, it is very plausible that after D<<r terms in the entanglement entropy, the Schmidt values are as good as zero. Therefore it suffices in many cases to only keep track of D Schmidt values instead of the full rank r! In other words, the bond dimension D matrix product state corresponds to doing a ’truncated’ SVD at each bipartitation, instead of doing a full SVD. In the truncated SVD you keep only the D column vectors of U and the D row vectors of V†_{and discard the remaining r}₋_{D singular values. The area law for entanglement} entropy now emerges because we fix D independent of the system size.

It is only for ground states of gapped, local Hamiltonians that we expect a quickly de-creasing entanglement spectrum. We don’t expect critical states to be represented well by a low D MPS. From the successive Schmidt decomposition above, we would already expect that the MPS with finite bond dimension will be a good approximation for the ground state of gapped local Hamiltonians, because they too obey an area law. However this was only rigorously shown by Hastings in 2007.[25]

The proof that MPS are indeed an efficient parametrization of natural occurring states can be found in [32] and [33]. 1.5.2 Observables & MPS A(1) . . . A(i) ˆ Oi . . . A(j) ˆ Oj . . . _A(N) ¯ A(1) . . . A¯(i) . . . A¯(j) . . . A¯(N)

Figure 9: Representation of the two point function evaluated in a periodic boundary MPS.

As we mentioned in1.1.2A good variational class should exhibit simple ways to calculate

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to calculate a two point functionhOˆiOˆji. This is represented in figure9. It is straightforward to see that (for a translation-invariant system)

h_Oˆ_i_Oˆ_ji =Tr[BEiEOiE

j−i−1_E OjE

N−j_] ₍₅₄₎

whereEI :=E is the transfer matrix, defined locally as A(i) ¯ A(i) d

∑

si=1 Asi₍_i₎_A¯si₍_i_{) =}E₍_i₎ E(i) . (55)

This is a sum over all physical indices, hence the transfer matrices work entirely on the virtual system (as is clear from the four red horizontal legs in (55)). Similarly

A(i) ˆ Oi ¯ A(i) d

∑

si,s0i=1 hs0_i|Oˆi|siiAs 0 i(i) ⊗A¯si₍_i_{) =}E Oi(i) EO(i) (56) is called Oi-transfer matrix. Those transfer matrices are rank-4 tensors, or D2×D2matrices. Let’s do an eigendecomposition of the transfer matrix

E(i) E=

∑

k λ_k|r_ki hl_k|, hlk| λk |rki (57) where|rkiandhlk|are the right and left eigenvectors ofE respectively. Suppose we work in the thermodynamic limit N→_{∞, which combined with translation invariance defines the} class of uniform matrix product states. In that case it is clear that, if we take the l-the power of the transfer matrix and order the eigenvalues from big to small, we obtain

El I =

∑

k λl_k|rki hlk| =λl₁

∑

k (λk λ1 )l|rki hlk| l→_{= |}∞ ri hl| (58)

where|riandhl|are the eigenvectors corresponding to λ1. We normalize the MPS so that the dominant eigenvalue|λ₁| =1 (all other eigenvalues thus lie in the unit disk). This is a very powerful result for (uniform) MPS, as it tells us that we can calculate the two point function (54) according to17

hOˆiOˆji = hl|EOi(i)E

j−i−1_E

O(j) |ri. (59)

17 One can also show that the two-point function falls of exponentially as a function of i−j by taking into account the second eigenvalue λ2The correlation length equals ξ = −_{ln λ}1₂_/λ₁.[2] This is a nice property because ground states of gapped non-critical 1d systems exhibit the same behaviour. Again, Hastings showed this rigorously.[32]

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1.6 holographic quantum states

1.5.3 Gauge freedom

The class of matrix product states form in fact an over-complete parametrization, as one can do transformations of the form g−1(i−1)A(i)g(i)at all sites (insert the identity between each site) and obtain the same state, but now parametrized by different matrices B(i). We can fix this gauge freedom by putting a certain constraint on the A matrices. The most useful form of the MPS is the canonical form that we derived in subsection1.5.1. In that case each bond index

αcorresponds to the labelling of the Schmidt vectors across that index. This is very useful if we want to calculate entanglement entropies and/or spectra. Apart from the canonical form we have another useful gauge, namely the left canonical form, where we absorb the Schmidt coefficientsΣ into the left unitary matrix Γ

A(n)

=

Γ(n) Σ(n)

. (60)

Due to unitarity of theΓ’s, this corresponds to A(n) ¯ A(n)

∑

si Asi†_Asi ₌1 ₍₆₁₎

being the identity matrix on the virtual level.[34] This then automatically normalizes the MPS

hΨ|Ψi =

∑

s1,...,sN

(AsN₎†_{. . .}₍_As1₎†_As1_{. . . A}sN ₌_1. ₍₆₂₎

Given this, the right canonical form is not very hard to guess.

1.6 Holographic quantum states

One may wonder why are matrix product states, and in general tensor network states, so succesful? The answer lies in the holographic principle. This idea has its origins in the study of black hole thermodynamics by Bekenstein and Hawking, who discovered that the entropy of a black hole scales as the area of the event horizon, instead of the volume.[35][36] The

fundamental degrees of freedom that describe the system are living on the boundary of that system. The same story holds for matrix product states. In that case the ’boundary’ theory can be given an explicit form. We discuss a scheme to ’generate’ matrix product states with a zero-dimensional ancilla system.[37]

Consider a system of N uncorrelated spins, and extend it with some ancilla D-level system. Initially the physical system thus is in the disentangled product state|0i₁⊗ · · · ⊗ |0i_N. The initial state of the ancilla system is|φi_I. In every timestep18

i the ancilla system interacts with

18 Of course time is not discrete, but ’timestep’ should be interpreted as ’time interval’[i−e

2, i+e2]where e is the

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Figure 10: The generation of entangled spin states on a lattice can be obtained by sequentially coupling the ancilla system to the previously disentangled spins. The ancilla system can be given the physical interpretation of a D-level atom, trapped in a cavity, and gradually leaking photons: if spin N−i is up, it corresponds to a photon emitted at timestep i. If it is down, no photon was emitted in that time interval. [38]

the spin at site N−i. This is represented by a joint unitary evolution operator working on both the spin system and the ancilla

ˆ A(i) = d

∑

si=1 D

∑

α,β=1 Asi α,β(i) |αi |sii hβ| h0|i, (63)

where{|αi,|βi}are the basis states of the ancilla system (in the cavity analogy of figure10,

the energy levels of the trapped atom) and|siiare the physical spin/qudit states. The rank-3 tensor Asi

α,βdescribes how the ancilla converts the state |0ii to a general spin |sii, as well as

how the ancilla system itself evolves from|βito|αi. After N timesteps, the obtained state is ˆ

A(1)Aˆ(2). . . ˆA(N) |φ_Ii |0i₁⊗ · · · ⊗ |0i_N. (64) The N spins are now entangled with each other ´and the ancilla system. We should thus as a final step disentangle the ancilla system. The final state of the spin system is obtained by projecting out the ancilla system to a state|φ_Fi

|A(1), . . . , A(N)i =

∑

s1...sN

hφ_F|As1₍₁₎_As2₍₂_{) · · ·}_AsN₍_N_)|φ

Ii |s1, s2, . . . , sNi. (65)

This is exactly a matrix product state (with open boundary conditions) as in (49). So indeed as

anticipated, the MPS is entirely described by an ancilla system living in a (0+1)-dimensional space.

1.7 Projected entangled pair states

We were not entirely honest when we said that the MPS originated as the variational class that is sweeped over in DMRG. Actually the MPS, although it was then called ’finitely correlated state’,19

appeared in 1987 when Affleck, Kennedy, Lieb and Tasaki studied a model that was an extension of the Majumdar-Gosh chain (a Heisenberg spin model with next-to-nearest-neightbour interactions).[39][40] The AKLT Hamiltonian reads

ˆ HAKLT = 1 2

∑

_i ˆSi· ˆSi+1+ 1 3(ˆSi· ˆSi+1) 2_. ₍₆₆₎

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1.7 projected entangled pair states

Figure 11: Ground state of the AKLT model. At each site two virtual 2-dimensional Hilbert spaces are introduced.[4]

Of course the term 1/3 is not so random, as the researchers first constructed the ground state, and afterwards the Hamiltonian from which it originated. They were interested in the construction of a one-dimensional spin-1 state, with at each site two virtual spin 1₂ states. Just as in the Majumdar-Gosh model, adjacent virtual spins are combined into a singlet state

|Ψ−i_i,i₊₁ √1 2 |1 2;+ 1 2ii| 1 2;− 1 2ii+1− | 1 2;− 1 2ii| 1 2;+ 1 2ii+1 , (67)

as is depicted in figure 11by the blue line. This is exactly the Bell state (24) with maximal

entanglement that we discussed in subsection1.2.3. At the end of the day we project every

two virtual spins at site i onto a physical spin at that same site with a projection operator20 ˆ

P(i):C2×C2→C3 (68)

on the physical spin-1 state. This is depicted in figure 11 with the ellipses at each site.

Specifically, the projection operator reads ˆ P(i) = 2

∑

si=0 1 2

∑

α,β=−1₂ Asi α,β(i) |sii (hα|i⊗ hβ|i), (69)

so that we arrive at a state |Ψi_AKLT =Pˆ(1)_|_Ψ−_i 1,2⊗Pˆ(1)|Ψ−i2,3⊗ · · · ⊗Pˆ(N)|Ψ−iN,1 (70) = 2

∑

s1,...,sN 1 2

∑

α1,...,αN As1 α0,α1(1)A s2 α1,α2(2). . . A sN αN−1,αN(N) |s1, . . . , sNi, (71)

in which we supposed periodic boundary conditions α0= αN, so that the right virtual spin at site N is maximally entangled with the left virtual spin at site 1, and no unpaired ancillas remain.21

How can we then derive the parent Hamiltonian that corresponds to this ground state? For every adjacent pair of spin-1 states, two of the four virtual spin 1₂-states are stuck in a state with total spin zero. Therefore, each pair of these spin-1 states is forbidden from being in a spin-2 state. In other words, the projector operator on the spin-2 subspace, annihilates (70). We

20 Remember that a spin-s state spans a 2s+1-dimensional Hilbert space.

21 However, not doing this, opens up a range of possibilities. The AKLT state with open boundary conditions has a fourfold degeneracy, corresponding to the two non-restricted degrees of freedom left and right of the chain. This observation eventually leads to the conclusion that the state is in a symmetry-projected topologically ordered phase.[4]

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can thus obtain the AKLT Hamiltonian from the sum of projectorsQ_i,i+1on the spin-2 state. Comparing the AKLT state to (49), we see that it corresponds to a bond dimension D=2 MPS

with periodic boundary conditions. This observation is very important, as we now see clearly that the bond dimension does not only parametrize the amount of entanglement22

between two sites, it also corresponds to the dimension of a ’virtual Hilbert space’ living on each site of the bond.23

States like (70) are called valence bond solids, and it is clear that it is not difficult to

generalize these to higher-dimensional (D) virtual Hilbert spaces and/or higher-dimensional (d) physical Hilbert spaces. The projection operator is then a map

ˆ

P(i):CD×CD →Cd. (72)

Generalization to higher dimensions is now also piece of cake. In one dimension we got two ancillas per site, which is the coordination number of a one-dimensional chain. In two dimensions we do the same thing, introduce as many ancillas as the coordination number. For the two-dimensional square lattice, this corresponds to putting 4 virtual spins on each site, subsequently making maximally entangled pairs between sites an projecting in the end to physical spins.

1.7.1 Parent Hamiltonians

A D = 2 MPS is thus the exact ground state of some local24

Hamiltonian. Fannes, Nachter-gaele and Werner showed in 1992 that any MPS is the ground state of some gapped local Hamiltonian.[41] Combining Hasting’s result with this, we see that in fact we can replace the

original Hamiltonian with the parent Hamiltonian of the MPS approximation to the ground state of that original Hamiltonian. This replacement can be done without closing the gap, so that all gapped Hamiltonians in a certain phase can be represented by a particular parent Hamiltonian. This observation permits us to classify all gapped quantum phases with the MPS formalism, one of the great successes of MPS.[42]

22 In this case the entanglement between two sites, is just the entanglement of a Bell state of which we argued in subsection1.2.3that it is S=ln 2. This is independent of the length of the chain, and hence the ground state of the

AKLT Hamiltonian obeys an area law. This was to be expected from Hasting’s proof.[25] 23 In this case the virtual Hilbert space would be that of one electron.

24 Local in the sense that there are only nearest-neighbour interactions. It can also be shown that the AKLT Hamiltonian exhibits a spectral gap.[40]

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2

C O N T I N U O U S M AT R I X P R O D U C T

S TAT E S

The introduction to continuous matrix product states is quite similar to the introduction we did for tensor network states. Weakly interacting quantum field theories are only a small part of the story, which we however understand very well thanks to perturbation theory. Start from the ground state of a free QFT, and add interactions order by order making use of the language of Feynman diagrams. Perturbation theory fails however for strongly interacting quantum field theories, of which of course QCD is the most famous. Perturbation theory is blind to effects like confinement or the quark-gluon plasma, which are intrinsically non-perturbative. A clear understanding of the mass gap of QCD is still one of the (hardest) ways you can earn a million dollars.1

We’re not entirely honest when saying that we know nothing about non-perturbative effects in relativistic QFT. Lattice gauge theory is actually remarkably succesful in extracting such properties.[43] We can even use the discrete tensor network states of the

previous chapter to do this.[44] However such methods require a lattice cut-off which inevitably

breaks Euclidean invariance. Furthermore, due to terms like _e12 in the discretization of the

kinetic energy, energy optimizations are very sensitive to small movements in parameter space. Before we explain how we could possibly solve this, we must mention that the solution did not come from the motivation of variationally optimizing relativistic QFT’s. Many models in condensed matter physics are also continuous and strongly interacting. These non-relativistic QFT’s are the ones we want to study in optical lattice systems, like the one shown in figure1.

It turns out that once again input from the quantum information community provided insights. In section2.2we will describe how one can take the continuum limit of the matrix product

state to obtain a variational ansatz state directly in the continuum. We then go on by describing its succesful victory over Feynman’s objections in section2.1to use the variational principle

for quantum field theories. We write down an expression of the continuous matrix product state (cMPS) in Fock state, to make clear it is a state with a non-definite number of particles. We make the link with field coherent states, which are used e.g. to describe the mean field theory of cold Bose gases. We give a brief introduction to (field) coherent states in the appendixA. In subsection2.2.2we show that how to calculate expectation values, and observe that this can

be done very efficiently. We introduce the cMPS transfer matrix and sketch how we can use the generating functional formalism to extract observables from the cMPS. The cMPS inherits much of the properties of its discrete counterparts, as e.g. the overcomplete parametrization (gauge freedom). We will describe the transformations of the variational parameters that leave the state invariant in subsection 2.2.3. We discuss the subspace of uniform cMPS, that are

translation-invariant (which suffice in the most cases) and argue that the dominant eigenvalue of the transfer matrix should have a real part of zero, and all others should have negative real parts. This nice property is equivalent with the transfer matrix formalism in MPS. We conclude