Abelian and Non-Abelian Quantum Hall Hierarchies

Yoran Tournois Abelian and non-abelian quantum Hall hierarchies

Abelian and non-abelianAbelian hierarchies quantum Hall Yoran Tournois

ISBN 978-91-7911-190-8

Department of Physics

Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020

Abelian and non-abelian quantum Hall hierarchies Yoran Tournois Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Friday 4 September 2020 at 10.00 in sal FD5, AlbaNova universitetscentrum, Roslagstullsbacken 21.

Abstract A core tenet of condensed matter physics has been that different phases of matter can be classified according to Landau's symmetry breaking paradigm. It has become clear, however, that phases of matter exist that are not distinguished by symmetry, but rather by topology. A paradigmatic example of this topological order are the fractional quantum Hall phases, which are the topic of this dissertation. Such phases exhibit the fractional quantum Hall effect, which occurs when electrons confined to two dimensions are subjected to a strong perpendicular magnetic field at very low temperatures. Characterized by a precise quantization of the Hall resistance and a concomitant vanishing of the longitudinal resistance, the fractional quantum Hall effect results from the formation of a strongly correlated quantum liquid of electrons. This quantum liquid supports remarkable quasiparticle excitations, which carry a fractional charge and are thought to obey fractional statistics beyond the familiar Bose-Einstein and Fermi-Dirac statistics. The theoretical understanding of the topological orders realized by the fractional quantum Hall states has progressed by the proposal of explicit trial wave functions as well as various types of effective field theories. This dissertation focuses on two series of trial wave functions, abelian and non-abelian hierarchy wave functions. We study the non-abelian hierarchy wave functions using conformal field theory techniques, by means of which the associated topological properties are studied. These include the fractional charges of the quasiparticles and their non-abelian fractional statistics. In addition, we study abelian hierarchy wave functions using effective Ginzburg-Landau theories in a way that connects to their known conformal field theory description.

Stockholm 2020 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-181624

ISBN 978-91-7911-190-8 ISBN 978-91-7911-191-5

Department of Physics

Stockholm University, 106 91 Stockholm

ABELIAN AND NON-ABELIAN QUANTUM HALL HIERARCHIES

Yoran Tournois

Abelian and non-abelian quantum Hall hierarchies

Yoran Tournois ©Yoran Tournois, Stockholm University 2020

ISBN print 978-91-7911-190-8 ISBN PDF 978-91-7911-191-5

Printed in Sweden by Universitetsservice US-AB, Stockholm 2020 Abstract

A core tenet of condensed matter physics has been that different phases of matter can be classified according to Landau’s symmetry breaking paradigm. It has become clear, however, that phases of matter exist that are not distinguished by symmetry, but rather by topology. A paradigmatic example of this topological order are the fractional quantum Hall phases, which are the topic of this dissertation. Such phases exhibit the fractional quantum Hall effect, which occurs when electrons confined to two dimensions are subjected to a strong perpendicular magnetic field at very low temperatures. Characterized by a precise quantization of the Hall resistance and a concomitant vanishing of the longitudinal resistance, the fractional quantum Hall effect results from the formation of a strongly correlated quantum liquid of electrons. This quantum liquid supports remarkable quasiparticle excitations, which carry a fractional charge and are thought to obey fractional statistics beyond the familiar Bose-Einstein and Fermi-Dirac statistics. The theoretical understanding of the topological orders realized by the fractional quantum Hall states has progressed by the proposal of explicit trial wave functions as well as various types of effective field theories. This dissertation focuses on two series of trial wave functions, abelian and non-abelian hierarchy wave functions. We study the non-abelian hierarchy wave functions using conformal field theory techniques, by means of which the associated topological properties are studied. These include the fractional charges of the quasiparticles and their non-abelian fractional statistics. In addition, we study abelian hierarchy wave functions using effective Ginzburg-Landau theories in a way that connects to their known conformal field theory description.

1 Samanfattning på Svenska

De fraktionella kvanthalltillstånden är paradigmatiska exempel på topologiskt ord- nade materiefaser. Dessa uppträder när elektroner, vars rörelse är begränsad till en tvådimensionell yta, påverkas av ett starkt, mot ytan vinkelrätt, magnetfält, och då bildar en starkt korrelerad kvantvätska. Denna vätska karaktäriseras av en mycket precist kvantiserad Hallresistans, och har kvasipartiklar med remarkabla egenskaper som fraktionell elektrisk laddning. Dessa kvasipartiklar anses också vara anyoner, med vilket menas att de har en fraktionell kvantstatistik som in- terpolerar mellan Bose- och Fermistatistik. Vissa fraktionella kvanthalltillsånd, som de med fyllnadsfaktor ν = 5/2 och ν = 12/5, har tilldragit sig stort intresse eftersom de tros ha kvasipartiklar som är ickeabelska anyoner, vilka eventuellt kan användas för att bygga topologiskt skyddade kvantbitar. Vi undersöker två klasser av försöksvågfunktioner för fraktionella kvanthalltill- stånd — abelska och ickeabelska hierarkivågfunktioner. För att studera de topol- ogiska egenskaperna hos de nyligen föreslagna ickeabelska hierarkivågfunktionerna använder vi två olika tekniker. Flera av dessa egenskaper, såsom kvasipartiklar- nas fraktionella laddning, kan beräknas genom att studera vågfunktionerna i en endimensionell gräns, medan den ickeabelska statistiken kan bestämmas med hjälp av konform fältteori. Vågfunktionerna i den abelska hierarkin, och speciellt då de mest prominenta tillstånden i den lägsta Landaunivån, studerar vi genom att konstruera eﬀektiva Ginzburg-Landauteorier. På så sätt har vi etablerat ett nytt samband mellan de eﬀektiva fältteorierna och den redan existerande beskrivningen av de abelska hierarkitillstånden baserad på konform fältteori.

2 List of accompanying papers and contributions

Paper I Conformal ﬁeld theory construction for nonabelian hierarchy wave functions Yoran Tournois and Maria Hermanns Phys. Rev. B 96, 245107 (2017)

Paper II Braiding properties of paired spin-singlet and non-abelian hierarchy states Yoran Tournois and Eddy Ardonne J. Phys. A 53, 055402 (2020)

Paper III Microscopic derivation of Ginzburg-Landau theories for hierarchical quantum Hall states Yoran Tournois, Maria Hermanns and Hans Hansson SciPost Phys. 8, 079 (2020)

Contributions

Paper I I participated in all parts of the project, in particular ﬁnding the fusion rules of the non-abelian hierarchy wave functions in the thin torus limit and contributing to the conformal ﬁeld theory description. I contributed to the writing of all sections.

Paper II I initiated the project and performed all calculations presented in the paper, except for those in Appendix E. I contributed to the writing of all sections.

Paper III I participated in all parts of the project. Primarily, my contribution has been to introduce the phase representation, which was crucial for the precise connection to conformal ﬁeld theory. I also wrote the proof for the displaced derivatives representation of the Jain wave function. I performed all calculations in the paper and contributed to the writing of all sections.

3 Note

This dissertation is in some parts based on my Licentiate thesis [1]. In particular, the structure of the discussion in Chapter 6 is based on Chapter 4 in Ref. [1], and two ﬁgures in Chapter 1 appear in the licentiate thesis as well.

Acknowledgements

First and foremost, I would like to thank Maria for being a great supervisor. Thank you for suggesting interesting projects to work on and for giving me the opportunity to continue my PhD with you, first in Gothenburg and later in Stockholm. I am very grateful to you for the many discussions we’ve had and for making me feel welcome to drop by and ask a lot of questions. You always managed to provide a solution to a problem I was stuck on, or at least provide several good ideas on how to proceed. I would also like to thank you for the time you dedicated to help me prepare for numerous talks, and for your invaluable feedback on the practice runs. Finally, I want to thank you for your detailed proofreading of this thesis. I would also like to thank my other collaborators, Eddy and Hans. Thank you both for taking the time on so many occasions to answer my questions and for your proofreading of this thesis. Eddy, thank you for mentioning an earlier paper of yours that started our collaboration on the braiding project, for explaining parts of that paper in detail, and for teaching me about conformal field theory. Hans, thank you for a wonderful collaboration as well. I can’t imagine the number of hours we spent discussing the first part of our project, but I am proud of the result that came out of it. Thank you for teaching me about composite boson theories, and for the discussions we’ve had on several other ideas following our project. Thank you also for your help with the Swedish summary. I am also very grateful to all the terrific colleagues I was fortunate enough to meet. First, I would like to thank my office mates over the past years: Jan, Henry and Emilio in Cologne, Oleksandr and Jonatan in Gothenburg, and Iman, Carlos, Kang and Axel in Stockholm. Thank you for the many discussions we’ve had, on physics and life in general, and for creating a great atmosphere. I also want to thank all my other colleagues at the instutite for theoretical physics in Cologne, the departement of phyiscs in Gothenburg and the KOF group in Stockholm for the fun lunch and coffee breaks, and for making the departments great environments to work in. I want to especially thank Flore for our many chit-chats and for sharing her thesis template with me, Iman for sharing some great music, Themis for his support and the very interesting conversations we’ve had, Anders and Sören for being a part of the pre-defense committee, and Anshuman, Jan, Oleksandr and

4 Jonatan for the fun nights out in Gothenburg. I also wish to express my gratitude to my friends and family. First, I want to thank Dustin and Tom for being terrific friends and for brightening my day on so many occasions. I owe Tom a special thanks for letting me work at the CWI in Amsterdam a few times. I would also like to thank Emilio, Henry, and Jan for their friendship and for making my experience in Cologne a memorable one. Henry, thank you for being a great flatmate as well as a fun office mate. I would like to thank Ahmed for his friendship as well, and for the fun times we had hanging out in Stockholm. I am deeply grateful to my parents and my brother for their unconditional support and for their encouragement to pursue my goals, as well as their advice and their interest in my research. I also want to thank Lydia, Twan, Lois and Albert for their support and taking an interest in my PhD. Last, but certainly not least, I want to thank Suzanne for being my best friend and for always being there for me, already way before I started my PhD.

Contents

1 Introduction 9 1.1 The integer quantum Hall eﬀect ...... 13 1.2 The fractional quantum Hall eﬀect ...... 18

2 Trial and model wave functions 21 2.1 Laughlin states ...... 21 2.2 Other plateaus ...... 27 2.3 The Moore-Read wave function ...... 31

3 Conformal ﬁeld theory and the FQHE 35 3.1 Conformal ﬁeld theory ...... 35 3.2 The FQHE-CFT connection ...... 41 3.3 FQH wave functions from CFTs ...... 45 3.4 The Moore-Read conjecture ...... 52 3.5 Quasielectron operator ...... 53

4 CFT description of non-abelian hierarchy states 57 4.1 The Hermanns hierarchy ...... 57 4.2 Conformal ﬁeld theory construction ...... 60 4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states ...... 63 4.A WZW current algebras ...... 70

5 Ginzburg-Landau theories for hierarchical quantum Hall states 75 5.1 The GLCS theory for Laughlin states ...... 75 5.2 The GLCS-CFT connection ...... 80 5.3 GLCS theory for hierarchy states ...... 83 5.A Path integral ...... 90

6 Matrix Product states and the FQHE 93 6.1 Matrix Product States ...... 94

7 Contents

6.2 FQH states as matrix product states ...... 95

7 Conclusion 105

8 Chapter 1

Introduction

The pursuit to determine what matter is made of, and how those constituents interact, has led to the formulation of the Standard Model of particle physics. It describes the elementary particles, such as electrons, quarks, and the Higgs boson, and by classifying them according to the various charges they carry, it describes how they interact through fundamental forces. Particles with electric charge interact via the electromagnetic force, described by quantum electrodynamics, while the weak and color charges are influenced by the weak and strong forces, respectively. Although it does not include a description of gravity, the Standard model is an incredibly successful description of subatomic physics: famously, within quantum electrodynamics, the calculated value of the anomalous magnetic moment of the electron [2] agrees with its measured value to 9 significant figures [3]. Yet this understanding of elementary particles does not extrapolate to many- body systems, where there are phenomena at the collective level that are not exhibited by the microscopic degrees of freedom. Examples include ferromagnetism, hydrodynamic phenomena, and superconductivity. Instead of reducing systems to their fundamental constituents, the approach taken in condensed matter physics is in some sense the reverse, namely asking which phenomena can emerge by organizing particles in different ways. A natural question that arises is one of classification: which possible organi- zations – or orders – of the constituent particles can occur, and how are these distinguished? For a long time, it was thought that all phases of matter, and the phase transitions between them, could be described in terms of local symmetries associated with the organization of the particles. That is, two orders are distinct if they respect or break different symmetries [4], such as continuous or discrete spatial translation symmetry, (spin) rotational symmetry, parity symmetry, time reversal symmetry, chiral symmetry, et cetera. For example, a liquid is distinct from a crystal, as the former has continuous translation and rotation symmetry,

9 Chapter 1 Introduction which are broken down to discrete symmetries in the crystal phase. Another example is the breaking of spin rotation symmetry in ferromagnets, which acquire a net magnetization below the Curie temperature. However, in the advent of the discovery of the fractional quantum Hall effect [5] it was realized that this symmetry breaking paradigm is not exhaustive. Observed in two-dimensional electron systems subjected to strong perpendicular magnetic fields and low temperatures, the fractional quantum Hall effect (FQHE) is characterized by a precise quantization of the Hall resistance and a concomitant vanishing longitudinal resistance of the sample, insensitive to microscopic details such as the precise amount and distribution of disorder and the sample shape. Crucially, this effect is exhibited by phases of matter that have identical symmetry breaking patterns, indiscernible by means of local order parameters, but that are nevertheless distinct. For such phases, the Landau symmetry breaking paradigm is supplanted by description rooted in topology. Topological phases are characterized and distinguished by certain topological invariants and come, broadly speaking, in two varieties: phases that have symmetry protected topological order, and phases that have (intrinsic) topological order. The former are only distinguished by topological invariants in the presence of certain symmetries, such as time-reversal symmetry. Contrarily, phases that are topologically ordered, such as the phases that exhibit the FQHE, are distinguished by topological invariants even in the absence of symmetries. One example of such a topological invariant for topologically ordered systems is the degeneracy of the ground state on a topologically non-trivial surface [6, 7], such as a torus. Usually, such degeneracies are due to a symmetry of the Hamiltonian, but the topological ground state degeneracy is fundamentally different: it occurs irrespective of symmetries, and cannot be lifted by local perturbations. Another example is the topological entanglement entropy [8], which manifests as a constant contribution to the entanglement entropy, i.e. the von Neumann entropy of a reduced density matrix. Independent of length scales, the topological entanglement entropy indicates the occurrence of long-range entanglement present in topologically ordered states [9]. Perhaps the most captivating feature of topologically ordered systems is that they support collective excitations known as anyons. To explain this concept, we revisit the Standard Model, which also classifies the elementary particles according to their statistics. All elementary particles are eithers bosons, such as photons or the Higgs boson, or fermions, such as electrons and quarks. Formally, this dichotomy can be understood from the symmetry with respect to permutations of the particle coordinates, which is a consequence of the particles being indistinguishable. The wave function describing such particles should therefore transform according to an

10 irreducible representation of the permutation group, and there are only two such representations of dimension one – the trivial and the “sign” representations – that represent bosons and fermions, respectively. However, this argument depends subtly on the dimension of the underlying space [10]. The above considerations, which were implicitly in three-dimensional space, apply equally well to any (hypothetical) higher-dimensional case. However, they fail when the space is two-dimensional, where the permutation group is not the relevant symmetry group of the wave function. To appreciate this, consider a double exchange of two identical particles, which is equivalent to one particle encircling the other as shown in Fig. 1.1a. In dimensions D ≥ 3, such a process is trivial. Namely, the closed loop can be deformed and shrunk to a point, as shown in Fig. 1.1b. This is in accordance with the fact that any permutation squares to the identity. In two dimensions, however, a well-defined winding number can be associated to the closed loop1. As long as the particles are not coincident, no continuous deformation of the loop can change the winding number, which implies that the double exchange is distinct from not exchanging the particles. As a result, the exchanges are not described by permutations, which square to the identity. Mathematically, this is expressed by the statement that the first homotopy group of the configuration space of identical particles is Z2 for dimensions D ≥ 3, while it is given by Z for D = 2. Consequently, the group of permutations does not describe

(a) (b)

Figure 1.1: (a): a double exchange of identical particles, equivalent to one particle encircling another, has associated to it a winding number (equal to +1 as shown) (b): in three dimensions, the path may be lifted into the third dimension and shrunk to a point. the symmetry of the wave function1 1 of identical particles in two dimensions.1 1 Instead,

1Here, a distinction could be made between the winding number of the path in real space and that of the path in conﬁguration space. In the conﬁguration space of identical particles, a single exchange is a closed loop with unit winding number, while a double exchange has winding number two.

11 Chapter 1 Introduction the relevant symmetry group is the braid group, which can be viewed as the group of permutations σ for which the constraint σ2 = 1 is relaxed. Identical particles in two dimensions then correspond to the irreducible representations of the braid group. The one-dimensional representations are classified by an angle θ, such that to an exchange of identical particles is associated the phase eiθ. The fact that this phase need not square to 1 as in the higher-dimensional case means that there is no additional constraint on the angle θ – the particles are referred to as anyons [11] for this reason. The phases that exhibit the FQHE are thought to harbor such anyonic excitations, which carry a fractional charge and obey fractional statistics. Several such quasiparticles combine to form one electron, or expressed differently, in these phases the electron can effectively split up into several independent quasiparticles. While the occurrence of fractional charge in the FQHE has been established [12–15], a direct observation of fractional statistics has proved more elusive. Most efforts have centered on interferometry experiments [16, 17], but despite numerous attemps have not provided conclusive evidence yet. A recent experiment [18] did show direct experimental evidence of fractional statistics by measuring current correlations resulting from anyon collisions, but it remains to see if these results can be reproduced. In certain FQH phases it is thought that even more exotic anyons, corresponding to higher-dimensional representations of the braid group, can occur. These come about when the space of wave functions for fixed quasiparticle configurations is degenerate. If Ψa denotes a basis of degenerate wave functions, with a = 1, . . . , d, Pd a quasiparticle wave function Ψ is some linear combination Ψ = a=1 caΨa. Thus, if quasiparticles are exchanged, Ψ generally transforms according to a d × d matrix within the degenerate subspace. Because such transformations generally do not commute, these anyons are referred to as non-abelian anyons. Non-abelian anyons constitute a possible avenue towards the realization of a topological quantum com- puter [19, 20]. Here, the core idea is that non-abelian anyons can be used to encode quantum information in a non-local way, such that it is inherently protected from decoherence due to local perturbations.

Outline In the remainder of this chapter, we briefly describe the integer and fractional quantum Hall effects. Chapter 2 introduces several trial and model wave functions for fractional quantum Hall states, specifically the Laughlin model wave function and its multi-component extensions, the non-abelian Moore-Read wave function, and the trial wave functions resulting from the composite fermion and abelian hierarchy pictures. In Chapter 3, the connection between the FQHE and conformal

12 1.1 The integer quantum Hall eﬀect

field theory is discussed, beginning with a brief overview of conformal field theory. Subsequently, the wave functions introduced in Chapter 2 are studied again in this context. Chapter 4 pertains to results obtained in Papers I and II, introducing a series of non-abelian hierarchy wave functions which are given a conformal field theory description, and the braiding properties of which a subsequently studied using conformal field theory techniques. Chapter 5 contains the results found in Paper III, concerning the Ginzburg-Landau-Chern-Simons effective field theory of several quantum Hall states, in particular the abelian quantum Hall hierarchy. It also discusses the connection of this effective field theory to conformal field theory. Finally, in Chapter 6 the matrix product state formulation of FQH wave functions is discussed, with an outlook for the implementation for composite fermion states. The core results obtained and collected in this thesis are as follows. For a series of non-abelian hierarchy wave functions, a conformal field theory description is found (paper I). This description hinges on the conformal field theory structure of closely related wave functions, the braiding properties of which are studied, and this analysis is subsequently applied to the non-abelian hierarchy wave functions (paper II). Further, an effective Ginzburg-Landau-Chern-Simons theory for abelian hierarchy states is found, and an explicit connection to their conformal field theory description is established (paper III).

1.1 The integer quantum Hall eﬀect

We begin with a brief description of the classical Hall effect. Consider electrons moving in a two-dimensional plane, with a magnetic field B = Bzˆ and an in-plane electric field E = Exˆ. The magnetotransport can be determined within the Drude model, with the equation of motion p p˙ = −eE − ev × B − . (1.1) τ This describes electrons under the influence of the Lorentz force, with a scattering time τ which gives the characteristic time of the exponential decay of the momentum in the absence of external fields. In the steady state p˙ = 0, the equation of eτ eτ motion is rewritten as v + m v × B = − m E, which is Ohm’s law J = σE, relating the current J = −nev to the electric field. The conductivity in this case is a tensor, given by 2 ne τ 1 −ωcτ σ = 2 (1.2) m(1 + ωc τ) ωcτ 1 eB in terms of the cyclotron frequency ωc = m . The off-diagonal components indicate a response transversal to the electric field, due to the magnetic field. The resistivity

13 Chapter 1 Introduction

−1 B tensor ρ = σ has the oﬀ-diagonal component ρxy = ne , which is linear in the magnetic ﬁeld strength. Likewise, the Hall resistance, given by the voltage Vy in the y direction divided by the current Ix in the x direction, is given by

Vy B Rxy = = . (1.3) Ix ne In 1980, Klaus von Klitzing discovered the integer quantum Hall eﬀect (IQHE) in a silicon MOSFET [21], a discovery which was awarded the Nobel prize in 1985 [22]. See Fig. 1.2. The measured resistances deviate from the classical prediction: as a function of the perpendicular magnetic ﬁeld strength the Hall resistance exhibits h plateaus, where it takes precisely quantized integer values in units of e2 : 1 h Rxy = (1.4) i e2 with i integer. Concurrently the longitudinal resistance vanishes in the zero- temperature limit, indicating dissipationless transport. In particular, it is acti- vated, showing the Arrhenius behavior

∆ − 2k T Rxx ∼ e B (1.5) which indicates a gap in the spectrum.

Landau levels It is well known that the spectrum of non-interacting electrons in a magnetic ﬁeld consists of highly degenerate Landau levels, separated by the cyclotron energy. Assuming the electrons are spin-polarized, the Hamiltonian reads

(p + eA)2 H = (1.6) 2m where B = ∇ × A, and where m denotes the band mass. Introducing the kinetic momentum π = p + eA, we ﬁnd that its components form a pair of conjugate variables, as 2 ~ [πx, πy] = −i~e(∇ × A) · zˆ = −i . (1.7) `2 Here the magnetic length deﬁned by r ~ ` = (1.8) eB

14 1.1 The integer quantum Hall eﬀect is the typical length scale of the problem, given by ` ∼ 25 nm for a magnetic ﬁeld of 1 Tesla. Drawing on the analogy to the quantum harmonic oscillator, we form the raising and lowering operators

` a = √ (πx − iπy) 2 ~ (1.9) † ` a = √ (πx + iπy) , 2~ which obey [a, a†] = 1. The Hamiltonian therfore takes the form † 1 H = ~ωc a a + , (1.10) 2 with a spectrum consisting of equidistant Landau levels n = 0, 1, 2,... , and separated in energy by the cyclotron energy ~ωc. We refer to the level n = 0 as the lowest Landau level (LLL). Each Landau level is highly degenerate, as can be seen from the fact that the Hamiltonian (1.6) has only one degree of freedom while (1.10) has two. A convenient way to conceptualize the degeneracy is to introduce the guiding center coordinates that, viewed classically, form the center (X,Y ) about which the electron performs the cyclotron motion in the presence of a magnetic ﬁeld. In a quantum mechanical description, they are given by

1 1 X = x − πy,Y = y + πx, (1.11) mωc mωc which commute with the Hamiltonian but do not mutually commute. In fact,

[X,Y ] = i`2. (1.12)

In a semi-classical picture we find that the radius of the cyclotron motion is quantized due to the Bohr-Sommerfeld condition, while the Landau level degeneracy is a result of the non-commutativity of the guiding center coordinates. That is, similar to how phase space is discretized into regions of size 2π~ because the position and momentum operators do not commute, an area 2π`2 is associated to each quantum state in a Landau level by virtue of (1.12). The total number of states in a Landau level is therefore given by N = A/(2π`2) = AB , where A is the area of φ φ0 the sample and φ0 = h/e is the flux quantum. Thus, with AB = Φ being the total flux, the Landau level degeneracy is the number of flux quanta Φ/φ0 penetrating the sample.

15 Chapter 1 Introduction

Due to the Pauli principle, each state can be occupied by at most one spin- polarized electron. Hence, the number of occupied states is captured by the ﬁlling fraction

Ne ν = . (1.13) Nφ

The degenerate states in a given Landau level are then labeled by a pair of conjugate operators b, b†, with b = √1 (X + iY ) and [b, b†] = 1. A general state is therefore ` 2 given by a† nb† m |n, mi = √ |0i. (1.14) n!m! To ﬁnd explicit wave functions in terms of electron coordinates, it is convenient to choose a gauge for A.

B Symmetric gauge In the symmetric gauge A = 2 (−y, x, 0), so that the Hamil- tonian reads 1 B 2 B 2 H = − i~∂x − y + − i~∂y + x 2m 2 2 (1.15) 2 ¯ ¯ 1 = ~ωc − 4` ∂∂ − z∂ +z∂ ¯ + zz¯ , 4`2 where we introduced the complex coordinates z = x − iy and z¯ = x + iy, as well 1 ¯ as the holomorphic and anti-holomorphic derivatives ∂ ≡ ∂z = 2 (∂x + i∂y) , ∂ ≡ 1 ∂z¯ = 2 (∂x − i∂y). We ﬁnd the ladder operators

1 z 1 z¯ a = √ ( + 2∂¯) b = √ ( + 2∂). (1.16) 2 2 2 2

Thus, the single particle basis of the LLL, with a|ψi = 0 are

m − 1 |z|2 ψ0,m (z, z¯) = z e 4`2 . (1.17)

Since H is rotationally invariant in the symmetric gauge (i.e. it is invariant under z → eiαz, z¯ → e−iαz¯), angular momentum is a good quantum number. In particular, the ψm are the eigenstates of the z-component of angular momentum ¯ L = i~∂φ = i~ z∂ +z ¯∂ with eigenvalue m:

Lzψ0,m (z, z¯) = ~mψ0,m (z, z¯) . (1.18)

16 1.1 The integer quantum Hall eﬀect

Landau gauge In the Landau gauge, we let Ax = −By and Ay = 0, so that the momentum in the x direction is a good quantum number. The Hamiltonian can be written as " 2 2# 1 ` y `px H = ~ωc py + − , (1.19) 2 ~ ` ~ so that the eigenfunctions are

1 y `px 2 y `px ipxx/~ − 2 ( ` − ) ψpx,n(r) = e e ~ Hn( − ) (1.20) ` ~ in terms of the Hermite polynomials Hn. In this gauge, the eigenfunctions are plane waves in the x-direction, and have a Gaussian proﬁle (in the LLL) centered 2 at the position y = ` px/~.

Explaining the integer quantum Hall effect The integer quantum Hall effect can be understood by only considering the Landau level structure and the presence of disorder [23]. At integer filling ν = n, the non-interacting electrons fill n Landau levels, and the ground state wave function is a Slater determinant of the single particle states (1.17). The wave function for the filled LLL may be evaluated exactly, and reads

1 P 2 Y − |zi| Φν=1 (z1, . . . , zN ) = (zi − zj) e 4`2 i . (1.21) i

With n Landau levels filled, it is straightforward to show that the Hall resistance h is given by Rxy = ne2 . But any variation of the magnetic field – which leads to a change in the filling factor – will change the resistance, whereas the plateaus indicate the Hall resistance is constant. What is missing in the above picture is disorder. In the absence of disorder the system has continuous translation invariance, which yields a Hall resistance linear in the magnetic field. Disorder breaks the translation symmetry, while also lifting the Landau level degeneracy. Weak disorder localizes some electrons to potential minima (or maxima), while other states are essentially unaffected. The spectrum 1 splits into extended states near the unperturbed energies ~ωc(n + 2 ), and localized states in the region between the bands of extended states. While the gaps between Landau levels disappear, there is a “mobility gap” between the extended states, as the localized states do not contribute to the transport. If we imagine increasing the Fermi energy in the region of localized states, only localized states are filled. In this process, therefore, the Hall resistance is unchanged, which explains the plateaus.

17 Chapter 1 Introduction

The remaining subtlety is to explain why the Hall resistance is quantized as though n Landau levels were formed without disorder. This was explained in an argument by Laughlin in Ref. [23], later reﬁned by Halperin [24], which shows that the Hall resistance is always quantized as long as the Fermi energy lies in the mobility gap, ensuring that all extended states are ﬁlled. Later, this precise quantization was elucidated by relating the Hall conductivity σH to a topological invariant [25], quantized in units of e2/h. In particular, it was shown that the Kubo formula for the Hall conductivity of a 2d band insulator can be formulated in terms of a Chern number, which is always an integer for topological reasons. The introduction of this topological invariant set the stage for the theoretical understanding of topological insulators [26–28].

1.2 The fractional quantum Hall eﬀect

In 1983, Tsui, Störmer and Gossard discovered the fractional quantum Hall effect (FQHE) in a GaAs-AlGaAs heterostructure [5]. Such a structure consists of a layer of the semiconductor GaAs, on top of which is grown a layer of Aluminum-doped GaAs where a fraction x of Ga atoms are replaced by Al atoms. These semiconduc- tors have the same (zinc-blende) crystalline structure and nearly identical lattice spacing, allowing for a very thin interface. Their band gap is different, however, which leads to a discontinuity in the valence and conduction bands at the interface and, in turn, to the formation of a triangular well. By n-doping the AlGaAs layer, electrons drop into the GaAs layer and, confined by the triangular well, form a two-dimensional electron gas (2DEG). Because the dominant contribution to electron scattering at low temperatures is scattering due to the donors, the electron mobility is enhanced further by physically separating the donors froms the interface, referred to as modulation doping. The current highest-mobility samples have µ = 3 × 107 cm2/Vs, with an electron mean free path of 0.3 mm [29]. Superficially, the FQHE is similar to the IQHE, characterized by a quantized Hall resistance and a concurrent vanishing longitudinal resistance. The Hall resistance takes the values 1 h Rxy = (1.22) f e2 1 2 2 with f a rational number, such as f = 3 , 3 , 5 , ... - see Fig. 1.2. Currently, the FQHE has been observed in other systems [31] – most notably in graphene [32] – a testament to the universality of the phenomenon. It is natural to presume that the fractional quantum Hall effect can be explained in a similar manner as the integer quantum Hall effect, by the formation of a gap in the spectrum and its subsequent division into localized and extended states due to

18 1.2 The fractional quantum Hall eﬀect

Figure 1.2: The Hall resistance Rxy and the longitudinal resistance Rxx versus the magnetic field strength, showing plateaus in in Rxy at both integer 1 2 3 (ν = 1, 2, 3, ...) and fractional (ν = 3 , 5 , 7 , ...) filling fractions, and a concurrent vanishing Rxx. Figure reprinted from Ref. [30]. disorder. However, since the FQHE involves a fractional filling of a Landau level, the origin of this energy gap is different: to explain it, an energy gap must form as a result of to electron-electron interactions. In general, the partial filling is in the highest occupied Landau level, with an integer number of Landau levels fully filled. In the case that the typical energy of interactions is much less than the Landau level spacing ~ωc, Landau level mixing is suppressed and it is reasonable to restrict to the partially filled Landau level alone. A minimal description of the Hamiltonian, ignoring e.g. the Zeeman term, corrections due to the thickness of the sample and Landau level mixing, is Z Z 1 X 2 1 2 2 0 0 0 H = (pi − eA) + d r d r : ρ(r)V (r − r )ρ(r ): . (1.23) 2m 2 e i

Here me denotes the band mass of the electrons. The experimental data is explained if it can be argued that the spectrum is gapped for certain filling fractions. Note, however, that in restricting to one Landau level the kinetic term is constant, so that it drops out of the problem. The remaining Hamiltonian is therefore strongly coupled, as the interaction is effectively infinitely large compared to the kinetic

19 Chapter 1 Introduction energy. In the absence of interactions, the partially filled Landau level is vastly degenerate2 – as interactions are turned on, the Hilbert space rearranges itself dramatically so that a single ground state is selected. The great difficulty is in searching for this ground state without the help of a small interaction parameter. This challenge has sparked various approaches to better understand the physics in the fractional quantum Hall regime. Collectively, these approaches explain the FQHE as resulting from the formation of an incompressible – or gapped – quantum liquid by the electrons, which is a topologically ordered state of matter. This quantum liquid supports fractionally charged excitations in the bulk, thought be anyons, as well as gapless edge modes on the boundary. From the microscopic point of view a core approach has been to propose representative wave functions, following a seminal paper by Laughlin [33], leading to the prediction that the quasiparticle excitations are fractionally charged and obey fractional statistics [34]. Generally, these have a high overlap with the exact ground state, calculated numerically, for small system sizes. In certain cases they are model wave functions, i.e. they are the exact ground state of some idealized model Hamiltonian, separated from excited states by a gap. Their purpose is to capture the topological properties of the actual ground state, meaning they is in the same topological phase, with the hope that the model Hamiltonian can be con- tinuously connected to the physically relevant one without closing the gap. More generally, however, the wave functions are trial wave functions for which no parent Hamiltonian is known, but that are nevertheless aimed at describing the relevant topological quantum numbers. From the field theory point of view, effective field theories have been proposed in terms of particles that are composites of charge and flux, composite fermions [35] and composite bosons [36]. The former are electrons bound to an even number of flux quanta, experiencing a reduced magnetic field and allowing for a mapping onto the IQHE. The latter are electrons bound to an odd number of flux quanta, such that the external magnetic field is canceled. Another type of field theory proposed are the low-energy, long-wavelength effective field theories. In the simplest cases, these are Chern-Simons theories [37]. These encode the topological properties of the quantum Hall liquids, such as the filling factor, the genus-dependent degeneracy of the ground state and the quasiparticle charges.

2The typical electron density is 1015m−2, so that the number of electrons on a 1mm × 1mm 9 1 sample is about 10 . At ﬁlling ν = 2 , the number of ground states without interactions is 2×109 2·109 then 109 ∼ 2 .

20 Chapter 2

Trial and model wave functions

A core approach to the theoretical understanding of the fractional quantum Hall eﬀect has been to introduce trial wave functions, which the aim of capturing several topological properties of the fractional quantum Hall states. We begin by introducing the Laughlin model wave function, after which we consider the composite fermion and hierarchy picture, which provide a systematic approach for writing down trial wave functions for many ﬁlling factors. Additionally, we introduce the multi-component wave functions and the Moore-Read wave function, the latter being the simplest example of a non-abelian wave function.

2.1 Laughlin states

Ground state In a seminal paper [33], Laughin proposed a model wave function to explain the 1 FQH plateaus at ν = q , with q odd. It reads

q Y q − 1 P |z |2 4`2 i i ΨL (z1, . . . , zN ) = (zi − zj) e , (2.1) i

1 2 2 M m − (|z1| +|z2| ) φM,m(z) = (z1 + z2) (z1 − z2) e 4`2 , (2.2) where M, m are the center of mass- and relative angular momentum, respec- Q tively. Motivated by this, Laughlin made the Jastrow ansatz Ψ = i

21 Chapter 2 Trial and model wave functions

1. The function λ should be a polynomial since Ψ should be a LLL wave function, i.e. a ﬁnite linear combination of the single particle orbitals (1.17);

2. The polynomial λ should be homogeneous, meaning all terms have the same degree, since the wave function can be chosen to be an eigenstate of angular momentum L = z∂z for a central potential; 3. It should be antisymmetric in order for Ψ to describe electrons. These requirements fix λ(z) = zm for some integer m, which should be odd due to the third criterion. It is straightforward to show that the resulting wave function 1 has filling fraction ν = m , so that m = q, giving Eq. (2.1). For q = 1, (2.1) reduces to the exact wave function (1.21) for the filled LLL. This model wave function builds in good correlations between the electrons, as it vanishes rapidly when two electrons are taken to the same point. Thus, it promotes configurations where electrons are kept apart, lowering the potential energy. Consequently, for small system sizes, the Laughlin wave function has a high overlap with the numerically calculated ground state for various central potentials 1 [33]. For low filling factors ν . 7 , it is expected that the Wigner crystal [38] is favored over the Laughlin state [39]. It is important to emphasize that the Laughlin wave function is not the exact ground state of the Coulomb Hamiltonian Eq. (1.23). It is, however, the densest zero-energy eigenstate of the model Hamiltonian [40]

q−1 X X Hq = vl Pm(ij) (2.3) m=0 i

22 2.1 Laughlin states

Excitations

The wave function Eq. (2.1) is a model wave function that describes the ground state. An excited state can be described by piercing a flux tube at a position w = (w, w¯) and changing the flux from zero to the flux quantum φ0 adiabatically [33]. This flux change induces an azimuthal electric field, which pushes away the liquid radially because of the Hall response, thus leaving a local deficit of charge – a quasihole. At the level of the wave function, this is achieved by multiplying with a factor Q i(zi − w): the general model wave function for m such quasiholes is

N m N q Y Y Y q − 1 P |z |2 4`2 i i ΨL({z}; {w}) = (zi − wj) (zi − zj) e . (2.4) i=1 j=1 i

Because the spectrum of the model Hamiltonian Hq is periodic with respect to inserted flux, with period φ0, the state obtained by inserting a quasihole at w = 0 1 is also a zero energy eigenstate of Hq, albeit at a lower density . The effect of inserting a flux quantum at w = 0 is to increase the angular momentum of all electrons by one as compared to the ground state. The quasiholes carry a fractional charge. A simple way to see this is to take q such quasiholes at positions w1, . . . , wq to the same point w. This leads to a many-quasihole wave function that looks like the usual model wave function, with an electron removed at w. In fact, removing an electron is precisely equivalent to creating q holes [42], so that the charge associated to one quasihole is qh = e/q. It is natural to expect that the quantum Hall liquid also supports localized excitations with an excess charge −e/q. Such excitations, called quasielectrons, have been harder to describe [43]. The original proposal by Laughlin [33] amounts to lowering the angular momentum of electrons by taking derivatives, i.e.

q Y ∂ w¯ Y − 1 P |z |2 q 4`2 i i ΨL({z}, w) = 2 − 2 (zi − zj) e (2.5) ∂zj ` i i

1The quasihole state is an excited state if we include a conﬁning potential

23 Chapter 2 Trial and model wave functions

Plasma analogy

Several key insights into the Laughlin model wave function are based on the so- called plasma analogy, formulated by Laughlin [33], where one relates the modulus q squared of ΨL to as a Boltzmann weight for a classical one-component plasma in two dimensions. In the general case with quasiholes, one writes

q 2 −βU |ΨL({z}, {w})| = e (2.6)

2 and setting β = q , the energy U reads

2 X X q X 2 U = −q log(|zi − zj|) − q log(|zi − wj|) + |zi| . (2.7) 4`2 i

This is the energy of a classical two-dimensional one-component plasma with impurities, interacting via the Coulomb interaction. To see this, recall the Poisson equation in two dimensions

− ∇2φ(r) = 2πρ(r). (2.8)

Using the identity ∇2 ln |r − r0| = 2πδ(r − r0) as well as ∇2r2 = 4, the potentials for a point charge q and a uniform charge density are

ρ = qδ(r): φ(r) = −q log(|r|) 2πρ¯ (2.9) ρ =ρ ¯ : φ(r) = − |r|2. 4

Thus, the first term in (2.7) is the potential energy U = qφ between point charges of charge q, and the last term is the potential energy of these charges due to the 1 constant charge density ρ¯ = − 2π`2 . The second term describes the interaction of the charge q particles with unit charge impurities, which reflects the fractional charge of the quasiholes. Equation (2.6) indicates that the most likely configurations of electrons are those for which the Hamiltonian U is minimized. In the absence of impurities, the plasma minimizes the energy by neutralizing the background charge density ρ¯, leading to 1 a uniform density n = 2πq`2 , which is the electron density at filling ν = 1/q. For q . 70, numerical results indicate that the plasma (with temperature and density dependent on q) is in a liquid phase [44]. In that case, if impurities are present, they are screened by the plasma so that the average density is restored on the Debye length scale.

24 2.1 Laughlin states

Fractional statistics The fractionally charged quasiparticles are expected to obey fractional statistics, meaning they are anyons [10, 11]. The statistics are found by exchanging the quasiparticles and finding the effect on the wave function, as was first performed in [45]. Imagine moving one quasihole at w1 along a closed loop C, encircling another quasihole at w2 in a counterclockwise fashion. This state is separated from other excited states, which include more quasiparticles, by a gap. Performing the exchange sufficiently slowly, the adiabatic theorem implies that the wave function returns to itself, up to a phase. Specifically, the gap to additional excitations sets the time scale with which the moving quasiparticle should traverse the path C in order for the adiabatic theorem to apply. The resulting phase is most generally expressed as the Berry holonomy I B = M exp i Awdw + Aw¯dw¯ (2.10) C which consists of two parts: the monodromy M and the well-known Berry phase, in terms of the Berry connection A = (Aw, Aw¯). Here

q d q Aw = −ihΨL(w1, w2)| |ΨL(w1, w2)i, (2.11) dw1 and a similar expression holds for Aw¯. The monodromy is the explicit transformation of the two-quasihole wave function, which can be non-trivial if it is multi-valued in the quasihole coordinates2. For the quasihole wave functions (2.4), the monodromy is trivial (M = 1), so the Berry holonomy is given by the Berry phase. However, there is a freedom in the choice of the quasihole wave functions, in particular in factors involving quasihole-quasihole correlations. Different choices yield different Berry phases, but also different monodromies, such that the combined Berry holonomy B is independent of the choice made. Wave functions with non-trivial monodromies follow naturally from the conformal field theory approach, which we consider in the next chapter. To calculate the Berry phase, the two-quasihole wave function has to be normal- ized. Writing q Y Y − 1 P |z |2 q 4`2 i i ΨL({z}, w1, w2) = N (w1, w2) (zi − wj) (zi − zj) e , (2.12) i,j i

1 1 2 2 3 − 2 (|w1| +|w2| ) the normalization factor N is written N = N |w1 − w2| q e 4q` . The plasma analogy then shows that if the plasma is in a screening phase, the remaining 2The wave function is only required to be holomorphic in the electron coordinates {z}. 3Doing so gives a treatment on equal footing of the impurities and the particles in the Coulomb plasma.

25 Chapter 2 Trial and model wave functions

factor N does not depend on w1, w2, as long as their separation is large compared R Q 2 q 2 to the Debye length. The partition function Z = i d zi |ΨL| takes the form 1 1 2 2 Z = Z0 exp log |w1 − w2| − (|w1| + |w2| ) (2.13) q 2q`2 with Z0 independent of the wi. Then, the Berry connection reads

i ∂ log Z i 1 iw¯1 Aw1 (w1, w¯1) = − = − + 2 , (2.14) 2 ∂w1 2q w1 − w2 4q` and similarly for Aw¯1 . Subsequently, the Berry holonomy reads

i 2π + eΦ B = e q q~ , (2.15) which captures the eﬀect of the presence of the quasihole w2 as well as the Aharonov- Bohm phase due to the enclosed ﬂux Φ of the loop C. Thus, the fractional phase i π associated to a single exchange is therefore given by e q . Such fractional statistics does not apply to the quasielectron described by the wave function Eq. (2.5). In particular, numerics has shown their braiding phase to be path-dependent [46]. Another proposal for a quasielectron wave function is discussed in Chapter 3, which does yield quasielectrons wave functions that encode the expected braiding properties [47].

Read operator Following an observation of Girvin and Macdonald [48], Read showed that the Laughlin state has oﬀ-diagonal long-range order in the operator [49]

† † φR(r) = ψ (r)Vq(z) R 2 0 † 0 0 (2.16) = ψ†(r)eq d r ln(z−z )(ψ (r )ψ(r )−ρ¯),

† a composite of an electron ψ and an order q vortex Vq(z). Here ρ¯ denotes the mean density. q 0 0 † q Speciﬁcally, Read showed that the expectation value hΨL|Vq(z )ψ(r )ψ (r)Vq(z)|ΨLi 0 † acquires a non-zero value in the limit of large separation |z − z | → ∞. Thus, φR may be viewed as a non-local order parameter of the Laughlin state. Accordingly, the Laughlin state may be constructed in analogy to a Bose condensate

1 Z N |Ψq i = d2rφ† (r) |0i (2.17) L N! R

26 2.2 Other plateaus

† in terms of φR, which is bosonic for q odd. Indeed, the fermionic wave function q h0|ψ(r1) ··· ψ(rN )|ΨLi reproduces (2.1). Moreover, the operators U1 create quasihole wave functions, and taking q of them to the same point amounts to removing an electron from the condensate. Thus, this perspective corroborates the description of the quasihole as an object with vorticity one and charge e/q.

2.2 Other plateaus

The Laughlin model wave function provides a good description of the filling factors ν = 1/q (q odd), as well as their particle-hole conjugates at ν = 1 − 1/q. However, the experimental data in Fig. 1.2 reveals many more plateaus occurring at filling factors that are not of this form. Two pictures have emerged, that explain the states at other plateaus by con- structing trial wave functions: the composite fermion picture [50] and the hierarchy picture [40, 51]. These two approaches differ substantially, in that the trial wave functions are constructed in very different ways. However, the trial wave function agree in some cases and generally describe the same topological order [52]. Be- sides these pictures, multi-component trial wave functions have been proposed for bilayer systems, or in systems where the electron spins are important.

Composite Fermions The composite fermion (CF) picture [50] is centered on the idea that the relevant particles of the FQHE are electrons bound to an even number of vortices, called composite fermions. This picture establishes powerful connection between the integer and fractional quantum Hall effects and provides numerically very accurate trial wave functions. Following [53], consider non-interacting electrons at integer filling factor ν∗ = n ∗ ∗ ∗ in a magnetic filed B , which is either positive or negative. Thus, ν = ρφ0/|B |. Now, imagine attaching thin flux tubes to the electrons, of strength 2p (with p integer) in units of the flux quantum φ0. This means there is an additional auxiliary magnetic field, which is zero everywhere except for the positions of the electrons, with the aforementioned associated flux. This procedure of attaching thin flux tubes is not observable: it does not affect the spectrum or the statistics of the particles. Next, imagine spreading the additional flux adiabatically – with the time scale set by the energy gap – until it forms a uniform auxiliary magnetic field. The total magnetic field is then

∗ B = B + 2pρφ0, (2.18)

27 Chapter 2 Trial and model wave functions with ρ the density of electrons. Thus, the electrons experience the ﬁeld B and have ρφ0 the ﬁlling factor ν = B , or

± ρφ0 n ν = ∗ = . (2.19) B + 2pρφ0 2pn ± 1 Here the signs depend on the sign of B∗, while B is always positive. This yields a series of filling fraction referred to as the positive and negative Jain series, depending on the sign. These filling fractions correspond to prominently observed FQH plateaus in the lowest Landau level. Referring to the data in Fig. 1.2, taking p = 1 gives the leading, i.e. most prominent, positive and negative Jain series: 1 2 3 ν+ = , , ,... 3 5 7 2 3 4 ν− = , , ,... 3 5 7 These sequences of filling fractions are particle-hole conjugates, and converge to ν = 1/2. Similar series follow by letting p = 2, 3, .... We turn to the trial wave functions for these filling fractions. The flux-attachment procedure outlined above yields a mean-field wave function, which has several un- satisfactory properties [53]. However, it may be modified slightly [50] in a way that amounts to having electrons bind to vortices instead of flux quanta, which yields very good trial wave functions. For the positive Jain series, these read

Y 2p Ψν+ ({z}) = PLLL (zi − zj) Φν∗=n, (2.20) i

+ where Φν∗=n is the wave function for n ﬁlled Landau levels, and ν implicitly depends on n, p. The (full) Jastrow factor attaches 2p vortices to all electrons, and the operator PLLL projects the wave function onto the LLL. This projection is achieved by replacing z¯ → 2∂z [54]. The negative Jain series has similar trial wave functions, denoted

Y 2p Ψν− ({z}) = PLLL (zi − zj) Φν∗=−n. (2.21) i

∗ Here Φν∗=−n refers to the wave function of n ﬁlled Landau levels for B < 0. Because the reversal of the sign of B∗ amounts to complex conjugation of the ∗ coordinates z, z¯, we have Φν∗=−n = (Φν∗=+n) . The CF picture is a very successful picture that accounts for most of the observed LLL plateaus. Generally, the CF trial wave functions have very high overlaps with exact ground states for small system sizes [53]. Additionally, CF theory explains

28 2.2 Other plateaus the absence of a plateau at ν = 1/2. At half-filling, the effective magnetic field B∗ vanishes, i.e. the composite fermions experience no net magnetic field. Thus, absent any interactions, they form a compressible CF Fermi sea [55]. This picture has been validated experimentally [56, 57] by measuring the radius of the semiclassical cyclotron orbit of the charge carriers around ν = 1/2. One example of an observed plateau that does not fit into the CF sequences given above, is the plateau at ν = 4/11. Within the CF paradigm, it occurs as a consequence of residual interactions between composite fermions. Thus, contrary to the positive and negative Jain series which are understood as integer quantum Hall effects of composite fermions, this state is to be understood as a FQHE of composite fermions [53] While the CF picture is very successful, it should be emphasized that it is still not as well understood as the Laughlin states. In particular, there is no plasma analogy for CF states, and there is no known model Hamiltonian for which the Ψν± are ground states (see however [58]).

Hierarchy picture The heuristic picture that motivates the hierachy description of fractional quantum Hall states is as follows. At the center of the plateau, the quantum Hall state is an incompressible liquid with a uniform density. If the magnetic field is changed uni- formly, localized quasiparticle excitations are created, corresponding to an excess or a deficit in the density. These quasiparticles are pinned to impurities for small deviations of the magnetic field but ultimately become itinerant as the magnetic field is further changed, leading to a loss of the quantized Hall response. Since these quasiparticles are charged particles in a magnetic field, the idea is that they may form a Laughlin-like state on top of the original “parent” liquid. This picture may be iterated, and it can be shown that any filling factor with an odd denominator can be obtained from successively condensing quasiparticles in Laughlin states [40, 51]. The idea behind the hierarchy picture is as follows [51]: the hierarchy wave function at level n + 1 is obained from the many-quasiparticle wave function Ψn at level n by Z Y 2 ∗ Ψn+1({z}) = d wi Φ ({w})Ψn({z}, {w}) (2.22) i which may be viewed as the formation of a correlated state of quasiparticles de- ∗ ∗ scribed by Φ [42] in the state Ψn. Referred to as a “pseudo wave function”, Φ is typically of Laughlin type and various choices yield different daughter wave functions at different densities.

29 Chapter 2 Trial and model wave functions

As a microscopic description, the hierarchy picture is not particularly realistic. It centers on the quasiparticle picture, viewing quasiparticle are largely independent objects, which is questionable when their density becomes large. Indeed, because of their finite extent, the individual nature of the quasiparticle breaks down before the density needed for them to condense into a Laughlin-like state is reached. Moreover, explicitly evaluating the integrals in (2.22) is exceedingly difficult, and has limited exact diagonalization studies to very small system sizes only [59]. Nonetheless, general arguments can be given that predict quasiparticles of charge e p ± q for a filling factor ν = q [40]. Additionally, it is expected that the energy to create such particles is proportional to their charge. Thus, from the Arrhenius behavior of the longitudinal resistance, we expect the Coulomb energy gap, the 1 energy needed to create a quasihole-quasielectron pair, to be proportional to q . In turn, this implies that as the denominator q increases, the states become pro- gressively less stable, which is indeed borne out in experiments (see Ref. [60] and references therein). Moreover, the trial wave functions correspond to a definite topological order, and the associated topological properties agree with that of the composite fermion picture [52] for those states at the same filling factor. An exception to the aforementioned difficulty in evaluating the integrals (2.22) is the series of states that result from the successive condensation of quasielectrons. In this case, the integrals can be evaluated exactly, and the resulting wave functions are identical to the CF wave functions in the positive Jain series [61, 62].

Multi-component states

Additional model wave function have been proposed for systems with multiple spin species [63], such as systems where electrons are not spin-polarized due to e.g. a weaker magnetic ﬁeld, or bilayer systems. Omitting the Gaussian factors, Halperin proposed wave functions of the form

(m,m,l) Y ↑ ↓ m Y ↓ ↓ m Y ↑ ↓ l Ψ ({z}) = (zi − zj ) (zi − zj ) (zi − zj ) , (2.23) i

30 2.3 The Moore-Read wave function

trix Kαβ, so that

n Y Y (α) (α) Kαα Y Y (α) (β) Kαβ ΨK ({z}) = (zi − zj ) (zi − zj ) (2.24) α=1 i

(α) where z denotes a coordinate in Mα. The K-matrix is a symmetric n × n matrix of positive integers: for (2.23), the components are the particle spins and m l K = l m . The n = 2 and n = 4 cases are of the most direct signiﬁcance, the former being relevant for bilayer systems and systems where electrons are not spin-polarized, while the latter applies to graphene where the electrons have spin as well as valley indices. The relevance of the general multi-component states is that they are closely related to certain composite fermion wave functions. For example, the ν = 2/5 CF wave function Ψν+=2/5 and the (3, 3, 2) Halperin wave function are related by

Y (3,3,2) Ψ2/5({z}) = A{ ∂ (2) Ψ ({z})} (2.25) zi i where A is the antisymmetrization operator. Additionally, derivatives with respect to the coordinates in group M2 are taken. Such relations are non-trivial, and were derived in Refs. [61, 62] from the hierarchy picture. Speciﬁcally, the right hand side of (2.25) is the densest daughter state of the ν = 1/3 Laughlin state, obtained by condensing quasielectrons, and was shown to exactly reproduce the ν = 2/5 CF wave function.

2.3 The Moore-Read wave function

The aforementioned composite fermion and hierarchy picture capture the topological properties and provide trial wave functions for states in the lowest Landau level. The CF picture is widely believed to accurately describe the most prominently observed states, while the hierarchy picture provides trial wave functions for any filling fraction. A common feature of the LLL states is that the filling fractions are always of the form ν = p/q with q odd, and p, q relatively prime. This “odd-denominator” rule is ascribed to the fermionic nature of the electrons, as is most clearly seen in the Laughlin model wave function. For this reason, the observation [64] of a plateau at half filling in the second Landau level, at ν = 5/2, was a major surprise. Initially, it was thought this was due to a spin-singlet quantum Hall state [65], motivated by the fact that the state vanishes upon applying a tilted magnetic

31 Chapter 2 Trial and model wave functions

field. Later numerical and experimental work showed, however, that the state is spin-polarized [66, 67]. The plateau at ν = 5/2 is in striking constrast to the half-filled lowest Landau level, described by a compressible CF Fermi sea. A better comparison can be made by using the fact that the LLL and second Landau level are in one-to-one correspondence, meaning the Coulomb interaction in the second Landau level can be mapped onto an effective Coulomb interaction in the lowest Landau level. Evi- dently, it is these effective interactions that give rise to the stark difference between both states. One of the leading candidates for the ν = 5/2 plateau, the Moore-Read wave function [68], has the interpretation that due to the effective interactions, the CF Fermi sea at half-filling has a p-wave BCS pairing instability. This pairing can result from the overscreening of the effective Coulomb interaction by the composite fermions [69], leading to attractive residual interactions between them. Originally written down using techniques in conformal field theory [68], the proposed model wave function reads 1 2 ΨMR({z}) = Pf ΨL({z}) (2.26) zi − zj 2 where ΨL denotes the bosonic Laughlin ν = 1/2 wave function (including the Gaussian factor). The “Pfaffian” Pf introduces the pairing correlations between electrons and agrees asymptotically with the pairing wave function for the BCS p-wave superconductor in the weak-pairing phase [70, 71]. This factor is defined by the antisymmetrized sum Pf( 1 ) = A{ 1 ··· 1 }, which modifies the zi−zj z1−z2 zN−1−zN 1 correlations due to the Jastrow factor. The filling factor is 2 : the filling ν = 5/2 = 2+1/2 is viewed as resulting from the filling of two additional, inert Landau levels. The Moore-Read wave function is the densest zero-energy eigenstate of a model Hamiltonian [71], similar to the Laughlin wave function. In the following, it will prove fruitful to consider the bosonic Moore-Read wave function, explicitly given by 1 1 P 2 bos Y − 2 i |zi| ΨMR({z}) = Pf (zi − zj)e 4` . (2.27) zi − zj i

bos X HMR = A δ(zi − zj)δ(zi − zk) (2.28) i

32 2.3 The Moore-Read wave function

An important corollary is that the bosonic Moore-Read wave function admits an equivalent representation [72]. Namely, the following wave function is also the densest zero-energy eigenstate of (2.28), meaning it equals (2.27). Up to a normalization, bos X 2 2 ΨMR({z}) = ΨL(S1)ΨL(S2). (2.29) S1,S2

Here S1,S2 are subsets of the coordinates {z} of equal size, which can be thought of as layers. The sum amounts to a symmetrization over all inequivalent ways of dividing the particles over these layers, and we refer to (2.29) as the symmetrized representation of the bosonic Moore-Read wave function. Consequently, the fermionic Moore-Read wave function has the equivalent expressions 1 2 ΨMR({z}) = Pf ΨL({z}) zi − zj (2.30) X 2 2 Y = ΨL(S1)ΨL(S2) (zi − zj). S1,S2 i

We turn to quasiholes in the Moore-Read state. Repeating Laughlin’s flux inser- tion argument by threading a unit flux quantum, one finds a model wave function for a “Laughlin quasihole” that amounts to multiplying the Moore-Read wave func- Q tion with a factor i(zi − w). The pairing picture described above, however, suggests there is a quasihole of smaller charge, associated with pair breaking. These fundamental quasiholes correspond to half a flux quantum, and can only be created in pairs. In the symmetrized representation, the model wave function for a single pair of quasiholes reads

X Y Y 2 2 ΨMR({z}, w1, w2) = (zi − w1) (zi − w2)ΨL(S1)ΨL(S2) S1,S2 zi∈S1 zi∈S2 Y (2.31) × (zi − zj) i

In this representation, the fundamental quasihole corresponds to an ordinary Laugh- lin quasihole in one of the layers. Taking w1, w2 → w, we see that the wave function for a Laughlin quasihole is reproduced. This also shows that the quasiholes carry a charge half of that of the Laughlin quasihole, i.e. their charge is e/4. The fundamental quasiholes are non-abelian anyons. As discussed in Chapter 1, they occur when the space of quasihole wave functions is degenerate for fixed quasihole configurations. Fixing quasiholes at w1, . . . , w4, one indeed finds two linearly

33 Chapter 2 Trial and model wave functions

independent four-quasihole wave functions, written Ψ12;34 and Ψ13;24, where

X Y Y 2 2 Ψab;cd({z}, {w}) = (zi − wa)(zi − wb) (zi − wc)(zi − wd)ΨL(S1)ΨL(S2) S1,S2 zi∈S1 zi∈S2 Y × (zi − zj). i

5 Other candidates for ν = 2 The Moore-Read, or “Pfaffian”, wave function is among the leading candidates to describe the plateau at ν = 5/2. Chief among the other candidates is its particle- hole conjugate, the “Anti-Pfaffian” wave function [75, 76]. In the absence of Landau level mixing these are equal in energy, but numerics has shown that when LL mixing is included, the Anti-Pfaffian is favored [77]. Another candidate is the particle-hole Pfaffian (PH-Pfaffian) state [78]. Although 5 numerics seem to indicate that the 2 state is either in the Pfaffian or Anti-Pfaffian phase, recent observations [79] of a half-integer quantized thermal conductance of the ν = 5/2 state, indicative of non-abelian fractional statistics, were found to be consistent with the PH-Pfaffian phase. However, there is an ongoing debate [80–82] regarding the interpretation of this experiment, the agreed upon conclusion at the time of writing being that more research is needed.

34 Chapter 3

Conformal ﬁeld theory and the FQHE

In this chapter we discuss the connection between conformal field theory (CFT) and the fractional quantum Hall effect. We begin with a brief overview of CFT in Section 3.1, emphasizing its most important concepts. Conformal field theory being a vast area on its own, this overview is by no means comprehensive, and the reader is referred to Refs. [83, 84] for excellent introductions. In Section 3.2, we discuss the connection between CFT and the effective field theory of the FQHE, Chern-Simons theory. Next, in Section 3.3, we discuss concretely how FQH wave functions are obtained using conformal field theory techniques. Finally, in Section 3.4 we introduce the Moore-Read conjecture.

3.1 Conformal ﬁeld theory

A conformal field theory is a quantum field theory with an action invariant under conformal transformations, defined as coordinate transformations that do not change angles. In terms of the metric, gµν (x) → Λ(x) gµν (x) under a conformal transfomration, with Λ a local scale factor. The set of invertible conformal transformations forms a group, consisting of translations, rotations, dilations and so-called special conformal transformations [83]. In the following, we are interested in conformal field theories in two dimensions. These are special, because in two dimensions, the infinitesimal conformal transformations form an infinite-dimensional algebra. Here, a distinction should be made between the infinitesimal transformations that generate the group elements, which form a finite-dimensional subalgebra, and the remaining generators, which generate conformal transformations that can not be defined globally. We consider the complex plane, introducing coordinates z = x0 + ix1 and z¯ =

35 Chapter 3 Conformal ﬁeld theory and the FQHE x0 − ix1. In these coordinates, the inﬁnitesimal conformal transformations are z → z + (z) and z¯ → z¯ + ¯(¯z). The generators of such transformations are

n+1 n+1 `n = −z ∂z, `¯n = −z¯ ∂z¯, (3.1) where n ∈ Z. The total algebra consists of two independent copies of the Witt algebra

[`n, `m] = (n − m)`n+m, [`¯n, `¯m] = (n − m)`¯n+m, [`n, `¯m] = 0 (3.2)

Together with their conjugates, the generators `−1, `0, `1 form the subalgebra which generates the conformal group, while the remaining `n generate the local conformal transformations. Upon quantization, this conformal symmetry is broken. This is referred to as the conformal anomaly: while the action is invariant under conformal transformations, the path integral is not. In the quantum theory the Witt algebra is replaced by its central extension1, the Virasoro algebra

c 2 [Ln,Lm] = (n − m) Ln+m + n n − 1 δn+m,0 (3.3) 12 with an isomorphic algebra for the L¯n. The central charge c is a constant, the value of which depends on the CFT in question. In conformal field theory, a central role is played by the energy momentum tensor Tµν. Conformal invariance implies that the energy momentum tensor is µ ν traceless, T µ = 0, and yields a set of conserved currents Jµ = Tµν associated with infinitesimal transformations xµ → xµ +µ. In complex coordinates, Tzz¯ = Tzz¯ = 0, and the conserved currents are Jz = T (z)(z) and Jz¯ = T¯(¯z)¯(¯z), where T (z) = Tzz n+1 and T¯(¯z) = Tz¯z¯. For (z) = z , the conserved charge associated to the current is precisely I 1 n+1 Ln = dz T (z)z , (3.4) 2πi so that X −n−2 T (z) = Lnz , (3.5) n∈Z in terms of the generators obeying (3.3). Thus, any conformal field theory has a symmetry algebra Vir⊕Vir of holomorphic and anti-holomorphic Virasoro algebras, generated by the modes of the energy

1Upon quantization, one considers projective representations of the Witt algebra – this is equivalent to considering the linear representations of its central extension

36 3.1 Conformal field theory momentum tensor. The Hilbert space of a unitary2 conformal field theory then corresponds to a unitary representation of this algebra. Important representations of the Virasoro algebra are the highest weight representations, determined by a highest weight state denoted |h, h¯i from which all other states in the representation are obtained by acting with the modes Ln, L¯n. In particular, L0|h, h¯i = h|h, h¯i, L¯0|h, h¯i = h¯|h, h¯i (3.6) Ln|h, h¯i = 0 for n ≥ 1 i.e. the Ln≥1 are raising operators. The Ln≤−1 are lowering operators, and each state |h, h¯i generates an infinite tower of descendant states, of the form ¯ L−k1 ··· L−kn |h, hi, with 1 ≤ k1 ≤ · · · ≤ kn. Such a highest weight state is created by a primary field. These are fields Φ(z, z¯) which transform covariantly with respect to the (local) conformal transformations z → f(z), z¯ → f¯(¯z), that is

h¯ df h df¯ Φ(z, z¯) = Φ0 f(z), f¯(¯z) . (3.7) dz dz¯

We say the field Φ has conformal dimensions h, h¯. The highest weight state |h, h¯i is simply the application of Φ on the vacuum |0i, in the infinite past t → −∞, which is referred to as the state-field correspondence. Similarly, descendant states are created by descendant fields. Together, the primary field Φ and its descendants form conformal families. As irreducible representations of the Virasoro algebra, these transform among themselves under conformal transformations. The objects of interest are typically the N-point correlation functions of primary fields, as they can be used to determine correlation functions of their descendants. These read

hΦ1(z1, z¯1) ··· ΦN (zN , z¯N )i = h0|RΦ1(z1, z¯1) ··· ΦN (zN , z¯N )|0i, (3.8) and have the following salient features:

1. They are radially ordered, as indicated by the symbol R. Radial ordering is the counterpart to time ordering in the complex coordinates z, z¯: smaller (larger) magnitudes |z| correspond to earlier (later) times.

2A CFT is unitary if there is positive deﬁnite inner product of states; in this case, the space of states is a Hilbert space.

37 Chapter 3 Conformal ﬁeld theory and the FQHE

2. They are constrained by the conformal symmetry. In general, these constraints are formulated as the conformal Ward identities (CWIs), in terms of the holomorphic and anti-holomorphic components of the energy momentum tensor. The holomorphic CWI reads

X hi ∂zi hT (z)Φ1 (z1, z¯1) ··· ΦN (zN , z¯N )i = + 2 z − z i (z − zi) i (3.9)

× hΦ1 (z1, z¯1) ··· ΦN (zN , z¯N )i.

3. They are built out of conformal blocks. These are purely holomorphic and purely anti-holomorphic functions F, F¯ that solve the holomorphic and anti- holomorphic CWIs. Generally, the correlation function decomposes as the sum X 2 hΦ1(z1, z¯1) ··· ΦN (zN , z¯N )i = |Fi(z1, . . . , zN )| . (3.10) i

The singular behavior of such correlators is captured by the operator product expansion (OPE), which gives the local, singular, behavior of a product of two local ﬁelds in a CFT in terms of other ﬁelds in the CFT. For example, the holomorphic CWI is encoded in the OPE

hi 1 T (z)Φi(zi, z¯i) ∼ 2 Φi(zi, z¯i) + ∂zi Φi(zi, z¯i). (3.11) (z − zi) z − zi

Such OPEs are to be thought of inside correlation function and include only the singular terms. Generally, the OPE for two ﬁelds takes the form

¯ ¯ ¯ X c hc−ha−hb hc−ha−hb Φa(z1, z¯1)Φb(z2, z¯2) ∼ Cab(z1 − z2) (¯z1 − z¯2) Φc(z2, z¯2), (3.12) k where the fields Φa, Φb and Φc may be primary fields, or descendants thereof, and c where the structure constants Cab are determined by the three-point functions hΦaΦbΦci. It is natural to ask which fields Φc can occur in the OPE of two fields Φa and Φb. This is encoded in the fusion rules, formulated in terms of the conformal families. Denote by [Φa] and [Φb] the conformal families to which the fields Φa, Φb belong. The OPE (3.12) of Φa and Φb contains fields Φc, which belong to conformal families [Φc]. Then, the fusion rules read

X c [Φa] × [Φb] = Nab[Φc], (3.13) c

38 3.1 Conformal ﬁeld theory

c where the Nab are non-negative integers, which are non-zero only if ﬁelds in the conformal familiy [Φc] appear in the OPE. The fusion rules may be viewed as the decomposition of a tensor product of two irreducible representations (conformal families) into irreducible representations, analogous to the addition of angular mo- P c menta. Often such fusion rules are simply written Φa × Φb = c NabΦc in terms of the primary ﬁelds that generate the relevant conformal families.

Example: the free boson CFT The simplest example of a conformal ﬁeld theory is arguably that of a free, massless boson theory in two dimensions, with c = 1. In complex coordinates z, z¯, the action reads Z 1 2 S = d z ∂zφ (z, z¯) ∂z¯φ (z, z¯) , (3.14) 8π and it is straightforward to determine the two-point function

hφ (z, z¯) φ (w, w¯)i = − log |z − w|2. (3.15)

The canonical quantization of φ is performed by considering the cylinder geometry, where periodic boundary conditions apply, before mapping the cylinder onto the plane. The resulting mode expansion of the ﬁeld φ is [83]

X an −n a¯n −n φ (z, z¯) = φ0 − iπ0(log z + logz ¯) + i z + z¯ , (3.16) n n n6=0 where an, a¯n are raising (lowering) operators for n < 0 (n > 0). Here [φ0, π0] = i, and writing π0 = a0, [an, am] = nδn+m. (3.17) An important feature of this action is that it has a U(1) symmetry. Namely, by virtue of the symmetry of the action under φ(z) → φ(z) + φ0, it has a conserved U(1) Noether current j (z) = i∂zφ, and a similar conserved current ¯j. Given the mode expansion (3.16), the holomorphic current reads

X −n−1 j (z) = i∂φ(z) = anz (3.18) n∈Z in terms of the modes an, which obey (3.17). An independent algebra is formed by the modes of the current ¯j. As usual, the U(1) symmetry leads a conserved Noether 3 1 H charge Q = 2πi dzj (z), and the mode expansion shows that Q = π0 = a0. 3This is the usual expression for the conserved Noether charge, as a ﬁxed length |z| indicates ﬁxed time

39 Chapter 3 Conformal ﬁeld theory and the FQHE

As a0 commutes with all other operators, the Hilbert space is built from states |αi with U(1) charge α, i.e. a0|αi = α|αi. These are the highest weight states, created by primary fields Vα in the infinite past. The primary fields are the vertex operators iαφ(z) Vα(z) = :e : , (3.19) with :: denoting normal ordering (see [83]). 1 The energy momentum tensor of the free boson theory reads T (z) = − 2 : ∂φ(z)∂φ(z):, from which the Virasoro generators Ln, expressed in terms of the operators an, follow. Explicitly

1 X Ln = : aman−m : . (3.20) 2 m

It can then be shown that these obey the commutation relations (3.3), with c = 1. In the following, a key role is played not precisely by the free boson CFT, but rather CFTs of free bosons with a slight twist. These bosonic fields obey an additional condition φ ∼ φ+2πR, meaning they can be thought of as angular variables. Here R is referred to as the compactification radius. For these CFTs, the primary fields Vα are only well-defined if the U(1) charge α is an integer multiple of 1/R

inφ(z)/R Vn(z) =: e : . (3.21)

Extended symmetries Generally, any CFT has the Virasoro algebra as its symmetry algebra, but they may have a larger, extended symmetry. CFTs having only the Virasoro algebra as the symmetry algebra are referred to as the “minimal” models, one example being the Ising CFT. As discussed above, the Virasoro algebra is generated by the modes Ln of the conserved current T (z), the holomorphic component of the energy momentum tensor. Similarly, conformal field theories having an extended symmetry have a larger symmetry algebra generated by modes of additional conserved currents j(z). Various possible extensions exists, loosely classified by the conformal dimension of the additional currents j: our interest here is for extensions where j has conformal dimension 1. The free boson CFT is an example of a CFT with an extended symmetry. The current j = i∂φ has conformal dimension 1, and its modes form the algebra (3.17). In terms of these modes, the Ln defined by (3.20) give the c = 1 Virasoro algebra. The symmetry encoded in the commutation relations of the modes of the conserved currents j(z) is often expressed in terms of operator product expansions.

40 3.2 The FQHE-CFT connection

This encodes the same information. The OPEs of the currents form a closed algebra, called the chiral algebra. For the free boson CFT, it is called the U(1) chiral algebra, and the relevant OPE is

1 j(z)j(z0) ∼ . (3.22) (z − z0)2

In the following, we consider non-abelian generalizations of this chiral algebra. These correspond to CFTs known as Wess-Zumino-Witten models [85, 86]. Gener- ally, their chiral algebras are denoted gk where g is the Lie algebra of a Lie group G, and k is the so-called level. Relevant for us here are the cases G = SU(N). The a chiral algebra of the su(N)k WZW model has conserved currents j that obey the general OPE kδab if abcjc(z0) ja(z)jb(z0) ∼ + . (3.23) (z − z0)2 z − z0 Here f abc denote the structure constants of the algebra su(N) which are zero in the abelian case (3.22).

3.2 The FQHE-CFT connection

The relevance of conformal field theory for the fractional quantum Hall effect warrants justification. Namely, CFT conventionally describes (quantum) critical points invariant under a change of length scale, while the FQHE is not a critical phe- p nomenon, and has an explicit length scale: the magnetic length ` = ~/eB. Instead, the connection to CFT comes about because of a deep relation between the low-energy effective field theory of the FQHE – Chern-Simons theory – and conformal field theory. We proceed by introducing Chern-Simons theory, before discussing this deep connection.

Chern-Simons theory The low-energy effective field theories that describe abelian fractional quantum Hall states are Chern-Simons (CS) theories. They are topological quantum field theories (TQFTs), which means all correlations of local fields do not depend on the metric, but only on the topology of the underlying manifold. That Chern-Simons theories are the effective field theories of quantum Hall states can be argued for in two ways. Firstly, field theories that give an exact microscopic description of the Laughlin and multi-component states can be formulated [36]. These are discussed in more detail in Chapter 5. In practice these field theories

41 Chapter 3 Conformal field theory and the FQHE can only be used in certain mean-field approximations, but within those approximations, Chern-Simons theory emerges as the low-energy effective theory upon integrating out the microscopic degrees of freedom. Secondly, it can be argued that, constrained by several physical conditions relevant to the fractional quantum Hall states, their low-energy effective theory must be of Chern-Simons type. These conditions include gauge invariance, locality and rotational symmetry, as well as the absence of time-reversal and parity symmetry due to magnetic field. Following [87, 88], in 2 + 1 dimensions the conservation µ µ 1 µνσ ∂µJ = 0 implies that the current may be written J = 2π ∂νaσ, where aσ is a gauge field. We then seek an action in terms of this gauge field consistent with the aforementioned constraints, in the long-distance limit. The possible terms arising in this action can be organized according to their mass dimension, and the most dominant term – in the renormalization group sense – yields the Chern-Simons action Z k 3 µνσ SCS = d x aµ∂νaσ. (3.24) 4π The Chern-Simons term obeys rotation symmetry and breaks time-reversal and parity symmetry, but the discussion regarding its gauge invariance is subtle. Under gauge transformations aµ → aµ + ∂µλ, the Chern-Simons action is gauge invariant only up to a boundary term. With respect to such gauge transformations, therefore, the CS action is gauge invariant on manifolds without a boundary, but not on manifolds with a boundary such as a typical FQH sample. This is discussed in more detail in the next section. Besides this, the behavior of the Chern-Simons theory with respect to large gauge transformations, which cannot be connected to 2 the identity, dictates that k should be an integer in units of e /~. To see that the action Eq. (3.24) indeed describes a quantum Hall state, we couple it to an external field Aµ. This field describes perturbations of the background magnetic field which has already been taken into account by the Chern-Simons µ term. Including a term AµJ , the total action reads Z 1 3 µνσ S = SCS − d x aµ∂νAσ. (3.25) 2π

We may then integrate out the CS gauge field aµ to obtain the effective action 1 R 3 µνρ in terms of Aµ, given by Seff [Aµ] = 4πk d x Aµ∂νAρ. From this follows the response of the current to the perturbation Aµ:

µ δSeﬀ 1 µνρ hJ i = = ∂νAρ. (3.26) δAµ 2πk The spatial components of this equation indicate the transversal response to the

42 3.2 The FQHE-CFT connection electric ﬁeld. Restoring factors of e and ~, we ﬁnd

2 i 1 e ij hJ i = Ej, (3.27) k h

1 e2 such that the Hall conductivity is σH = k h . Thus, the CS action captures the Hall response for the Lauhglin states at ν = 1/k. The general abelian quantum Hall states have been classified according to extensions of the action (3.24) [37], referred to as the Wen-Zee classification. The i action is given in terms of multiple gauge fields aµ, i = 1, . . . , n, coupled to one another via the so-called K-matrix. The CS gauge fields can couple to an exter- µ nal field Aµ, and to currents j which describe the quasiparticle excitations. We consider the simplest minimal coupling where each gauge field couples only to one current. In addition, for curved surfaces, the gauge fields can couple to the spin 4 connection ωi. The associated coupling constants are collected in a charge vector t, a quasiparticle charge vector l, and a spin vector s. The general action reads Z 3 µνρ 1 i j 1 i 1 i j µ j S = d x Kija ∂νa − tiAµ∂νa − siωµ∂νa + l j a . (3.28) 4π µ ρ 2π ρ 2π ρ i i µ

This action encodes the topological data for an abelian quantum Hall state. Specif- ically, the topological ground state degeneracy is d = (det K)g where g is the genus of the surface; the ﬁlling fraction is given by ν = tT K−1t; the electric charge of T −1 quasiparticle i is qi = −et K li; and the clockwise braiding phase of distinct T −1 T −1 quasiparticles i, j is θi,j = 2πli K lj, while if i = j θi = πli K li. The shift 1 S, deﬁned by S = ν Ne − NΦ, is the correction to the naively expected relation Ne = νNΦ on curved surfaces such as the sphere. This is also a topological number, 2 T −1 given by S = ν t K s.

Conformal field theory A deep connection between Chern-Simons theory and conformal field theory exists, first elucidated by Witten [89]. For the simple Laughlin states ν = 1/k, this connection can be appreciated by considering the Chern-Simons theory (3.24) on a manifold with a boundary. As discussed in the previous section, the Chern-Simons action is not gauge invariant on such a manifold: indeed, letting aµ → aµ + ∂µλ, the Lagrangian density changes according to

k µνρ LCS → LCS + ∂µ(λ∂νaρ), (3.29) 4π

4 ij The spin connection is related to the Ricci scalar by R = 2 ∂iωj .

43 Chapter 3 Conformal field theory and the FQHE which does not vanish if a boundary is present. This violation of gauge invariance µ 1 µνσ indicates that the current J = 2π ∂νaσ is not conserved on the boundary – this suggests that an additional current must be present on the boundary, such that the total current is conserved. Indeed, such an additional current is present, carried by the gapless FQH edge modes on the boundary, as discussed briefly in Chapter 1. These edge modes occur because the Landau levels cross the Fermi energy as a result of the confining potential. Importantly, the sum of the edge current and the current J µ, is conserved – put another way, the total action, consisting of the CS action and an action that governs the edge modes, is gauge invariant. The action that governs the edge has been shown by Wen [90], using classical hydrodynamics considerations, to be given by the so-called chiral Luttinger liquid model, with the Lagrangian k L = (∂xφ∂tφ − v∂xφ∂xφ) . (3.30) edge 4π 1 Here φ is related to the density at the edge by ρ = 2π ∂xφ, and obeys the constraint ∂tφ = v∂uφ. This describes a massless (chiral) boson CFT. Thus, the Chern- Simons theory for Laughlin states is closely related to the conformal field theory of a massless chiral boson. A more direct connection was established by Witten [89], which relates the bulk properties of more general Chern-Simons theories to certain conformal field theories. In particular, (3.24) is a U(1) Chern-Simons theory, as the gauge field aµ is a U(1) gauge field. More generally, one may consider Chern-Simons theories based on any Lie group G, Z k µνρ a a 2 abc a b c SCS = (aµ∂νaρ + f aµaνaρ), (3.31) 4π M 3 a where aµ is a gauge field valued in the Lie algebra of G. Here M = Σ × R, where Σ is two-dimensional and compact, and R is interpreted as time. Then Witten showed that, quantizing the Chern-Simons theory, one obtains a finite-dimensional Hilbert space HΣ which can be identified with the space of conformal blocks of the corresponding Wess-Zumino-Witten CFT. This connection may be envisioned as follows. In the low-energy limit of the FQH state on Σ, described by Chern-Simons theory, the particles correspond to sources that carry quantum numbers, such as charge and spin. These sources are represented by Wilson loops in the CS theory, which trace out (closed) worldlines in 2+1 dimensions. Fixing a time, these worldlines intersect the constant-time surface at a set of distinct points z1, . . . , zn, carrying quantum numbers λ1, . . . , λn. Then, the Hilbert space of physical states on the constant-time surfaces is isomorphic to a vector space of conformal blocks. In particular, the quantum numbers

44 3.3 FQH wave functions from CFTs

λ1, . . . , λn correspond to representations of a chiral algebra A, which characterizes the CFT. The sources in the Chern-Simons theory therefore correspond to ﬁelds in the conformal blocks.

3.3 FQH wave functions from CFTs

Given the connection between Chern-Simons theory and CFT, Moore and Read suggested to use CFT to analyze and propose new FQH trial wave functions in a highly inﬂuential paper [68]. We discuss the Laughlin states, the multi-component states, and the Moore-Read state.

General discussion The manifestation of the CFT-FQHE connection is that the trial wave functions for the ground state and excited states are represented as conformal blocks, or holomorphic correlation functions, in a CFT. Schematically, the ground state wave function is given by Ψ({z}) ∼ hV (z1) ··· V (zN )iCFT (3.32) where the ﬁelds V in the CFT represent the electrons. Similarly, quasihole wave functions are schematically given by

Ψ({z}, {w}) ∼ hH(w1) ··· H(wm)V (z1) ··· V (zN )iCFT, (3.33) where the fields H represent the quasiholes. There are several conditions that must be met for the CFT to represent an acceptable FQH wave function. We mention several crucial conditions here, following the discussion in Ref. [91] to which the reader is referred for a more complete list. The conditions are as follows 1. The CFT has to be unitary [92], and rational. Unitarity is necessary (but not sufficient) to describe a gapped state, while rationality is equivalent to there being a finite number of particle types.

2. The chiral algebra A of the CFT must contain a U(1) chiral algebra, which means the particles carry a quantum number that corresponds to electric charge.

3. The operators H that represent the quasiholes must be relatively local to the electron operators V , which means that their operator product expansion obeys V (z)H(w) ∼ (z − w)lH˜ , where l ≥ 0 is integer and H˜ is another quasihole ﬁeld. Physically, this means that any braiding of an electron and

45 Chapter 3 Conformal ﬁeld theory and the FQHE

a quasihole is trivial, which implies that the quasihole wave functions are single-valued in the electron coordinates. The chiral algebra is assumed to be a product A = C ⊗ u(1) of a u(1) chiral algebra that describes the electric charge, and a neutral chiral algebra C. In the simplest case the neutral algebra is trivial, and the particles carry only an electric charge. Generally, the ﬁeld V and H factor into neutral ﬁelds and vertex operators for the u(1) chiral algebra

V (z) = Φ(z)eiαφ(z) (3.34) H(w) = Σ(w)eiβφ(w), where α, β describe the electric charges. A salient feature of the correlation functions is that they are vacuum expectation values. For the charge sector u(1), this means that all charges must add to zero in order for the correlation function to be non-zero, or equivalently that all vertex operators must fuse to the identity. This condition is referred to as charge neutrality. A similar requirement holds for the correlator in the neutral sector C, which is only non-zero if the fields Φ and Σ fuse to the identity. However, charge neutrality is explicitly violated in the charge sector. Namely, if α denotes the charge associated to one electron, then N electron operators have charge Nα. To resolve this issue, a background charge operator Obg with a total charge −Nα is added. One choice is to place the appropriate charges at infinity, which amounts to changing the “out state” h0| to hNα|. Another choice, which we make in this thesis, amounts to spreading out the compensating charge in a uniform way, which has the additional benefit of reproducing the Gaussian factors.

Laughlin states The Laughlin states for ν = 1/q are characterized by the chiral u(1) algebra only, i.e. the neutral algebra is trivial. The electron and quasihole operators are the vertex operators √ V (z) = ei qφ(z) (3.35) √i φ(w) H(w) = e q , with U(1) charges q and 1, respectively. Here, normal ordering is implicit. We proceed to show that the ground state wave function is given by

√ √ √ R 2 q i qφ(z1) i qφ(zN ) −iρ¯ q d z φ(z) ΨL({z}) = he ··· e e i. (3.36) √ R 2 Here Obg = exp[−iρ¯ q d z φ(z)] is the spread-out background charge operator, where ρ¯ = (2πq`2)−1, which ensures charge neutrality as discussed above. Without

46 3.3 FQH wave functions from CFTs the background charge operator, the following identity for the correlator holds (see [83]): P α α h0|φ(z )φ(z )|0i h0|eiα1φ(z1) ··· eiαN φ(zN )|0i = δP e i

√ √ √ R 2 1 P 2 i qφ(z1) i qφ(zN ) −iρ¯ q d z φ(z) Y q − |zi| he ··· e e i = (zi − zj) e 4`2 i . (3.38) i

Before proceeding with the quasihole wave functions, we discuss brieﬂy the √ U(1) charge. The U(1) charge of the operator ei qφ(z) is q, and has the interpretation of vorticity. The association of a vorticity q to each electron is in accordance with the non-local order parameter given in Eq. (2.16). The conservation of this U(1) charge corresponds to particle number conservation, which is explained by the incompressibility of the state – it takes a ﬁnite amount of energy to add a particle, and therefore to change the total vorticity. Although the vorticity is not the same as the electric charge, the two are closely related. We turn to the quasihole wave functions, which result from inserting the operators H in the correlator. The calculation is a straightforward generalization of the one outlined above for the ground state wave function, and yields

√i √i √ √ q q φ(w1) q φ(wm) i qφ(z1) i qφ(zN ) ΨL({z}, {w}) = he ··· e e ··· e Obgi 1 P 2 1 P 2 Y 1/q Y Y q − i |zi| − i |wi| = (wi − wj) (zi − wj) (zi − zj) e 4 4q . i

5An interesting perspective on this disregard of background-background correlations, resulting from the connection between CFT and the eﬀective Ginzburg-Landau-Chern-Simons theories, is given in Appendix B.

47 Chapter 3 Conformal ﬁeld theory and the FQHE

To see this, we consider the exchange of w1 and w2 and evaluate the associated monodromy of Eq. (3.39). We recall that due to radial ordering, the wave function is strictly speaking only defined on the wedge |w1| > |w2| > ··· > |wm|. Thus, in order to determine the monodromy, the wave function must be analytically continued. Doing so, a counterclockwise exchange gives the monodromy M = eiπ/q, which is indeed non-trivial. At the same time, the Berry phase associated with this exchange is given only by the Aharonov-Bohm phase. Therefore, although the specific contributions of the monodromy and the Berry phase are different as compared to their contributions for the Laughlin many-quasihole wave function, the total Berry holonomy is the same. Evidently, the CFT representation of the Laughlin state naturally selects a form of the many-quasihole wave function for which the statistics is given by the monodromy. This feature, referred to as “holonomy = monodromy”, is conjectured to hold more generally and will play a crucial role in Chapter 4.

Multi-component states The CFT description of the multi-component states is a straightforward generalization of that of the Laughlin state. The chiral algebra is A = u(1)n, where n denotes the number of components. We restrict to those multi-component states 6 that have a description in terms of a positive definite K-matrix Kαβ, of full rank, where α = 1, . . . , n. In those cases, K = QQT , (3.40) where Q is an n × n matrix. The CFT description involves a set of n independent 0 0 compactified bosons φα, which obey hφα(z)φβ(z )i = −δαβ ln(z − z ). In terms of these fields, the n vertex operators Vα are given by

iQαβ φβ (z) Vα(z) = e , (3.41) where summation is implied by repeated indices. The decomposition (3.40), and therefore the speciﬁc form of the vertex operators (3.41), is not unique. In light of the decomposition of A into a charge and a neutral sector, however, it can be convenient to choose Q such that one ﬁeld carries the U(1) charge. For example, for the Halperin (m, m, l) wave function we may choose

 q m+l q m−l  2 2 Q =   (3.42) q m+l q m−l 2 − 2 ,

6In Chapter 5, we consider negative deﬁnite K-matrices in the context of the negative Jain series.

48 3.3 FQH wave functions from CFTs

in which case the ﬁeld φ1 describes the U(1) charge, while φ2 describes spin. In the n-component case, we denote the n groups by Mα and the number of particles in group Mα by Nα. Then,

n Nα Y Y (α) ΨK ({z}) = h Vα(zi )Obgi (3.43) α=1 i=1

R 2 reproduces (2.24). Here, Obg = exp[−iρ¯α d z Qαβφβ(z)] is the background charge operator, where ρ¯α denotes the mean density of the component α.

The Moore-Read state The Moore-Read wave function (2.26) was originally written down using CFT techniques [68], before its interpretation in terms of composite fermions, discussed in Section 2.3, was given. Its CFT description is in terms of the su(2)2 Wess- Zumino-Witten CFT. The associated neutral chiral algebra√ is the Ising CFT, and the charged sector has compactification radius R = 2. Specifically, the electron operator is √ V (z) = ψ(z)ei 2φ(z) (3.44) where φ is the compactified boson and ψ is a free chiral fermion. It is a Majorana fermion, i.e. ψ† = ψ, and it is primary field of the Ising CFT. The correlator factors into correlators of the neutral and charged sectors. For an even number of electrons, √ √ i 2φ(z1) i 2φ(zN ) ΨMR({z}) = hψ(z1) ··· ψ(zN )ihe ··· e Obgi

1 P 2 1 Y 2 − |zi| (3.45) = Pf (zi − zj) e 4`2 i . z − z i j i

The second factor is obtained as in the Laughlin case. The Pfaffian factor is obtained by first observing that by virtue of the fusion rule ψ×ψ = 1, the correlator in the fermion sector is zero unless the number of electrons is even. This is an example of a charge neutrality constraint, where we may envision a “topological charge” associated to the fermions ψ (such that a pair is charge neutral). For an even number of electrons, Wick’s theorem applied to the two-point function 0 1 1 1 1 hψ(z)ψ(z )i = 0 yields an antisymmetrized product A{ ··· }, z−z z1−z2 z3−z4 zN−1−zN which is precisely the Pfaffian factor. The quasihole operator H has a similar form, and is given by √ H(w) = σ(w)eiφ(w)/2 2. (3.46)

49 Chapter 3 Conformal ﬁeld theory and the FQHE

Here σ is another primary field of the Ising CFT, referred to as a spin field. This quasihole operator is the operator with smallest charge for which, when inserted in the correlator, the many-quasihole wave function is single-valued in the electron coordinates. In particular, its U(1) charge is 1/4 that of the electron. Within the su(2)2 theory, H is a primary field with respect to the chiral algebra, while the electron operators V are closely related to the currents themselves7 – this is worked out in Appendix 4.A. Inserting two quasihole operators and omitting the Gaussian factors, we obtain

ΨMR({z}, w1, w2) = hH(w1)H(w2)V (z1) ··· V (zN )Obgi 1 h ··· i − 8 = σ(w1)σ(w2)ψ(z1) ψ(zN ) (w1 w2) (3.47) Y 1 Y 2 × (zi − wj) 2 (zi − zj) . i,j i

(p) (p) ΨMR({z}, {w}) = hσ(w1) ··· σ(w4)ψ(z1) ··· ψ(zN )i Y 1/8 Y 1/2 Y 2 (3.48) × (wi − wj) (zi − wj) (zi − zj) , i

(p) (p) (p) ΨMR({z}, {w}) = A ({w})Ψ12;34({z}, {w}) + B ({w})Ψ13;24({z}, {w}). (3.49) 7The operators V are Virasoro primaries, but WZW descendants. 8The alternative case, where the spin ﬁelds and the fermions fuse to ψ, has the same counting.

50 3.3 FQH wave functions from CFTs

Here the expansion coeﬃcients A(p),B(p) depend only on the quasihole coordinates, and contain information about the monodromy of the conformal block. In this sense, the relation between the symmetrized wave functions Ψ12;34, Ψ13;24 and the (p) conformal blocks ΨMR mirrors the relation between the two representations of the quasihole wave functions in the Laughlin case.

CFT representation for the symmetrized representation A CFT representation that directly yields the symmetrized representation of the Moore-Read state also exists, and will in fact be used in the next chapter. First, the “electron” operator for the bosonic Moore-Read state is given by

V (z) = ψ(z)eiφ(z), (3.50) which yields the correct wave function as can checked by comparing to Eq. (3.45). Then, the idea is to rewrite the fermion ψ, expressing it in terms of a free boson ϕ. Since ψ† = ψ, one ﬁnds 1 V (z) = (eiϕ(z) + e−iϕ(z))eiφ(z) 2 (3.51) = V1(z) + V2(z).

1 −1 Here the factor 2 is chosen so that hψ(z)ψ(w)i = (z − w) . It is convenient to de- ﬁne new ﬁelds φ±(z) = √1 (φ(z)±ϕ(z)) associated to the layers in the symmetrized 2 √ √ i 2φ+(z) i 2φ−(z) description, so that V1 = e and V2 = e . Then, a correlator of an even number N of operators V = V1 + V2 reads

hV (z1) ··· V (zN )Obgi = S{hV1(z1) ··· V1(zN/2)V2(zN/2+1i) ··· V2(zN )} 1 X 2 2 (3.52) = Ψ (S1)Ψ (S2), 2N−1 L L S1,S2 which is indeed the symmetrized representation. Eq. (3.52) follows by considering N all 2 possible terms involving insertions of V1 and V2, noting that only those terms with N/2 insertions of each contributes, by virtue of charge neutrality in the sector. This yields the second line, where P is the sum over all inequivalent ϕ S1,S2 ways of dividing particles over the subsets S1,S2, where permutations diﬀering by S1 ↔ S2 are considered equivalent. Thus, this sum is half the full symmetrization S. In the symmetrized representation, quasihole wave functions are obtained by inserting Laughlin quasiholes in the two layers. These quasiholes are represented by

51 Chapter 3 Conformal ﬁeld theory and the FQHE

i √1 φ+(w) i √1 φ−(w) the operators H1 = e 2 and H2 = e 2 . It should be noted that using such operators yield quasihole wave functions with the correct electron-electron and electron-quasihole correlations, but which do not encode quasihole-quasihole correlations correctly. This can be seen already for the two-quasihole wave function, which does not include any factor dependent on (w1 − w2) in the symmetrized representation. As in the original Laughlin quasihole wave function compared to its CFT counterpart, the diﬀerence lies in the monodromies of the quasihole wave function.

3.4 The Moore-Read conjecture

In the previous sections, we have illustrated the connection between the FQHE and conformal field theory, and used this connection to obtain and analyze trial wave functions. In their seminal paper [68], Moore and Read conjectured that every FQH trial wave function can be expressed as a correlator in a CFT and that, by virtue of the close connection between CFT and topological quantum field theories, such a representation allows for the identification of the topological properties of the FQH state. One example is the braiding statistcs of the quasiholes, which we have shown to be manifest in the CFT representation for the Laughlin wave function. Following work has led to a series of conjectures made about this framework, which we refer to as the “Moore-Read conjecture”, following Ref. [93]: 1. Representative wave functions for quantum Hall states, both the ground state and the excited states, can be represented as conformal blocks, or sums of conformal blocks, in a rational and unitary CFT. 2. The monodromies of the conformal blocks equal the statistics of the quasiholes, both in the abelian and non-abelian case. This is referred to as the “holonomy=monodromy” conjecture. 3. The same CFT yields a minimal description of the edge properties associated with the FQH state. Regarding the first point, besides being rational and unitary, the CFT should obey additional conditions gven in Ref. [91] , some of which are listed in Section 3.3. In the examples shown in the previous section, the model wave functions were conformal blocks of primary fields in a CFT. More generally, trial wave functions are antisymmetrized sums of conformal blocks, such as the hierarchy and composite fermion wave functions. In the CFT framework, the trial wave functions give rise to an infinite family of wave functions, as there are infinitely many descendant fields for each primary

52 3.5 Quasielectron operator

field9. This family of wave functions have the same topological properties, but differ in their short-distance properties. It is expected that representative wave functions for quantum Hall states fall in such families. The “holonomy= monodromy” conjecture has been shown to hold in several cases, but is not proven in general. It holds for the Laughlin state, where braiding statistics was first calculated in Ref.[45], and later in Ref. [94] by using the plasma analogy. The conjecture was directly verified using matrix product state techniques in Ref.[95]. It holds also for the Moore-Read state, shown using a plasma analogy [74] and verified numerically using Monte Carlo techniques[96]. Finally, the conjecture was shown to hold also for the non-abelian Z3 Read-Rezayi state [97] using a matrix product state implementation. In the following chapter, this conjecture is used to determine the braiding properties in more complicated non-abelian states.

3.5 Quasielectron operator

In Section 2.1, Laughlin’s proposal for the quasielectron wave function in the Laughlin state was given and subsequently shown to have several problems, most notably a braiding phase that is path dependent. Here, we focus on another proposal to describe the quasielectron, rooted in conformal ﬁeld theory. This operator and the insights leading to its formulation are discussed in Refs. [61, 62, 98], which we follow in the ensuing discussion. We consider the Laughlin case ν = 1/q for simplicity. Given a quasihole operator H, a naive guess would be to use the operator H−1. But although it has the correct topological quantum numbers, inserting into correlators H−1 leads to a wave function singular in the electron coordinates, which is not acceptable as a LLL wave function. Recall that the quasihole operator H, with vorticity 1, leads to an expansion of the quantum Hall liquid by increasing the angular momenta of electrons by one with respect to the vortex. Its inverse has vorticity −1, and corresponds to an attempt to decrease the angular momenta of electrons, thus leading to a contraction of the liquid. The singularities occur when a vortex is created at a position where an electron is already present, in which case the angular momentum of the electron cannot be lowered without the electron leaving the LLL [99]. On a side note, the operator H−1 does constitute a good quasielectron operator on the lattice [100], although recent results indicate that it does not yield a proper quasielectron upon taking the continuum limit [101]. Refs. [61, 62] proposed an alternative way to create localized excess charge. Since each electron carries vorticity q, which is an order q zero of the wave function, it

9In the hierarchy and composite fermion states, it should be noted that the electron operators themselves are already descendant ﬁelds.

53 Chapter 3 Conformal ﬁeld theory and the FQHE creates around it a correlation hole. A possible contraction of the liquid can thus occur if this correlation hole is shrunk. The simplest way to achieve this is to fuse the electron with the inverse quasihole H−1, reducing its vorticity. The proposal for the quasielectron operator P [98] which creates a quasielectron when inserted into correlators, is

Z q − h (|z|2+|w|2−2wz ¯ ) 2 4`2 −1 ¯ P(w ¯) = d z e H ∂j n . (3.53) √ Here j(z) = i∂φ(z)/ q is the U(1) current, and its divergence ∂j¯ (z) is only non- zero on the positions of the electrons. Inserting this operator in a correlator, ∂j¯ can thus be replaced by a sum over delta functions δ(z − zi), where zi denote the electron positions. For each such electron position, H−1 is fused with the operator V (zi), which needs to be regularized, denoted by (·)n. This fusion is the shrinking of the correlation hole of the electron at zi, and the total operator performs such fusions at all positions zi in a weighted fashion. Thus, a quasielectron is inserted at w¯ by performing a weighted superposition of shrinking the correlation hole around each of the electrons. Although this is clearly not a local operator, it does yield a quasi-local quasielectron in that the total excess charge is localized on the magnetic length scale. Inserting one such operator in the correlator, we find q ΨL({z}, w¯) = hP(w ¯)V (z1) ··· V (zN )Obgi qh 2 2 X − 2 |zi| +|w| −2wz ¯ i = e 4` hV (z1) ··· V (zi−1)V˜ (zi)V (zi+1) ··· V (zN )Obgi i (3.54) −1 −1 where V˜ = (H V )n(z) = ∂z:H V :(z) denotes the regularized fusion of the inverse quasihole H−1 and V . This single-quasielectron wave function is in fact identical to the proposal based on the composite fermion quasielectron, although the many-quasielectron wave functions differ slightly. There is an important freedom in the precise choice of inverse quasihole, −1. √ H For the Laughlin state, we would write H−1(w) = :e−iφ(w)/ q:. But the quasi- −1 electron described by (H V )n is anyonic. Thus, inserting several such operators generally gives wave functions that are not single-valued in the electron coordinates. The solution is to ensure that the statistics of H−1 is trivial, i.e. bosonic or fermionic, which is achieved by adding an auxiliary field. Thus, the choice √ H−1 = :e−iφ(w)/ q+αχ(w): is made, with α suitably chosen. This only modifies the quasielectron-quasielectron correlations, i.e. it does not affect the correlations between the quasielectons and the electrons or quasiholes. However, recent work [47] investigating the quasielectron wave functions using matrix product states techniques has shown that the quasielectron operator still has an issue: it does not

54 3.5 Quasielectron operator yield quasielectrons localized to the correct locations if other quasiparticles are present. This was found to be due to the lack of the screening of the charge associated to the auxiliary ﬁelds. A solution to this issue was given in Ref. [47] but it is unclear how this should be applied in a more general setting. The operator P is applicable generally to both abelian and non-abelian states, where the correct choice of the inverse quasihole must be made. The appropriate such choice for the abelian states is a straightforward generalization of the Laugh- lin case the non-abelian states, where the inverse quasihole is determined by the quasihole operator for one layer in the symmetrized description.

Chapter 4

CFT description of non-abelian hierarchy states

In this chapter, we introduce trial wave functions that belong to a non-abelian hierarchy. These were ﬁrst introduced by Hermanns in [102], based on a generalization of the hierarchy picture to non-abelian states. We refer to these states as Hermanns hierarchy states, and introduce them in Section 4.1. We proceed by giving the CFT description of these trial wave functions in Section 4.2, which is the main result of Paper I. This CFT description hinges on a close relation between the Hermanns hierarchy states and a series of paired spin-singlet states [103, 104], 1 which are described by the su(n + 1)2 WZW CFTs . The same CFT describes the Hermanns hierarchy states, and encodes several topological properties. Among these are the non-abelian statistics of the quasiholes, which we discuss in Section 4.3. The determination of the non-abelian statistics for the paired spin-singlet states and the application to the Hermanns hierarchy states are the core results of Paper II.

4.1 The Hermanns hierarchy

The hierarchy picture provides an organizing principle for the plateaus observed in the lowest Landau level. However, the presence of the plateau at ν = 5/2 indicates a qualitative diﬀerence can occur between states in the LLL and the second Landau level. It is natural to seek a similar hierarchy picture for states in the second Landau level, with the Moore-Read state as the parent state, and indeed several proposals exist. One is the Levin-Halperin hierarchy [105], which is an abelian hierarchy in

1Paper I deals with more general states in the Hermanns hierarchy in terms of clustered spin-

singlet states descirbed by su(n + 1)k models, but we specify to the case k = 2 here.

57 Chapter 4 CFT description of non-abelian hierarchy states the second Landau level. Another, the Bonderson-Slingerland hierarchy [106] may be viewed as an abelian hierarchy on top of the Moore-Read state. As a result, the states are all of “Ising-type”, by which is meant in CFT parlance that the neutral sector is the Ising CFT. In Paper I, the focus was instead on the Hermanns hierarchy (referred to as non- abelian hierarchy states in the paper), for which the topological properties were not known. This hierarchy picture involves the condensation of non-abelian quasiparticles. Interestingly, this leads to daughter states with non-abelian topological orders different from the parent states. In the following we outline the derivation of the states, referring the reader to the original paper [102] and the review [93] for details. The Moore-Read state supports non-abelian quasiparticles. In the Hermanns hierarchy picture, these form a correlated state on top of the Moore-Read state, which is encoded mathematically in Eq. (2.22). Obtaining trial wave functions seems infeasible at first sight. Namely, because of the non-abelian statsitics, the many-quasiparticle wave function has a non-trivial monodromy. Thus, the procedure of performing integrals over the quasiparticle coordinates is ill-defined. The insight in Ref. [102] was that such integrals can be performed by using the symmetrized representation of the Moore-Read state, (2.30). The many-quasiparticle wave function in this representation has only trivial monodromies, the non-abelian statistics being given by the Berry phase. To simplify the ensuing discussion we consider bosonic wave functions, multiplying with an overall Jastrow factor at the end. To streamline the notation we denote the bosonic Laughlin wave function by

1 P 2 Y 2 − 2 i |zi| Ψ(2,1)(Sa) = (zi − zj) e 4` (4.1) i

Here the notation Ψ(2,k) refers to a wave function with su(2)k symmetry, described by the su(2)k Wess-Zumino-Witten CFT (for the Laughlin case, u(1)2 and su(2)1 are equivalent). We consider daughter states that result from the condensation of quasielectrons, the strategy being as follows:

1. We use the CFT expression for Ψ(2,2) in terms of V (z) = V1(z)+V2(z), which

58 4.1 The Hermanns hierarchy

was given in Section 3.3. Then, we obtain the quasielectron wave function by inserting the quasielectron operators Pa(w) in the layers a = 1, 2.

2. With a wave function Φ∗ for the quasielectrons of Laughlin type in each layer, we perform the integrals over the quasiparticle coordinates to obtain the daughter state.

For the ﬁrst step, many linearly independent quasielectron wave functions exist. These are determined by the ordering of the quasielectron operators. We ﬁx one ordering – all orderings give the same daughter wave function [102] – considering the wave function with 2m quasielectrons

m 2m Y Y Y Ψ(2,2)({z}, {w}) = h P1(wi) P2(wi) V (zi)Obgi, (4.3) i=1 i=m+1 i where V (z) = V1(z)+V2(z) in the symmetrized representation. With the appropriate choice of the wave function Φ∗ that describes the correlated state formed by the quasielectrons, the integrals over the quasiparticle coordinates can be performed exactly. The daughter wave function at the highest density is given by

Ψ[2]({z}) = S{D1Ψ(3,1)(S1)D2Ψ(3,1)(S2)}. (4.4)

Here Ψ(3,1) is the bosonic Halperin (2, 2, 1) wave function, with su(3)1 symmetry. Explicitly,

Y 2 Y 2 Y Ψ(3,1)(Sa) = (zi − zj) (zi − zj) (zi − zj), (4.5) (a) (a) (a) (a) i

(a) where Mα denotes the group α of coordinates in layer a – to avoid clutter in the notation, we omit additional superscripts (α, a) for the coordinates. Further, S denotes a full symmetrization, and Da denotes a product of derivatives in layer a Y Da = ∂zi . (4.6) (a) i∈M2

This procedure may be iterated by writing quasielectron wave functions for the daughter state. The daughter state at the highest density at level n in the hierarchy is described by the wave function

Ψ[n]({z}) = S{D1Ψ(n+1,1)(S1) D2Ψ(n+1,1)(S2)}, (4.7)

59 Chapter 4 CFT description of non-abelian hierarchy states where in genreal n Y Y 2 Y Y Ψ(n+1,1)(Sa) = (zi − zj) (zi − zj) α=1 (a) α<β (a) (a) i

4.2 Conformal ﬁeld theory construction

Instead of ﬁrst performing a symmetrization over the components, which leads to the “bipartite composite fermion state” Eq. (4.9), the basic insight that leads to the CFT description of the Hermanns hierarchy state is to instead ﬁrst symmetrize (1) (2) over the “layers” S1,S2. Denoting D = D D , which is invariant with respect to this operation, this yields for the bosonic wave function (4.7)

Ψ[n]({z}) = S{D Ψ(n+1,2){z}} (4.10)

60 4.2 Conformal ﬁeld theory construction where we have identiﬁed the paired spin-singlet state X Ψ(n+1,2)({z}) = Ψ(n+1,1)(S1)Ψ(n+1,1)(S2). (4.11) S1,S2

The wave function Ψ(n+1,2) may be viewed either as the generalization of the Moore-Read state to a state with multiple components2, or as the non-abelian generalization of Ψ(n+1,1). The n = 2 paired spin-singlet states have been studied extensively as non-abelian spin-singet (NASS) states [103], where the components are spins. Our interest in the general paired spin-singlet states is their application to the Hermanns hierarchy states [102, 104], analogous to the the relation between the multi-component states and the composite fermion states.

The paired spin-singlet states are described by the su(n + 1)2 Wess-Zumino- Witten CFT. This statement can be motivated by noting that for n = 1, the bosonic Moore-Read wave function is obtained by symmetrizing two copies of the bosonic Laughlin wave function, and that their CFT descriptions are su(2)2 and su(2)1, respectively. In a similar way, the (bosonic) paired spin-states are obtained by symmetrizing two copies of generalized Halperin states. Since the latter are described by su(n + 1)1 WZW models, the former are described by the corresponding models su(n + 1)2 at level k = 2. The precise relation between the WZW current algebra and the electron and quasihole operators is outlined in Appendix 4.A; here, we mention only the results. There are n electron operators Vα and n + 1 fundamental quasihole operators Hµ, represented by operators √ ivαφ(z)/ 2 Vα(z) = ψα(z)e √ (4.12) iqµφ(z)/ 2 Hµ(w) = σµ(w)e , where α = 1, . . . , n, µ = 0, . . . , n, and ψα and σµ denote parafermion and spin ﬁelds in the pertinent parafermion CFT. Finally, φ = (φ1, . . . , φn) denotes a collection of independent chiral bosons and vα and qµ are vectors. As in the multi-component case, there is a freedom in the precise choice of these vectors; they are only required to obey certain inner products (see Paper II for more details). The ground state is given by Y Y Y Ψ(n+1,2)({z}) = h V1(zi) V2(zi) ··· Vn(zi)Obgi i∈M1 i∈M2 i∈Mn 1 (4.13) Y Y 2 = h ψα(zi)i Ψ(n+1,1)({z}) . α i∈Mα

2We refrain from thinking of the components as layers, to avoid confusion with the layers that appear in the symmetrized representations

61 Chapter 4 CFT description of non-abelian hierarchy states

Several quasihole wave functions exist, because various Hµ can be inserted. The simplest example is a pair of quasiholes H1, which gives Y Ψ(n+1,2)({z}, w1, w2) = hH1(w1)H1(w2) Vα(zi)Obgi α,i 1 Y 2 = hσ1(w1)σ1(w2) ψα(zi)i Ψ(n+1,1)({z}) (4.14) α,i 1 1 Y q1·q1/2 × (zi − w1) 2 (zi − w2) 2 w12 . i∈M1

In the next section, we delve deeper into the equivalence between such wave functions and their symmetrized representations, which is crucial to determine the braiding properties of the fundamental quasiholes. Returning to the Hermanns hierarchy states, the relation Eq. (4.10) paves the way for their CFT representation. In particular, for the ground state we have

Y Y Y n−1 Ψ[n]({z}) = S {h V1(zi) ∂zV2(zi) ··· ∂z Vn(zi)i}, (4.15) i∈M1 i∈M2 i∈Mn in terms of the primary ﬁelds Vα. Similarly, we may consider quasihole wave functions by inserting appropriate operators Hµ: for example,

Y α−1 Ψ[n]({z}, w1, w2) = S {hH1(w1)H1(w2) ∂zi Vα(zi)i}. (4.16) α,i∈Mα

One might expect that the various types of quasiholes become indistinguishable due to the total symmetrization S. However, the electrons are not treated symmetrically within the symmetrization, due to the presence of the derivatives. Thus, it is expected that the quasiholes remain distinguishable even after the symmetrization is performed, analogous to the situation for the composite fermion wave functions, where quasiholes correspond to holes in diﬀerent CF Landau levels. In Paper I, this CFT description was checked by employing the thin torus limit [108]. When placed on a torus, taking the limit where one of the circumferences becomes small, the degenerate ground states of a FQH state become simple patterns of occupation numbers. These are determined by the ﬁlling fraction and the vanishing properties of the state in question. In addition, quasiparticle excitations correspond to domain walls between these ground state patterns. Insofar as no phase transition occurs when the limit is taken, this limit retains several topological properties of the quantum Hall state, including the degeneracy of the ground state, the fractional charges of the quasiparticles [108], and their fusion rules [109].

62 4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states

The analysis performed in Paper I shows that the degenerate ground states of the Hermanns hierarchy state Ψ[n] on the torus are in one-to-one correspondence with the irreducible representations of the su(n + 1)2 WZW model. In addition, the domain wall structures associated to the quasiparticles are in one-to-one correspondence with the su(n + 1)2 fusion rules. Naively, braiding properties are inaccessible in this limit, precisely because the problem becomes one-dimensional. Instead we utilize the CFT description directly for the braiding properties of the quasiholes in the next section, but mention also Ref. [110].

4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states

In Paper II, the braid statistics of the fundamental quasiholes in the general paired spin-singlet state Ψ(n+1,2) were determined. This calculation is technical, so the emphasis here is on the general strategy and the key ideas rather than the details of the calculation, which are presented in Paper II. This calculation generalizes the calculation performed for the non-abelian spin-singlet states in Ref. [111], and the Moore-Read state in Ref. [73]. We discuss ﬁrst the “master formulas”, which are used to explicitly determine the wave functions for four quasiholes. In the section thereafter, we outline the calculation of the braiding properties for the case n = 3, listing the results for general n. Finally, we turn to the Hermanns hierarchy states, arguing that their braiding properties should agree with those of the paired spin-singlet states.

Master formulas To determine the braiding properties of the quasiholes, we make use of the Moore- Read conjecture, discussed in Section 3.4. Recall that this states that the (non- abelian) statistics of the quasiparticles is given by the monodromy of the correlator that represents the many-quasiparticle wave function. This assumes that the Berry phase is trivial, i.e. equal to the Aharonov-Bohm phase. Ostensibly, the strategy is simple: write down the correlator that gives the many- quasihole wave function, and explicitly determine how it transforms as quasiholes are exchanged. The difficulty lies in finding the explicit expressions for the correlators needed to determine the monodromy. Even in the Moore-Read case, where such explicit expressions are known, the evaluation of the relevant correlators is not straightforward [112]. A first step to resolve this issue is to make use of the symmetrized representation of the paired spin-singlet states. To begin, we have seen that the model wave

63 Chapter 4 CFT description of non-abelian hierarchy states function for the ground state is given by Y Ψ(n+1,2)({z}) = h Vα(zi)Obgi α,i (4.17) 1 X = Ψ (S )Ψ (S ). N (n+1,1) 1 (n+1,1) 2 S1,S2 Recall that this follows because the correlator and the symmetrized representation obey the same vanishing properties. Here we have included a normalization factor N , which is ﬁxed by taking limits on either side of the equation. This provides an explicit expression for the correlators of parafermions ψα. Turning to the two-quasihole wave function, the symmetrized wave functions have one quasihole per layer. Taking the simplest example of two identical quasiholes of type µ = 1, Y Ψ(n+1,2)({z}, w1, w2) = hH1(w1)H1(w2) Vα(zi)Obgi α,i

A(w1, w2) X Y Y = (zi − w1) (zi − w2) (4.18) 2N S1,S2 (1) (2) i∈M1 i∈M1

× Ψ(n+1,1)(S1)Ψ(n+1,1)(S2). Again, this relation holds because of the vanishing properties obeyed by the correlator and the symmetrized wave function. In particular, both have the same vanishing properties when either electrons are taken to the same point (z → z0), or an electron is taken to the same point as a quasihole (z → w). The additional factor A depends only on w1, w2, and ensures that the right hand side also has the same vanishing properties when quasiholes are taken to the same point. Similar to the factor N , it can be fixed by taking the limit w1 → w2. Thus, we have been able to find explicit expression for the ground state wave function and the two-quasihole wave function by using the symmetrized representations. For more than two quasiholes, however, the situation is more complicated. In those cases, the correlator has multiple components, because the quasiholes can fuse to the identity in different ways. Similar to the Moore-Read case, the four-quasihole wave function has two components

(p) Y (p) Ψ(n+1,2)({z}, {w}) = hH1(w1) ··· H1(w4) Vα(zi)Obgi , (4.19) α,i labeled by a fusion channel p = 0, 1. Additionally, similar to the Moore-Read case, there are two linearly independent symmetrized wave functions with four

64 4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states

quasiholes, denoted Ψ12;34 and Ψ13;24. Explicitly, X Y Y Ψab;cd({z}, {w}) = (zi − wa)(zi − wb) (zi − wc)(zi − wd)

S1,S2 (1) (2) i∈M1 i∈M1 (4.20)

×Ψ(n+1,1)(S1)Ψ(n+1,1)(S2).

Consequently, on grounds of the identical vanishing properties as z → z0 or z → w, (p) each conformal block Ψ(n+1,2) is related to the symmetrized wave functions up to factors that only depend on the {w}. Thus,

(p) (p) (p) Ψ(n+1,2)({z}, {w}) = A ({w})Ψ12;34({z}, {w}) + B ({w})Ψ13;24({z}, {w}). (4.21) Relations like (4.21), relating conformal blocks to symmetrized wave functions, are referred to as “master formulas” (similar master formulas hold for other four- quasihole wave functions). The master formulas do not constitute an explicit expression, because the co- eﬃcients A(p),B(p) are unknown. However, for four quasiholes precisely, these coeﬃcients, and therefore the braiding properties, can be determined. Overall, the strategy is as follows:

1. Write down the master formula relating the conformal blocks to the symmetrized wave functions.

2. Take limits on both sides of the master formula, where either electrons are taken to same point (z → z0), or electrons are taken to the positions of the quasiholes (z → w). This reduces the correlator to a four-point function of spin ﬁelds σµ only. Importantly, such four-point functions can be determined independently [111, 113].

3. Taking independent limits, yielding independent equations, solve for the co- eﬃcients A(p),B(p). 4. With an explicit expression for the conformal blocks, determine their monodromy by exchanging quasiholes.

Braiding properties of paired spin-singlet states Here we outline the calculation for the case n = 3, referring to Paper II for the discussion for general n. The master formula Eq. (4.21) reads

(p) (p) (p) Ψ(4,2) ({w}, {z}) = A ({w})Ψ12;34 ({z}, {w}) + B Ψ13;24 ({z}, {w}) . (4.22)

65 Chapter 4 CFT description of non-abelian hierarchy states

Using the explicit representation of the electron and quasihole operators Eq. (4.12), the conformal block is proprtional to

(p) Y (p) Ψ(4,2) ∝ hσ1(w1)σ1(w2)σ1(w3)σ1(w4) ψα(zi)i . (4.23) α,i

At this stage in the calculation, both the coeﬃcients A(p),B(p) as well as the conformal block in Eq. (4.23) are unknown. Turning to the second point we consider two limits, I and II, of the master formula. These are straightforward to evaluate for the symmetrized wave functions. Applied to the conformal block, these limits reduce it to a correlator involving four spin ﬁelds:

Y (p) I (p) hσ1(w1)σ1(w2)σ1(w3)σ1(w4) ψα(zi)i → hσ1(w1)σ1(w2)σ1(w3)σ1(w4)i α,i Y (p) II (p) hσ1(w1)σ1(w2)σ1(w3)σ1(w4) ψα(zi)i → hσ1(w1)σ1(w2)σ2(w3)σ2(w4)i . α,i (4.24) In particular, in limit I pairs of electrons of the same type (ψα, ψα) are taken to the same point, using the fusion rule

ψα × ψα = 1. (4.25)

In limit II, two pairs of electrons of diﬀerent type (ψ1, ψ2) are taken to the same point, and each is subsequently taken to the position of a quasihole. In this case, the relevant fusion rules are

ψ1 × ψ2 = ψ12, ψ12 × σ1 = σ2. (4.26) Taking the same limits of the symmetrized wave functions, we ﬁnd the following equations for the coeﬃcients A(p),B(p):

3 3 3 3 3 3 (p) (p) (p) 8 8 8 8 8 8 A ({w}) + B ({w}) =hσ1(w1)σ1(w2)σ1(w3)σ1(w4)i w12w13w14w23w24w34 3 7 1 1 7 3 (p) (p) 8 8 − 8 − 8 8 8 B ({w}) =hσ1(w1)σ1(w2)σ2(w3)σ2(w4)i w12w13w14 w23 w24w34. (4.27) Here wij ≡ wi − wj. Importantly, the four-point functions of spin ﬁelds can be found independently, and are explicitly given in Paper II. We list here the explicit expression of the coeﬃcients that follow

7 1 3 p p x p (p) p 8 − 4 A ({w}) = (−1) [w12w34] x 8 (1 − x) h 2 F2 (x) − F1 (x) 1 − x (4.28) 7 1 1 p p (p) p 8 − − 4 B ({w}) = (−1) [w12w34] x 8 (1 − x) h 2 F1 (x) .

66 4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states

Here x denotes the anharmonic ratio x = w12w34 , the functions F p are given in w13w24 1,2 √ 1 1 − 6 terms of certain hypergeometric functions, and the factor h = 4 2 . The solutions for A(p),B(p) yield an explicit expression for the conformal blocks, via the master formula (4.22). Turning to the third step, we now determine the monodromy of the conformal block. We consider the exchanges w1 w2 , w2 w3, and w1 w3. In general, the transformation of the conformal block is

(p) 0(p) 0 0(p) 0 Ψ(4,2)({z}, {w}) → A ({w})Ψ12;34({z}, {w}) + B ({w})Ψ13;24({z}, {w}), (4.29) which is a combined transformation of the coeﬃcients A, B and that of the sym- (p) (p) metrized wave functions Ψab;cd. The transformation of the coeﬃcients A ,B p depend on the transformation properties of the functions Fi , which are straightforward to determine. The transformation of the symmetrized wave function is simple: for example,

w1w2 w1w2 1 x Ψ12;34 → Ψ12;34, Ψ13;24 → Ψ14;23 = Ψ13;24 − Ψ12;34, (4.30) 1 − x 1 − x where in the second case we have re-expanded into the basis Ψ12;34, Ψ13;24. Com- bined, the eﬀect on the conformal block is a matrix multiplication

(p) p (q) Ψ(4,2) → (Uij)q Ψ(4,2), (4.31) where Uij is the 2×2 matrix associated with the exchange wi wj. These matrices are explicitly given by ! 1 1 0 U12 = (−1) 8 2 (4.32) 0 (−1) 3 5 1 √ ! (−1) 8 1 − (−1) 3 2 √ U23 = 1 √ − 1 (4.33) 3 − (−1) 3 2 (−1) 3 7 √ (−1) 8 1 2 U13 = √ √ . (4.34) 3 2 −1

Several comments are in order. Firstly, the matrices are unitary, and obey U13 = U12U23U12 = U23U12U23, i.e. they constitute a two-dimensional unitary representation of the braid group. This is also true for the matrices for general n, listed below, which additionally reduce to the braid matrices previously found for the Moore-Read (n = 2) case [73] and the non-abelian spin-singlet (n = 3) case [111].

67 Chapter 4 CFT description of non-abelian hierarchy states

Secondly, the braid properties of the diﬀerent types of quasiholes are identical. This is due an su (4) symmetry between the quasiholes for the bosonic wave function. For the fermionic wave functions, this symmetry is broken down to an su(3) symmetry between the quasiholes µ = 1, 2, 3. Similar statements hold for general n. Thirdly, the matrices (4.32)-(4.34) are closely related to the braid matrices for the k = 4 Read-Rezayi wave functions, which were obtained in Ref. [111]. This is a consequence of a deep relation between the respective WZW models su(4)2 and su(2)4 called “rank-level duality” (see [83]). In general, the matrices for the paired spin-singlet states based on su(n + 1)2 are closely related to the k = n + 1 Read-Rezayi wave functions based on su(2)n+1. Finally, the matrices for the fermionic wave functions diﬀer from the bosonic matrices only by global phases. We refer to Paper II for more details, listing here the salient results. The braid matrices for the bosonic paired spin-singlet states, for general n, read ! n 1 0 (n+1,2) (n+1)(n+3) U12 = (−1) n+1 (4.35) 0 (−1) n+3 2∆ 2 ! n+3 p 2 (n+1,2) (−1) 1 − (−1) dn − 1 U23 = 2 4 (4.36) n+3 p 2 n+3 dn − (−1) dn − 1 − (−1) n(n+4) (n+1)(n+3) p 2 (n+1,2) (−1) 1 dn − 1 U13 = p 2 , (4.37) dn dn − 1 −1

π n(n+2) where dn = 2 cos n+3 , and 2∆ = (n+1)(n+3) . By multiplying Eq. (4.22) with a Q M full Jastrow factor i

−M (n+1,2),M (n+1)(n+1+2nM) (n+1,2) U12 = (−1) U12 (n+1,2),M −M (n+1,2) (n+1)(n+1+2nM) (4.38) U23 = (−1) U23 −3M (n+1,2),M (n+1)(n+1+2nM) (n+1,2) U13 = (−1) U13 . for the quasiholes µ = 1, . . . , n, and

−n2M (n+1,2),M (n+1)(n+1+2nM) (n+1,2) U12 = (−1) U12 2 (n+1,2),M −n M (n+1,2) (n+1)(n+1+2nM) (4.39) U23 = (−1) U23 −3n2M (n+1,2),M (n+1)(n+1+2nM) (n+1,2) U13 = (−1) U13 .

68 4.3 Braiding properties of paired spin-singlet and Hermanns hierarchy states for the µ = 0 quasihole.

Application to Hermanns hierarchy states In the previous section we determined the braiding properties of the paired spin- singlet states, using their CFT description. We now turn to the Hermanns hierarchy states, which are described by the same CFT as discussed in section 4.2. In particular, the Hermanns hierarchy state Ψ[n] is obtained by performing a symmetrization S (or anti-symmetrization A in the fermionic case) over the components of the paired spin-singlet state Ψ(n+1,2). This is analogous to the symmetrization performed over the components of abelian multi-component states to yield the composite fermion states, as discussed in Section 2.2. Such a symmetrization is expected not to change the statistics of the quasiholes. For the CF states, this amounts to the statement that the statistics should be determined by the K-matrix – an important step towards verifying this expectation would be to calculate the statistics numerically, using matrix product states introduced in Chapter 6. For the Hermanns hierarchy states, the symmetrization is likewise not expected to change the non-abelian statistics of the quasiholes in the paired spin-singlet states. This is in stark contrast with the symmetrization performed over identical layers. In the Moore-Read case, and the paired spin-singlet case in general, the symmetrization over identical layers renders the quasiholes non-abelian. This occurs because the dimension of the Hilbert space of quasihole states is reduced. However, such a reduction is expected not to occur for the symmetrization over components, because the components are not treated symmetrically. Consequently, we expect that the braiding properties of the Hermanns hierarchy states are the same as that of their paired spin-singlet counterparts. For the fermionic Hermanns hierarchy state, therefore, the two possible braid matrices are expected to be given by (4.38) and (4.39). This argument, if true, has an interesting consequence. By virtue of rank-level duality, the braiding properties of the n = 2 and n = 3 Hermanns hierarchy states agree with the braiding properties of the k = 3 and k = 4 Read-Rezayi states, respectively. The braid properties of the Read-Rezayi states are denoted Zk, and the expected braiding properties of the Hermanns hierarchy states and the related Read-Rezayi states are listed in Table 1. However, the k = 3 and k = 4 Read-Rezayi states occur at different filling factors. In Table 1, we list the filling factors νf of the fermionic M = 1 model wave functions and list the experimentally relevant filling factor ν which differs from νf by a particle-hole conjugation and the addition of inert Landau levels. Note that the n = 3 Hermanns hierarchy state and k = 3 Read-Rezayi state occur at the same

69 Chapter 4 CFT description of non-abelian hierarchy states

state HHn=2 HHn=3 RRk=3 RRk=4 4 3 3 2 νf 7 5 5 3 3 2 2 1 ν˜ (PH) 2 + 7 2 + 5 2 + 5 2 + 3 WZW CFT su (3)2 su (4)2 su (2)3 su (2)4 braiding Z3 Z4 Z3 Z4

Table 4.1: The ﬁlling factor νf for M = 1, the PH conjugated ﬁlling factor ν˜ and expected braid properties of the n = 2, 3 Hermanns hierarchy (HH) states and the k = 3, 4 Read-Rezayi (RR) states. The rank-level duality is manifested in the braid properties of states whose CFTs are dual:

Z3 for the dual su (3)2 and su (2)3 theories, Z4 for the dual su (4)2 and su (2)4 theories.

filling factor, but have different expected braiding properties. In particular, if the ground state at ν = 12/5 is in the universality class of the Hermanns hierarchy state as opposed to the k = 3 Read-Rezayi state, the braiding properties are of type Z4 instead of Z3. This is interesting, as Z4 braiding properties are not universal for topological quantum computing, while Z3 braiding properties are. Additionally, we point out that the braid behavior differs also from that of quasiholes in the Bonderson-Slingerland hierarchy state at ν = 12/5, which is of Ising (Z2) type.

4.A WZW current algebras

Current algebra

The WZW models are characterized by a chiral algebra generated by current operators, which are the conserved currents of the action. For the su(n + 1)k model, these currents correspond to the generators ta of the algebra su(n + 1), denoted J a. The current algebra, represented in terms of operator product expansions, is

k c δab X ifabcJ (w) J a(z)J b(w) ∼ 2 + (4.40) − 2 − (z w) c z w where the f abc are the structure constants of su(n + 1) and k is the level of the model. The currents have conformal dimension ∆J = 1. To connect to the CFT description of the paired spin-singlet states, we consider explicit representations of the currents J a that realize the OPEs (4.40). To explain these representations, we start from the simplest case, n = 1, k = 1.

70 4.A WZW current algebras

The case n = 1, k = 1 The OPEs for n = 1, k = 1 read

1 c δab X iabcJ (w) J a(z)J b(w) ∼ 2 + (4.41) − 2 − (z w) c z w where a = 1, 2, 3 labels the currents J a which correspond to the Pauli matrices a σ that generate su(2) (fabc = abc). The currents can be represented as vertex operators in the following way 1 1 √ √ J 1(z) = (J +(z) + J −(z)) = (ei 2φ(z) + e−i 2φ(z)) 2 2 1 1 √ √ J 2(z) = (J +(z) − J −(z)) = (ei 2φ(z) − e−i 2φ(z)) (4.42) 2i 2i i J 3(z) = √ ∂φ(z), 2 which follows from the OPEs for the bosonic ﬁeld φ (we remind the reader that the regular parts are ignored). Note that the conformal dimension of the currents are 2 indeed ∆J = 1, as the conformal dimension of a vertex operator Vα is ∆Vα = α /2. The raising operator J + is precisely the operator V that creates a boson in the CFT representation of the bosonic Laughlin state Ψ(2,1). That is, √ V (z) = J +(z) = ei 2φ(z). (4.43)

The ﬁeld V is a primary ﬁeld with respect to the Virasoro algebra, but not with respect to the current algebra.

The case n = 1, k = 2 For n = 1 and k = 2 the OPEs read c δab X iabcJ (w) J a(z)J b(w) ∼ + (4.44) − 2 − (z w) c z w where again a = 1, 2, 3. At ﬁrst sight, it seems straightforward to slightly modify the representation (4.42) by multiplicative factors. Indeed, taking J 3(z) = i∂φ(z) gives the correct OPE J 3(z)J 3(w). To ensure the OPEs with J 1 and J 2 are correct, we need to modify √ ei 2φ(z) → eiφ(z). 1 1 2 However, this operator has conformal dimension 2 , not 1. Thus, the currents J ,J cannot be represented this way.

71 Chapter 4 CFT description of non-abelian hierarchy states

1 This issue is resolved by including another ﬁeld with conformal dimension 2 , namely the free fermion ﬁeld ψ, so that the total conformal dimension is 1. The following operators yield the correct OPEs √ J +(z) = 2ψ(z)eiφ(z) √ J −(z) = 2ψ(z)e−iφ(z) (4.45) J 3(z) = i∂φ(z).

The raising operator J + is precisely the operator V that creates a boson in the CFT representation of the bosonic Moore-Read state Ψ(2,2): √ V (z) = J +(z) = 2ψ(z)eiφ(z). (4.46)

As in the Laughlin case, V is Virasoro primary, but not a primary w.r.t the current algebra.

The case n > 1, k = 1 The algebra su(n + 1) has dimension (n+1)2−1 = n(n+2), so there are many more currents J a. Motivated by the observation that the CFT operators correspond to the raising operators J + for su(2), we consider su(2) subalgebras of su(n + 1), + − 3 formed by triplets of generators {tα , tα , tα}. Here α = 1, . . . , n is a shorthand for a set of n roots in the root lattice of su(n+), deﬁned explicitly below. The associated currents obey the OPEs

1 2J 3(w) J +(z)J −(w) ∼ + α α α (z − w)2 z − w ±J ±(w) J 3(z)J ±(w) ∼ α (4.47) α α z − w 1/2 J 3(z)J 3(w) ∼ . α α (z − w)2

Now, these OPEs are represented by a collection of vertex operators. Deﬁning n independent free boson ﬁelds φ1, . . . , φn, denoted by φ = (φ1, . . . , φn), the explicit representation is + ivαφ(z) Jα (z) = e − −ivαφ(z) Jα (z) = e (4.48) 3 i J (z) = vα∂φ(z). α 2

72 4.A WZW current algebras

Here vα is a vector, where we omit vector superscripts to avoid clutter. If α1, α2, . . . , αn denote the simple roots of su(n + 1), then n X v1 = α1, v2 = α1 + α2, . . . , vn = αi. (4.49) i=1

In abuse of notation, the labels α given above correspond to these n roots vα. From this follows that the vα obey the inner products vαvβ = 1 + δαβ, and in turn that the currents in (4.52) obey the OPEs (4.47). In fact, this representation can be used to generate the representation of the full current algebra in terms of the free bosons φi. + The raising operators Jα are precisely the operators Vα that create the bosons of type α in the CFT representation of the generalized Halperin states Ψ(n+1,1):

+ ivαφ(z) Vα(z) = Jα (z) = e . (4.50)

The case n > 1, k = 2 For k = 2, we seek to represent the OPEs of the set of currents 2 2J 3(w) J +(z)J −(w) ∼ + α α α (z − w)2 z − w ±J ±(w) J 3(z)J ±(w) ∼ α (4.51) α α z − w 1 J 3(z)J 3(w) ∼ . α α (z − w)2 As in the n = 1 case, we attempt to modify the k = 1 representation, and 1 in a similar wat additional ﬁelds with conformal dimensions 2 are needed. The operators that represent the OPEs (4.51) are given by √ + ivαφ(z)/ 2 Jα (z) = ψα(z)e √ − −ivαφ(z)/ 2 Jα (z) = ψα(z)e (4.52) 3 i Jα(z) = √ vα∂φ(z). 2

Here, the vα are the same as in the k = 1 case. + The raising operators Jα are the operators that create the bosons of type α in the paired spin-singlet states Ψ(n+1,2): √ √ + ivαφ(z)/ 2 Vα(z) = Jα (z) = 2ψα(z)e . (4.53)

The ﬁeld Vα is a Virasoro primary, but a WZW descendant.

Chapter 5

Ginzburg-Landau theories for hierarchical quantum Hall states

In this chapter, we turn to an effective field theory description of abelian quantum Hall states. In Chapter 3, we indicated how the Chern-Simons theories describe the topological features of the Laughlin state and, more generally, multi-component states. Here, we focus on the Ginzburg-Landau Chern-Simons theories, which are effective field theories of fractional quantum Hall states interpolating between microscopic wave functions and Chern-Simons theories. In Section 5.1, we introduce the GLCS theory for the Laughlin states. Next, in Section 5.2, we turn to the connection of this GLCS theory to the CFT description of the Laughlin states, and generalize the discussion to the multi-component states. Finally, in Section 5.3, the GLCS theory for hierarchical quantum Hall states is given, which is the most important result of Paper III.

5.1 The GLCS theory for Laughlin states

Introduction Broadly speaking, effective field theories are aimed at describing the low-energy properties of a system, usually at the mean-field level, such as the symmetry breaking patterns in conventional systems or the relevant topological numbers for topologically ordered states. Commonly, such effective field theories result from the introduction of auxiliary variables and the subsequent integrating out of the microscopic degrees of freedom. In this sense, the effective field theories considered in this chapter are different: instead of introducing auxiliary variables, a singular gauge transformation is performed. To motivate this transformation, we recall the Read operator for the Laughlin

75 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states state, introduced in Section 2.1. This operator describes composites of charge and vorticity, and is given by [49]

† † q R d2z0 ln(z−z0)ρ(r0) φR(r) = ψ (r)e , (5.1) where ψ† creates an electron, and the second operator creates a vortex of strength † q. For q odd, φR is bosonic, and the Laughlin state can be viewed as a Bose condensate in this operator, cf. Eq. (2.17). However, it is challenging to extract a mean-field theory: this can be appreciated by noting that the integral in the vortex P operator is only well-defined when the density is of the form ρ(r) = i δ(r − ri). This is used, for example, when calculating the Laughlin wave function. A mean- field treatment, however, amounts to replacing this density by a mean density ρ¯, in which case the integral is ill-defined. This issue can be traced back to the imaginary part of the logarithm, motivating the decomposition

R 2 0 0 0 R 2 0 0 0 R 2 0 0 0 eq d z ln(z−z )ρ(r ) = eiq d z α(z−z )ρ(r )eq d z ln |z−z |ρ(r ), (5.2) where α(z − z0) denotes the angle with z − z0 and a fixed axis. Importantly, the first factor is a singular gauge transformation that, applied to the fermion operator ψ†, changes its statistics. Specifically, for q odd, the composite operator

R 2 0 0 0 φ†(r) = ψ†(r)eiq d z α(z−z )ρ(r ) (5.3) is a bosonic operator. Such a transformation can be thought of as attaching thin flux tubes to particles, with flux equal to q in units of the flux quantum φ0. From this perspective, the statistics transmutation is realized by virtue of the Aharonov- Bohm phase that is accumulated upon exchanging the charge-flux composites. The Ginzburg-Landau-Chern-Simons (GLCS) theory [36] (see Ref. [114] for an excellent review) describes the composite bosons φ† and can be treated in a mean- field approximation, as the remaining integral in (5.2) is well-defined for smooth densities. The bosons are described by a Ginzburg-Landau term, while the binding of the charge and the flux is effectuated by the Chern-Simons action. Crucially, the bosons experience a field given by the total of the external field and the statistical field strength associated to the statistical fluxes. The GLCS theory captures several essential features of the FQHE. Firstly, the matter fields and the statistical gauge field can be integrated out to yield an effective field theory of Wen-Zee type, Eq. (3.28) [36]. As discussed in Section 3.2, it describes the topological properties of the system, such as the Hall response. Secondly, as indicated above, the theory can be treated in a mean-field approximation, which leads to a beautiful description of the quantum Hall state: in the

76 5.1 The GLCS theory for Laughlin states mean-field, the statistical magnetic field cancels against the external magnetic field and the bosons form a charged superfluid. This captures the incompressibility of the FQHE, and describes the fractionally charged quasiparticles as vortices in the superfluid. Finally, as we will see, the theory retains microscopic wave functions, such as Laughlin’s model wave function, within a perturbative expansion in the density fluctuations.

The GLCS action

The time-independent Schrödinger equation Hf Ψf = EΨf for two-dimensional interacting electrons in a magnetic ﬁeld is determined by the Hamiltonian

1 X 2 X H = (p − eA(ri)) + V (|ri − rj|), (5.4) f 2m i i

1 X 2 X H = (p − eA(ri) + a(ri)) + V (|ri − rj|) (5.5) b 2m i i

−1 Hf = ΦqHbΦ q (5.6) Ψf = ΦqΨb where q is an odd integer and Φq is the phase transformation

q P 2 iq αij Y zi − zj Φq = e i

z −z Here α = 1 ln i j , which denotes the angle between r − r and the x axis. ij 2i z¯i−z¯j i j Φq amounts to a Fermi-Bose transformation, and is the ﬁrst quantized version of the singular gauge transformation discussed above. Equation (5.5) follows, where the statistical gauge ﬁeld reads X a(ri) = q ∂ri αij. (5.8) j6=i

77 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states

In second quantization, introducing a pair of conjugate bosonic fields φ(r), φ†(r), the bosonic Hamiltonian is given by Z Z Z 1 2 2 1 2 2 0 0 0 Hb = d r|(−i~∇ − eA + a)φ(r)| + d r d r δρ(r)V (|r − r |)δρ(r ) 2m 2 (5.9) where the interacting term is written in terms of the density fluctuation operator δρ = ρ − ρ¯ in order to define a thermodynamic limit for long-range interactions. The second quantized form of the statistical gauge field is Z 0 (r − r )β aα(r) = qαβ d2r0 ρ(r0). (5.10) |r − r0|2

The effect of the gauge field is to attach thin flux tubes to the bosons. Namely, the statistical field strength is

b(r) = ∇ × a = 2πqρ(r). (5.11)

Consequently, for a particle at ri the field strength is b(r) = 2πqδ(r − ri), with the associated flux Φ = R bd2r = 2πq. In the Lagrangian formalism, the constraint (5.11) is implemented by a multiplier 1 field a0 – that is, a term a0(−ρ+ 2πq b) is added, which yields the constraint equation 1 as the a0 equation of motion. The term 2πq a0b is precisely the Chern-Simons term in the Coulomb gauge ∇ · a = 0 (restoring it in the usual form is possible by a “reverse gauge fixing” [36]). Hence, the total Lagrangian L is the sum of a Ginzburg- Landau term for the bosonic fields, the interaction term, and the Chern-Simons term: † 1 2 LGLCS = φ (−i∂0 − a0)φ − |(−i~∇ − eA + a)φ| 2m (5.12) 1 0 0 1 − δρ(r)V (|r − r |)δρ(r ) + a0b. 2 2πq Referred to as Ginzburg-Landau-Chern-Simons theory, this theory is an exact rewriting of the problem of interacting fermions in a magnetic field.

The mean-field solution and Gaussian fluctuations The equations of motion of the GLCS theory have the mean-field solution √ φ = ρ,¯ a = eA, a0 = 0, (5.13) provided the mean density ρ¯ minimizes the potential term. In addition, the constraint (5.11) sets 2πkρ¯ = ¯b, where the mean statistical field strength is ¯b = eB.

78 5.1 The GLCS theory for Laughlin states

The mean-field solutions describe the formation of a charged superfluid by the bosons, at mean density ρ¯, as a consequence of the cancellation of the statistical gauge field and the external gauge field. We turn to fluctuations around this mean-field solution. To avoid confusion later on, we explicitly denote operators with hats from now on. Using the decomposition of the bosonic fields into density and phase variables,

ˆ p p ˆ φˆ(r) = eiθ(r) ρˆ(r), φˆ†(r) = ρˆ(r)e−iθ(r) (5.14) we consider a perturbative expansion of the Hamiltonian in the density ﬂuctuation δρˆ =ρ ˆ − ρ¯. In particular, we ignore derivatives of the density and terms proportional to δρˆ2, including the potential term. The potential term is important for inter-Landau level excitations, which are discussed in Paper III. The resulting Hamiltonian is a collection of harmonic oscillators, Z 2 † Hb = ωc d r aˆ (r)ˆa(r) (5.15)

in terms of the cyclotron frequency ωc and the raising and lowering operators

r 1 h i aˆ(r) = ∂¯ θˆ(r) − iχˆ(r) πq (5.16) r 1 h i aˆ†(r) = ∂ θˆ(r) + iχˆ(r) . πq

Here ∂, ∂¯ are the holomorphic and anti-holomorphic derivatives, introduced by using ∇2 = 4∂∂¯, and the ﬁeld χˆ is deﬁned by the relation −∇2χˆ(r) = 2πkδρˆ(r). Drawing on the analogy to the single-particle harmonic oscillator, we determine the ground state wave functional by solving aˆ(r)|Ψbi = 0 in the density and the phase representations. By virtue of the commutation relation [ˆρ(r), θˆ(r0)] = iδ(r − r0), these representations are conjugate: either ρˆ or θˆ is diagonal, while the other amounts to a functional derivative, precisely analogous to the position and momentum representations of single-particle quantum mechanics. Additionally, the overlap between the density and phase eigenstates is given by1

R 2 hρ(r)|θ(r)i = ei d r ρ(r)θ(r). (5.17)

1This holds up to a normalization: the overlap may be derived as usual by solving hθ|ρˆ(r)|ρi = iδ/δθ(r)hρ|θi = ρ(r)hρ|θi and a similar equation using the operator θˆ. This shows the overlap hρ|θi is as indicated, up to a term in the integral independent of ρ and θ.

79 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states

ˆ δ Density representation In the density representation, θ(r) = −i δρ(r) and the ground state wave functional reads

q Z Z hρ(r)|Ψ i = exp d2r d2r0ρ(r) ln |r − r0|ρ(r0) b 2 Z Z (5.18) × exp −qρ¯ d2r d2r0 ln |r − r0|ρ(r0) .

Specifying to the density eigenstate hρ(r)| = h0|φˆ(r1) ··· φˆ(rN ), with eigenvalue P 2 ρ(r) = i δ(r − ri), this yields the composite boson wave function

1 P 2 Y q − |ri| Ψb(r1,..., rN ) = hρ(r)|Ψbi = |ri − rj| e 4 i , (5.19) i

δ Phase representation In the phase representation, ρˆ(r) = i δθ(r) . In this case, we ﬁnd the ground state wave functional

1 Z Z hθ(r)|Ψ i = exp d2r θ(r)∇2θ(r) exp −iρ¯ d2r θ(r) . (5.20) b 4πq

The second term results from writing out δρˆ =ρ ˆ − ρ¯ in the deﬁnition of χˆ.

5.2 The GLCS-CFT connection

A core result in Paper III is a connection between the GLCS theory and the conformal ﬁeld theory description for the Laughlin states, as well as the multi-component states. This connection hinges on the phase representation of the composite boson ground state introduced above. We motivate this connection here and give the main results, referring to the paper for a detailed discussion. To motivate this connection we make the observation that in the phase representation Eq. (5.20), the ﬁrst term takes the form of a weighting factor e−S in a Euclidean path integral, where S is the free boson action

1 Z S[θ] = d2r∇θ(r) · ∇θ(r). (5.21) 4πq

2 R 2 0 0 1 2 1 02 0 2 To obtain the Gaussian factors, we use −kρ¯ d r ln |r−r | = − 4`2 |r| by writing 1 = 4 ∇ |r | and integrating by parts. In addition, we assume that the short distance singularities (coming from the diagonal terms in the integrals) are regularized.

80 5.2 The GLCS-CFT connection

The second term, meanwhile, is very similar to the spread out CFT background charge operator Obg. Therefore, we anticipate that the phase field θ will play the role of the compactified boson in the CFT description. Motivated by this observation, we seek a relation between the composite boson wave function (5.19) and the phase representation. Inserting a resolution of identity and using (5.17), we find

Z R iθ(r1) iθ(rN ) −iρ¯ θ(r) −S[θ] Ψb(r1,..., rN ) = [Dθ(r)]e ··· e e e (5.22) R 2 iθ(r1) iθ(rN ) −iρ¯ d rθ(r) = Z[0]he ··· e e iS.

Here Z[0] = R Dθe−S[θ]. Thus, we ﬁnd an expression of the composite boson wave function as a two-dimensional Gaussian path integral. This result is explicitly veriﬁed by evaluating the path integral in Appendix 5.A. We now pass to the operator formalism, recognizing Eq. (5.22) as the path integral version of the correlator

ˆ ˆ R 2 ˆ iθ(r1) iθ(rN ) −iρ¯ d rθ(r) Ψb(r1,..., rN ) = Z[0]h0|e ··· e e |0i, (5.23) which is a radially ordered vacuum expectation value in the free boson CFT. From 0 0 hθ(r)θ(r )iS = −q ln |r − r | follows that the the phase operator θˆ has the (radially ordered) two-point function hθˆ(r)θˆ(r0)i = −q ln |r − r0|. In the following, as in Paper III, the factor Z[0] will be dropped, being loosely viewed as a normalization of the composte boson wave function. However, we show in Appendix 5.A that it admits a nice interpretation. √ Performing a rescaling θˆ → qθˆ and factoring the correlator into conformal blocks, we ﬁnd

√ ˆ √ ˆ √ R 2 ˆ i qθ(z1) i qθ(zN ) −i qρ¯ d r θ(z) 2 Ψb(r1,..., rN ) = | h0|e ··· e e |0i | (5.24) where θˆ(z), θ¯ˆ(¯z) are the holomorphic and anti-holomorphic phase field operators. ˆ ˆ 0 1 0 The holomorphic phase field obeys hθ(z)θ(z )i = − 2 ln(z − z ), with a similar two- point function for θ¯ˆ. Thus, we have expressed the composite boson wave function in terms of the free boson CFT of the phase field θˆ. The fermionic wave function is obtained by multiplying with the phase factor Φq. That this yields the correct result can be seen by working out the conformal blocks in Eq. (5.24) and multiplying with Φq as given in Eq. (5.7), which cancels the anti-holomorphic terms. Alternatively, we can write Φq as a correlator in terms of auxiliary fields ϕˆ(z), ϕ¯ˆ(¯z), which yields

√ ˆ √ ˆ √ R 2 ˆ i qφ(z1) i qφ(zN ) −i qρ¯ d zφ(z) Ψf (z1, . . . , zN ) = h0|e ··· e e |0i (5.25)

81 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states in terms of a new field φˆ(z) = θˆ(z) +ϕ ˆ(z). These results are generalized straightforwardly to the multi-component case. Re- ferring the reader to the paper for more details, in the multi-component case several √ iθˆα bosonic fields φˆα = e ρˆα are introduced. Because the particles are distinguishable, different phase transformations are performed for the different components. In particular, in terms of the K matrix, the phase transformation is

K Kαα αβ Y zi − zj 2 Y zi − zj 2 ΦK ({r}) = . (5.26) z¯i − z¯j z¯i − z¯j α,i

α=1 i∈Mα where the action is given by Z 1 2 −1 2 S = d r K θα(r)∇ θ (r). (5.29) 4π αβ β To proceed, we decomposite the K matrix as the product K = QQT , and perform a transformation of the ﬁelds θ → Q θ . This generalizes the transformation √ α αβ β θ → qθ made in the Laughlin case, and explicitly serves to bring the action in the form of a sum of free boson actions. This decomposition of the n×n K matrix, into a symmetric n × n matrix Q and its transpose, holds when K is positive deﬁnite; in Section 5.3, we discuss the negative Jain series which are related to multi-component states for which this is not true. With the decomposition, the composite boson wave function reads n ˆ R 2 ˆ Y Y iQαβ θβ (ri) −iρ¯αQαβ d rθβ (r) Ψb({r}) = h0| e e |0i, (5.30)

α=1 i∈Mα which may then be decomposed into conformal blocks as in the Laughlin case. Multiplying with the phase factor ΦK ﬁnally yields the expected fermionic multi- component wave function, which admits a CFT representation akin to (5.25).

82 5.3 GLCS theory for hierarchy states

5.3 GLCS theory for hierarchy states

The main result in Paper III is the formulation of a GLCS theory for hierarchy states. It results from a modification of the GLCS theory for the multi-component states, motivated by the close connection, discussed in Section 2.2, that exists between the hierarchy and multi-component wave functions. Complementing the paper, the specific modification to the multi-component GLCS theory is motivated, after which the GLCS theory for the negative Jain series is discussed, which did not appear in generality in Paper III.

Motivation As discussed in Section (2.2), the states in the postive and negative Jain series are closely related to certain multi-component states. As an example, we recall the 2 relation between the ν = 5 Jain wave function and the (3, 3, 2) Halperin state:

Y (3,3,2) Ψ2/5({z}) = A{ ∂zi Ψ ({z})} (5.31) i∈M2

3 2 Both states are described by the same matrix K = ( 2 3 ), but differ in that Ψ2/5 includes a description of orbital spin, while Ψ(3,3,2) does not. Indeed, the ν = 2/5 wave function is obtained by filling two CF Landau levels and the particles in higher Landau levels have higher orbital spin. This is reflected in (5.31) by the presence of the derivatives: they serve to increase the orbital spin and may be viewed as resulting from the LLL projection z¯ → 2∂z applied to the particles in the first CF Landau level. (3,3,2) So, although both Ψ and Ψ2/5 have the same K matrix, their Chern-Simons effective field theories differ. In particular, the s vectors differ, which manifests as a difference in the associated shifts on curved surfaces. A proper GLCS theory for the CF wave functions – and generally, states in the hierarchy – should therefore include a description of orbital spin. Motivated by this, we seek a modification to the phase factor ΦK which endows a subset of particles with orbital spin. Recalling that the composite bosons are charge-flux composites, a guiding heuristic picture is that of a charge separated from the flux, performing an orbital motion around it. Mathematically, the separation of charge and flux is implemented by the following point-splitting of the bosonic field: p ˆ φˆ†(r) = ρˆ(r)e−iθ(r+) (5.32) ˆ Namely, the operator e−iθ creates a charge in the density representation, which we displace by a vector from the position r. Because the Chern-Simons gauge field

83 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states couples to the density, a thin ﬂux tube is attached at the position r, thus realizing the charge-ﬂux separation. The phase factor should contain an additional term that selects the correct orbital spin, according to the given topological data of the state in question. For the example (5.31) above, the phase transformation should increase the orbital spin of the particles in group M2 by one. The proposal in Paper III is to perform the phase transformation −l/2 l Y i ΦK ({r}, {ξ}) = ΦK ({r}, {ξ}) , (5.33) ¯i i∈M2 where i, ¯i denote the holomorphic and anti-holomorphic components of , ξ = r + , and l is the orbital spin. This phase transformation is straightforwardly generalized to the general multi-component case described by the matrix K and set of orbital spins {l} for the various components.

Positive Jain series 2 Paper III shows in detail how the ν = 5 wave function results from the phase transformation (5.33), with l = 1. We outline this result here, referring the reader to the paper for the details and the generalization to other states in the positive Jain series. Starting from the bosonic Hamiltonian for the (3, 3, 2) wave function, it is unchanged when multiplying with the additional factor in (5.33); as a result the composite boson wave function is the same, except that the charges in group M2 are now displaced. Explicitly,

Y 3 Y 3 Y 2 Ψb({r}, {ξ}) = |zi − zj| |ξi − ξj| |zi − ξj| , (5.34) i

l=1 Y (3,3,2) Ψf ({z}) = ∂zi Ψ ({z}), (5.35) i∈M2 in terms of the fermionic Halperin (3, 3, 2) wave function. Finally, antisymmetrizing over the groups to obtain a state with indistinguishable particles, one obtains precisely the CF wave function for ν = 2/5.

84 5.3 GLCS theory for hierarchy states

Making use of the phase representation, we also establish the connection to conformal ﬁeld theory in this case. The composite boson wave function (5.34) can be rewritten as the correlator expression

ˆ ˆ Y iQ1β θβ (ri) Y iQ2β θβ (ξ ) Ψb({r}, {ξ}) = h e e i Obgi. (5.36)

i∈M1 i∈M2

Here, we have used the decomposition K = QQT , which may be used for the positive Jain series in general. Again, to obtain the fermionic wave function, we l multiply by ΦK and integrate over the angles of the displacement vectors. The end result is equivalent to (5.35), with the CFT representation of Ψ(3,3,2). Thus, we reproduce the CFT representation of the CF wave function: Y Y Y Ψ2/5({z}) = h ∂zi V1(zi) ∂zi V2(zi)i i∈M2 i∈M1 i∈M2 (5.37) ˆ iQαβ φβ (z) Vα = e .

n These results generalize to the positive Jain series, with ν = 2n+1 .

Negative Jain series

n The states in the negative Jain series, with filling fractions ν = 2n−1 , are the particle-hole conjugates of states in the positive Jain series. In terms of the Wen- Zee classification, states in the positive Jain series are characterized by positive definite K matrices, while the K matrices for negative Jain series have negative eigenvalues, which are associated with anti-chiral edge modes. As a result, from the CFT perspective, the negative Jain series cannot be written in terms of purely holomorphic correlators. α−1 iQαβ φβ T For the positive Jain series, we have Vα(z) = ∂z e , where K = QQ . For the negative Jain series, however, this decomposition does not hold: instead, the proposal Refs. [115, 116] is to decompose the K matrix as

K = κ − κ¯ (5.38) where both κ, κ¯ are positive definite. The decomposition of the K matrix is not unique, but different choices are expected to have the same topological properties [116]. As before, the positive definite matrices are decomposed as κ = qqT and κ¯ =q ¯q¯T , and the CFT representation is in terms of vertex operators of the form ¯ sα s¯α iqαβ φβ (z) iq¯αβ φ(¯z) Vα(z, z¯) = ∂z ∂z¯ e e . Here the powers of the derivatives are related to the spin vector, as in the positive Jain series.

85 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states

The ν = 2/3 example

We consider the example ν = 2/3. Its K matrix is written

1 2 3 1 1 1 1 −1 K = = − , (5.39) 2 1 2 1 1 2 −1 1 from which follow the vertex operators

q 3 q 1 i φ1(z) i φ¯2(¯z) V1(z, z¯) = e 2 e 2 (5.40) q 3 q 1 i φ1(z) −i φ¯2(¯z) V2(z, z¯) = ∂e¯ 2 e 2 .

This yields an explicit model wave function that is not purely holomorphic. As explained in Refs. [115, 116] however, it may be interpreted as a wave function for the guiding centers of the electrons, which may be projected onto the LLL to yield an explicitly holomorphic wave function. Below, we re-derive this wave function using the GLCS theory, and establish the connection to the CFT representation (5.40). We now turn to the GLCS theory for ν = 2/3. Similar to Eq. (5.33), the proposal for the phase transformation in the negative Jain series results from modifying the phase transformation for the corresponding multi-component state ΦK by adding a term that incorporates the orbital spin. In this case, however, the decomposition −1 l (5.38) suggests writing ΦK = ΦκΦκ¯ , so that the total phase transformation ΦK is

3 1 3 1 4 4 4 ¯ ¯ 4 l Y zi − zj z¯i − z¯j Y ξi − ξj ξi − ξj ΦK ({r}, {ξ}) = z¯i − z¯j zi − zj ξ¯i − ξ¯j ξi − ξj i

Here the gauge ﬁelds read a (r ) = P K P α , and the associated ﬁelds α i β αβ j∈Mβ ij strengths are constrained by 2πKαβρβ = bα. Using the K matrix decomposition,

86 5.3 GLCS theory for hierarchy states

we write a1 = a + c, a2 = a − c, with the ﬁeld strengths 3 ba = 2π (ρ1 + ρ2) 2 (5.43) 1 bc = 2π (ρ2 − ρ1). 2

The mean-field conditions for the original fields, ¯b1 = ¯b2 = eB, becomes ¯ba = eB and ¯bc = 0. These imply that ρ¯1 =ρ ¯2 =ρ/ ¯ 2, with ρ¯ set by the mean-field condition for ¯ba. Defining ρ ≡ ρ1 + ρ2 and ρc ≡ ρ2 − ρ1, we wish to re-express the bosonic Hamiltonian (5.42) using their conjugate phase variables θa, θc. These are

1 θa = (θ1 + θ2) 2 1 θc = (θ2 − θ1) . 2 Generalizing the expression Eq. (5.27), the bosonic Hamiltonian is then a sum of decoupled terms Z ρ¯ 2 X 2 2 H = d r θi(r)(−∇ )θi(r) + χi(r)(−∇ )χi(r) . (5.44) b 2m i=a,c

2 3 2 1 Here the fields χâ, χˆc obey −∇ χa = 2π 2 δρ and −∇ χc = 2π 2 δρc. With the bosonic Hamiltonian at hand, it is straightforward to determine the composite boson wave function. Writing it as hρ1, ρ2|Ψbi, we proceed by inserting a resolution of identity by performing a path integral over the phase field θ = (θa, θc). Using the phase representation of the ground state

1 R 2 2 2 2 R 2 d r [ θa(r)∇ θa(r)+2θc(r)∇ θc(r) −iρ¯ d rθa(r) hθ(r)|Ψbi = e 4π 3 e , (5.45) which deﬁnes the action S, we use eiθ1(r) = ei(θa−θc)(r) and eiθ2(r) = ei(θa+θc)(r) to ﬁnd

R 2 i(θa−θc)(r1) i(θa−θc)(rN ) i(θa+θc)(ξ ) i(θa+θc)(ξ ) −iρ¯ d rθa Ψb({r}, {ξ}) = he ··· e e 1 ··· e N e iS Y 2 Y 2 Y 1 = |ri − rj| |ξi − ξj| |ri − ξj| . i

87 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states

which amounts to taking derivatives in the group M2. In this case, the fermionic wave function becomes ¯ Y Y Y Ψf ({z, z¯}, {ξ, ξ}) = ∂zi (zi − zj)|zi − zj| (zi − zj)|zi − zj| i∈M2 i

i∈M1,j∈M2 (5.47) This is precisely the wave function for ν = 2/3 obtained by using the CFT description given above. Although the powers are fractional, the total expression is single-valued and, as explained in Ref. [115], can therefore be properly projected onto the lowest Landau level. In fact, we can establish the connection to this CFT representation by represent- ing the path integral expression in Eq. (5.46) as the correlator

ˆ ˆ ˆ ˆ ˆ ˆ ˆ iθa(r1) iθa(rN ) iθa(ξ ) iθa(ξ ) −iθc(r1) −iθc(rN ) iθc(ξ ) Ψb({r}, {ξ}) = h0|e ··· e e 1 ··· e N e ··· e ··· e N |0i (5.48) −1 and multiply with the phase factor ΦκΦκ¯ , similarly expressed as ratio of correlators. For simpiicity, we have omitted the background charge operator. Then, multiplying with the orbital spin factor and performing the same steps as above to obtain the fermionic wave function, we ﬁnd

q q q q 3 ˆ 3 ˆ 1 ˆ¯ 1 ˆ¯ Y Y i 2 φa(zi) Y i 2 φa(zi) Y −i 2 φc(¯zi) Y i 2 φc(¯zi) Ψf ({z, z¯}) = ∂zi h e e ih e e i i∈M2 i∈M1 i∈M2 i∈M1 i∈M2

= hV1(z1, z¯1) ··· V1(zN , z¯N )V2(zN+1, z¯N+1) ··· V2(z2N , z¯2N )i. (5.49) Here φâ(z) = θâ(z)+ϕ â(z) in terms of the holomorphic field θˆ(z) and the auxiliary field ϕ(z) that comes from the phase factor. To obtain the last line, we recognized the vertex operators from Eq. (5.40).

The full negative Jain series The generalization to the full negative Jain series is straightforward. Here, we n present a simpliﬁed derivation for the general case ν = 2n−1 . For the general negative Jain series, the n × n K matrix reads

 1 2 ··· 2   2 1 ··· 2    (5.50) K =  . .. .   . . .  2 2 ··· 1

88 5.3 GLCS theory for hierarchy states

We diagonalize K = ODOT by an orthogonal transformation O, where D contains T negative eigenvalues. Then, we define new gauge fields a˜α = Oαβaβ, where aα is determned by the K matrix as before. Consequently, the gauge fields b˜α = ∇ × a˜α obey the constraint equation b˜α = 2πDααρ˜α (5.51) T ˜ T where ρ˜α = Oαβρβ. The conjugate phase fields are θα = Oαβθβ, which means that the general bosonic Hamiltonian (5.27) is the sum of decoupled terms Z X ρ/n¯ 2 2 2 Hb = d r θ˜α(−∇ )θ˜α +χ ˜α(−∇ )˜χα . (5.52) α 2m T Here χ˜α = Oαβχβ, and we used ρ¯α =ρ/n ¯ , which follows from the mean-field conditions. For the new variables, the mean-field condition reads ρ˜¯1 =ρ ¯ and ρ˜¯α = 0 for α = 2, . . . , n. These follow because K has an eigenvector (1, 1,..., 1), P which shows ρ˜1 = α ρα. This gives the first mean-field condition, and the second follows by noting that O is orthogonal. The resulting phase transformation is " Z Z # ˜ 1 2 X ˜ −1 2 ˜ 2 ˜ hθ|Ψbi = exp d r θα(r) Dαα ∇ θα(r) − iρ¯ d r θ1(r) , (5.53) 4π α where the absolute value results from taking the square root in Hb. The procedure to obtain the composite boson wave function is now a straightforward generalization of the steps outlined for the ν = 2/3 example: the resulting wave function is, in compact notation, ˜ Y Y (α) (β) Kαβ Ψb({ξ}) = |ξi − ξj | (5.54) α≤β i∈Mα,j∈Mβ with ξα the displaced vectors (by convention {ξ(1)} = {r}), and K˜ = O|D|OT . For n = 2, this reproduces Eq. (5.46). As before, the fermionic wave function follows by multiplying with ΦK , in the general form (5.26). One finds κ κ¯ Y Y (α) (β) αβ ¯(α) ¯(β) αβ Ψf ({ξ}) = ξi − ξj ξi − ξj , (5.55) α≤β i∈Mα,j∈Mβ 1 ˜ 1 ˜ where κ = 2 (K + K), κ¯ = 2 (K − K) are precisely the generalizations of the decomposition (5.39). The final result is obtained by also multiplying with the orbital spin factor, and performing the integrals over the displacement vectors. Although the fermionic wave function that results is not purely holomorphic and is characterized by correlations having fractional powers, it is single-valued and yields a well-defined LLL wave function as in the ν = 2/3 case.

89 Chapter 5 Ginzburg-Landau theories for hierarchical quantum Hall states

5.A Path integral

We verify the relation (5.22). This follows from the expression Z −S[θ]+i R d2r J(r)θ(r) q R d2r R d2r0J(r) ln |r−r0|J(r0) Z[J(r)] = [Dθ(r)]e = Z[0]e 2 . (5.56)

Here we should remember that the field θ is an angular variable.. The measure of the path integral is defined by expanding the field in a basis which respects the periodicity, such that the propagator of the field θ is the inverse of the Laplacian ∇2 with periodic boundary conditions. Using ∇2 ln |r−r0| = 2πδ(r−r0), this yields the right hand side, where Z[0] = R [Dθ(r]e−S[θ]. Clearly, the path integral expression PN for the CB wave function corresponds to J(r) = i=1 δ(r − ri) − ρ¯. Writing it as J(r) = δρ(r) = ρ(r) − ρ¯, where ρ denotes the density operator eigenvalue, we find

q R d2r R d2r0(ρ(r)−ρ¯) ln |r−r0|(ρ(r0)−ρ¯) Z[J(r)] = Z[0] e 2 . (5.57) Working out the products in the exponential yields precisely the density representation (5.18) but includes also a divergent term proportional to ρ¯2. This term is precisely canceled by Z[0]. Namely, Z[0] is the path integral representation of hρ =ρ ¯|Ψbi, and the expression for the latter is easily found via (5.18). Thus, we conclude that Z[J], for J = δρ(r), equals the density representation (5.18), and therefore the composite boson wave function (5.19). Turning to the correlator expression (5.23), we note that it contains the divergent factor Z[0] as a multiplicative factor. Given the above calculation, we anticipate it is canceled by a formally divergent term in the correlator. In particular, this term must originate from the self-interactions of the background charge operator. Usually such self-interactions are ignored [68], but the present discussion allows for a diﬀerent perspective: they contribute a formally divergent piece which is canceled by Z[0]. Thus, we may drop the factor Z[0], and continue with the correlator expression, with the understanding that the background charge does not produce any additional correlations with itself. The cancellation of Z[0] and the background-background contributions of the correlator can be shown by regularizing the background charge operator, using the proposal of Ref. [62], and calculating their self-interactions in the continuum limit. This regularization procedure amounts to discretizing the background charge, plac- ing charges on a (square) lattice. More precisely, the background charge operator is reg Y 2 √ 2 Obg = exp −ia φ(zn)/( q2π` ) , (5.58) n where a is the lattice constant, and n labels the sites. This yields a background 4 P 2 2 charge-background charge contribution exp(a i

90 5.A Path integral we have performed a singular gauge transformation to remove the imaginary part. 2 4 P 2 This contribution equals exp(qρ¯ a i

Chapter 6

Matrix Product states and the FQHE

Chapters 3 and 4 have shown the utility of conformal ﬁeld theory in proposing and studying trial wave funtions for fractional quantum Hall states. A crucial role in this regard is the “holonomy=monodromy” conjecture discussed in Section 3.4, which was used to ﬁnd the braiding properties of paired spin-singlet states, and by extension the Hermanns hierarchy states. It is important to emphasize that, strictly speaking, it remains a conjecture. While it has been shown to hold in some cases, it is not a proven statement in general. Absent a direct proof, verifying this conjecture numerically is of considerable interest.

A promising approach is the numerical implementation of trial wave functions as matrix product states (MPS), pioneered in Ref. [95]. MPS allows for the implementation of FQH wave functions on larger system sizes and for the determination of several properties of interest, including entanglement properties of the ground state, such as the entanglement spectrum and the (topological) entanglement entropy, as well as the braiding properties of the quasiparticles.

In the following, we introduce matrix product states in Section 6.1, before discussing the matrix product state structure of trial wave functions. We discuss the MPS representation of the Laughlin state in detail, and generalize the discussion to the Halperin (3, 3, 2) wave function [117]. In Section 6.2, we turn to the ν = 2/5 composite fermion wave function and indicate that its MPS implementation via the Halperin (3, 3, 2) state converges slowly, meaning it is not very well suited to calculate observables eﬃciently. We propose an alternative implementation that might prove useful in this regard.

93 Chapter 6 Matrix Product states and the FQHE

6.1 Matrix Product States

1 Consider a pure many-body state |Ψi of a one-dimensional system of N spin- 2 particles. This state may be expanded as

X s1···sN |Ψi = C |s1i ⊗ · · · ⊗ |sN i, (6.1)

{si}=±1 with respect to the many-body basis |s1, . . . , sN i = |s1i⊗· · ·⊗|sN i. Evidently, the Hilbert space has dimension 2N , exponential in the number of spins. As a result, finding the ground state with respect to some Hamiltonian H, which amounts to finding the 2N coefficients C{s}, is intractible as N becomes large. Fortunately, many Hamiltonians of interest are local, i.e. involve interactions between nearest or next-nearest neighbors, which has important ramifications for the expansion (6.1). To explain this, imagine dividing the chain of spins into two regions A and B. To region A we can associate the reduced density matrix ρA, obtained by tracing out the degrees of freedom of region B in the density matrix

ρ = |ΨihΨ of the full system. The entanglement entropy is deﬁned as the von Neumann entropy of ρA, i.e.

S = −Tr ρA log ρA. (6.2) One would expect the entanglement entropy to be extensive, i.e. to scale with the size or “volume” of the region A. However, the entanglement entropy for ground states and low-lying excited states of local and gapped Hamiltonians in general dimensions obeys an area law instead (see Ref. [118]). For these states the entanglement entropy scales not with the size of A, but rater with the size of its boundary, which separates it from B. In the one-dimensional case, this means the entanglement entropy is constant, while in the two-dimensional case the entanglement entropy scales with the size of the one-dimensional boundary of A. Crucially, the states that obey the area law for the entanglement entropy constitute an exponentially small subset of the total Hilbert space. In other words, expansions like (6.1) over the entire Hilbert space consider many states which do not obey the area law and are thus irrelevant. On the contrary, matrix product states [119] obey the area law, and therefore target this subset of relevant states. A matrix product state is of the form X X | ˜ i s1 s2 ··· sN | i ⊗ · · · ⊗ | i (6.3) Ψ = Bα0,α1 Bα1,α2 BαN−1αN s1 sN , {si} {αi} s i.e. its coeﬃcients are products of matrices B i associated to spin si. These are matrices with respect to the bond indices αi = 1, . . . , χ, where χ is the bond

94 6.2 FQH states as matrix product states

dimension. In this context, the indices si are called the physical indices. The bond indices α0, αN reflect the imposed (open) boundary conditions – closed boundary conditions amounts to taking a trace, i.e. α0 = αN . The bond dimension is related to the entanglement: for χ = 1, |Ψ˜ i is a product state with C{s} = Bs1 Bs2 ··· BsN , while χ > 1 gives an entangled state obeying the area law. Matrix product state are dense, in the sense that |Ψ˜ i can represent any state |Ψi simply by increasing sufficiently the bond dimension χ. Importantly, relatively small values of χ often suffice to describe low-energy states |Ψi, which can therefore be efficiently numerically simulated.

6.2 FQH states as matrix product states

In Ref. [95], it was shown that the Laughlin and Moore-Read wave functions can be studied efficiently using MPS. This warrants justification: matrix product states are suited for one-dimensional systems as they encode entanglement along one direction, while the fractional quantum Hall effect takes place in two dimensions. The resolution to this paradox is that in restricting to one Landau level, neglect- ing Landau level mixing and treating filled Landau levels as inert, the projected position operators xˆ and yˆ do not commute. Hence, only one of x and y is a good quantum number and the system can be treated as effectively one-dimensional. In the following we consider fractional quantum Hall states on the cylinder geometry, which has distinct advantages such as the absence of curvature, the possi- bility to obtain entanglement properties by making a single cut, and the absence of boundary effects if the cylinder is infinite [95]. On a cylinder with circumerence L, we parametrizing the coordinates as z = x + iτ with x the coordinate around the cylinder (i.e. x ≡ x + L) and τ the coordinate along the cylinder. The wave functions are conveniently studied in the Landau gauge, described in Section 1.1, where the single particle orbitals are localized along the cylinder but completely delocalized in the compact dimension. A natural basis for a state |Ψi is provided by the orbital occupation numbers, X m1···mNΦ |Ψi = C |m1i ⊗ · · · ⊗ |mNΦ i, (6.4)

{mi} where mi is the occupation number of orbital i. For a fermionic Laughlin state, mi ∈ {0, 1}, while bosonic Laughlin states have mi ∈ {0, 1, 2,... }. More complicated states such as the Halperin state involve occupations of diﬀerent electron types, such as mi = {0, ↑, ↓, ↑↓}. n

95 Chapter 6 Matrix Product states and the FQHE

xAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N2IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB5jmM/A==

⌧AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NJlmyu3fszgnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelEhh0fe/vdLa+sbmVnm7srO7t39QPTxq2zg1jLdYLGPTiajlUmjeQoGSdxLDqYokf4wmt7n/+MSNFbF+wGnCQ0VHWgwFo5hLPaRpv1rz6/4cZJUEBalBgWa/+tUbxCxVXCOT1Npu4CcYZtSgYJLPKr3U8oSyCR3xrqOaKm7DbH7rjJw5ZUCGsXGlkczV3xMZVdZOVeQ6FcWxXfZy8T+vm+LwOsyETlLkmi0WDVNJMCb542QgDGcop45QZoS7lbAxNZShi6fiQgiWX14l7Yt64NeD+8ta46aIowwncArnEMAVNOAOmtACBmN4hld485T34r17H4vWklfMHMMfeJ8/IS2OSA== 2⇡`2

AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPJWkCHosevHgoYL9gCaWzXbSLt1swu5GKCHgX/HiQRGv/g5v/hu3bQ7a+mDg8d4MM/OChDOlHefbKq2srq1vlDcrW9s7u3v2/kFbxamk0KIxj2U3IAo4E9DSTHPoJhJIFHDoBOPrqd95BKlYLO71JAE/IkPBQkaJNlLfPvJCSWhW9xKGPeD8oZ5nt3nfrjo1Zwa8TNyCVFGBZt/+8gYxTSMQmnKiVM91Eu1nRGpGOeQVL1WQEDomQ+gZKkgEys9m5+f41CgDHMbSlNB4pv6eyEik1CQKTGdE9EgtelPxP6+X6vDSz5hIUg2CzheFKcc6xtMs8IBJoJpPDCFUMnMrpiNi8tAmsYoJwV18eZm06zXXqbl359XGVRFHGR2jE3SGXHSBGugGNVELUZShZ/SK3qwn68V6tz7mrSWrmDlEf2B9/gDQw5Vh L

Figure 6.1: The cylinder geometry, where L is the circumference and x, τ param- eterize the surface. Here x ≡ x + L is the coordinate in the compact direction, and τ the coordinate along the cylinder. The single-particle orbitals are plane waves around the cylinder, and Gaussians of width 2π`2 ∼ ` along the cylinder, separated by δτ = L .

The Laughlin state Following the derivation in Ref. [120], we determine the MPS representation of the fermionic Laughlin wave function by ﬁrst considering only its polynomial part in the planar geometry. Subsequently, we map the plane onto the cylinder, taking into account also the Gaussian factors.

MPS formulation of the polynomial The polynomial part of the fermionic Laughlin wave function is given by the CFT expression N Y q (zi − zj) = hNq|V (z1) ··· V (zN )|0i, (6.5) i

N Y q X (zi − zj) = cλmλ(z1, . . . , zN ), i

96 6.2 FQH states as matrix product states

Here λ = (λ1, . . . , λN ) denotes a partition of the total degree of the polynomial, q q 0 0 q such that λN > λN−1 > ··· > λ1 ≥ 0. For example, (z1 −z2) = cq,0(z1z2 −z1z2)+ q−1 1 1 q−1 cq−1,1(z1 z2 − z1z2 ) + ... . Comparing these two expressions, using the mode expansions of the vertex operators, we ﬁnd √

cλ = N! hNq|V−λ1 ··· V−λN |0i. (6.8)

We can now ﬁnd an MPS description for the coeﬃcients cλ by inserting resolutions P of identity 1 = α |αihα| in the CFT Hilbert space. This yields

cλ X mNΦ mNΦ−1 m0 √ = hNq|A |αNΦ ihαNΦ |A |αNΦ−1i · · · hα1|A |0i N! {α} (6.9) X mN = A Φ ··· Am0 αNΦ+1,αNΦ α1,α0 {α}

mi mi Here we have defined A = (V−i) for mi = 0, 1, and α0, αNΦ+1 correspond to the in- and out states. Seemingly, we have not gained much. Firstly, one might object that the CFT Hilbert space is infinite-dimensional. On a finite cylinder, however, only finitely many states in the CFT Hilbert space contribute. Moreover, in general, the polynomial part of the wave function can be approximated well by truncating the Hilbert space to a finite subset. One might also be dissuaded by the fact that, ostensibly, the matrix elements of many matrices V−i need to be calculated. However, this last issue can be resolved by spreading out the background charge. The background √ charge operator is e−iNqφ0/ q, and the modes enjoy the property

√n √n i q φ0 −i q φ0 V−n = e V0e , (6.10) where V0 denotes the zero mode. This yields

cλ X mN √ = B Φ ··· Bm0 (6.11) αout,αNΦ α1,αin N! {α} where the charges of the in- and out states are modiﬁed, and

√i m m − q φ0 B = (V0) e . (6.12)

Consequently, only the matrix elements of the zero mode V0 and that of the dis- tributed background charge operator need to be determined. We do not consider the calculation of the matrix elements in detail, referring the reader to the original paper [95] and Refs. [47, 120].

97 Chapter 6 Matrix Product states and the FQHE

MPS formulation for the cylinder geometry Having obtained the MPS representation of the polynomial part of the Laughlin wave function in terms of CFT, we turn to the full wave function on the cylinder geometry. Its MPS description can be obtained from that of the polynomial part on the plane as follows. First, we ensure that the orbitals have the correct normalization factors, which depend on the geometry. The Slater coeﬃcients for the wave functions on the cylinder are given by

cyl Y cλ = cλ Nλi , (6.13) i

p √ 1 2π`2j 2 where Nj = L` π exp 2`2 ( L ) . It turns out that these additional factors can be obtained by redeﬁning the matrices Bm. In particular, these additional factors are reproduced by multiplying the Bm with the operators [120]

2 −δτH − 2π` L U(δτ) = e = e L 0 (6.14) where L0 is the Hamiltonian for the (chiral) CFT. Thus, the updated B matrices are √ m m −iφ0/ q B = V0 e U(δτ). (6.15) The factor U(δτ) can be thought of as a time-propagating operator [95], where τ plays the role of imaginary time. With the MPS form of the Laughlin state at hand, several observables can be calculated. These include the entanglement properties of the ground state, such as the entanglement entropy (EE), the associated topological entanglement entropy (TEE), and the entanglement spectrum, as well as the correlation length and the braiding statistics of the quasiholes. In Ref. [47], the braiding statistics of quasielectrons was found, using the quasielectron operator discussed in Section 3.5. In the following we show reproduced results for the ν = 1/3 Laughlin state. The entanglement entropy can be used to estimate the system sizes where MPS is useful. For an MPS with a bond dimension χ, one finds that the EE S obeys −1 the area law S = αL − γ + O L for a region Lmin ≤ L ≤ Lmax,. Here Lmax depends on χ, which is explained in the following. For L < Lmin the MPS is converged, but the area law is not obeyed as deviations occur due to finite-size effects. In the deep limit L → 0 the thin torus limit is approached, for which S → 0. For L > Lmax, the MPS is not fully converged and the EE saturates, violating the area law. The important system sizes are therefore Lmin ≤ L ≤ Lmax, for which the MPS is converged and the EE obeys the area law. Fig. 6.2 shows the entanglement entropy as a function of the circumference L, from which follows the

98 6.2 FQH states as matrix product states value γ ≈ 0.545 for the topological entanglement entropy by extrapolating from the region where the area√ law holds. This value is in good agreement√ with the predicted value γ = log 3 ≈ 0.549... [8], given generally by log D with D the so-called quantum dimension, which is D = 3 for the ν = 1/3 Laughlin state. An accurate simulation of a larger system size requires a higher bond dimension of the MPS. This can be motivated as follows. In the B matrices, the operators 2π`2 2 U(δτ) = exp(−( L ) L0) are diagonal in the CFT Hilbert space, because L0 is diagonal. In particular on any CFT Hilbert space it evaluates to L0|αi = ∆α|αi, where ∆α is the conformal dimension of the state |αi. Thus, states having higher conformal dimension have contributions exponentially suppressed compared to states with lower conformal dimensions. As L → 0 only the highest weight states, which have the lowest conformal dimensions, are kept – this produces the thin cylinder limit. As L increases, the relative contributions of the states at higher ∆ become more important. Therefore, an accurate simulation at larger systems sizes requires a larger cut-oﬀ in the conformal dimension, i.e. a higher bond dimension. In fact, the bond dimension required for an implementation at a ﬁxed precision is exponential in the circumference L [95]. Quasiholes can be studied by inserting

2 S

0 5 10 15 20 25 30 L ( )

1 Figure 6.2: The entanglement entropy S of the ν = 3 Laughlin state obtained by the orbital cut, as a function of the circumference of the cylinder L. Here the bound on the conformal dimension is ∆max = 12, and the extrapolated TEE is given by γ ≈ 0.545. Small L correspond to the thin torus limit, causing a deviation from the area law. the appropriate vertex operator H in the correlator, which amounts to adding a matrix on the bond between two B matrices. Inserting a pair of quasiholes, their braiding statistics can be determined by moving one quasihole on a closed loop

99 Chapter 6 Matrix Product states and the FQHE encircling the other and calculating the Berry holonomy. Denoting the MPS by ˜ q ΨL, the Berry phase is given by I ˜ q d ˜ q γ = hΨL(w1, w2)| |ΨL(w1, w2)i, (6.16) C dw1 which is numerically calculated by discretizing a closed loop C into Ns steps

{u1, . . . , uNs−1}: Ns X ˜ q ˜ q γ ≈ hΨL(ui, w2)|ΨL(ui+1, w2)i. (6.17) i=1

Here uNs+1 = u1, which encodes the monodromy acquired during the closed loop. Hence, (6.17) actually determines the full Berry holonomy. It is convenient to choose C as in Ref. [47], consisting of two circles at ﬁxed τ, connected by two counterpropagating lines that do not contribute to the Berry holonomy. To obtain the statistics of the quasihole, one substracts from γ the phase obtained by moving the quasihole w1 along the loop C without the quasihole at w2, and divides by two. In Fig. 6.4, the density proﬁle of a single quasihole is shown, and is seen to be radially symmetric, with vanishing density at the core. Fig. 6.3 shows the braiding statistics ∆γ/2π of the quasiholes as a function of their separation. The result is 2π the expected result 3 .

Outlook: the ν = 2/5 composite fermion wave function An MPS implementation of composite fermion wave functions is also of considerable interest. One reason is that the theoretical tools to describe the composite fermion states, and generally abelian hierarchy states, are much less powerful in the absence of a known (model) parent Hamiltonian and a simple plasma analogy. Although strong numerical evidence exists regarding the properties of composite fermion states (see [53] and references therein), using MPS techniques would allow for a numerical study on larger system sizes. In addition, such an MPS implementation would pave the way for the implementation of the Hermanns hierarchy states when viewed as “bipartite composite fermion states” as explained in Section 4.1. The simplest candidate to implement using MPS is the ν = 2/5 composite fermion wave function and a natural way to perform this implementation is to utilize its close connection to the Halperin (3, 3, 2) state, which has been invoked on several occasions already. We recall

Y ∂ (3,3,2) Ψ2/5({z}) = A{ ↓ Ψ ({z})}, (6.18) i ∂zi

100 6.2 FQH states as matrix product states

1.0

0.8

0.6 2 /

0.4

0.2

0.0 0 2 4 6 8 10 d ( )

Figure 6.3: The quasihole statistics, given by 2∆γ = γ − γAB, as a function of the separation d of the quasiholes in units of the magnetic length, on a cylinder of size L = 15. Here γ is the Berry phase obtained by moving a quasihole around a closed loop containing another quasihole, and γAB is the Aharonov-Bohm phase associated to the same loop without the other quasihole present. where the Halperin (3, 3, 2) wave function is explicitly given by

3 3 2 1 P σ 2 Y ↑ ↑ Y ↓ ↓ Y ↑ ↓ − 2 σ,i |zi | Ψ(3,3,2)({z}) = zi − zj zi − zj zi − zj e 4` i

q 5 i i φc(z)± √ φs(z) V± (z) = e 2 2 (6.20) described by two independent chiral bosonic fields for charge φc and spin φs. If the number of insertions of V+ and V− is equal, the background charge Obg depends only on the charge field φc, as charge neutrality in the spin sector automatically holds. The MPS implementation for the Halperin states was performed in Ref. [117]. As in the Laughlin case, the auxiliary space can be identified with the CFT Hilbert space, which in this case is a tensor product of the Hilbert spaces of the two chiral boson fields φc and φs. The derivation of the MPS description on the cylinder mirrors that given above, and we refer the reader to the details in Ref. [117]. The B

101 Chapter 6 Matrix Product states and the FQHE

⇢(x, ⌧)

AAAB83icbVBNSwMxEM36WetX1aOXYBEqSNkVQY9FLx4r2A/oLiWbZtvQbLIkE7GU/g0vHhTx6p/x5r8xbfegrQ8GHu/NMDMvzgQ34Pvf3srq2vrGZmGruL2zu7dfOjhsGmU1ZQ2qhNLtmBgmuGQN4CBYO9OMpLFgrXh4O/Vbj0wbruQDjDIWpaQvecIpASeFoR6oytN5CMSedUtlv+rPgJdJkJMyylHvlr7CnqI2ZRKoIMZ0Aj+DaEw0cCrYpBhawzJCh6TPOo5KkjITjWc3T/CpU3o4UdqVBDxTf0+MSWrMKI1dZ0pgYBa9qfif17GQXEdjLjMLTNL5osQKDApPA8A9rhkFMXKEUM3drZgOiCYUXExFF0Kw+PIyaV5UA78a3F+Wazd5HAV0jE5QBQXoCtXQHaqjBqIoQ8/oFb151nvx3r2PeeuKl88coT/wPn8AUFqRMg==

Figure 6.4: The density proﬁle of a quasihole on a cylinder of circumference L = 16.

↑ ↓ matrices in this case are characterized by two occupation numbers, mi = (mi , mi ) σ with mi = 0, 1, and are explicitly given by

q 2 (m↑,m↓) m↑ m↓ −i (φc,0) B = (V+,0) (V−,0) e 5 U(δτ). (6.21)

Here φc,0 and Vσ,0 denote zero modes. Returning to the CF wave function, we are led to an MPS implementation using the operators q 5 i i φc(z)+ √ φs(z) V+ (z) = e 2 2 (6.22) q 5 i i φc(z)− √ φs(z) V− (z) = ∂ze 2 2 , where the final wave function Eq. (6.18) is obtained by an antisymmetrization. This antisymmetrization can be implemented at the CFT level by including appropriate phase factors in the B matrices [95, 117]. However, preliminary results indicate that this MPS implementation of the ν = 2/5 state is problematic. The convergence of the ground state density to the expected value is slow for finite systems, and asymmetries in the density are observed at the edge. This is thought to be due to the fact that the wave function is not a zero angular momentum (L = 0) state. Indeed, this is the chief difference between the CF wave functions and the Halperin wave functions. A possible resolution is to use the insight from Ref. [121], where it was shown that to write L = 0 Jain wave functions on a spherical geometry, a conformal field theory representation in terms of three chiral boson fields, instead of two, is needed.

102 6.2 FQH states as matrix product states

Using this description and changing the normalization factors of the single-particle orbitals on the sphere to the those on the cylinder, we obtain an MPS for the L = 0 state on the cylinder geometry. The hope is that such an implementation ensures a better convergence. One might expect that the CFT HIlbert space for the three-field implementation is much larger for a given cut-off in conformal dimension compared to its counterpart in the two-field implementation. However, two technical features of the three-field implementation ameliorate this bleak view. Firstly, there are still only two electron operators, which gives rise to relations between the quantum numbers associated with the three fields. Hence, the Hilbert space growth should be similar to that in the two-field implementation. Secondly, the compactification radii of the chiral bosons needed are small. In particular, the three field implementation√ iφ (z) iφ (z) i 2χ(z) makes use of the vertex operators V˜1(z) = e 1 , V˜2 = e 2 , and V˜χ = e , in terms of which the electron operators are

V1 = V˜1(z)V˜χ(z),V2(z) = α(∂V˜2(z))V˜χ(z) + βV˜2(z)∂V˜χ(z) (6.23)

Here α, β are coefficients that ensure that the resulting state has angular√ momentum zero. Their compactification radii, R1 = R2 = 1 and√ Rχ = 2,√ are small compared to those in the two-field implementation Rc = 10 and Rs = 2. Such small compactification radii also have a positive effect on the growth of the Hilbert space1. Although these considerations are optimistic, detailed numerical simulations using this three-field implementation have not been performed at the time of writing. This is a goal for future research, with the hope that such an MPS implementation may also be generalized to the Hermanns hierarchy states.

1Specifically, the conformal dimension of a highest weight state |αi is inversely proportional to the square of the compactification radius. For a given cut-off, more states should therefore be taken into account for larger compactification radii.

103

Chapter 7

Conclusion

To conclude, we summarize the core results obtained and described in this thesis, and indicate several ways in which these results could be extended. In Paper I, the conformal field theory description of a series of non-abelian hierarchy wave functions was given. These are the Hermanns hierarchy wave functions, which could be relevant in describing certain quantum Hall states, most notably the state at filling ν = 12/5. Their conformal field theory description is in terms of the su(N)2 Wess-Zumino-Witten models, which are expected to encode the topological properties in a manifest way. The fusion rules, ground state degeneracy and quasiparticle charges were verified by utilizing the thin torus limit.

The su(N)2 WZW models also describe closely related paired spin-singlet states, which were analyzed in more detail in Paper II. In particular, the non-abelian braiding properties of quasiholes in paired spin-singlet states were determined by making use of conformal ﬁeld theory techniques, hinging on the “holonomy=monodromy” conjecture. Due to their close connection, it was argued that Hermanns hierarchy states and the paired spin-singlet states should have the same braiding properties. This led to an interesting observation on the braiding properties predicted by various trial wave functions for the observed fractional quantum Hall state at ν = 12/5. While the well-known Read-Rezayi k = 3 wave function predicts non- abelian statistics that is universal for topological quantum computing, this is not the case for the Hermanns hierarchy (as well as the Bonderson-Slingerland hierarchy wave function). A more detailed numerical study comparing these trial wave functions would be interesting, to see which, if any, describes the topological order associated to the FQH state. In addition, it is possible in principle to generalize the calculation of the braiding properties to clustered spin-singlet states, based on su(N)k WZW models. In particular, the calculation for su(3)3 would be interesting, as this is the simplest case with a fusion multiplicity. In Paper III, Ginzburg-Landau theories for abelian quantum Hall hierarchy

105 Chapter 7 Conclusion wave functions were found, by modifying the Ginzburg-Landau theories for multi- component states. In particular, utilizing a generalized flux attachment procedure, orbital spin was implemented and shown to yield a Ginzburg-Landau theory that retains the hierarchy wave functions in a mean-field treatment. Beisdes this, the paper established a connection between the effective Ginzburg-Landau theories and conformal field theory, both for the Laughlin and multi-component cases that were already understood, as well as for the states in the positive and negative Jain series. We believe these results carry over naturally to the full chiral hierarchy, where an interesting and experimental relevant case would be the hierarchy wave function for ν = 4/11. Several extensions of these ideas are possible. A straightforward extension is to find the Ginzburg-Landau theories for more general abelian hierarchy states, namely those that result from successive condensation of both quasiholes and quasielectrons. Also, for the states in the negative Jain series, it was shown that the GL theories yield a particular decomposition of the K matrix into positive definite matrices. Thus, it naturally leads to a particular series of representative wave functions. It would be interesting to see if these are numerically better than other candidate wave functions, corresponding to other K matrix decompositions, that differ in short-distance properties. Another interesting extension would be to see if the appropriate Wen-Zee effective field theory can be obtained from the GL theories for hierarchy states. In particular, the term of interest in the Wen-Zee Lagrangian is the “orbital spin term”, involving the spin connection. Thus, one needs to consider the proposed Ginzburg-Landau theories on a curved space, and integrate out the bosonic degrees of freedom. The Ginzburg-Landau theories for the Laughlin and multi-component states have been analyzed on curved spaces in Ref. [122], which showed that the charge-flux composites minimally couple to the spin-connection, where the coupling constant is the expected orbital spin. It would be interesting to see if the generalized flux attachment for the hierarchy states, i.e. the point-splitting and the projection into a definite angular momentum state, also yields the correct orbital spin term upon integrating out the boson degrees of freedom. Finally, it is natural to attempt to extend these ideas to non-abelian states. Such Ginzburg-Landau theories have been proposed for the bosonic Moore-Read state (with a generalization to the Read-Rezayi states) [123], and for more general non- abelian states in Ref. [124] on the basis of non-trivial rank-level dualities. Here, it would be interesting to see if, similar to the abelian hierarchy cases, a connection to the CFT description of the non-abelian states can also be made. In the last chapter, a description of the ν = 2/5 composite fermion wave function using matrix product state techniques was given, but its implementation was found

106 to have poor convergence. A slightly different implementation using three fields was suggested. It remains to see if this implementation yields better convergence, although heuristic reasons supporting this can be given. Performing the implementation of the ν = 2/5 wave function is important, because it would pave the way for the calculation of the fractional statistics of quasiparticles in this state. It is expected the fractional statistics is the same as that for quasiparticles in the corresponding multi-component state, and if the fractional statistics is indeed unaffected by the antisymmetrization procedure performed, this would lend credibility to a similar equivalence of non-abelian statistics between the Hermanns hierarchy wave functions and the paired spin-singlet states. Moreover, implementing this wave function serves as a stepping stone towards the MPS implementation of the Her- manns hierarchy wave functions themselves, viewed as bipartite composite fermion states.

107

Bibliography

[1] Y. Tournois, “Properties of non-abelian hierarchy wave functions in the fractional quantum Hall eﬀect.” Licentiate thesis, Gothenburg University, May 29, 2018.

[2] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, “Tenth-order electron anomalous magnetic moment: Contribution of diagrams without closed lep- ton loops,” Phys. Rev. D, vol. 91, p. 033006, Feb 2015.

[3] D. Hanneke, S. Fogwell, and G. Gabrielse, “New measurement of the electron magnetic moment and the ﬁne structure constant,” Phys. Rev. Lett., vol. 100, p. 120801, Mar 2008.

[4] L. Landau, “On the theory of phase transitions,” Zh. Eksp. Teor. Fiz., vol. 7, pp. 19–32, 1937.

[5] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett., vol. 48, pp. 1559– 1562, May 1982.

[6] X. G. Wen, “Vacuum degeneracy of chiral spin states in compactiﬁed space,” Phys. Rev. B, vol. 40, pp. 7387–7390, Oct 1989.

[7] X. G. Wen and Q. Niu, “Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces,” Phys. Rev. B, vol. 41, pp. 9377–9396, May 1990.

[8] A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys. Rev. Lett., vol. 96, p. 110404, Mar 2006.

[9] X. Chen, Z.-C. Gu, and X.-G. Wen, “Local unitary transformation, long- range quantum entanglement, wave function renormalization, and topological order,” Phys. Rev. B, vol. 82, p. 155138, Oct 2010.

109 Bibliography

[10] J. M. Leinaas and J. Myrheim, “On the theory of identical particles,” Il Nuovo Cimento B, vol. 37, no. 1, pp. 1–23, 1977.

[11] F. Wilczek, “Quantum mechanics of fractional-spin particles,” Phys. Rev. Lett., vol. 49, pp. 957–959, Oct 1982.

[12] R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, “Direct observation of a fractional charge,” Nature, vol. 389, pp. 162–164, 1997.

[13] M. Reznikov, R. d. Picciotto, T. G. Griﬃths, M. Heiblum, and V. Umansky, “Observation of quasiparticles with one-ﬁfth of an electron’s charge,” Nature, pp. 238–241, 1999.

[14] L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, “Observation of the e/3 fractionally charged Laughlin quasiparticle,” Phys. Rev. Lett., vol. 79, pp. 2526–2529, Sep 1997.

[15] J. Martin, S. Ilani, B. Verdene, J. Smet, V. Umansky, D. Mahalu, D. Schuh, G. Abstreiter, and A. Yacoby, “Localization of fractionally charged quasiparticles,” Science, vol. 305, no. 5686, pp. 980–983, 2004.

[16] C. de C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, and X. G. Wen, “Two point-contact interferometer for quantum hall systems,” Phys. Rev. B, vol. 55, pp. 2331–2343, Jan 1997.

[17] P. Bonderson, A. Kitaev, and K. Shtengel, “Detecting non-abelian statistics in the ν = 5/2 fractional quantum hall state,” Phys. Rev. Lett., vol. 96, p. 016803, Jan 2006.

[18] H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Boc- quillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, and G. Fève, “Fractional statistics in anyon collisions,” Science, vol. 368, no. 6487, pp. 173– 177, 2020.

[19] A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics, vol. 303, no. 1, pp. 2 – 30, 2003.

[20] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non- abelian anyons and topological quantum computation,” Rev. Mod. Phys., vol. 80, pp. 1083–1159, Sep 2008.

110 Bibliography

[21] K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the ﬁne-structure constant based on quantized hall resistance,” Phys. Rev. Lett., vol. 45, pp. 494–497, Aug 1980.

[22] “The Nobel Prize in Physics 1985.” https://www.nobelprize.org/prizes/ physics/1985/summary/. Accessed: March 4, 2020.

[23] R. B. Laughlin, “Quantized hall conductivity in two dimensions,” Phys. Rev. B, vol. 23, pp. 5632–5633, May 1981.

[24] B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B, vol. 25, pp. 2185–2190, Feb 1982.

[25] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett., vol. 49, pp. 405–408, Aug 1982.

[26] F. D. M. Haldane, “Model for a quantum hall eﬀect without landau levels: Condensed-matter realization of the "parity anomaly",” Phys. Rev. Lett., vol. 61, pp. 2015–2018, Oct 1988.

[27] C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin hall eﬀect,” Phys. Rev. Lett., vol. 95, p. 146802, Sep 2005.

[28] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall eﬀect and topological phase transition in hgte quantum wells,” Science, vol. 314, no. 5806, pp. 1757–1761, 2006.

[29] M. J. Manfra, “Molecular beam epitaxy of ultra-high-quality algaas/gaas heterostructures: Enabling physics in low-dimensional electronic systems,” Annual Review of Condensed Matter Physics, vol. 5, no. 1, pp. 347–373, 2014.

[30] H. Stormer, “Two-dimensional electron correlation in high magnetic ﬁelds,” Physica B: Condensed Matter, vol. 177, no. 1, pp. 401 – 408, 1992.

[31] S. F. Nelson, K. Ismail, J. J. Nocera, F. F. Fang, E. E. Mendez, J. O. Chu, and B. S. Meyerson, “Observation of the fractional quantum Hall eﬀect in Si/SiGe heterostructures,” Applied Physics Letters, vol. 61, no. 1, pp. 64–66, 1992.

111 Bibliography

[32] K. Bolotin, F. Ghahari, M. Shulman, H. Stormer, and P. Kim, “Observation of the fractional quantum hall eﬀect in graphene,” Nature, vol. 462, pp. 196–9, 11 2009.

[33] R. B. Laughlin, “Anomalous quantum Hall eﬀect: An incompressible quantum ﬂuid with fractionally charged excitations,” Phys. Rev. Lett., vol. 50, pp. 1395–1398, May 1983.

[34] D. Arovas, J. R. Schrieﬀer, and F. Wilczek, “Fractional statistics and the quantum Hall eﬀect,” Phys. Rev. Lett., vol. 53, pp. 722–723, Aug 1984.

[35] A. Lopez and E. Fradkin, “Fermionic Chern-Simons theory for the fractional quantum Hall eﬀect in bilayers,” Phys. Rev. B, vol. 51, pp. 4347–4368, Feb 1995.

[36] S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum hall effect,” Phys. Rev. Lett., vol. 62, pp. 980–980, Feb 1989.

[37] X. G. Wen and A. Zee, “Classiﬁcation of abelian quantum Hall states and matrix formulation of topological ﬂuids,” Phys. Rev. B, vol. 46, pp. 2290– 2301, Jul 1992.

[38] E. Wigner, “On the interaction of electrons in metals,” Phys. Rev., vol. 46, pp. 1002–1011, Dec 1934.

[39] P. K. Lam and S. M. Girvin, “Liquid-solid transition and the fractional quantum-hall eﬀect,” Phys. Rev. B, vol. 30, pp. 473–475, Jul 1984.

[40] F. D. M. Haldane, “Fractional quantization of the hall eﬀect: A hierarchy of incompressible quantum ﬂuid states,” Phys. Rev. Lett., vol. 51, pp. 605–608, Aug 1983.

[41] F. D. M. Haldane and E. H. Rezayi, “Finite-size studies of the incompressible state of the fractionally quantized hall eﬀect and its excitations,” Phys. Rev. Lett., vol. 54, pp. 237–240, Jan 1985.

[42] S. M. Girvin, “Particle-hole symmetry in the anomalous quantum hall eﬀect,” Phys. Rev. B, vol. 29, pp. 6012–6014, May 1984.

[43] R. Prange and S. Girvin, The Quantum Hall eﬀect. Graduate texts in con- temporary physics, Springer-Verlag, 1987.

112 Bibliography

[44] J. M. Caillol, D. Levesque, J. J. Weis, and J. P. Hansen, “A monte carlo study of the classical two-dimensional one-component plasma,” Journal of Statistical Physics, vol. 28, no. 2, pp. 325–349, 1982.

[45] D. Arovas, J. R. Schrieﬀer, and F. Wilczek, “Fractional statistics and the quantum hall eﬀect,” Phys. Rev. Lett., vol. 53, pp. 722–723, Aug 1984.

[46] H. Kjønsberg and J. Leinaas, “Charge and statistics of quantum hall quasiparticles — a numerical study of mean values and ﬂuctuations,” Nuclear Physics B, vol. 559, no. 3, pp. 705 – 742, 1999.

[47] J. Kjäll, E. Ardonne, V. Dwivedi, M. Hermanns, and T. H. Hansson, “Ma- trix product state representation of quasielectron wave functions,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2018, p. 053101, may 2018.

[48] S. M. Girvin and A. H. MacDonald, “Off-diagonal long-range order, oblique confinement, and the fractional quantum hall effect,” Phys. Rev. Lett., vol. 58, pp. 1252–1255, Mar 1987.

[49] N. Read, “Order parameter and Ginzburg-Landau theory for the fractional quantum Hall eﬀect,” Phys. Rev. Lett., vol. 62, pp. 86–89, Jan 1989.

[50] J. K. Jain, “Composite-fermion approach for the fractional quantum hall eﬀect,” Phys. Rev. Lett., vol. 63, pp. 199–202, Jul 1989.

[51] B. I. Halperin, “Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States,” Phys. Rev. Lett., vol. 52, pp. 1583–1586, Apr 1984.

[52] N. Read, “Excitation structure of the hierarchy scheme in the fractional quantum hall eﬀect,” Phys. Rev. Lett., vol. 65, pp. 1502–1505, Sep 1990.

[53] J. K. Jain, Composite fermions. Cambridge University Press, 2007.

[54] S. M. Girvin and T. Jach, “Formalism for the quantum Hall eﬀect: Hilbert space of analytic functions,” Phys. Rev. B, vol. 29, pp. 5617–5625, May 1984.

[55] B. I. Halperin, P. A. Lee, and N. Read, “Theory of the half-ﬁlled landau level,” Phys. Rev. B, vol. 47, pp. 7312–7343, Mar 1993.

[56] R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiﬀer, “Experimental demonstration of a fermi surface at one-half ﬁlling of the lowest landau level,” Phys. Rev. Lett., vol. 71, pp. 3846–3849, Dec 1993.

113 Bibliography

[57] W. Kang, H. L. Stormer, L. N. Pfeiﬀer, K. W. Baldwin, and K. W. West, “How real are composite fermions?,” Phys. Rev. Lett., vol. 71, pp. 3850–3853, Dec 1993.

[58] N. Regnault, B. A. Bernevig, and F. D. M. Haldane, “Topological entanglement and clustering of jain hierarchy states,” Phys. Rev. Lett., vol. 103, p. 016801, Jun 2009.

[59] M. Greiter, “Microscopic formulation of the hierarchy of quantized hall states,” Physics Letters B, vol. 336, no. 1, pp. 48 – 53, 1994.

[60] E. J. Bergholtz, T. H. Hansson, M. Hermanns, and A. Karlhede, “Microscopic theory of the quantum hall hierarchy,” Phys. Rev. Lett., vol. 99, p. 256803, Dec 2007.

[61] T. H. Hansson, C.-C. Chang, J. K. Jain, and S. Viefers, “Conformal ﬁeld theory of composite fermions,” Phys. Rev. Lett., vol. 98, p. 076801, Feb 2007.

[62] T. H. Hansson, C.-C. Chang, J. K. Jain, and S. Viefers, “Composite-fermion wave functions as correlators in conformal ﬁeld theory,” Phys. Rev. B, vol. 76, p. 075347, Aug 2007.

[63] B. Halperin, “Theory of the quantized hall conductance,” Helv. Phys. Acta, vol. 56, pp. 75–102, 1983.

[64] R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, “Observation of an even-denominator quantum number in the fractional quantum Hall eﬀect,” Phys. Rev. Lett., vol. 59, pp. 1776–1779, Oct 1987.

[65] F. D. M. Haldane and E. H. Rezayi, “Spin-singlet wave function for the half- integral quantum hall eﬀect,” Phys. Rev. Lett., vol. 60, pp. 956–959, Mar 1988.

[66] R. H. Morf, “Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the ν = 5/2 Enigma,” Phys. Rev. Lett., vol. 80, pp. 1505–1508, Feb 1998.

[67] W. Pan, H. Stormer, D. Tsui, L. Pfeiﬀer, K. Baldwin, and K. West, “Exper- imental evidence for a spin-polarized ground state in the ν = 5/2 fractional quantum hall eﬀect,” Solid State Communications, vol. 119, no. 12, pp. 641 – 645, 2001.

114 Bibliography

[68] G. Moore and N. Read, “NonAbelions in the fractional quantum Hall eﬀect,” Nuc. Phys. B, vol. 360, pp. 362–396, 1991.

[69] V. W. Scarola, K. Park, and J. K. Jain, “Cooper instability of composite fermions,” Nature, vol. 406, no. 6798, pp. 863–865, 2000.

[70] N. Read and D. Green, “Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall eﬀect,” Physical Review B, vol. 61, no. 15, p. 10267, 2000.

[71] M. Greiter, X.-G. Wen, and F. Wilczek, “Paired Hall state at half ﬁlling,” Phys. Rev. Lett., vol. 66, pp. 3205–3208, Jun 1991.

[72] A. Cappelli, L. S. Georgiev, and I. T. Todorov, “Parafermion hall states from coset projections of abelian conformal theories,” Nuclear Physics B, vol. 599, no. 3, pp. 499 – 530, 2001.

[73] C. Nayak and F. Wilczek, “2n quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states,” Nuclear Physics B, vol. 479, pp. 529–553, 1996.

[74] P. Bonderson, V. Gurarie, and C. Nayak, “Plasma analogy and non-abelian statistics for ising-type quantum hall states,” Phys. Rev. B, vol. 83, p. 075303, Feb 2011.

[75] M. Levin, B. I. Halperin, and B. Rosenow, “Particle-hole symmetry and the Pfaﬃan state,” Phys. Rev. Lett., vol. 99, p. 236806, Dec 2007.

[76] S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, “Particle-hole symmetry 5 and the ν = 2 quantum Hall state,” Phys. Rev. Lett., vol. 99, p. 236807, Dec 2007.

[77] E. H. Rezayi, “Landau level mixing and the ground state of the ν = 5/2 quantum hall eﬀect,” Phys. Rev. Lett., vol. 119, p. 026801, Jul 2017.

[78] D. T. Son, “Is the composite fermion a dirac particle?,” Phys. Rev. X, vol. 5, p. 031027, Sep 2015.

[79] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, “Observation of half-integer thermal hall conductance,” Nature, vol. 559, no. 7713, pp. 205–210, 2018.

[80] S. H. Simon, “Interpretation of thermal conductance of the ν = 5/2 edge,” Phys. Rev. B, vol. 97, p. 121406, Mar 2018.

115 Bibliography

[81] D. E. Feldman, “Comment on “interpretation of thermal conductance of the ν = 5/2 edge”,” Phys. Rev. B, vol. 98, p. 167401, Oct 2018. [82] S. H. Simon, “Reply to “comment on ‘interpretation of thermal conductance of the ν = 5/2 edge’ ”,” Phys. Rev. B, vol. 98, p. 167402, Oct 2018. [83] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory. New York: Springer, 1997. [84] P. H. Ginsparg, “APPLIED CONFORMAL FIELD THEORY,” in Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phe- nomena Les Houches, France, June 28-August 5, 1988, pp. 1–168, 1988. [85] J. Wess and B. Zumino, “Consequences of anomalous ward identities,” Physics Letters B, vol. 37, no. 1, pp. 95 – 97, 1971. [86] E. Witten, “Global aspects of current algebra,” Nuclear Physics B, vol. 223, no. 2, pp. 422 – 432, 1983. [87] J. Frohlich and A. Zee, “Large scale physics of the quantum Hall fluid,” Nucl. Phys., vol. B364, pp. 517–540, 1991. [88] E. Ardonne, A conformal field theory description of fractional quantum Hall states. PhD thesis, University of Amsterdam, 2002. [89] E. Witten, “Quantum field theory and the jones polynomial,” Comm. Math. Phys., vol. 121, no. 3, pp. 351–399, 1989. [90] X. G. Wen, “Chiral luttinger liquid and the edge excitations in the fractional quantum hall states,” Phys. Rev. B, vol. 41, pp. 12838–12844, Jun 1990. [91] J. Fröhlich, B. Pedrini, C. Schweigert, and J. Walcher, “Universality in quantum Hall systems: Coset construction of incompressible states,” Journal of Statistical Physics, vol. 103, no. 3-4, pp. 527–567, 2001. [92] N. Read, “Conformal invariance of chiral edge theories,” Phys. Rev. B, vol. 79, p. 245304, Jun 2009. [93] T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, “Quantum hall physics: Hierarchies and conformal field theory techniques,” Rev. Mod. Phys., vol. 89, p. 025005, May 2017. [94] H. Kjønsberg and J. M. Leinaas, “On the anyon description of the Laughlin hole states,” International Journal of Modern Physics A, vol. 12, no. 11, pp. 1975–2001, 1997.

116 Bibliography

[95] M. P. Zaletel and R. S. K. Mong, “Exact matrix product states for quantum hall wave functions,” Phys. Rev. B, vol. 86, p. 245305, Dec 2012.

[96] Y. Tserkovnyak and S. H. Simon, “Monte carlo evaluation of non-abelian statistics,” Phys. Rev. Lett., vol. 90, p. 016802, Jan 2003.

[97] Y.-L. Wu, B. Estienne, N. Regnault, and B. A. Bernevig, “Braiding non- abelian quasiholes in fractional quantum Hall states,” Phys. Rev. Lett., vol. 113, p. 116801, Sep 2014.

[98] T. H. Hansson, M. Hermanns, and S. Viefers, “Quantum hall quasielectron operators in conformal ﬁeld theory,” Phys. Rev. B, vol. 80, p. 165330, Oct 2009.

[99] T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, “Quantum hall physics: Hierarchies and conformal ﬁeld theory techniques,” Rev. Mod. Phys., vol. 89, p. 025005, May 2017.

[100] A. E. B. Nielsen, I. Glasser, and I. D. Rodríguez, “Quasielectrons as inverse quasiholes in lattice fractional quantum hall models,” New Journal of Physics, vol. 20, p. 033029, mar 2018.

[101] A. Patra, B. Hillebrecht, and A. E. B. Nielsen, “Continuum limit of lattice quasielectron wavefunctions,” arXiv e-prints, p. arXiv:2004.12205, Apr. 2020.

[102] M. Hermanns, “Condensing non-abelian quasiparticles,” Phys. Rev. Lett., vol. 104, p. 056803, Feb 2010.

[103] E. Ardonne and K. Schoutens, “New class of non-abelian spin-singlet quantum hall states,” Phys. Rev. Lett., vol. 82, pp. 5096–5099, Jun 1999.

[104] Y. Tournois and M. Hermanns, “Conformal ﬁeld theory construction for non- abelian hierarchy wave functions,” Phys. Rev. B, vol. 96, p. 245107, Dec 2017.

[105] M. Levin and B. I. Halperin, “Collective states of non-abelian quasiparticles in a magnetic ﬁeld,” Phys. Rev. B, vol. 79, p. 205301, May 2009.

[106] P. Bonderson and J. K. Slingerland, “Fractional quantum hall hierarchy and the second landau level,” Phys. Rev. B, vol. 78, p. 125323, Sep 2008.

[107] G. J. Sreejith, C. Tőke, A. Wójs, and J. K. Jain, “Bipartite composite fermion states,” Phys. Rev. Lett., vol. 107, p. 086806, Aug 2011.

117 Bibliography

[108] E. J. Bergholtz and A. Karlhede, “Quantum hall system in tao-thouless limit,” Phys. Rev. B, vol. 77, p. 155308, Apr 2008.

[109] E. Ardonne, “Domain walls, fusion rules, and conformal ﬁeld theory in the quantum hall regime,” Phys. Rev. Lett., vol. 102, p. 180401, May 2009.

[110] J. Flavin and A. Seidel, “Abelian and non-abelian statistics in the coherent state representation,” Phys. Rev. X, vol. 1, p. 021015, Dec 2011.

[111] E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum regis- ters,” Annals of Physics, vol. 322, no. 1, pp. 201 – 235, 2007. January Special Issue 2007.

[112] E. Ardonne and G. Sierra, “Chiral correlators of the Ising conformal ﬁeld theory,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 50, p. 505402, 2010.

[113] V. Knizhnik and A. Zamolodchikov, “Current algebra and Wess-Zumino model in two dimensions,” Nuclear Physics B, vol. 247, no. 1, pp. 83 – 103, 1984.

[114] S. C. ZHANG, “The chern–simons–landau–ginzburg theory of the fractional quantum hall eﬀect,” International Journal of Modern Physics B, vol. 06, no. 01, pp. 25–58, 1992.

[115] J. Suorsa, S. Viefers, and T. H. Hansson, “A general approach to quantum hall hierarchies,” New Journal of Physics, vol. 13, p. 075006, jul 2011.

[116] J. Suorsa, S. Viefers, and T. H. Hansson, “Quasihole condensates in quantum Hall liquids,” Phys. Rev. B, vol. 83, p. 235130, Jun 2011.

[117] V. Crepel, B. Estienne, B. A. Bernevig, P. Lecheminant, and N. Regnault, “Matrix product state description of the halperin states,” 2018.

[118] J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys., vol. 82, pp. 277–306, Feb 2010.

[119] J. I. Cirac and F. Verstraete, “Renormalization and tensor product states in spin chains and lattices,” Journal of Physics A: Mathematical and Theoreti- cal, vol. 42, p. 504004, dec 2009.

[120] B. Estienne, N. Regnault, and B. A. Bernevig, “Fractional quantum hall matrix product states for interacting conformal ﬁeld theories,” 2013.

118 Bibliography

[121] T. Kvorning, “Quantum hall hierarchy in a spherical geometry,” Phys. Rev. B, vol. 87, p. 195131, May 2013.

[122] G. Y. Cho, Y. You, and E. Fradkin, “Geometry of fractional quantum Hall ﬂuids,” Physical Review B, vol. 90, no. 11, p. 115139, 2014.

[123] E. Fradkin, C. Nayak, and K. Schoutens, “Landau-Ginzburg theories for non- abelian quantum Hall states,” Nuclear Physics B, vol. 546, no. 3, pp. 711–730, 1999.

[124] H. Goldman, R. Sohal, and E. Fradkin, “Landau-ginzburg theories of non- abelian quantum hall states from non-abelian bosonization,” Phys. Rev. B, vol. 100, p. 115111, Sep 2019.

119