Diss. ETH N 19813 Non-Abelian Braiding Statistics In

Diss. ETH N◦19813

Non-Abelian braiding statistics in the fractional quantum Hall state at ﬁlling factor ν = 5/2? Exact diagonalization investigations

A dissertation submitted to ETH ZURICH

for the degree of Doctor of Sciences

presented by Maurizio Luigi Storni Dipl. Phys. ETH born April 28, 1976 citizen of Capriasca (TI)

accepted on the recommendation of Prof. Dr. J¨urgFr¨ohlich, examiner Prof. Dr. Nicholas d’Ambrumenil, co-examiner Dr. Rudolf Morf, co-examiner

2011

Per mamma e pap`a.

Contents

Abstract 3

Riassunto 5

Aknowledgments 7

1 The quantum Hall effect 9 1.1 The integer quantum Hall effect ...... 9 1.2 The fractional quantum Hall effect ...... 11 1.3 Excitations in fractional quantum Hall systems: quasiparticles and quasiholes ...... 13

2 ν = 5/2: what is known and what is possible 15 2.1 Enter 5/2 ...... 15 2.2 5/2: polarized or unpolarized? ...... 16 2.3 The Moore-Read (“Pfaffian”) state: the field theoretical point of view ...... 20 2.4 The Moore-Read (“Pfaffian”) state: the microscopic point of view 21 2.5 The “Pfaffian” at ν = 5/2? First hints...... 24 2.6 Nonabelions in the Moore-Read state ...... 27 2.7 The “weak paired” phase ...... 29 2.8 “Pfaffian” numerics ...... 30 2.9 The “Pfaffian” on a torus ...... 31 2.10 The others ...... 32 2.11 Topological quantum computation ...... 35 2.12 The “Pfaffian” (or others) at ν = 5/2? Experiments...... 37 2.13 The “Pfaffian” (or others) at ν = 5/2? Numerics...... 45

3 ν = 5/2: our numerical investigations 53 3.1 Introduction ...... 53 3.2 The ground state ...... 55 3.3 Quasihole excitations ...... 60

Conclusions 70

Bibliography 71

Curriculum Vitae 87

1 2 Abstract

The subject of this dissertation is the fractional quantum Hall effect at the filling fraction ν = 5/2. Nearly 25 years after its experimental discovery, its nature is still unclear: it is the strongest fractional quantum Hall effect in the second Landau level but, because of its even denominator, it does not fit in the “odd denominator rule” of the hierarchy scheme. For this reason, to describe it, some alternative states has been proposed: between them of particular interest is the (Moore-Read) “Pfaffian” state. The elementary excitations above this state are e/4 charged quasiparticles that obey non-Abelian braiding statistics: the exchange of the position of two quasiparticles results in a rotation in the multidimensional Hilbert space of the many-quasiparticle states, and not merely in a 1 or a phase factor (as for fermions/bosons or Abelian anyons). For example,± the Hilbert space for a system containing four Moore-Read quasiholes is two-dimensional (whereas there is no such degeneracy in a two-quasiholes system). Because of this property, the “Pfaffian” state has been proposed for the realization of a topological quantum computer. There are two possible ways to address the question whether the experimentally realized fractional quantum Hall effect at ν = 5/2 is indeed in the same universality class of the “Pfaffian” state: by experimental or by numerical investigations. Experimental tests of the non-Abelian braiding statistics have been proposed and some results are already available, but they are not yet conclusive (and other experiments are in progress). In this dissertation we take the other way, studying, by exact diagonalization, quantum Hall systems with a small number of spin-polarized electrons (Nel 20) in the sphere geometry. At first we consider the ground state≤ at ν = 5/2. By calculating the energy spectrum of the few lowest-energy states, we show that the ground state for Coulomb interaction is adiabatically connected with the Moore-Read state: interpolating between the two limits, the ground state is protected by a large gap, with no sign of phase transition for all examined system sizes. We also modify the (two-body) electron interaction, by varying the Haldane pseudopotentials v1 and v3 (keeping all others at their Coulomb value), drawing a phase diagram in the (v1, v3)-plane. (For example the finite thickness of the quantum Hall sample causes a variation of these pseudopotentials.) We find that in the (v1, v3)-plane the quantum Hall energy gap and the overlap of the exact ground state with the “Pfaffian” state form two hills, whose positions and extents coincide: the energy gap is large there where the overlap is large. We interpret this as a sign that the fractional quantum Hall (ground) state at ν = 5/2 is indeed in the “Pfaffian” phase. During these investigations we also look at the system at ν = 1/2, finding that it is in a compressible phase, but near to a phase transition to the quantum Hall (gapped) phase. The variation of the system thickness could lead to a phase transition into the quantum Hall phase, however from our calculations this seams unlikely to happen. Then we look at systems at ν = 5/2 containing a small number of localized e/4 charged quasiholes. We first show that it is indeed possible to localize such quasiholes on the surface of the sphere, using δ-function pinning potentials. Using a smooth pinning potential we also show that it is possible to reduce the radius of the quasihole to a minimum of about three magnetic lengths. We then

3 perform the adiabatic connection investigations for systems containing two and four localized quasiholes. For two quasiholes, the lowest energy state evolves adiabatically between the “Pfaffian” and the Coulomb limit, without mixing with higher lying energy states. For four quasiholes we find that the lowest two states in the “Pfaffian” limit, corresponding to the degenerate Moore-Read doublet, remain the lowest-energy states even for pure Coulomb interaction. We conclude that the adiabatic continuity holds also for systems containing quasiholes. Finally, in the Coulomb limit (only slightly modifying the first Haldane pseudopotential), we perform quasiholes braidings in systems containing four quasiholes: we keep two of them fixed and exchange the positions of the other two by stepwise changing the location of their pinning potentials. We find that under such a braiding, the system goes from one of the states of the Moore-Read doublet to the other: a sign of their non-Abelian braiding statistics.

4 Riassunto

Il tema di questa dissertazione èl’effetto Hall quantistico frazionario per il fattore di riempimento ν = 5/2. Quasi 25 anni dopo la sua scoperta in un es- perimento, la sua natura èancora misteriosa: èl’effetto Hall frazionario più evidente nel secondo livello di Landau ma, a causa del suo denominatore pari, non obbedisce alla “regola del denominatore dispari” dello schema gerarchico. Per questa ragione, per descriverlo, sono stati proposti alcuni stati alterna- tivi: tra questi di particolare interesse èlo stato “Pfaffiano” di Moore e Read. Le eccitazioni elementari di questo stato sono quasiparticelle con carica e/4 che obbediscono ad una statistica di treccia non-Abeliana: lo scambio della posizione di due quasiparticelle risulta in una rotazione nel multidimensionale spazio di Hilbert contenente gli stati con molteplici quasiparticelle, e non solamente in un fattore o in una fase (come per fermioni/bosoni o anioni Abeliani). Per esempio lo± spazio di Hilbert per un sistema contenente quattro quasibuchi di Moore e Read èbidimensionale (mentre non c’èuna tale degenerazione per un sistema con due quasibuchi). A causa di questa proprietà,lo stato “Pfaffiano” èstato proposto per la realizzazione di un computer quantistico topologico. Ci sono due possibili approcci per affrontare la questione se l’effetto Hall frazionario per ν = 5/2 realizzato sperimentalmente èdavvero nella stessa classe di universalitàdello stato “Pfaffiano”: con investigazioni sperimentali o numeriche. Prove sperimentali della statistica di treccia non-Abeliana sono state proposte e alcuni risultati sono giàdisponibili, ma non ancora conclusivi (e altri esperimenti sono in corso). In questa dissertazione scegliamo l’altra via, studiando, per mezzo di diagonalizzazioni esatte, sistemi Hall quantistici con un piccolo numero (Nel 20) di elettroni (spin-polarizzati) nella geometria sferica. Dapprima consideriamo≤ lo stato fondamentale per ν = 5/2. Calcolando lo spettro energetico degli stati energeticamente piùbassi, mostriamo che lo stato fondamentale per l’interazione di Coulomb èconnesso adiabaticamente con lo stato di Moore e Read: interpolando tra i due limiti, lo stato fondamentale è protetto da un grande gap, con nessun segno di transizione di fase per tutti i sistemi esaminati. Inoltre modifichiamo l’interazione (a due corpi) tra gli elettroni, variando gli pseudopotenziali di Haldane v1 e v3 (tenendo tutti gli altri al loro valore di Coulomb), disegnando un diagramma di fase nel piano (v1, v3). (Per esempio lo spessore finito del campione Hall quantistico causa una variazione di questi pseudopotenziali.) Troviamo che nel piano (v1, v3) il gap energetico Hall quantistico e la sovrapposizione dello stato fondamentale esatto con lo stato “Pfaffiano” formano due colline, le cui posizioni e estensioni coincidono: il gap energetico ègrande làdove la sovrapposizione ègrande. Interpretiamo questo come un segno che lo stato Hall quantistico frazionario (fondamentale) per ν = 5/2 èin effetti nella fase “Pfaffiana”. Nel corso di queste investigazioni guardiamo anche al sistema per ν = 1/2, trovando che èin una fase compressibile, ma vicino ad una transizione di fase verso la fase Hall quantistica (con gap). La variazione dello spessore del sistema potrebbe portare ad una transizione di fase verso la fase Hall quantistica, ma dai nostri calcoli questo sembra improbabile. In seguito consideriamo sistemi a ν = 5/2 contenenti un piccolo numero di quasibuchi (con carica e/4) localizzati. Dapprima mostriamo che èin effetti possibile localizzare tali quasibuchi sulla superficie della sfera, usando potenziali di localizzazione a funzione δ. Usando un potenziale di localizzazione regolare,

5 mostriamo inoltre che èpossibile ridurre il raggio del quasibuco fino ad un minimo di circa tre lunghezze magnetiche. Poi eseguiamo le investigazioni di connessione adiabatica per sistemi contenenti due e quattro quasibuchi localizzati. Per due quasibuchi, lo stato energetico piùbasso evolve adiabaticamente tra il limite “Pfaffiano” e il limite di Coulomb, senza mescolanza con altri stati con energia superiore. Per quattro quasibuchi troviamo che i due stati piùbassi, che corrispondono al doppietto degenerato di Moore e Read, rimangono gli stati energetici piùbassi anche per la pura interazione di Coulomb. Concludiamo che la continuitàadiabatica èvalida anche per sistemi contenenti quasibuchi. In conclusione, nel limite di Coulomb (unicamente lievemente modificando il primo pseudopotenziale di Haldane), eseguiamo degli scambi di quasibuchi in sistemi contenenti quattro quasibuchi: manteniamo fissi due di loro e scambi- amo la posizione degli altri due, cambiando passo dopo passo la posizione dei loro potenziali di localizzazione. Troviamo che a causa di un tale scambio di posizione, il sistema evolve da uno degli stati del doppietto di Moore e Read all’altro: un segno della statistica di treccia non-Abeliana.

6 Aknowledgments

I want to thank ...... Rudolf Morf, for all his help, support and kindness (...and for the Sprüngli Truffles) ... JürgFröhlich, for the useful discussions ... all the (past and present) condensed matter theory colleagues at the Paul Scherrer Institute, for the pleasant atmosphere in the group ... my friends, for the enjoyable time spent together during these years ... my parents (ciao mamma), who always supported me with their love ... and the swans on the Aare near to PSI, for the silent smile they cause in me

7 8 Chapter 1

The quantum Hall eﬀect

1.1 The integer quantum Hall eﬀect

In 1980, K. Von Klitzing, G. Gorda and M. Pepper made a remarkable discovery [1]: In doing Hall measurements on a two-dimensional electron system at the interface of a silicon MOSFET (metal-oxide-semiconductor field effect transistor) at low temperatures, they found that the Hall resistance RH = VH /I (see Fig. 1.1 for a sample setup) did not follow the classical behaviour, which would be linear in the applied perpendicular magnetic field B: B R = , (1.1) H ne where e is the elementary charge and n the electron surface density. Instead, they found that, at certain values of the magnetic field, plateaux were formed, 1 where the Hall conductance σH = RH− was very precisely quantized to an integer times the fundamental unit e2/h (h the Planck’s constant). At the values of the magnetic field where the plateaux in the Hall conductance are observed, the longitudinal resistance RL = VL/I goes to zero. This effect is called the integer quantum Hall effect: see Fig. 1.2 for recent experimental results. This observation was completely unexpected, but it can be explained in terms of non-interacting electrons confined in a two-dimensional system, with a strong magnetic field perpendicular to it. In these systems, the electronic states are extended and organize themselves in evenly spaced Landau levels, which are highly degenerate. The Landau levels are separated by large energy gaps ~ωc (ωc = eB/me is the cyclotron frequency), in comparison to the other energy scales in the problem, which are the Zeeman and interaction energy. With p (an integer) Landau levels completely filled, the Hall conductance is 2 quantized to σH = p e /h. To explain the plateau behaviour, the consequences of disorder have to be taken into account. The effect of disorder is to localize some of the extended states, while their energy is slightly shifted. By changing the magnetic field, one changes the cyclotron energy and thus the filling of the Landau levels and the Fermi-level. If the Fermi-level is in a region where only localized states (which do not contribute to the conductance) are present, changing the magnetic field does not change the conductance and hence one observes a plateau. The regions where the Hall conductance changes from one

9 I

Figure 1.1: Schematic experimental setup. The magnetic field B is perpendicular to the (almost) two-dimensional sample, in which flows the electric current History of the (Quantum) Hall Effect I. VL is the longitudinal potential drop, VH the Hall voltage.

3.0 2.0 Ix Vy

Vx 2.5

1.5 2.0

)

) 6 5 4 3 2 1 1.5

Ω

(h/e

1/2 ρ xx 1.0 2/3 3/5 3/7 2/5

ρ 3/2 3/4 4/7 4/9 1.0 5/9 5/11

6/11 6/13 4/3 0.5 0.5 5/3 5/7 8/5 7/5 0.0 8/15 7/15 4/5 7/13 0 0 4 8 12 16 magneticMagne tifieldc Field B[T]B (T)

Fig. 1.3 Typical signature of the quantum Hall effect (measured by J. Smet, MPI-Stuttgart). FigureE 1.2:ach pla Integerteau in the andHall fractionalresistance is acc quantumompanied by Halla vani effectshing lon experimentalgitudinal resistance results:. The classical Hall resistance is indicated by the dashed-dotted line. The numbers label the longitudinalplateaus and: integ Hallral n d resistivityenote the IQHE asand functionn = p/q, w ofith theintegr perpendicularal p and q, indicate t magnetiche FQHE. field (measurements by J. Smet, MPI-Stuttgart). The (integer and fractional) numbers inin figureterms o aref an in theteger correspondingn. The plateau in t Landauhe Hall resi levelstance fillingis accom factors.panied by a (vRanish-= ρ , ing longitudinal resistance. This is at first sight reminiscent of the Shubnikov-de HHaas xy R ρ .) L ∝ exxffect, where the longitudinal resistance also reveals minima although it never van- ished. The vanishing of the longitudinal resistance at the Shubnikov-de Haas minima may indeed be used to determine the crossover from the Shubnikov-de Haas regime to plateauth toe IQ anotherHE. correspond to magnetic fields where the Fermi-level lies in It is noteworth to mention that the quantisation of the Hall resistance (1.12) is the regiona uni ofver thesal ph extendedenomenon, i.statese. indep (whichendent of contributethe particula tor pr theopert conductance).ies of the sample, 2 To explainsuch as it thats geom theetry, Hallthe h conductanceost materials use isd quantizedto fabricate t preciselyhe 2D electr aton pga es /hand,also in the presenceeven mo ofre disorder,importantly, one its i canmpuri usety co ance gaugentratio argumentn or distribut (givenion. Thi bys un Laughliniversality [2]). is the reason for the enormous precision of the Hall-resistance quantisation (typically Due to gauge−9 invariance and to the presence of a (mobility) gap,3 adiabatically ∼ 10 ), which is nowadays – since 1990 – used as the resistance standard, changing the magnetic flux by one flux2 quantum will just result in the transfer of charge from one edge of theRK sample−90 = h/e to= another.25 812.807 Ω If, p electrons are transferred,(1.13) which is also called the Klitzing constant (Poirier 2and Schopfer, 2009a; Poirier and Schopfer, 2009b). this leads to a Hall conductance σH = p e /h, regardless of the presence of Furthermore, as already mentioned in Sec. 1.1.2, the vanishing of the longitudinal re- disorder in the sample. The quantization of the Hall conductance can be so 3The subscript K honours v. Klitzing and 90 stands for the date since which the unit of resistance is defined by the IQHE. 10 precise as found in the experiments because it is based only on gauge invariance and the presence of a mobility gap.1

1.2 The fractional quantum Hall eﬀect

Two years after the discovery of the integer quantum Hall effect an even more remarkably effect was discovered [4]: in extremely clean GaAs/AlxGa1 xAs het- erostructures, D. C. Tsui, H. L. Störmerand A. C. Gossard observed a− quantum Hall effect at values of the magnetic field which correspond to a fractionally filled Landau level. The value of the Hall conductance at these plateaux was a simple fraction of the fundamental conductance: p e2 σ = , (1.2) H q h where p and q are small integers, while q is (mostly) odd. The rational number ν = p/q is the Landau levels filling fraction. This effect is called the fractional quantum Hall effect. Figure 1.2 shows typical resistivity measurements on a clean sample: the trace of the Hall resistance RH (= ρxy) shows plateaus and at the same places the longitudinal resistance RL( ρxx) has deep minima. This fractional quantum Hall effect can not∝ be explained using the non- interacting picture introduced above to explain the integer quantum Hall effect, because in that picture no gap can arise within a Landau level. The presence of a plateau in the Hall conductance and the vanishing longitudinal resistance imply the presence of a (mobility) gap. The interactions between the electrons are crucial in the formation of such a gap. The first step in explaining the fractional quantum Hall effect was made by Laughlin [5], who proposed a set of quantum states in the form of trial wavefunctions for the filling fractions ν = 1/q. These wavefunctions were shown to capture the basic features of the fractional quantum Hall states, such as the gap and the low energy excitations. In Laughlin’s picture it is assumed that all the electrons are in the lowest Landau level and that they are fully spin-polarized by the strong magnetic field. In the planar geometry, for the symmetric gauge, the single-electron wavefunctions have then the form z 2/4 Ψ(r) = p(z)e−| | , (1.3) where p(z) is a polynomial in z = (x iy)/`0, a complex number representing the 2-dimensional coordinates r = (x,− y) in the plane and

~ `0 = (1.4) reB is the magnetic length.2 The polynomial p(z) = zm represents a state with angular momentum component Lz = m~. Since electrons are fermions, the many-particle wavefunctions Ψ(r1, r2,...) = P 2 zi /4 f(z1, z2,...)e− i | | have to be antisymmetric under the exchange of any

1For a more thoroughly discussion of the integer quantum Hall eﬀect see for example the ﬁrst chapters of [3]. 2For a derivation see for example Chapter 2, “Two-Dimensional Electrons in a Magnetic Field” in [3].

11 two particles; furthermore, in the disk geometry, they may be chosen to be eigenstates of the total angular momentum operator, because the interaction between the electrons is rotational symmetric. Thus f(z1, z2,...) has to be an homogeneous, antisymmetric polynomial in the variables z1, z2,... In addition it is assumed that the (strong repulsive) interactions are taken into account via a Jastrow factor g(zi zj), which is a two-body correlation keeping the electrons apart. With these− constraints one obtains the unique solution g(z) = zq, with q an odd integer,Q and the Laughlin’s wavefunctions are:

q P z 2/4 Ψ (r , r ,...) = (z z ) e− i | i| . (1.5) q 1 2 i − j i>j Y They describe a quantum Hall system with filling factor ν = 1/q: the maximum power in zi, that is the maximum angular momentum of the i-th electron (also equal to Nφ, the number of magnetic flux quanta in the system), is M = (N 1)q, where N is the number of particles; the num- el − el ber of orbitals Norb in the lowest Landau level is equal to M + 1 (the states with higher angular momentum are unoccupied) and this gives the filling factor ν = N /N = N /[(N 1)q + 1] 1/q in the thermodynamic limit el orb el el − → Nel . These→ ∞ wavefunctions are not the exact ground states for Coulomb interaction, but they are a good approximation of them [6,7]: they were shown to have very large overlap with the numerically obtained ground state wavefunctions for small numbers of electrons and for a large class of repulsive interactions. In particular Ψq is the exact ground state of the short-range interaction with pos- 3 itive Haldane pseudopotentials vm for m < q and all others equal to zero [8]. The Haldane pseudopotential vm is defined as the interaction energy between two particles (in a given Landau level) with relative angular momentum m~. Before we go on to discuss the properties of the excitations over the quantum Hall systems, we will first briefly discuss the other fractional quantum Hall states, at filling ν = p/q, with p > 1. Note that all the fractions presented in the experimental plot in Fig. 1.2 have an odd denominator. First, it is clear that the filling factors ν = (q 1)/q (such as 2/3, 4/5,...) can be explained as − Laughlin states Ψq describing holes instead of electrons. To explain the other quantum Hall systems there are two different, but equivalent [10], pictures: the hierarchy of quantum Hall states and the composite fermions approach. In the viewpoint of the hierarchy [8, 11, 12] the quasiparticles that are the elementary excitations of Laughlin ground state (see the following section) condense themselves in a Laughlin-like state, if the particle density is appropriate, giving rise to a new quantum Hall state. The quasiparticles of this “daughter” state can also condense and so on: in this way one can costruct a quantum Hall state for each odd-denominator fraction. The realization of it depends on the interactions between the quasiparticles and on the size of the gap in the parent state. The fact that only odd-denominator filling fractions can be realized this way, has its roots in the antisymmetric condition for the Laughlin wavefunctions, that is in the fermionic nature of the electrons. In the composite fermions approach [13] an even number of magnetic flux quanta is bound to each electron, forming so-called “composite fermions”, which

3See also Chapter 8, “The Hierarchy of Fractional States and Numerical Studies” by F. D. M. Haldane in [9].

12 effectively feel a reduced magnetic field. These composite fermions can condense in an integer quantum Hall state, that for the original electrons corresponds to a fractional quantum Hall effect with filling fraction ν = p/(2mp 1), where p is the number of occupied Landau levels for the composite fermions± and 2m is the number of attached flux quanta. Almost all the observed fractions can be obtained in this way, but there are exceptions [14]. Until now we have considered only fractions with odd denominators: the prominent exception to this “odd denominator rule”, the quantum Hall state at ν = 5/2 [15], is the central topic of this work and will be discussed at length in the following chapter.

1.3 Excitations in fractional quantum Hall systems: quasiparticles and quasiholes

The Laughlin wavefunction Ψq (1.5) introduced in the previous section describes an incompressible fluid with a uniform electron density and a finite energy gap between the ground state and the first excited states in the bulk. It can be shown [5] that the elementary excitations above it are localized excesses (or deficiencies) of the electron density (from its mean value) and can be described as quasiparticles (or quasiholes). The size of them is of the order of few magnetic lengths (1.4) and the excess (deficiency) in charge is such that they carry a fractional charge e/q. This prediction has been confirmed experimentally for ν = 1/3 by means± of shot noise experiments [16, 17]. One can introduce a quasihole in a uniform quantum Hall system at ν = 1/q by (adiabatically) inserting a magnetic flux quantum φ0 = h/e in the system, decreasing its filling fraction. This insertion has the effect of expelling some charge from the region of the flux quantum and is described by a factor p(z0) = Nel i=1(zi z0) multiplying the wavefunction Ψq (1.5), where z0 is the quasihole position:− Q Nel Ψ = (z z )Ψ . (1.6) q,qh i − 0 q i=1 Y The factor p(z0) add a zero at z0 for each particle coordinate zi, such that the electron density there decrease. One can show that the missing charge at z0 is exactly e/q: adding q quasiholes and then an electron at z0 corresponds to a multiplication of Ψ by a factor Nel (z z )q, where now z is the coordinate q i=1 i − 0 0 of the new electron. What we obtain is a Laughlin wavefunction for Nel + 1 electrons, that is again a uniformQ system: the q quasiholes are neutralized by the electron. In analogy, one can introduce a quasiparticle (a quasielectron) by removing a flux quantum, increasing the filling fraction. These quasiholes (and quasiparticles) are anyons: their exchange statistics is fractional [11, 18], in the sense that it interpolates between Fermi and Bose statistics. Taking two identical quasiholes and exchanging their positions, the wavefunction for the new configuration is obtained multiplying the old wavefunction by a phase factor eiθ (and not merely by a 1 factor as for fermions or bosons). ± This is possible only in systems with a (spatial) dimension d < 3: in three (and higher) dimensions, different exchange paths of two identical particles can

13 be continuously deformed into each other, thus, after two successive exchanges, the system comes back to the original situation, described by the same wavefunction. The phase factor corresponding to the exchange of two particles has to be +1 or 1, corresponding to Bose or Fermi statistics. In lower dimensions the situation− is diﬀerent and “exotic” statistics are allowed, as ﬁrst recognized in 1+1-dimensional space-time [19,20] and then in two spatial dimensions 4 [22–29]. Concretely, in the latter case, not all particle exchange paths can be deformed into each other, because one would have to pull a path of a particle through the position of another particle. Thus the result of a particle exchange is not restricted to a 1-factor and fractional statistics is allowed: in two dimensions the topology of± the exchange paths is relevant and the exchange properties of a system with N particles is described by the braid group (while in three BN dimensions it is the permutation group SN ). The Laughlin quasiparticles described above transform according to a one- dimensional representation of the braid group. There are attempts to measure the fractional statistics of these quasiparticles, by means of interferometric experiments, but the results are not yet conclusive [30]. Another interesting consequence of the fact that particles in lower spatial dimensions have to form a representation of the braid group, rather than of the permutation group, is that higher dimensional representations can be possible [21,28,31–35], that is the exchange of particles is described by matrices. If these matrices do not commute with each other, this particles are called non-Abelian anyons, while the quasiparticles that transforms according to one-dimensional representations are Abelian anyons.5 In this work we will study the quantum Hall system at ν = 5/2, where this non-Abelian braiding statistics could be realized [37].

4For a historical overview of this issue, see the introduction of [21]. 5See [36] for a pedagogical review on Abelian and non-Abelian anyons in the framework of the fractional quantum Hall eﬀect.

14 VOLUME 83, NUMBER 17 PHYSICAL REVIEW LETTERS 25OCTOBER 1999

electron system (2DES) and its host lattice decreases pre- mounted in the low-field region, at the top of the nuclear Chapter 2 cipitously with decreasing temperature [4,16]. In fact, at stage. The accuracy of the thermometry is estimated to be very low temperatures cooling of the electrons proceeds better than 0.05 mK. All measurements were performed in largely via the electrical contacts. Electrons diffuse to an ultraquiet environment, shielded from electromagnetic the contacts, where they cool in the highly disordered re- noise. RC filters with cutoff frequencies of 10 kHz were gion formed by the “dirty” alloy of GaAs and indium. employed to reduce rf heating. The data were collected us- = 5 2: whatTherefore, is knowncooling of the contacts is of andparamount impor- ing a PAR-124A analog lock-in with an excitation current ν / tance in low-temperature transport experiments and our of 1 nA at typically 5 Hz. At this current level electron cooling system was designed to cool specifically the con- heating was undetectable for a temperature greater than what is possibletact areas of the 2DES specimen. Eight sintered silver 8 mK. This was deduced from a series of heating experi- heat exchangers each having an estimated surface area of ments, in which Rxx was measured at different currents 2 ϳ0.5 m and formed around a 10 mil silver wire were sol- from 0.5 to 100 nA at Tb ෇ 8.0 mK. The resistance of dered directly to the indium contacts of the sample using the strongly T-dependent Rxx peak at 3.75 T in Fig. 1 was indium as a solder. They provide electrical contact and si- used as an internal thermometer for the electron tempera- multaneously function as large area cooling surfaces. The ture, Te, assuming Te ෇ Tb at higher Tb and in the limit 5 back side of the sample was glued with gallium to yet an- of low current. The Te vs I data fit the expected T rela- 2 ෇ 1 5 2 5 2.1 Enter 5/2 other large surface area heat exchanger for efficient cool- tionship [4,16], ln͑I ͒ const ln͑Te Tb ͒, where I is ing of the lattice of the specimen. The inset of Fig. 1 in nA, T in mK, and const ෇ 27.5.AtTb ෇ 8.0 mK and ෇ 8.05 All the quantum Hall states discussedshows so fara sketch hadof athis fillingarrangement. factorSample withand anheat oddex- 1 nA current it yields Te mK, sufficiently close to changers were immersed into a cell made from polycar- Tb for the difference to be negligible. 3 denominator, which was explained viabonate the hierarchyand filled with schemesliquid He. startingThe liquid fromis cooled theby Figure 1 shows the Hall resistance Rxy and the longi- the PrNi5 nuclear demagnetization stage of a dilution re- tudinal resistance Rxx between Landau level filling fac- Laughlin states. In 1987 a first experimental indication was found of a quantum3 frigerator via annealed silver rods that enter the He cell. tors n ෇ 3 and n ෇ 2 at Tb ෇ 2 mK. Using the above Hall effect at a filling fraction with anThis evenbrings denominator:the system’s baseν temperature= 5/2 [15].to 0.5 InmK, thiswell equation for an excitation current of 1 nA, the deduced state the lowest Landau level of both spinsbelow the is8 fullmK andof the adilution secondunit. Landau level is electron temperature, Te,isϳ4 mK. Ultimately it is un- half occupied. A cerium magnesium nitrate thermometer, a 3He melt- clear, whether the T 5 law continues to hold. However, if ing curve thermometer, and/or a Pt NMR thermometer are anything, we would expect Te ϳ 4 mK to be a conserva- tive estimate, since electron cooling via the well heat-sunk a) b) contacts should lower Te below the limit set by lattice cooling. Strong minima emerge in Rxx in the vicinity of filling factors n ෇ 5͞2, n ෇ 7͞3, and n ෇ 8͞3. Satel- lite features can be made out that are probably associated with filling factors n ෇ 19͞7, 13͞5, 12͞5, and 16͞7.If this identification is correct, apart from the n ෇ 5͞2 state, the successive development of FQHE states is not unlike the one observed in the lowest Landau level. However, as long as such minima do not show concomitant Hall plateaus, one cannot be certain as to their quantum numbers. Strikingly different from the Rxx pattern around n ෇ 1͞2 is not only the existence of the central n ෇ 5͞2 state, but also the emergence of very strong maxima flank- ing this minimum. The primary minima at n ෇ 5͞2, n ෇ 7͞3, and n ෇ 8͞3 show well developed Hall plateaus. In particular, the 5͞2 plateau is extensive, allowing its value to be measured to high precision. This was performed with a current of 20 nA, which raises Te to ϳ15 mK, but increases the signal to noise, while keeping the plateau intact. The lock-in operated at 23 Hz and its output was averaged for 20 min. The neighboring integral quantum Hall effect (IQHE) plateaus at n ෇ 2 and n ෇ 3 were used as standard resistors to which the value at n ෇ 5͞2 was FIG. 1. Hall resistance Rxy and longitudinal resistance Rxx at compared. In the center of the plateau, as determined by an electron temperature T ഠ 4.0 mK. Vertical lines mark the e the midpoint between the sharply rising flanks on either Landau level filling factors. The inset shows a schematic of 2 Figure 2.1: Experimental evidence forthe thesample quantumwith attached Hallsintered effectsilver atheatν exchangers= 5/2:(gray) a) side, Rxy was found to be quantized to h͑͞5͞2͒e to better 2 the first experiment (from [15]) and b)to thecool exactthe 2DES. quantization (from [38]). than 2 3 10 6 over a range of at least 0.01 T, equivalent

In this ﬁrst experiment [Fig. 2.1a)] the plateau in the Hall resistivity and the 3531

15 deep minimum in the longitudinal resistivity were not yet fully developed, but nevertheless they were a strong evidence for a new quantum Hall state. Indeed with the improvement of the quantum Hall samples and of the experimental techniques it has been possible to obtain the exact quantization at ν = 5/2 [38] and this state is now routinely observed [Fig. 2.1b)]. Similar results are obtained for the ﬁlling factor ν = 7/2 [39], corresponding to a half ﬁlled second Landau level for holes. The activation gaps ∆ are usually extracted from transport experiments (in [40–45] the most recent ones) through the temperature dependence of the longitudinal resistivity ρxx via the Arrhenius law

ρ exp( ∆/2k T ) (2.1) xx ∝ − B

(where T is the temperature and kB the Boltzmann’s constant). The gaps extracted this way are much smaller than the theoretical expectations, calculated in exact diagonalization [43, 46, 47] or density-matrix renormalization group [48,49] studies: even the biggest experimental activation gaps for ν = 5/2 are a factor 5-6 smaller than the predictions for a clean system. Inclusion of finite thickness and Landau level mixing effects [43,47] can reduce the difference by about a factor of two, but a big discrepancy remains. This discrepancy has to be (eventually) explained by the effects of disorder. Recently d’Ambrumenil et al. [50] proposed a model for the dissipative conductance, taking into account the thermally assisted tunneling of the quasiparticles through the saddle points of the potential caused by the disorder. In the framework of this model the agreement of the experimentally obtained gaps at ν = 5/2 with the theoretical calculations is as good as for the (hierarchical) ν = 4/9 state. The (conventional) composite fermion and hierarchy approaches can not explain the quantum Hall effect at ν = 5/2, as they only cover states with an odd denominator filling. From the beginning it was clear that this quantum Hall system was different from all the previously observed states and needed a different explanation. Various quantum Hall states were proposed to account for this effect: we will discuss some of them in the following sections, particularly focussing on the (weak) paired quantum Hall state proposed by Moore and Read [51, 52], described by the “Pfaffian” wavefunction, that supports quasiparticle excitations obeying non-Abelian braiding statistics.

2.2 5/2: polarized or unpolarized?

A central issue for an explanation of the quantum Hall effect at ν = 5/2 is the spin polarization of the electrons in the second Landau level: they may be fully polarized and aligned with the external magnetic field, or unpolarized, half of them with spin up, half with spin down, or something in between (partially polarized). Note that the electrons in the full first Landau level have both spin orientations and give no contribution to the total polarization. All the quantum Hall systems discussed so far were considered to be fully spin polarized by the strong external magnetic field. But if the magnetic field is sufficiently low, with a corresponding low Zeeman energy, the two spin bands (of the same Landau level) can mix and cause an unpolarized, or only partially polarized, ground state.

16 The odd-denominator rule for the Laughlin states (and their hierarchical descendants) was a consequence of the spin polarization, that forced the wavefunctions to be fully antisymmetric in the spacial coordinates. Being the ν = 5/2 state an exception to this rule, it was natural to speculate that this peculiar- ity could reflect a breakdown of the assumption of full spin polarization. The relatively low magnetic fields (B 5T ) at which the ν = 5/2 state was first observed supported this speculation.≈ To gain access to the spin degree of freedom the tilted magnetic field tech- nique has been used: the Landau level spacing ~ωc depends only on the perpendicular component B of the magnetic field, while the Zeeman energy is ⊥ proportional to the total field BTOT . Thus, by tilting the quantum Hall sample in a magnetic field, it is possible to vary the two energies independently. For example experiments at ν = 8/5 in a tilted magnetic field (keeping B constant) showed a phase transition from an unpolarized state at small tilt angles⊥ (lower Zeeman energy) to a polarized state at larger angles (larger Zeeman energy) [53].

Figure 2.2: Disappearence of the ν = 5/2 quantum Hall state in a tilted ﬁeld: energy gap ∆5/2 as function of the total magnetic ﬁeld BTOT (and of the tilting angle θ). The collapse of the gap was interpreted to be caused by a Zeeman energy term with g-factor g = 0.56. (From [54].)

Shortly after its discovery, tilted field experiments were performed also on the ν = 5/2 state [54,55]. Examining the activation gaps from these experiments, it was found that the gap ∆ decreases with increasing tilt angle (see Fig. 2.2) and that the Hall plateau disappeared beyond some critical tilt angle. These results were interpreted as a signal that the quantized Hall state is unpolarized (or at most only partially spin-polarized) with a first excitation involving a spin flip, which is responsible for the linear decrease of the gap as function of BTOT :

∆ = ∆ gµ B (2.2) 0 − B TOT

(∆0 the gap in absence of Zeeman energy, g the GaAs g-factor, µB the Bohr magneton), until at some critical tilt angle the increasing Zeeman energy produces a phase transition to a gapless polarized state. From this experiment a g-factor g = 0.56 was extracted, compatible with the bulk GaAs g-factor g 0.44. ≈

17 This interpretation agreed with the proposal by Haldane and Rezayi of an unpolarized spin-singlet wavefunction [56], possibly describing a quantum Hall state at half ﬁlling. This wavefunction is the exact (and unique) ground state of an “hollow-core” model, in which all Haldane pseudopotentials (including v0, which corresponds to the contact interaction of two particles) vanish, with the only exception of v1 = 0. But it became soon clear that this model was not representative of the real6 Coulomb interactions in the second Landau level. Early numerical works with small numbers of particles (Nel 7) showed that the Coulomb ground state for ν = 5/2 has zero overlap with≤ the spin-singlet 1.2 wavefunction of Haldanea) unpolarized and RezayiGS: energy and gaveper electron the ﬁrst hints that the system spin-singlet GS in second LL

-0.34 g(R) 1.0 could be spin] polarized [57, 58]. Later [59] it became evident that this wave- 0

/ -0.35 N=12 N =22

function does2 not describe a quantum Hall state, having gapless excitations and 0.8 belonging[e to a (critical) transition point between a weak and a strong paired

-0.36 function phase. /N

) 0.6 An important-0.37 step in the spin polarization issue was then a work by Morf 0.4 [60], in which-0.38 the non-polarization picture of the 5/2-state was strongly chal- E(N,N N =2N-4 lenged. Comprehensive exact diagonalization results for small systems (Nel N =2N-3 ≤ correlation 0.2 12) on a sphere-0.39 for bothN =2N-2 spin-unpolarized and fully polarized states showed that Energy the latter almost alwaysN =2N-1 have the lowestL=0 energyGS (cf. Fig. 2.3), indicating that pair 0.0 -0.4 0 2 4 6 the ground state is fullyN =2N spin polarized, even for vanishing Zeeman energy. An great circle distance R experimental-0.41 investigation of the transport gap at ν = 5/2 as function of the

(perpendicular)] magneticb) polarized field,GS: inE/N a sample in which it was possible to vary the FIG. 2. Pair correlation function g(R) 0 -0.36 / electron density2 [61], agreed with this result: no sign of phase transition was ob- for N=12, NΦ=22 in the second LL. served in passing[e -0.37 from the low-field to the high-field regime, where the system is /N

almost surely) spin polarized. In contrast in the same sample a phase transition -0.38 was observed at ν = 8N/5,=2N-5 revealed by a collapse of the gap and a sharp change lously high energy, cf. shaded area in F N =2N-3 of its slope-0.39 at the transition point (see Fig. 2.4). E(N,N N =2N-1 L=0 To complement this picture, we sho pair correlation function g(R) for the

(Eunpolarized-Epolarized)/N at N=12, NΦ=22, and its components 0.03 ] 0 gup−up(R), for electrons with unlike a / 2 0.02 N =2N-4 [e N =2N-2 spectively. Clearly, g(R) is close to ze 0.01 E/N N =2N zero. In an unpolarized state, the nu 0.0 contributing to gup−down(R) is N/2 wh local -0.01 it is one less. In a spin-singlet, t 0.0 0.05 0.1 0.15 0.2 0.25 system size 1/N is close to the electron at the origin state, it is as far away as possible on FIG. 1. For systems with 4≤N≤18 electrons and flux 2. A system whose GS is polarized, bu FigureN 2.3:Φ= Energy2N-S differencesGS energ betweenies E/N unpolarizedare show andn polarizedfor unpo stateslarize ford the wavelength spin-excitation establishes and polarized systems. a) Unpolarized system: E/N for same particle number N and number of flux quanta Nφ, from exact diagonal- metry, would show such behavior. izations.S= (From0,1,2,3 [60].),4. At even NΦ, GS have angular momentum L=0. Results in the shaded area suffer from strong finite size effects, In Fig. 1b, the GS energy E/N o If thecf. quantumtext. b) HallE/N systemat S= is1 already,3,5 for spinpol polarizedarized sy atste lowms. magneticAt S=3 fields, is shown for systems with NΦ=2N-S, one neededand ev aen newN, explanation i.e. the qua forntu them nu disappearencembers of the ofPf theaffia quantizationn, all GS in with L=0 are marked with circles. For a tiltedha magneticve angula fieldr mo [54,men 55].tum AL= solution0. c) E tone thisrgy d questionifference camebetw witheen new typically L>0 and are not shown. Onl tilted field experiments at ν = 5/2 [62, 63] that showed a transition from the unpolarized and polarized state for the same N, NΦ. quantum Hall state to a compressible phase with strongly anisotropic elec- for even N are rotation invariant and tronic transport, as the parallel field component B exceeded a critical value. for FQH states. Their energy increas || have in almost all cases higher energy than polarized size, extrapolating to a bulk limit of ≈ states at the same N, NΦ, cf. Fig. 1c. The exception is agrees with predictions based on pair f 18 the unpolarized state at N=6, NΦ=10 (see Fig. 1c). At In Fig. 3a, we show the energy spec first, before investigating larger systems, we were hope- electron system for different pair intera ful, that this observation might help to explain the ν=5/2 the coupling strength V1 in the Lrel= Hall plateau. Our larger system studies do not support momentum channel, but keeping all the this hope: For systems with up to N=12 electrons, no values for Coulomb interaction in the Coul similar unpolarized state exists and there is no hint that can see, around V1=1 (in units of V1 in the bulk limit, the GS would be unpolarized [20]. ∆ in the excitation spectrum ∆≈0.02. In fact, there is evidence that the properties of the GS small and large V1, the gap disappears at N=6, NΦ=10 are not related to ν=5/2: Similar ‘cusps’ In Fig. 3b, we show the overlap of in E/N occur at N=8, NΦ=13 and N=10, NΦ=16, cf. Pfaffian wf. Clearly, the overlap is clo Fig. 1a. These appear on the line NΦ=3N/2+1, which V1 has the value for Coulomb interacti extrapolates to ν1=2/3 for large N, and have nothing to lap and gap have their maxima roughly do with the behavior at ν1=1/2. We believe that the V1≈1.1. These results are consistent w cusps reflect a property that for values of NΦ below this Greiter et al. [22] that the ν=5/2 FQ line (corresponding to the shaded area of Fig. 1a) it related to the Pfaffian. However, this o seems to be impossible to construct a spin-singlet wf for not be overstated: Indeed, the GS has which the pair correlation function g(R) vanishes at R=0, overlap with a pair wf [19], setting para whereas it is possible to do so for NΦ on or above this s=2 in eq.(1) of ref.[19]. In view of the line [21]. At filling ν1=1/2, NΦ=2N-S exceeds this limit wf’s, we cannot be sure that in the b for large enough N. For smaller N, the GS have anoma- will exhibit the characteristics of the P n. 41, 707 (1982). D.J. Bishop, J.P. Eisenstein, J.H. English, R. Ruel, and H.L. Stormer, Phys. Rev. B 0.0 4.5 5.0 5.5 88). 0.5 n, H.L. Stormer, L.N. Pfeiffer, and K.W. n=5.3×1011cm-2 62 ) ev. Lett. , 1540 (1989); Phys. Rev. B Ω k (

0). xx

.R. Haynes, A.M. Suckling, J.R. Mallett, R

J.J. Harris, and C.T. Foxon, Phys. Rev. 0.0 (1989). 8 9 10 , D.A. Syphers, and A.G. Swanson, Phys. 11 -2 n=7.6×10 cm , 1098 (1989). (c) .W. Hwang, T. Sajoto, D.C. Tsui, and M. 0.5 ys. Rev. B 45, 3418 (1992). von Klitzing, and G. Weimann, Phys. Rev. 988). 0.0 of experimental results on CF’s, see, for 11 12 13 14 . Stormer and D.C. Tsui, in Perspectives MAGNETIC FIELD [T] all Eﬀect, S. Das Sarma and A. Pinczuk FIG. 2. Magneto-resistance around ν = 5/2 at three den- New York (1996). sities at T ∼ 50mK. A low-frequency (∼ 7Hz) lock-in tech- rivate communication. nique with excitation current I = 10 nA is used. The vertical lines mark the positions of the FQHE states at ν = 8/3, 5/2, and 7/3.

1.0 n=3.0 (B=5T) 0.8 n=4.2 (B=7T)

n=5.3 (B=8.8T) ) 8

2 0.6 e - v a m (a) c R R / 11 4 0.4 ave 2 /

5 10 (

n 0 0 2 4 R V (V) 5/2 g 0.2 5 10 15 4 6 8 -1 1/T (K ) n (1011cm-2) n mobility as a function of density, n. The 0.3 ν e layer structure of the HIGFET. The right =8/5 s Vg for the HIGFET. The solid circles (•) )

ata. K 0.2 (

i s qua ∆ 0.1 ν=5/2

0.0 6 8 10 B (T) FIG. 3. (a) Arrhenius plot for R5/2/Rave at three den- sities, in units of 1011 cm−2. ( •) Smooth variation of Figurequ 2.4:asi- Energyenergy gapgap aso functionf the ν of= the5/2 appliedFQH (perpendicular)E state as a f magneticunction field B ino af samplemagne withtic fie variableld (i.e. electronelectro density:n-densit smoothy). (◦) behaviourCollapse forof tνhe= 5/2 and phase transition for ν = 8/5. (From [61].) ν = 8/5 quasi-energy gap due to the well-documented transition in its spin-polarizations. This anisotropic phase was also observed in higher Landau levels at half filling (ν = 9/2, 11/2,...), in this cases even in the absence of a parallel field component [64, 65]. These results were interpreted as the onset of a “stripe” phase [66–68], previously theoretically predicted for half-filled Landau levels of high index. The ν = 5/2 system appears to be at the border to this phase and B can drive a transition into it. This in-plane field component can indeed affect|| the interactions between the electrons, because of the finite thickness of the electron system in the direction perpendicular to it (a purely 2-dimensional system would be insensitive to B ). Thus the tilted field experiments at ν = 5/2 were not a probe for the spin polarization|| of the state, but for a transition from an4 incompressible quantum Hall state to a compressible “stripe” phase. This transition was also confirmed by a numerical work by Rezayi and Haldane in the torus geometry [69]. From these experimental and numerical results a coherent picture of a fully polarized quantum Hall state at ν = 5/2 emerges, near to the border of a “stripe” phase. But a direct measurement of the spin polarization is still missing (but in progress) and this issue is still matter of discussion and the subject of recent and current studies. From the theoretical side, Dimov et al. [70] showed by Monte Carlo sim- ulations that, assuming that the quantum Hall effect at ν = 5/2 is described by a paired state, it will exhibit spontaneous ferromagnetism. In agreement with this (and with previous numerical works) Feiguin et al. [71] obtained by exact diagonalization and density matrix renormalization group methods that for Coulomb interaction the finite-system ground states are fully polarized, also in absence of the Zeeman energy. From the experimental side, new in-plane field transport measurements at low magnetic fields [72] showed opposite behaviors for the systems at ν = 5/2 and ν = 7/3, casting doubts about the magneto-orbital coupling discussed above, that should give the same gap suppression in the tilted field for both systems. In contrast more recent experiments at high magnetic fields [73] show a similar behavior for the two states and give support to the full polarization

19 at ν = 5/2: the weakening and disappearence of the gapped phase in a tilted magnetic field is too abrupt and cannot be explained by a Zeeman energy term with a GaAs bulk g factor. The different behaviour for ν = 7/3 could maybe be explained by a different nature of the 7/3 state in the two regimes. The first results from direct spin polarization measurements, using optical methods [74,75], seem to be consistent with an unpolarized quantum Hall state at ν = 5/2. However these results are not yet conclusive 1 and could eventually be reconciled with full spin polarization of the 5/2 quantum Hall state: due to disorder, the experimental sample breaks up into incompressible and compressible domains [50,75]. However the optical experiments (photoluminescence spectroscopy [74] and resonant light scattering [75]) were not able to discriminate the contributions coming from these domains: the measured non-polarization could come from the compressible puddles (where there are no strong exchange effects that could enhance the spin polarization), while the incompressible fila- ments could still be fully polarized (in agreement with the theoretical results). Furthermore, charge disorder may also induce skyrmions in the incompressible domains [77, 78], which will tend to decrease the spin polarization. 2 In a recent analysis [81] of the energy gaps (obtained by transport experiments) as function of perpendicular and parallel magnetic fields, Das Sarma et al. argued that the results are more consistent with a spin unpolarized than with a spin polarized ν = 5/2 state. However the model they used did not include a complete description of the disorder effects, which is essential for a discussion of the energy gaps. In all our numerical investigations we vill assume that the electrons in the considered half filled Landau level (the lowest for ν = 1/2, the second for ν = 5/2) are fully spin polarized. (For ν = 5/2 we will consider the lowest Landau level as inert and completely filled with electrons of both spins.)

2.3 The Moore-Read (“Pfaﬃan”) state: the ﬁeld theoretical point of view

The issue of spin polarization is important because it is the basic assumption for some of the states proposed for the description of the quantum Hall effect at ν = 5/2: if the system is not fully spin polarized, these proposals have to be rejected (or at least modified). A particular interesting proposal is the so-called (weak paired) “Pfaffian” state, introduced by Moore and Read [51], describing a system at half filling. But before of a concrete description of it, we shortly present a general remark about the field theoretical description of a quantum Hall fluid (see for example [82] (and references therein) or the review [35] for a direct connection with the 5/2 state). In the scaling limit, the bulk of such a system is described by a topological (Chern-Simon) field theory. However in a sample with an edge, the charge current from this theory is not conserved (and not gauge invariant): one has to introduce a (chiral) current at the edge, in order to assure that the total charge current is conserved. This edge current also exactly cancels the gauge

1In [74] the measured polarization values for the integer quantum Hall eﬀect states are slightly oﬀset, casting doubts on the reliability of the results for ν = 5/2, cf. [76]. 2Very recent experimental investigations using NMR techniques [79, 80] found fully spin polarization for the ν = 5/2 quantum Hall state.

20 “anomaly” of the bulk theory. It exists thus a deep connection between the edge of a quantum Hall system, described by a (chiral) conformal field theory, and its bulk, described by a topological field theory: the first theory determine the second and vice versa (provided the boundary between bulk and vacuum is sharp, without reconstructed edges in between). For the Laughlin states, the chiral conformal field theory describing the edge contains only a free boson ϕ, from which it is possible to construct the electron sector: the (normal ordered) vertex operator : eiqϕ : for charge q = 1 (in elementary charge units) corresponds indeed to an electron. However this works only for filling factors with an odd denominator: the vertex operator written above indeed describing a fermion (as the electron is). For half filling the result would instead be a boson: in this case a single bosonic field is not enough to get the “desired” electron operator. To overcome this problem, one has to introduce other fields into the theory and it turns out that there are several possibilities for doing this. The “Pfaffian” state is one of these possibilities: in this case one introduce, beside the boson field ϕ, the fields of the conformal field theory that describes the two-dimensional Ising model at its critical point: the identity 1, the spin field σ and the (neutral) Majorana fermion ψ. From these fields one can construct all (physical) sectors: the electron is here described by the vertex operator iϕ ψel =: e : ψ, which is indeed a fermion, because of the Majorana fermion ψ. All other physical fields have to commute (or anticommute) with this electron field and it turns out that the sector with the smallest charge in this theory is iϕ/4 given by the vertex operator ψqh =: e : σ, corresponding to a quasihole with charge e/4. Of particular importance are the fusion rules of the Ising fields (analogous to the decomposition of a tensor product of representations in a direct sum of irreducible representations): σ σ = 1 + ψ, σ ψ = σ, ψ ψ = 1, (2.3) × × × which describe the result of the fusion of two objects. Of particular interest is the first one: fusing two e/4 charged quasiholes (whose vertex operators contain the spin field σ), there are two possible outcoming channels: the identity 1 and the Majorana fermion ψ. As we will see in section 2.6, from these different channels will follow a degeneracy for the states containing multiple quasiholes, leading to their non-Abelian braiding statistics.

2.4 The Moore-Read (“Pfaﬃan”) state: the microscopic point of view

Moore and Read [51, 52] constructed their “Pfaffian” (Moore-Read) wavefunction, interpreting the conformal blocks coming from the 1 + 1-dimensional conformal field theory describing the edge, as functions on the 2-dimensional bulk of the system. They first observed that the Laughlin wavefunction (and the wavefunctions describing quasiholes above it) can be indeed obtained as conformal blocks of the conformal field theory with the bosonic field ϕ of the preceding section. They then applied the same procedure to the conformal field theory with the Ising fields and from the conformal block containing an even number Nel

21 iϕ of “electronic” vertex operators ψel(zi) =: e (zi): ψ(zi) (with zi the complex coordinate of the i-th electron), they got the Moore-Read wavefunction:

1 2 P z 2/4 Ψ (z , . . . , z ) = Pf (z z ) e− i | i| , (2.4) MR 1 Nel z z i − j i j i>j − Y where the Pfaﬃan Pf denotes the antisymmetrized sum over pairs

1 1 1 Pf = ... , (2.5) z z A z z z z i − j 1 − 2 3 − 4 such that ΨMR is antisymmetric under the exchange of any two particles. The Pfaffian of an antisymmetric matrix can also be expressed as the square root of its determinant: Pf(Mij) = √det M. The wavefunction ΨMR describes a system with an even number Nel of spin polarized electrons in the half filled lowest Landau level (in the symmetric gauge for a planar geometry) and can be considered as a “trial wavefunction” (like the Laughlin wavefunctions): a prototype independent from the details of the electron interaction, but (possibly) defining a phase of matter. The actual wavefunction of the quantum Hall system would be somehow different from ΨMR and sensitive to the details of the interaction, but would be in the same phase as ΨMR, having a large overlap with it and sharing the same quantum numbers (for ground state and excitations). The “Pfaffian” wavefunction describes a system at half filling because, in analogy with the Laughlin state discussed in section (1.2), the maximum power in the complex coordinate zi is M = 2(Nel 1) 1 (the additional 1 coming from the Pfaffian) and thus the filling factor− is ν =−N /N = N /(N− 1)2 el orb el el − → 1/2 in the thermodynamic limit Nel . To obtain the wavefunction for the electrons in an half filled second Landau→ ∞ level, that is for ν = 5/2, it suffices to act on (2.4) with the Landau level raising operator for each particle. As expected from conformal field theory, the elementary quasihole excitations above this state are e/4-charged and obey non-Abelian braiding statistics (Moore and Read coined for them the term “nonabelions”), realizing the possibilities described in [21]; we will focus on these excitations in section 2.6. In their seminal work Moore and Read, did not discuss the fractional quantum Hall effect at ν = 5/2: at that time it was still widely believed that the ν = 5/2 state was spin unpolarized (as discussed in section 2.2), in contrast with the basic assumption for the “Pfaffian” state (however recently a non-polarized generalization of this state was investigated [83]). It was first a study by Gre- iter, Wen and Wilczek [84, 85] that mentioned the possibility that the state at ν = 5/2 could be related to the “Pfaffian” wavefunction (but without an explicit discussion of this issue). In their work, by simultaneously changing the applied magnetic field and the statistics of the particles, they showed that the Moore-Read state is adiabatically connected to a system of electrons in zero magnetic field with a BCS pairing instability. In fact the “BCS wavefunction” in real space for p-wave paired electrons is the pure Pfaffian Pf(1/(zi zk)) and the adiabatic changes in magnetic field and statistics give then rise− to the wavefunction in (2.4). This adiabatic connection with the BCS state made also plausible that the “Pfaffian” wavefunction could in fact describe a gapped state (Moore and Read did not discuss this issue).

22 This sort of pairing is often called “weak pairing”, to distinguish it from the “strong pairing” scenario, where two electrons are considered as really paired to form a particle with double charge and bose statistics. In the Moore-Read wavefunction instead the “weak pairing” is realized through the Pfaffian factor, that cancels some of the zeros (zi zj), such that two electrons can come nearer to each other. Greiter, Wen− and Wilczek argued that these two types of pairing belong to the same universality class, in particular claiming that the excitations of the “Pfaffian” state obey Abelian braiding statistics, but it is not the case: they have a different edge structure and different (topological) quantum numbers, as shown by Wen [86] a year later. In their work Greiter, Wen and Wilczek [84, 85] also pointed out that the “Pfaffian” state is the unique highest density zero energy eigenstate of a particular three-body interaction, in analogy with the Laughlin states that are zero energy eigenstates of certain two-body interactions defined in terms of Hal- dane pseudopotentials. If the electrons were bosons, this three-body interaction would simply require that three particles cannot be at the same position, and is thus expressed in real space as the product of two δ-functions. In the fermionic case this condition is already realized because of Pauli exclusion principle, but the interaction imposes the further requirement that the wavefunctions have to vanish sufficiently fast as three particles approach each other. The three-body interaction for which the “Pfaffian” wavefunction is an exact zero energy eigenstate is thus expressed in real space (and then projected on a specific Landau level) as the product of appropriate derivatives of δ-functions:

Nel V = V S ∆ δ(z z ) ∆2 δ(z z ) , (2.6) 3b ijk j i − j k i − k i

Nel V3b = v3,3 P3,3(i, j, k), (2.7) i 0 is the corresponding interaction energy (vL,N denotes the N-body pseudopotential corresponding to relative angular momentum L~). In other words the three-body interaction (2.6) is exactly the interaction that gives a positive energy to the three-body states with relative angular momentum

23 3~ (the minimum possible value for three particles, corresponding to the nearest possible approach), and no energy to all other states with higher relative angular momentum. Any three particles in the Moore-Read wavefunction have in fact at least relative angular momentum 5~ and thus this state is a zero energy eigenstate of (2.7). It is the highest density state with this property because the polynomial in front of the exponential has the lowest possible degree and thus belongs to the largest possible filling factor. In their original paper Greiter, Wen and Wilczek gave a wrong form of the three-body interaction in the fermionic case. The first correct expression was given in [86], where also first exact diagonalization calculations for small systems (Nel 10) involving the three-body interaction V3b (2.6) were presented. All these calculations≤ showed the presence of a gap in the energy spectrum, although it was difficult to estimate it in the thermodynamic limit because of the finite size effects. Neverthless the results suggested that the Moore-Read wavefunction describes in fact an incompressible state. In the same work, by exact diagonalizations in systems interacting via V3b in the planar geometry, Wen showed that the counting of the numbers of states of the low-lying edge excitations as function of the total angular momentum is consistent with the conformal field theory that describes the edge of the “Pfaffian” state, containing a free boson and the fields of the Ising model (in particolar the Majorana fermion ψ, see section 2.3).

2.5 The “Pfaﬃan” at ν = 5/2? First hints.

The first indications that the fractional quantum Hall effect at ν = 5/2 could be in fact related to the Moore-Read state came with the numerical work by Morf [60]. Exact diagonalization results for electrons on a spherical surface showed that the ground state at ν = 5/2 is spin-polarized (as discussed in section 2.2) and that its overlap with paired wavefunctions like the “Pfaffian” is quite large. An important point in this work is related to the so-called “shift” S, a topological quantum number [88, 89]. Because of the surface curvature [90], on a sphere the number of electrons Nel and the number of flux quanta Nφ for an half filled Landau level are related through the relation [91]

N = 2N S (2.8) φ el −

(instead of the canonical relation Nφ = 2Nel), where the shift S depends on the character of the fractional quantum Hall state: different states at the same filling factor can have different shifts. For example the “Pfaffian” state of Moore and Read on a sphere has a shift S = 3 (as already noted in [84, 85]) and the exact diagonalizations clearly showed that the quantum Hall state at ν = 5/2 belongs to this shift value. Indeed only for S = 3 the ground states for all (even) system sizes (Nel 18) are rotation invariant (having total angular momentum L = 0), as it should≤ be for incompressible states in the spherical geometry. Furthermore all this states show a gap and the corresponding ground state energies behave smoothly with the system size, allowing an extrapolation to the thermodynamic limit. The overlap of the ground state for Nel = 8 with the “Pfaffian” wavefunction

24 VOLUME 80, NUMBER 7 PHYSICAL REVIEW LETTERS 16FEBRUARY 1998

phase transitions to gapless states for small and large V1 appears firm. The compressible state at large V1 is the CF liquid [9,23]. This becomes clear from Fig. 4(b). At N ෇ 16, the CF state occurs at NF ෇ 30, one flux unit higher than for the FQH state. As a reference CF liquid wf, we use the GS for Coulomb interaction in the lowest LL [9]. As V1 is increased, its overlap with the GS approaches unity when the system becomes gapless. As incompressible and CF states do not exist at the same flux NF, a bias exists in favor of the FQH state at NF ෇ 29, whereas the CF liquid is favored at NF ෇ 30. The critical V1 value will thus be either overestimated or underestimated, depending on NF. In Fig. 4(b), we also show the overlap of the GS C0 at N ෇ 16, NF ෇ 29 with a trial state jpair͘, which is the GS at V1 ෇ 1.1 where the gap is maximal. The rapid drop of FIG. 3. (a) Energy of low-lying states of polarized system the overlap ͗pairjC0͘,asV1 is reduced below one, very of 8 electrons in second LL at flux NF ෇ 13 vs the L ෇ 1 ෇ Figure 2.5: a) Low-lying energy spectrum for NelCoul=o 8mb and Nφ = 13 (onsimilar the to the one observed for ͗PfaffianjGS͘ at N 8 pseudopotential V1, measured in units V1 . The state sphere) as function of the Haldane pseudopotential v (vCoulomb is its Coulomb becomes gapless for small and large V1. (b)1 Overlap1 of the [cf. Fig. 3(b)], is another indicator for the phase transition 2 value inGS thewf secondwith LandauPfaffian level).and pair-wf Energiestrial states. are measuredGap and inoverlap units e /`0to. b)the compressible state at small V1. This transition Overlapsboth of thehave exacttheir groundmaximum stateat withV1 theഠ 1.1 “Pfaffian”. and another paired wave-is associated with a small wave vector instability in the function for the same system. (From [60].) excitation spectrum. In our spherical system, it occurs at L ෇ 2 both for N ෇ 8 and 16. This compressible state is exhibit the characteristics of the Pfaffian, e.g., excitations 3 not a CF liquid. It might be the charge density wave state is quite large,with non-Abelian however the statistics[16]. overlap with another paired wavefunction (also proposed by Koulakov et al. [24]. To study such states, belonging toInS Fig.= 3)4(a), has awe similarshow sizethe in thisexcitation small system,spectrum such thatfor a it was not clear which state would “win” in the thermodynamic limit. the torus geometry may be more appropriate. much larger system, N ෇ 16, NF ෇ 29. The spectrum Modifying the particle interaction, by varying the Haldane pseudopotentialIt is instructive to study the system at the nearby looks similar with a gap that vanishes when V1 is be- v1 from its Coulomb value in the second Landau level, while keeping alln the෇ 7͞3 filling since Hall plateaux at 5͞2 and 7͞3 have low 0.9 or larger than 1.3. For Coulomb interaction, the other vm constant, one finds that the gap has a maximum near the Coulombbeen observed in the same experiment [5]. The results gap is again D ഠ 0.02, and its maximum still occurs at point and that for both small and large v1 the gap disappears (see Fig. 2.5).for energy gap and GS overlap with the Laughlin state C V ഠ 1.1. Similar excitation spectra are also seen for sizes 3 Interestingly1 the overlap also shows a maximum value (of almost unity) quiteshown in Fig. 5 are evidence for a phase transition from a at the sameN ෇ place10 and as the14, maximumwhile the ofsystem the gap:with weN will෇ encounter12, NF ෇ this21 feature in gapless at small V1 to an incompressible state at around our workis too.“aliased” [12] with a n1 ෇ 3͞5 state and its interpre- V1 ഠ 0.96. The energy gap for Coulomb interaction, The compressibletation as a n systems1 ෇ 1͞2 atstate largeis anddubious. small v1 Theare differentevidence infor nature: the V1 ෇ 1,isD73 ഠ0.02 which is close to the calculated state at large v1 has a big overlap with the composite fermion liquid state (the ͞ 5͞2 state that is realized at ν = 1/2), while for small v1 Morf proposed the possibilityvalue at . In the activation studies of Eisenstein et of a charge density wave state. al. [5], it was found that the gap at n ෇ 7͞3 decreases This scenario was indeed confirmed two years later by Rezayi and Haldanewith increasing tilt angle and disappears in much the [69] by exact diagonalizations (Nel 16) in the torus geometry, where all statessame way as at n ෇ 5͞2 [25]. As the FQH state at ≤ at half filling are realized at the same flux Nφ = 2Nel (that is, there is no “shift”7͞3 is almost certainly spin polarized, and according as on the sphere) and can thus be studied simultaneously. Varying v1 Rezayito our numerical results at 5͞2 likewise, a common and Haldane (see Fig. 2.6) found a sharp (first-order) phase transition fromorigin the for the reduction of the gaps with increasing tilt incompressible state to a stripe state at smaller v1. This transition is veryangle near and for their disappearance may be expected. Our to the Coulomb point: varying the system width (in the direction perpendicularresults imply that a reduction of V would simultaneously to the 2d-system) or tilting the magnetic field can drive the transition. On the 1 other hand the transition to the composite fermion liquid at larger v is notreduce so both gaps and eventually lead to compressible 1 states. Besides increasing the Zeeman energy, which 3Obtained by setting the parameters m = 1, t = 0 and s = 2 in equation (1) of [92].

FIG. 4. (a) Same as Fig. 3(a) but for N ෇ 16, NF ෇ 29. This larger system becomes gapless too for small and large V1. (b) Overlap of the GS C0 at N ෇ 16, NF ෇ 30 with the CF-liquid wf vs V1 (dotted line). For large V1, the overlap approaches unity. Overlap of the GS C0 at N ෇ 16, NF ෇ 29 with the “trial” state jpair͘, deﬁned as GS for maximal gap FIG. 5. Overlap of the GS with the Laughlin state C3 and (full line). excitation energy of 10 electron energy vs V1 at n ෇ 7͞3. 1507 VOLUME 84, NUMBER 20 PHYSICAL REVIEW LETTERS 15MAY 2000

͑n෇1͞2͒ has an opposite effect to dV3. A study using spherical jCCF ͕͑ki͖͒͘ ෇ det͓exp͑iki ? Rj͔͒ jCL ͘ , (4) i,j geometry [17] also identifies the phase at large dV1 with the CF liquid. A first-order phase transition from a com- where the ͕ki͖ are distinct [and belong to the usual set of wave vectors allowed by the periodic boundary conditions pressible state to an incompressible paired state is clearly seen. The transition is very close to the Coulomb value (PBC’s)] and are clustered together to form a filled “Fermi d ෇ d ෇ k ෇ k ͑ V1 V3 0͒. We obtain similar results in the low- sea” centered on av P i͞N. The total momentum quantum number K [25] is determined by the value of k est Landau level, except that the transition point occurs at av d ෇ 20.092, d ෇ 0 d ෇ 0, d ෇ 0.048 relative to the set of allowed k’s (the CF state is essentially V1 V3 and at V1 V3 . Details of these studies will be given elsewhere. For both left invariant by a uniform “boost” ͕ki͖ ! ͕ki 1 k͖), and takes one of N2 distinct values [25]. There are four dis- Landau levels, we observe only the strongly paired state tinct values of k which are invariant under 180± rota- in a narrow window. The projection of the MR state on av the exact ground state does not exceed 73% in this re- tion: kav ෇ 0 and kav halfway between allowed k vectors (three distinct values which correspond to the three distinct gion. However, if the MR state is first PH symmetrized, values of K for the MR state on the torus). this projection becomes 97%. The two-particle correlation ͑r͒ The n ෇ 1͞2 spin-polarized electron eigenstates of (3) function g of the states before and after symmetrization have particle-hole symmetry [26]; the CF state is almost is shown in Fig. 2. The paired character of the MR state is (99.935%) PH symmetric and also has a large projection essentially unaltered (Fig. 2 shows that each electron has ͑r͒ (99.25%) on the exactly PH-symmetric GS of the Coulomb one particularly close partner); the near isotropy of g potential in the lowest Landau level. is characteristic of the incompressible states, and should The periodic MR states [16] can be obtained as the improve with increasing system size. zero-energy ground states of a 3-body short-range potential An interesting feature in Fig. 1 is the absence of any [16], the corrected form of which is obvious sharp transition from the paired state to the compressible Fermi-liquid-like CF state as V1 is increased fur- ෇ 2 =4=2 d2 r 2 r d2 r 2 r ther. This is also seen in the excitation spectrum. Figure 3 H3 X Si,j,k͕ i j ͖ ͑ i j͒ ͑ j k͒ , i,j,k shows the low-lying excitation spectrum as a function of V1. Again, there is only one first-order level crossing tran- where Si,j,k is a symmetrizer. Note that [in contrast to (3)] sition (shown by up arrows). The levels that cross have the H3 has no PH symmetry and the MR state does not possess ± definite parity under PH transformations. same translational and 180 -rotation symmetry but belong The nature of the ground state of (3) depends on the to opposite parities under PH transformation. The MR state has a finite overlap with the exact GS on both sides relative strengths of the pseudopotentials, in particular V1 and V (even-m pseudopotentials do not affect polarized of the transition as it has components with both PH sym- 3 d states). Figure 1 and Fig. 4 (below) show the projection metries. As V1 increases further, the excitation spectrum of the CF and MR states on the exact GS in two dif- gradually evolves from having a clear gap to the compressible CF Fermi-liquid-like spectrum [23,24]. The ferent PBC geometries, as V1 and V3 are varied relative to their Coulomb values in the first excited Landau level crossover is approximately at the point where the spectrum begins to change at the level crossings of the excited states ͑n ෇ 1͒. Varying V3 alone (the inset of Fig. 1), or vary- V V dV (down arrows). Similar crossover behavior is also seen sharp and wasing interpretedboth 1 and as a crossover3, yields similar or a second-orderresults, though phase1 transition.

FIG. 1. The projection of the exact GS of the Coulomb in- FIG. 2. The real-space pair-correlation function for the MR teraction in the n ෇ 1 Landau level, plus an extra short-range state and its PH-symmetrized counterpart, evaluated in the sec- d d ෇ r Figure 2.6: Overlapspseudopotential of theV exact1 ( V3 groundin the inset), stateon fortheNCF, =MR, 10and and Nond ͑=n 201͒ Landau level; their difference is also shown. g͑ ͒ PH-symmetrized MR model states. The GS PH parityel changes inφa square unit cell is shown along a path from the origin O to (on the torus)at witha lev someel crossing modelnear states:dV ෇ 0 composite. fermion liquid (CF),the Moore-midpoint of a side S, to a corner C, and back to O. Read (MR Paired) and particle-hole symmetrized Moore-Read (Symm. MR). The Coulomb interaction in the second Landau level is modiﬁed by varying the 4686 Haldane pseudopotential v1 (v3 in the inset). (From [69].)

The incompressible phase in the central v1 interval was identified as the Moore-Read paired state, but to obtain an high overlap between the exact ground state and the “Pfaffian” wavefunction Rezayi and Haldane had to particle- hole symmetrize the latter. This has to do with the “Antipfaffian” state [93,94], the “Pfaffian” state of holes (we will describe it in section 2.10), that on the torus is realized at the same flux number Nφ = 2Nel as the “Pfaffian” state of electrons. Exact diagonalizations of particle-hole symmetric systems cannot decide which of this two states is realized and the ground state contains contributions from both. The situation in the sphere geometry is different: there the “Antipfaffian” state belongs to another shift value (S = 1) and thus the calculations at S = 3 are not directly influenced by it. − At the same time, there were also Monte Carlo studies in the spherical geometry [95], comparing the energies for some trial wavefunctions and indicating that at ν = 5/2 for Coulomb interaction the Moore-Read state has the lowest energy. In this and in a following work [96] 4, the “Pfaffian” state was also interpreted as a BCS instability of the composite fermion liquid state: in this picture composite fermions (rather than the “bare” fermions, as discussed by Greiter, Wen and Wilczek [85]) form Cooper pairs in the p-wave channel, open- ing a gap at the Fermi level. In this work it is claimed that this sort of pairing becomes possible because of the formation of composite fermions (at half filling each carrying 2 flux quanta and feeling no external magnetic field) that at ν = 5/2 overscreens the repulsive Coulomb interaction. Summarizing, around the year 2000 there were plenty of numerical and theoretical hints (reviewed in [98]) indicating that the state at ν = 5/2 could be described by the Moore-Read wavefunction. But the “puzzle” of the even denominator quantum Hall state was not yet fully resolved, particularly still

4This work has also been criticized, cf. [97].

26 missing an experimental evidence of the proposed non-Abelian braiding statistics.

2.6 Nonabelions in the Moore-Read state

The elementary excitations above the “Pfaffian” state at half filling are quasiparticles with charge e/4 that occur in pairs: because of the paired nature of the state the elementary± flux quantum is halved, that is, inserting a unit of flux quantum into the system two quasiholes with charge e/4 are generated (instead of a single quasihole with charge e/2). The exact diagonalization work by Morf [60] was consistent with this prediction: the energy gap obtained from charged excitations (by changing the flux Nφ by 1) was about twice the gap obtained from neutral excitations, indicating that± the insertion (exclusion) of a flux quantum corresponds to a creation of two quasiholes (quasiparticles). But this is not enough to establish the “Pfaffian” as the state realized at ν = 5/2: other candidate states support quasiparticles with charge e/4. In their seminal paper [51] Moore and Read described these excitations from the point of view of conformal field theory. They were not able to give explicit wavefunctions (except for two quasiholes), but from the properties of the conformal blocks they recognized that the wavefunctions describing four or more quasiholes span a vector space of dimension greater than one. The vector space for four quasiholes is namely two-dimensional: there are two linearly independent states describing the same configuration of localized quasiholes. One can prepare the system in one of these states; if then one transports a quasihole once around another (or interchanges two of them) one gets another state in the vector space (and not only a phase factor): this is the realization of non-Abelian 38 braiding statistics. Chapter 3. Clustered states

ψ �� σ � �� ...� �� 1 � �� 1 0 2 4 6 n-2 n

Figure 2.7: 3.1: The The Bratteli Bratteli diagram d foriagram the “Pfaffian” for the state: Moore-Re on the leftad the state. fields of the Ising model, under the diagram the number of fused spin fields σ.

As anticipated, this degeneracy of the states containing quasiholes has its rules of the parafermionorigin in the fields fusionψ rulesis trivial (2.3): σ σ = 1 + ψ, σ ψ = σ, ψ ψ = 1. Following Moore and Read, to construct× a wavefunction× for a system containing× 4 quasiholes (and an even number of electrons) one has to take the conformal block with the corresponding vertexψ ψ operators=1. for electrons and quasiholes. This (3.10) leads to a correlator containing 4× spin fields σ and an even number of Majorana For the spin fieldsfermionσ, the fields fusionψ: after rule fusing is moreall these c fieldsomplicated one has to end with the identity 1, otherwise the correlator vanishes. There are an even number of ψs, that fuse to σ σ=1+ψ. (3.11) × 27 The consequence is that upon calculating a correlator which contains a certain number of spin field, on in general has a choice of many different fusion paths which fuse the (spin) fields to the identity. In the end, after fusing all the fields, one has to end with the identity, in order to obtain a non-zero correlator. The number of ways in which this can be done can be obtained from a so called Bratteli diagram. In such a diagram, the fusion of fields is encoded in arrows, see figure 3.1. Each arrow stands for fusing with a certain field (in this case, the spin fieldσ). The field which is fus ed with the field corresponding to the arrow is at the starting point of the arrow, while the arrow points at a position corresponding to a field in the fusion. Taking the fusionσ ψ=σ into account, one finds th e diagram 3.1. × From the diagram in figure 3.1, one easily determines that the number of spin fields in the correlator has to be even. Only after the fusion of an even number of spin fields, one can end up in either the1 or theψ-sector. In the first case, the number of electrons need to be even as well, to end up in the ‘identity sector’. In the second case the number of Majorana fermions, and accordingly, the number of electrons, has to be odd. In both cases, the number n 1 of fusion paths which lead to the identity is determined to be2 2 − . The fact that there is more than one fusion channel makes the conformal correlator in (3.9) stand for a set of wave functions, or better, a wave vector. If one now takes a MR quantum Hall state, in which four quasiholes are present, and one braids these quasiholes, this will result in a phase, which depends on a phase matrix, instead of a simple phase factor. These phase matrices have been calculated [71, 74, 95], and it was found that they do not commute. Thus the ordering of the braiding is essential, explaining the nomenclature non-abelian statistics. At this point, it is useful to spend a few words on the underlying Lie algebra structure. That is, the electron operator for the MR state can be viewed as an su(2) current whenM=0. The Majorana fermion is viewed as the simplest parafermion related to the su(2)2/u(1) parafermion theory, which is in fact just the Ising model. In the other clustered quantum Hall states, which we will address in the following sections, parafermion fields will be present in the electron operators. Again, as a consequence, the quasiholes with smallest possible charge will contain spin fields. These spin fields also have non-trivial fusion rules, which are generalizations of eq. (3.11). Thus one can again argue, that the quasiholes over the identity, thus the same has to happen for the 4 σs. However because of the two channels in σ σ = 1 + ψ, there are two different ways (two fusion paths) to do this, as shown× in the Bratteli diagram in Fig. 2.7. In such a diagram, the fusion of fields is encoded in arrows: each arrow stands for fusing with a spin field σ. The arrow starts from the field (1, σ or ψ) that is fused with σ and points to a position corresponding to a fusion channel. Following the arrows in the diagram, it is clear that for 2 quasiholes there is only a possibility to reach the identity (this means that there is no degeneracy), whereas for 4 quasiholes there are two (fusion) paths, corresponding to two linearly independent wavefunctions N 1 describing this system. In general, for 2N quasiholes, there are 2 − fusion N 1 paths, leading to a 2 − degeneracy. From the diagram it is also clear that the number of spin fields in the correlator has to be even, since an odd number of them always fuse to an unpaired σ, indicating that the e/4 charged quasiholes only occur in pairs. For the spin fields that at the end fuses to the Majorana fermion ψ, the number of electrons in the system has to be odd (since in this case there is an unpaired ψ coming from the electron vertex operators allowing an overall fusion to the identity), N 1 also leading to a 2 − degeneracy. Explicit wavefunctions for quantum Hall systems at ν = 5/2 containing quasiholes were given later by Nayak and Wilczek [99], obtaining them as zero- energy states of the 3-body interaction V3b (equation 2.6). For 2 quasiholes at (complex) positions η1 and η2 it reads:

(zi η1)(zj η2) + (zi η2)(zj η1) 2 P z 2/4 Ψ = Pf − − − − (z z ) e− i | i| 2qh z z i − j i j i>j − Y (2.9) where the zi’s are electron coordinates and Pf denotes the Pfaffian (2.5). The extra flux quantum is halved between the two quasiholes, each carrying e/4 charge. When the two quasiholes are taken together at the position η, a single (Laughlin) quasihole results, carrying a full flux quantum and a e/2 charge:

1 2 P z 2/4 Ψ = Pf (z η) (z z ) e− i | i| . (2.10) 2qh,together z z i − i − j i j i i>j − Y Y

For 4 quasiholes at positions η1, η2, η3, η4 the situation becomes more interesting: one needs to modify the Pfaﬃan factor in the manner

(z η )(z η )(z η )(z η ) + (z η )(z η )(z η )(z η ) Pf i − 1 i − 2 j − 3 j − 4 j − 1 j − 2 i − 3 i − 4 , z z i j − (2.11) grouping two quasiholes with the electron coordinate zi and the other two with zj. There are apparently three possibilities to do this, but it turns out that only two of them are linearly independent; for example

(z η )(z η )(z η )(z η ) + (i j) Ψ = Pf i − 1 i − 3 j − 2 j − 4 ↔ (2.12) (13)(24) z z i − j 2 P z 2/4 (z z ) e− i | i| × i − j i>j Y

28 and (z η )(z η )(z η )(z η ) + (i j) Ψ = Pf i − 1 i − 4 j − 2 j − 3 ↔ (2.13) (14)(23) z z i − j 2 P z 2/4 (z z ) e− i | i| . × i − j i>j Y A particularly useful choice of basis in this 2-dimensional vector space is given by 1/4 (η13η24) Ψ4qh, = Ψ(13)(24) √xΨ(14)(23) , (2.14) ± (1 √x)1/2 ± ± 5 where ηij = ηi ηj and x = η14η23/η13η24. This basis wavefunctions have the property that there− is no Berry phase [100] as one braids the quasiholes: all the transformation properties are manifest in the wavefunctions [101,102]. Namely, as one starts with Ψ4qh,+ and takes the quasihole η2 once around the quasihole η3 and brings it back to its original position, then one gets the state Ψ4qh, (because of the branch cuts), and the other way round: the braiding yields− a rotation in the 2-dimensional space and the non-Abelian braiding statistics of the quasiholes is explicit in this basis. We also note that taking for example the quasihole η3 around both the fixed quasiholes η1 and η2 and then taking it back to its original position, results in a factor i in front of the original wavefunction if the quasiholes are in the state Ψ4qh,+, but in a factor i if they are in the state Ψ4qh, . The sign obtained with this operation can− thus be used to identify in which− particular state the system (with four quasiholes) is. On the other hand the wavefunctions Ψ(13)(24) and Ψ(14)(23) have no manifest transformation properties: the braiding is completely described by the Berry phase matrix, but the end effect of braiding must be the same in both bases, since it is physically observable. In the same work Nayak and Wilczek [99] generalized the construction of the wavefunctions to 2N quasiholes and showed that the vector space spanned by N 1 them is 2 − -dimensional, in agreement with the predictions based on conformal field theory. Independently Read and Rezayi [103] got the same result and confirmed it with numerical diagonalizations for moderate system sizes in the (rotation invariant) spherical geometry, for electrons interacting via the three- body interaction (2.6). (The non-Abelian braiding properties were also recov- ered a few years later by Slingerland and Bais [104] using a quantum group description of the quasiholes.)

2.7 The “weak paired” phase

In the following Read and Green [59] looked at the “Pfaﬃan” from the viewpoint of BCS paired states in two dimensions, with breaking of parity and time- reversal symmetry: they found that the Moore-Read state corresponds to a “weak paired” phase (in contrast to a “strong paired” phase of the Halperin type, see section 2.10). The “Pfaﬃan” wavefunction is obtained as an asymptotical limit and they argued that its properties are characteristic of the whole phase,

5Nayak and Wilczek [99] took a slightly diﬀerent anharmonic ratio x.

29 in particular its non-Abelian braiding statistics (while the “strong paired” phase is Abelian). The transition between the two phases involves the appearence of a Majorana fermion in the “weak paired” phase. These Majorana fermions were further studied by Ivanov [105], finding that their presence or absence in the vortex cores (the BCS analogue of quasiholes) N 1 leads exactly to the 2 − -degeneracy of the Moore-Read state with 2N quasiholes in it. Furthermore, by braiding the vortices, the occupation numbers of the Majorana fermions in the cores change, giving the same non-Abelian braiding statistics as that discussed by Nayak and Wilczek [99]. An important point in this discussion is that the Majorana fermions are stable with respect to local perturbations, implying the topological stability of the non-Abelian braiding statistics associated with them. A few years later Stern et al. [106] gave a more physical picture, but equivalent to that of Ivanov, of the non-Abelian braiding statistics of the vortices. N 1 The main difference in the new picture is that the 2 − degenerate states are not given by the occupation numbers of the Majorana fermions, but are rather entangled superpositions of all possible occupations of single-particle states near the vortex cores, in which the probability for all occupations is equal and only the relative phases changes from one state to another. The braiding does not affect the occupation numbers (a problematic point in Ivanov’s picture, because there is no tunneling if the vortices are kept far enough one from the other) but precisely changes the relative phases, bringing the system from a state to another and realizing the non-Abelian braiding statistics. Recently Möllerand Simon [107] constructed a family of BCS paired wavefunctions that generalize the Moore-Read state and remain in the same “weak paired” phase. They also showed that these wavefunctions can give a good description of the exact ground states for systems interacting via a realistic two-body interaction, obtained by varying the first Haldane pseudopotential v1 from its Coulomb value: indeed in a v1-interval the overlaps are quite high (larger than those with the original “Pfaffian” wavefunction), showing the finite extent of the “weak paired” phase in the interaction parameter spaces for these systems (with up to 16 particles).

2.8 “Pfaﬃan” numerics

The non-Abelian braiding statistics of the quasiholes above the “Pfaffian” state was originally deduced from conformal field theory [51, 99] and then confirmed by the BCS approach described in the preceding section [105,106]. With the goal to obtain an explicit demonstration of it in the microscopic “Pfaffian” model, Tserkovnyak and Simon [108] performed in the lowest Landau level a direct numerical (Monte Carlo) calculation of the evolution of the Berry matrix under interchange of the positions of two Moore-Read quasiholes, in the presence of other two quasiholes at fixed positions. The results showed strong oscillations, which however become smaller for larger system sizes and for larger quasiholes separations; in this limit they could confirm the predictions for the braiding of four quasiholes above the “Pfaffian” state. In the following Simon [109] investigated the effect of Landau level mixing on the braiding statistics: he found that, although the Abelian part of it is perturbed, the non-Abelian properties are robust to an accuracy that is ex-

30 ponentially small in the distance between the Moore-Read quasiholes. These results seem natural, since there is a continuum of possible Abelian statistics, whereas the character of non-Abelian braiding statistics is discrete and cannot be continuously perturbed (although there is in principle also the possibility that the whole concept of braiding statistics loses its meaning, the result of quasihole braiding becoming path dependent). The quasiholes are not point-like objects (in fact they are quite large) and they acquire their non-Abelian properties only if they are enough far apart from each other. Baraban et al. [110] numerically examined this dependence on the distance and found out that in a system with four quasiholes the exact two-fold degeneracy necessary for non-Abelian braiding statistics is reached in an exponential way; similarly the unitary transformation that describes the quasiparticle braiding converges exponentially towards its asymptotic limit: in both cases the length scale of the convergence is about 2-3 magnetic lengths `0, which suggests that the spacing between quasiholes should not represent a problem for an experimental realization of non-Abelian braiding statistics. At last, Prodan and Haldane [111] using suitable pinning potentials (in the “Pfaffian” interaction limit) were able to localize Moore-Read quasiholes at specified locations (in the spherical geometry) and to move them adiabatically along different paths. By means of exact diagonalizations (restricted to some zero modes spaces) they showed that the braiding properties of quasiholes are indeed as described by the underlying conformal field theory. In their geometri- cal representation of the braiding, the “source” of the non-Abelian properties is strongly localized near to the quasiholes, confirming the topologic character of these properties (that is, their independence on the precise shape of the braiding path, as long as the quasiholes are far enough from each other). Furthermore, they also studied the fusion of quasiholes: measuring the electron density as two quasiholes are taken together (in the presence of other two quasiholes) one can in fact determine in which of the two degenerated states the system originally was.

2.9 The “Pfaﬃan” on a torus

It is interesting to look at the “Pfaffian” state on the surface of a torus (instead of on a planar or spherical surface, as considered so far): indeed it turns out that, because of the different geometry, the “Pfaffian” state is six-fold degenerated. At half filling there is always (for each state) a two-fold degeneracy on the torus, due to a center of mass translational symmetry [112], but the additional threefold degeneracy is a characteristic peculiar to the “Pfaffian” state: other states have in general different degeneracies. Indeed the degeneracy on the torus is a topological quantum number characterizing a state, just like the “shift” S in the spherical geometry. The six “Pfaffian” wavefunctions fulfilling the (magnetic) boundary conditions on the torus are explicitely given in [103] (Section V.), in terms of the four theta functions ϑi (i = 1, 2, 3, 4): concretely, the three-fold degeneracy comes from the Pfaffian factor Pf(ϑa(zi zj)/ϑ1(zi zj)), where a can be chosen to be 2, 3 or 4; the overall two-fold− degeneracy− is realized through a factor that depends only on the center of mass coordinate.6

6These wavefunctions were already given by Greiter, Wen and Wilczek [85], but they also

31 Recently Oshikawa et al. [113] and Chung and Stone [114] gave a physical derivation of the topological degeneracy of the “Pfaffian” state on a torus (and on general Riemann surfaces with genus g), relating it to the fractional and non-Abelian nature of the Moore-Read quasiholes. In this picture, starting from one of the ground states, one obtains another ground state by creating a quasiparticle-quasihole pair, moving the quasihole around the torus and annihi- lating it again with the quasiparticle. Using this procedure, eight states can be generated; however, it turns out, exactly because of the non-Abelian properties of the quasiholes, that two of them are not ground states of the system, giving the six-fold degeneracy. Another interesting work (by Bergholtz et al. [115]) investigated the thin torus limit of the “Pfaffian” state. In this limit the system becomes essentially a one-dimensional (circular) chain and the states (in a given Landau level) are labeled by the occupation numbers on this chain. The six “Pfaffian” states at half filling correspond to the patterns 010101 ... 01 (two-fold degenerated) and 01100110 ... 0110 (four-fold degenerated); the (quarter-charged) quasiholes and quasiparticles are realized as domain walls between these degenerated states. It is interesting to note that this picture is manifestly particle-hole symmetric, allowing a unified description of quasiparticles and quasiholes, whereas in the original two-dimensional system much is known about quasiholes but only few about quasiparticles. By counting the number of states with fixed 2N domain N 1 walls, one easily gets the 2 − degeneracy for localized Moore-Read quasiparticles. Furthermore, Seidel [116] showed (for the bosonic ν = 1 case) that also in this one-dimensional limit the braiding statistics of the quasiparticles is non- Abelian and in agreement with the predictions of conformal field theory.

2.10 The others

The “Pfaffian” wavefunction of Moore and Read [51] represents an interesting possibility for the description of the quantum Hall effect at ν = 5/2, but it is not the only one. In this section we briefly review some of the other proposed possibilities.

K=8 As discussed by Halperin [117], assuming that there is some mechanism that causes the formation of electron pairs (the so-called “strong pairing”), then the so formed 2e-charged bosons can condense in a Laughlin state at (pair) filling factor νB = 1/8: 8 P Z 2/4 Ψ = (Z Z ) e− i | i| , (2.15) 8 i − j i>j Y where the Zi’s are the (complex) coordinates of the pairs. The filling factor for the original electrons is indeed ν = 1/2: the number of bosons NB is half the B number of electrons Nel and the number of available “bosons orbitals” Norb is el about twice the number of electronic orbitals Norb = Nφ + 1 (for the 2e charged el B bosons the flux quantum is halved) and thus ν = Nel/Norb 2 2 NB/Norb = 1/2. The elementary excitations of this state are Laughlin≈ quasiparticles· · with argued that the “Pfaffian” state should in reality be eight-fold degenerated.

32 charge e/4 (1/8 of the boson charge 2e), obeying an Abelian fractional statistics. ± The denomination “K = 8” comes from an effective theory introduced, independently, by Fröhlich et al. [118–122] and by Wen and Zee [88, 89, 123] that allows the classification of Abelian quantum Hall systems in terms of a quadratic matrix K. The above state corresponds to the 1 1 matrix K = 8 in this scheme. × 331 In the same work [117] Halperin also discussed the possibility of a two-component quantum Hall system, that is a system with two sorts of electrons, labeled for example by their spin orientation (if the system is not polarized) or by a layer index (in a bilayer system). If the two sorts are equally populated, the Laughlin wavefunction can be generalized to

Nel/2 Nel/2 Nel Ψ = (z(1) z(1))m (z(2) z(2))m (z(1) z(2))n mmn i − j s − t i − t × i>j s>t i,t Y Y Y P z(1) 2/4 P z(2) 2/4 e− i | i | e− t | t | , (2.16) × (1) (2) where zi and zt are the (complex) coordinates for the two sorts of electrons. The correlations between particles of the same sort are described by the exponent m, while n describes the correlations between particles of different sorts. This wavefunction corresponds to a filling factor ν = 2/(m + n) and its elementary excitations are (Abelian) quasiparticles with charge e/(m + n): the number of flux quanta in the system, that is the maximum power± in each coordinate (for each sort), is N = (m + n)N /2 m and thus in the thermodynamic φ el − limit the filling factor is ν Nel/Nφ 2/(m + n). Furthermore, a quasihole ≈ → Nel/2 (1) Nel/2 (2) at position z0 corresponds to a factor i=1 (zi z0) (or t=1 (zt z0)) inserted into the wavefunction, in analogy with the quasiholes− in Laughlin− wave- Q Q function. Comparing the maximum power of the quasihole coordinate z0 (Nel/2) with the maximum power of the electrons coordinates ((m + n)Nel/2 m) it is evident that one needs m + n quasiholes to neutralize an electron, that− is, the quasihole charge is e/(m + n). For the choice m = 3, n = 1, one thus gets a candidate for the quantum Hall effect at half filling, with quarter-charged quasiparticles. By means of the K- matrix classification one can also show that this state is in the same universality class of the state obtained by the condensation of 2e/3 charged quasiparticles above the ν = 1/3 Laughlin state, into a second level hierarchical fractional quantum Hall state [124–126]. The wavefunction Ψ331 is obviously antisymmetric as function of the coor- (1) (2) dinates zi and, separately, of the coordinates zt ; interestingly, if one in addition performs the full antisymmetrization of Ψ331, the result is the “Pfaffian” wavefunction ΨMR. This antisymmetrization is clearly a non-trivial operation: for example it transforms the braiding statistics of the quasiparticle excitations from Abelian to non-Abelian. Ho [127] showed that it is also possible to continuously transform Ψ331 into ΨMR, by varying the spin configurations: indeed both states belong to a family of triplet paired wavefunctions Ψd, parametrized by a complex vector d. Fur- thermore Ho constructed a family of model Hamiltonians H(d) that have Ψd as

33 ground state, thus allowing a continuous interpolation between the “331” and the “Pfaffian” wavefunction. This interpolation seemed to contradict the fact that the two states belong to different universality classes, but in reality there is no contradiction (cf. [103]): it is always possible to interpolate between any two states in the same Hilbert space. To show that two states belong to the same class, one has to perform such an interpolation without any phase transition, that is, the energy gap between the ground state and the first excitations has to remain finite over the whole parameter interval. For the Hamiltonians proposed by Ho, however, this is not the case because the “Pfaffian” point is “patholog- ical”: at this point the gap closes and there is a big degeneracy of zero-energy states.

Antipfaffian Levin et al. [93] and Lee et al. [94] noticed that the Moore-Read state is not symmetric under particle-hole conjugation: the “Pfaffian” state of holes is (topologically) different from the “Pfaffian” state of electrons and belongs to another universality class. In particular, the edge structure is different, whereas the bulk physics is similar: for this reason our calculations for electrons on the compact (edgeless) surface of the sphere, interacting via the (particle-hole symmetric) Coulomb interaction in a single Landau level can not discriminate between the “Pfaffian” and the “Antipfaffian”. To find out which of the two is favoured over the other, one has to consider the three-body interactions that arise in real samples from Landau level mixing and break particle-hole symmetry: the decisive factor appears to be the sign of this three-body interaction [128] (that is, whether it is attractive or repulsive), with a sharp (possibly first order) phase transition between the two phases at the particle-hole symmetry point. Studies investigating the effects of Landau level mixing were performed [129– 131], but the results are unclear, and partly contradictory. On one side Rezayi and Simon [129] did calculations in truncated Hilbert spaces, taking in account some Landau levels (and not only the “valence” Landau level) but allowing only a certain number of particle or hole excitations. From exact diagonalizations in the torus geometry in systems with up to 20 particles they found that the ground states (often) have a bigger overlap with the “Antipfaffian” than with the “Pfaffian” state, particularly for the larger systems. This effect is particularly magnified if the first Haldane pseudopotential v1 is slightly increased from its Coulomb value. On the other side Bishara and Nayak [130] did a perturbative expansion in the “small” dimensionless parameter κ, defined as the Coulomb interaction scale 2 e /4πε`0 divided by the cyclotron energy ~ωc. This parameter describes the Landau level mixing and in real systems it is actually not so small: κ 0.6 1.8 for magnetic fields in the range 15 2T . In the lowest order of this≈ expan-− sion, Bishara and Nayak calculated− the effective 3-body interaction (and the renormalization of the 2-body interaction) generated by Landau level mixing, expressing them in terms of the N-body pseudopotentials vL,N [87] which give the interaction energies of N particles in a state with relative angular momentum L~. The obtained 3-body pseudopotential for the lowest possible angular momentum, v3,3, corresponding to three particles at closest approach, is small but negative, thus in principle favorizing the “Antipfaffian” state. However,

34 from exact diagonalizations for small systems (with up to 18 electrons) in the spherical geometry, using the effective interaction of Bishara and Nayak, Wójs et al. [131] found that in the experimentally relevant κ-range the ν = 5/2 ground state and its (charged and neutral) low lying excitations are better described by the “Pfaffian” model. They explained this result with the influence of higher order vL,3 pseudopotentials, that dominate over the negative v3,3 and select the “Pfaffian phase”. Thus from the theoretical (numerical) viewpoint it is not yet clear which of these two states is favoured and further investigations are needed to identify which of the two approaches described above is the appropriate one to correctly take into account the effects of Landau level mixing. Experimentally the discrimination between “Pfaffian” and “Antipfaffian” can (in principle) be made by measuring properties that involves the edge physics, such as tunneling and thermal Hall conductance [93,94] (see also section 2.12). and more... These were only a few, the most “prominent”, candidates. Other possibilities are for example the “NAF” state proposed by Wen [132–134], a quaternate generalization of the Moore-Read state [135], or “Pfaffian” and “Antipfaffian” states with modified edge structures [126]. Furthermore, Boyarsky et al. [136] systematically investigated and listed the conformal field theories that, from physical arguments, could be candidates for the description of the quantum Hall effect at ν = 5/2. For each theory they gave the values of quasiparticle charge e∗ and scaling tunneling dimension g for a comparison with the experiments [137] (see also section 2.12).

2.11 Topological quantum computation

In recent years there has been a (big) renewed interest in the quantum Hall state at ν = 5/2, because of the possible exploitation of its non-Abelian braiding statistics for topological quantum computation (see [35] for a review). A quantum computation consists of three steps: first an initialization of the system in a known state Ψ , then a controlled unitary evolution to some final | 0i state U(t) Ψ0 and at the end the measurement of this state. The biggest source of problems| fori the realization of a quantum computer comes from the errors that occur during one of these steps, causing decoherence and eventually destroying the quantum information: these errors are not only mere discrete errors (like a spin flip) but also continuous (like an arbitrary change of a relative phase). Some error corrections schemes are available, but to be applicable they require a low basis error rate: practically one has to be able to perform about 104 106 operations without errors. − Topological quantum computation allows to circumvent this difficulty [138, 139]. Here the quantum information is stored in a system of localized non- Abelian anyons: the system is initially prepared in one of the degenerated ground states and the quantum computations are realized through braidings of the anyons, thus transforming the initial state in another (degenerated) one. Indeed, if the temperature is low enough and the anyon motions are sufficiently

35 slow, the system evolves only within the subspace of degenerated ground states, because of the energy gap that separates it from the rest of the spectrum. This scheme is naturally immune to errors, because here (in contrast to most proposed realizations of a quantum computer) the quantum information is stored non-locally, in a wavefunction describing many anyons well separated from each other: all local perturbations can not cause a transition from one of the ground states to another, only the (highly non-local) operation of braiding can do this. Furthermore, the result of a braiding operation depends only on the topology of the anyon path (and not on its precise geometry), thus unitarity errors during the quantum computations are also unlikely. 18

gate can be applied, by the passage of a single quasiparticle

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¡ from one edge to the other, provided that its trajectory passes

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¡ in between the two localized quasiparticles. This is a sim-

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¡ t t t tions of multi-quasiparticle states of non-Abelian quasiparti-

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¡ 1 1S 2 2 cles, which we discuss in more detail in section III. If we

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¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ fore and after applying this NOT gate, we will observe different values according to (16). For this operation to be a NOT gate, it is important that FIG. 3 If a third constriction is added between the other two, the cell just a single quasiparticle (or any odd number) tunnel from is broken into two halves. We suppose that there is one quasiparticle one edge to the other across the middle constriction. In or- Figure 2.8:( Aor a qubitny odd nu realizedmber) in eac withh half.T non-Abelianhese two quasipar anyons.ticles (labeled (Fromder t [35].)o regulate the number of quasiparticles which pass across 1 and 2) form a qubit which can be read by measuring the conduc- the constriction, it may be useful to have a small anti-dot in tance of the interferometer if there is no backscattering at the middle the middle of the constriction with a large charging energy so The Moore-Readconstri quasiholesction. When a si ofngle thequasi quantumparticle tunnels Hallfrom on effecte edge to at νth=at on 5l/y2a s couldingle quasiparticle can pass through at a time. If the other at the middle constriction, a σx or NOT gate is applied to we do not have good control over how many quasiparticles thus supply a systemthe qubit for. the realization of a topological quantum computer. et al. tunnel, then it will be essentially random whether an even or Das Sarma [140] described a qubit on which a logical NOTodd nu operationmber of quasiparticles tunnel across; half of the time, can be performed by quasiparticle braiding, as shown in Fig. 2.8:a NOT thegate qubitwill be applied and the backscattering probability netic field reflects the non-Abelian nature of the quasiparticles (hence the conductance) will change while the other half of (a two-level system)(Ilan iset a formedl., 2007). by two quasiholes, each localized on an antidot, labeled with 1 and 2; other quasiholes can move around them,th alonge time, thethe bac (inkscattering probability is unchanged. If the constriction is pinched down to such an extreme that the 5/2 blue depicted) edge states, and can eventually tunnel from onesta edgete is dis torupt theed between the quasiparticles, then when it is 4. A Fractional Quantum Hall Quantum Computer other at one of the constrictions, depending on the tunable tunnelingresto amplitudesred, there will be an equal probability for the qubit to be in either state. t , t , t . The state ofWe thenow qubitdescribe canhow bethe read constri outcted byHall anbar interferencemay be uti- measurement: 1 2 s This qubit is topologically protected because its state can lized as a quantum bit (Das Sarma et al., 2005). To that end, injecting a current at the lower left corner and allowing backscatteringonly be aff onlyected byata charge e/4 quasiparticle braiding with it. an even number of e/4 quasiparticles should be trapped in the If a charge e/4 quasiparticle winds around one of the antidots, the constrictions cet1ll andbetweetn2,the the cons currenttrictions, a thatnd a ne exitsw, tuna atble, theconst upperric- left corner is it effects a NOT gate on the qubit. The probability for such given by the interferencetion should b betweene added betw theeen th quasiholese other two so thatthat the tunnels cell is at t and the an ev1ent can be very small because the density of thermally- broken into two cells with an odd number of quasiparticles in others that goes around the qubit, tunneling at the quantum pointexcit contacted charge et/24.quasiparticles is exponentially suppressed each (See Fig. (3)). One way to tune the number of quasipar- at low temperatures, n ∼ e−∆/(2T ). The simplest estimate This interference dependsticles in eac onh ha thelf is t stateo have t ofwo theantido qubit,ts in the becauseHall bar. B they quasiholes that qp of the error rate Γ (in units of the gap) is then of activated go around it readt outuning ath sign,e volta dependingge on the antido onts, whichwe can c statehange th thee nu qubitmber is (as discussed form: in section 2.6). Onof qu theasiho otherles on eac hand,h. Let ifus aass singleume tha quasiholet we thereby fi goesx the around only one − number of quasiparticles in each half of the cell to be odd. Γ/∆ ∼ (T/∆) e ∆/(2T ) (17) of the antidots, byFor allowingconcreteness backscattering, let us take this odd at nu thember middleto be one constriction(i.e. ts, then this operation flipslet theus ass stateume th ofat w thee are qubit,in the id realizingealized situa ation NOTin wh gate.ich The most favorable experimental situation (Xia et al., 2004) Unfortunatelyth theere ar Moore-Reade no quasiparticle states in the isbulk not, and capable one quasiho ofle on universalconside quantumred in (Das Sarma et al., 2005) has ∆ ≈ 500 mK each antidot). These two quasiholes then form a two-level and T ∼ 5 mK, producing an astronomically low error rate computation, that is, the transformations that can be generated by− quasihole system, i.e. a qubit. This two-level system can be understood ∼ 10 15. This should be taken as an overly optimistic es- braidings are notin sufficientseveral ways, towh implementich we discuss i alln det unitaryail in section transformations.III. In timate. A How-more definitive answer is surely more compli- ever, it can be shownbrief, the thattwo sta onetes corr canespond realizeto wheth universaler the two σs quantumfuse to ca computationted since there are multiple gaps which can be relevant in 1 in this system by addingor ψ or, in athe littlelanguag numbere of chiral p of-wa topologicallyve superconductiv unprotectedity, a disordere opera-d system. Furthermore, at very low temperatures, the presence or absence of a neutral (‘Majorana’) fermion; or, we would expect quasiparticle transport to be dominated by tions [141–143], whichequival requireently, as th somee fusion error of two correction, quasiparticles orcarr someying th topology-changinge variable-range hopping of localized quasiparticles rather than operations [142, 144].spin-1 Some/2 repre othersentation non-Abelian of an SU(2) gaug states,e symme proposedtry in the tothe describermal activati theon. Indeed, the crossover to this behavior may fractional quantumspin Hall-0 or s effectpin-1 ch atann otherels. filling factors (such as thealr Read-Rezayieady be apparent (Pan et al., 1999b), in which case, the er- The interference between the t1 and t2 processes depends ror suppression will be considerably weaker at the lowest tem- state [145] proposedon the forstateνof t=he tw 12o-/le5)vel s supportystem, so th universale qubit can be quantumread by pe computationratures. Although the error rate, which is determined by the through quasiholea braidingsmeasurement alone.of the four-terminal longitudinal conductance probability for a quasiparticle to wind around the anti-dot, is not the same as the longitudinal resistance, which is the prob- 2 2 ∗ iφ GL ∝ |t1| + |t2| ± 2Ret1t2e (16) ability for it to go from one edge of the system to the other, 36 the two are controlled by similar physical processes. A more where the ± comes from the dependence of hξ0|Uˆ|ξ0i on the sophisticated estimate would require a detailed analysis of the state of the qubit, as we discuss in section III. quasiparticle transport properties which contribute to the er- The purpose of the middle constriction is to allow us to ma- ror rate. In addition, this error estimate assumes that all of nipulate the qubit. The state may be flipped, i.e. a σx or NOT the trapped (unintended) quasiparticles are kept very far from 2.12 The “Pfaffian” (or others) at ν = 5/2? Ex- periments.

To answer the question whether the fractional quantum Hall effect at ν = 5/2 is indeed described by the “Pfaffian” state of Moore and Read [51] or by another state, many possible experimental tests were proposed, mostly of them involving transport measurements (tunneling or interference of edge states, as in the qubit described in the previous section), but also including some examination of the bulk properties. The first results for such experiments have been recently published, giving some hints, but a clear answer to the question is still lacking. Here we present some of the proposals and their realizations.

Quasiparticle charge (and tunneling exponent) The first point to clarify is whether the elementary excitations in the quantum Hall effect at ν = 5/2 are indeed e/4-charged quasiparticles and quasiholes. This prediction is not unique to± the “Pfaffian” state, on the contrary, it is common to all possible states described in section 2.10 (and also to others), NATURE | Vol 452 | 17 April 2008 ARTICLES but it is a necessary condition: if it were found that the elementary excitations were e/2-charged objects, the “Pfaffian” (and the other proposals) should be rejected.± Shot noise experimentsa at a quantum point contact are sensitivec to the charge e 80 80 20 carried by the quasiparticles responsibleT = for 10 mK the current flow at the edges T = 10 of mK a T = 40 mK quantum Hall sample: from the dependencen = 5/2 of the tunneling shot noisen = 5/2 on 15 n = 5/2

the edge current one can (%) indeed70 deduce the charge of the60 quasiparticles that tunnel across the constriction. Such measurements allowed to determine the 10 5/2−2 quasiparticle charges in somet Laughlin and hierarchical states, like at ν = 1/3 5 and 2/5 [16, 17, 146]. 60 40 0 d f * * 2.5 e = e/4 e = e/4

) * –1 e = e/2 1.5 t Hz 2.0 aver

2 2 tdiff

(A e* = e/4 tdiff –29 1.5 1.0

1 1.0 0.5 0.5 Shot noise × 10 0 0 taver 0 −5 0 5 −5 0 5 −10 −5 0 5 10

Impinging current, Iimp (nA) FigureFigure4 | 2.9:Conducν =tance 5/2:and Shotshot noisenoise atmeasuremen a quantumts pointof partition contacted as functionobtained of after averaging. a,, Measurements at 10 mK, at weak particlesthe edgeat currentthe 5/2 state, (measuredwith a infillingnA).factor The measurementsin the bulk of n are5 5/2. consistent withbackscattering a , where both models of the transmission coincide (see text). Transmissionquasiparticleand chargeshot noisee∗ = e/were4. (Frommeasured [147].)as function of the impinging c, d, Measurements at 10 mK, transmission ,0.5, where the two models for 2 current Iimp 5 Vgi, with gi 5 2.5e /h. The effective transmission of the the transmission provide the two limits of the expected noise. partitioned channel was calculated assuming that the lower state below the e, f, Measurements at 40 mK, strong backscattering, where the two models n 5 5/2 is the n 5 2 state, determined by a similar method to that described in for the transmission provide the two limits of the expected noise. Fig. 2. Measurements were done in a similar37fashion to that described in Surprisingly, the charge remains e* 5 e/4, with no evident ‘bunching’ to e Fig. 3, but some of the data are represented by fewer points, which were when the QPC is nearly pinched. range nQPC 5 5/3…..3, identified on the two-terminal Hall conduc- present them in Supplementary Information). The panel at top left in tance of the actual sample on which the measurements were made. In Fig. 5 was measured at n 5 3, with relatively weak backscattering the top row, we show the effective transmission of the state under induced by the QPC. The transmission drops with increasing current study with the identification of the next lower lying state, which was in a ‘mound-like’ fashion and saturates around 5 nA. The saturation identified by the method shown in Fig. 2 (in some of the cases we did at higher currents (or voltage) is typical (see, for example, ref. 26), not plot the conductance and shot noise due to lack of space, but and is accompanied by a saturation of the shot noise. It could be

100 90 18 35 94 (%)

30 (%) 16 (%) 80 (%) (%) 25 85 92 60 20 14 3−8/3 2–5/3 3−8/3 t t 5/3–4/3

8/3–5/2 90 t t t 80 40 −10 0 10 −10 0 10 −10 0 10 −10 0 10 −10 0 10 Impinging current, Iimp (nA)

/h) 3.0 2 (e xy g 2.5

2.0

Hall conductance, 1.5

) 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

–1 Magnetic field, (T) Hz 2 e* = e e* = e/3 e* = e/3 e* = e/3 e* = e/4 e* = e e* = e/3 e* = e/3 (A 2.0 1.0 1.5 1.0 –29 4 1.5 1.0 0.5 0.5 2 1.0 0.5 0.5 0 0 0 0 −0.5 0 −10 0 10 −10 0 10 −10 0 10 −10 0 10 −10 0 10 Impinging current, Iimp (nA) Shot noise x 10

Figure 5 | Conductance and shot noise at different filling factors in the main plot. Except for the 5/2 state, measurements in the range nQPC 5 3…2 QPC. The main plot shows the two-terminal Hall conductance of the sample were performed at n 5 3 and at T 5 40 mK, and measurements in the range as a function of magnetic field; measurement points are indicated on this nQPC 5 2…5/3 were performed at n 5 2 and at T 5 90 mK. Near integer curve. For each measurement point, the top row shows the transmission as a fillings (2 and 3) the measured charge was e, whereas near (above and below) function of impinging current, and the bottom row shows the shot noise as a the fractional fillings (8/3 and 5/3) the measured charge was e/3. function of impinging current. Measured charge is shown boxed below the 833 © 2008 Nature Pu lishing Group First results for ν = 5/2 were presented by Dolev et al. [147]: from their shot noise measurements the deduced quasiparticle charge was consistent with a charge value e∗ = e/4, and inconsistent with a multiple of it, like e/2, as shown in Fig. 2.9. However, in the following, more detailed (and sensitive) investigations [148, 149] showed that the situation is more complicated: the tunneling quasiparticle charge e∗ deduced from the shot noise experiments is highly dependent on the temperature and on the tunneling strength at the quantum point contact. Indeed only in an intermediate regime, e∗ is found to be about e/4; for weaker or stronger tunneling and for lower temperatures, e∗ is significantly larger. This behaviour, particularly at low temperature and weak tunneling, is not yet fully understood, probably signaling a “bunching” of the tunneling quasiparticles [150], but the fact that the smallest observed value for e∗ is e/4, is at least a strong indication that the fundamental quasiparticles in the system carry this charge. Another method to determine the tunneling quasiparticles charge e∗ is the investigation of the temperature scaling of the bias-dependent tunneling at a quantum point contact, as done by Radu et al. [137] (more recent and better data in [151]). By fitting the experimental data with the theoretical curves for weak tunneling [86,152–156] one can indeed extract the quasiparticle charge e∗ and a scaling exponent g (describing the temperature dependence of the zero- 2g 2 bias tunneling conduction peak as T − ). The latter is of particular interest, because it has a specific value for each of the proposed states for the fractional quantum Hall effect at ν = 5/2; thus a precise determination of g could allow to recognizeRESEARCH whichARTICLES of these states is actually realized.

References and Notes 1. D. C. Tsui, H. L. Stormer, A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 2. B. I. Halperin, Phys. Rev. B 25, 2185 (1982). 3. X. G. Wen, Phys. Rev. B 44, 5708 (1991). 4. R. Willett et al., Phys. Rev. Lett. 59, 1776 (1987). 5. F. D. M. Haldane, E. H. Rezayi, Phys. Rev. Lett. 60, 956 (1988). 6. G. Moore, N. Read, Nucl. Phys. B 360, 362 (1991). 7. X. G. Wen, Phys. Rev. Lett. 66, 802 (1991). 8. M. Levin, B. I. Halperin, B. Rosenow, Phys. Rev. Lett. 99, 236806 (2007). 9. S.-S. Lee, S. Ryu, C. Nayak, M. P. A. Fisher, Phys. Rev. Lett. 99, 236807 (2007). 10. A. Y. Kitaev, Ann. Phys. 303, 3 (2003). 11. S. Das Sarma, M. Freedman, C. Nayak, Phys. Rev. Lett. 94, 166802 (2005). 12. C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, http://arxiv.org/abs/0707.1889 (2007). 13. R. H. Morf, Phys. Rev. Lett. 80, 1505 (1998). 14. B. I. Halperin, Helv. Phys. Acta 56, 75 (1983). 15. B. I. Halperin, P. A. Lee, N. Read, Phys. Rev. B 47, 7312 (1993). 16. X. G. Wen, Q. Niu, Phys. Rev. B 41, 9377 (1990). 17. X.-G. Wen, Phys. Rev. Lett. 70, 355 (1993). 18. W. Bishara, C. Nayak, Phys. Rev. B 77, 165302 (2008). 19. P. Fendley, M. P. A. Fisher, C. Nayak, Phys. Rev. Lett. 97, 036801 (2006). 20. M. Milovanović, N. Read, Phys. Rev. B 53, 13559 (1996). 21. X.-G. Wen, Adv. Phys. 44, 405 (1995). Fig. 5. Map of the fit quality. Normalized fit error is the residual from the least-squares fit, divided by the 22. E. Fradkin, C. Nayak, A. Tsvelik, F. Wilczek, Nucl. Phys. B number of points and by the noise of the measurement. Also marked on the map are proposed theoretical 516, 704 (1998). pairs of (e*, g). 23. P. Bonderson, A. Kitaev, K. Shtengel, Phys. Rev. Lett. 96,

016803 (2006). on November 17, 2010 Figure 2.10: ν = 5/2: mean squared fit error of the tunneling data at a quantum24. A. Stern, B. I. Halperin, Phys. Rev. Lett. 96, 016802 (2006). 25. C.-Y. Hou, C. Chamon, Phys. Rev. Lett. 97, 146802 (2006). point contact,Fig. 6. ( asA) R functionD (device 2) of the quasiparticle charge e∗ and of the exponent g. as a function of dc bias at 26. D. E. Feldman, A. Kitaev, Phys. Rev. Lett. 97, 186803 (2006). (Black for the best fit, yellow for the worst; from [137].) 27. R. de Picciotto et al., Nature 389, 162 (1997). fixed magnetic field (B = 28. L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, Phys. Rev. 5 4.31 T, middle of n = =2) Lett. 79, 2526 (1997). and fixed gate voltage (Vg = 29. V. J. Goldman, B. Su, Science 267, 1010 (1995). From their−2.4 V) fullat seve dataral tem- set for the tunneling at a constriction at ν = 5/230. DolevM. Dolev, M. Heiblum, V. Umansky, A. Stern, D. Mahalu, peratures. At the lowest Nature 452, 829 (2008). et al. extracted the best fit values e∗ = 0.17e and g = 0.35. Figure 2.10 shows31. S. Roddaro a et al., Phys. Rev. Lett. 90, 046805 (2003). temperature, the peak de-

32. S. Roddaro, V. Pellegrini, F. Beltram, G. Biasiol, L. Sorba, www.sciencemag.org color mapve oflops thea flat top (meanat a value squared) ﬁt error as function of e∗ and g (black forPhys. theRev. Lett. 93, 046801 (2004). of resistance consistent 33. S. Roddaro, V. Pellegrini, F. Beltram, L. N. Pfeiffer, with the resistance at n = K. W. West, Phys. Rev. Lett. 95, 156804 (2005). 7 34. J. B. Miller et al., Nat. Phys. 3, 561 (2007). =3 .(B) Zero-bias peak height as a function of 35. C. P. Milliken, C. P. Umbach, R. A. Webb, Solid State 38 Commun. 97, 309 (1996). temperature. The peak 36. Materials and methods are available on Science Online. height saturates at the 37. C. W. J. Beenakker, H. van Houten, Solid State Phys. 44, lowest temperatures. (C) 1 (1991). Peak width as a function 38. J. S. Xia et al., Phys. Rev. Lett. 93, 176809 (2004). Downloaded from of temperature. The red 39. P. Fendley, A. W. W. Ludwig, H. Saleur, Phys. Rev. B 52, 8934 (1995). line is the best fit of the 40. We acknowledge helpful discussions with W. Bishara, high-temperature data C. Chamon, C. Dillard, P. Fendley, M. Fisher, B. Halperin, with a line going through E. Levenson-Falk, D. McClure, C. Nayak, B. J. Overbosch, zero. Below ~30 mK the B. Rosenow, A. Stern, X.-G. Wen, and Y. Zhang and peak width no longer follows this line. experimental help from S. Amasha and A. Klust. This work was supported in part by Army Research Office (W911NF-05-1-0062), the Nanoscale Science and We are not aware of quantitative predictions sistent with previous strong-tunneling studies Engineering Center program of NSF (PHY-0117795), NSF n ¼ 5 n 32 39 R (DMR-0701386), the Center for Materials Science and for the strong tunneling regime for =2. for <1( , ). The value of D at the Engineering program of NSF (DMR-0213282) at MIT and However, qualitative comparisons with strong peak is consistent with full backscattering of by the Microsoft Corporation Project Q and the Center for 5 7 tunneling theory (39) and experiment (31–33)at the =2 edge and a n ¼ =3 underlying edge Nanoscale Systems at Harvard University. other FQH states (n < 1) can be made. For strong state. tunneling, the edge states associated with the Outlook. Beyond enabling investigations in Supporting Online Material topmost fractional state (n ¼ 5= in the present the fundamental physics toward a demonstra- www.sciencemag.org/cgi/content/full/1157560/DC1 2 Supporting Text case) are backscattered almost entirely so that tion of nonabelian statistics, these experi- Figs. S1 to S6 the quasi-particle tunneling takes place along ments demonstrate a high degree of control of References 5 the QPC rather than across it (19, 39). The interedge tunneling of the =2 edge state, a 10 March 2008; accepted 11 April 2008 flat-top peak shape and strong side dips (Fig. prerequisite for quasi-particle braiding opera- Published online 17 April 2008; 6A), much stronger than that expected from tions needed for related schemes of topological 10.1126/science.1157560 weak tunneling (Eq. 2), are qualitatively con- quantum computing. Include this information when citing this paper.

902 16 MAY 2008 VOL 320 SCIENCE www.sciencemag.org best ﬁt, yellow for the worst); the (e∗/e, g)-values for diﬀerent candidate states are also indicated: 7

(1/4, 1/8) for the “K = 8” state, • (1/4, 1/4) for the “Pfaffian” state, • (1/4, 3/8) for the “331” state, • (1/4, 1/2) for both the “Antipfaffian” and “NAF” states • The latter parameter pair has the smallest fit error, indicating that these states (both non-Abelian) are the most consistent with the weak tunneling experiments; however the (Abelian) “331” state can not be completely ruled out. In any case the data appear to be consistent with e∗ = e/4-charged tunneling quasiparticles and inconsistent with e∗ = e/2 (at least for the tunneling conditions considered in this experiment), signaling that this is the fundamental charge present in the system. The more recent (and better) results [151] yield e∗ = 0.23e and g = 0.44 for the best fit, confirming these conclusions. Other methods for the determination of the tunneling charge using current measurements (also involving photo-assisted currents) have been proposed [158], but not yet realized, for any filling fraction. Recently an experiment by Venkatachalam et al. [157] for the first time investigated the charge of quasiholes localized in the bulk of the sample (and not flowing at the edges or tunneling at constrictions like in the previous experiments). The quasiparticles are localized in puddles by the disorder potential: by measuring the (stepwise) charging process of these puddles at ν = 5/2 and ν = 7/3 as the electron density n is varied, Venkatachalam et al. could determine the ratio of the fundamental charges for the two filling fractions. The measurements were done using a single electron transistor, directly measuring 1 the local incompressibility κ = (∂µ/∂n)− , which at a charging event shows a (negative) spike because of the discontinuity in the local chemical potential µ. The spike frequency is inversely proportional to the quasiparticle charge e∗ = 1/q: a full electron corresponds to q charging events. Comparing the spectra for the two filling fractions (as shown in Fig. 2.11) and averaging over many measurements, a ratio of about 4/3 was found, consistent with the quasiparticle charges e/4 for ν = 5/2 and e/3 for ν = 7/3.

(Fabry-Pérot)Interferometry To directly detect the non-Abelian braiding statistics of quasiparticles in the ν = 5/2 state, several experimental setups involving interference of propagating edge modes has been proposed. The first proposals came, simultaneously and independently, by Bonderson et al. [159] and Stern and Halperin [160] (see also [36]), proposing an interferometry experiment (in the Fabry-Pérotgeometry), which symplified the setup for the qubit introduced by Das Sarma et al. [140] (discussed in section 2.11). The basic idea is still the interference between two quasiparticle currents that flow at the edges of the quantum Hall sample and eventually tunnel from one edge to the other at some constriction; but in this simplified version there is only a

7The values for other candidates are listed in [136].

39 a 2 5/2

.) 7/3

U 0 A. ( y

t -2

-4 essibili r

omp -6 nc I -8

0 1 2 3 4 5 6 7 8 9 13 ∆n at 7/3 (x 10 m-2 ) b

0.5 n o i

t 0.4 a l e rr o C

s 0.3 e B

0.2 0.4 0.8 1.29 1.6 2 2.4 Relative Scale (β)

Figure 3: Comparison of spectra at 5/2 and 7/3. a, To determine the charge, we first choose a relative scale betweenFigurethe tw 2.11:o densit Exampley axes of(β comparison), and determin of thee localthe offset incompressibilitybetween the spectratwo sp duringectra that maximizes the cross-covtheariance. chargingHere processthe densit ofy afor quasiparticlesthe spectrum puddleat 5/2 (fromis scaled [157]).up by a)a Localfactor incom-of 1.29 and shifted to match up withpressibilitythe spectrum as functionat 7/3. ofThe theguide densitylines forshoνw=the 7/densit3 (iny red)change andrequiredν = 5/2to (inadd 1 electron to an area of 100blue);nm x the500 spectrumnm, appro forximatelyν = 5/2the is scaledsize of upour (inSET. density)We w byould atherefore factor β =exp 1.ect,29 very roughly, 3 e/3 chargingandev shifted,ents in a inwindo orderw tothis maximizesize. b, Rep thee correlation.ating this for b)man Correlationy values of asβ functionsuggests that a relative scale of 1.29 ofbest thedescrib scalinges factorthis daβta, whichset. can be interpreted as the ratio of the quasiparticle charges for the two states. For this disorder configuration the maximal correlation is at β = 1a .29 Measu4/3.rement c Model ≈

single antidot, that can localize a variable number of quasiparticles, and only two constrictions, one at the left and one at the right of the antidot. Injecting a current in the sample, this can be backscattered in two ways, either at the first constriction (not encircling the antidot) or at the second (encircling the antidot): the interference of the two backscattered currents will then depend on the magnetic field B 0.4and on1.1 the area1.8 A2.5enclosed0.4 1.1 between 1.8 the 2.5 two constrictions, 0.44 showing oscillations0.43 ifbB or A are varied, withd a period reflecting the e/4 charge ∆=U of the elementary excitations. But the visibility of∆=0 this interference pattern n o i

will depend on thet number of quasiparticles localized on the antidot: indeed a l e

one can show that forrr the “Pfaﬃan” state, because of the non-Abelian braiding o C

statistics, the interferencet pattern will be visible only if the number of encircled s e quasiparticles is even;B for an odd number the interference is suppressed. This odd-even effect in the edge states interference has been investigated by 0.32 1.31 0.26 2/3 4/3 Willett et al. [161,162]:0.4 their1.1 results1.8 show2.5 an0.4 alternating1.1 1.8 pattern2.5 of e/4 and e/2 Relative Scale (β) Relative Scale (β) period oscillations as the area or the magnetic field are varied, thus changing the Figure 4: Summarynumberof ofData enclosedand quasiparticlesModel. a, Rep (seeeating Fig.the 2.12).measuremen Assumingt thatover theman quasiparti-y disorder configurations and samples clesshows thatthat tunnelsthe peak at theat 4/3 constrictionis usually arepresen indeedt. b,e/A4-chargedveraging o (aver pointall measuremen that is notts yields a clear peak at β = yet1.31 clear,, 3.8σ asab discussedove the uncorrelated above, commentingbackground thefor resultsthat ofscale Dolev(Pet= al.7 ×[147–149]),10−5), suggesting a local ± ± charge ratio of 4/3. c, d, Running our model with parameters = 0 .3, V = 0.3 0.2, and ∆5/2 = 0.01, 0.1, and 1.0 (all in units of U, the on-site charging energy). We simulated charging of four puddles, of which two 40 were capacitively coupled to the SET.

6 sample preparation j

0 40 80 120 160 200 240 280 320 360 near filling factor 5/2 e/4 period = 10mV 0.05 e/4 e/4 ) Ω 100

x 0.00 (

L R ∆ -0.05 e/2 e/2e/2e/2 e/2 e/2 e/2 e/2 e/2

0.3

ude e/2 t

Figure 2.12: Edgeli states interferometry (in the Fabry-P´erotgeometry) at ν =

p 0.2 m

5/2: alternatinga pattern of e/4 and e/2 period oscillations as the area containing the localized quasiholes is varied. (From [161].) FFT 0.1 e/4

this behaviour can0.0 be consistent with the theoretical predictions: the e/4 oscillations pattern corresponds14 to an even number of encircled quasiparticles; for an on

i 12 t i ) odd number theses e/4 oscillations are suppressed, as expected for the “Pfaﬃan” V 10 po m

( state, and onlyk e/28 oscillations survive. The latter are not yet fully understood, e c but they couldpea for6 example be caused by e/4 quasiparticles encircling twice en t t r e en

ff 4 i

the area betweenc the constrictions, or by the tunneling of e/2 charged “double d a quasiparticles”j at2 the constrictions. These effects were not considered in the ad 0 first theoretical proposals,0 40 but 80 were 120 the 160 object 200 of 240 study 280 in 320 subsequent 360 works, where also other issues (such aschange the interactionin side gate voltage of Vs the (mV) localized quasiparticles with the edges) and the signatures for other (Abelian and non-Abelian) states in the same experimentalsample prepa setupration werek investigated [163–172]: indeed some of the proposed states exhibit0 the 40 same 80 qualitative 120 160 behavior 200 240 (in particular 280 320 an odd-even effect for the non-Abelian states) and only quantitative dependences can help e/4 period = 10mV near filling factor 5/2 to discriminate between0.05 them. e/2 e/2 e/2 )

Summarizing,Ω the results of Willett et al. [161,162] seem to show the presence e/4 e/4 e/4 of an odd-even100 eﬀect in the (Fabry-P´erot)interferometry at ν = 5/2, possibly

x 0.00 ( a sign of non-AbelianL braiding statistics, although some important issues and R features still need∆ an explanation and the exact nature of the underlying state -0.05 can not yet be identiﬁed, needing further experimental investigations.

Neutral edge modes0.2 ude t e/4 li

An important featurep of the proposed non-Abelian states at ν = 5/2 is the m

a presence of neutral 0.1 chiral edge modes at the boundaries of the sample [51,86,93, 94,126,173], carrying energy but no charge. These neutral edge modes are absent

FFT e/2 in the most plausible Abelian states for ν = 5/2, thus their detection could be an indication of non-Abelian0.0 braiding statistics. Moreover, detecting number, chirality and other 14 properties of these modes could also help to determine which on i ) t 12 i V of the possible statess is realized (or at least could eliminate some of them).

m 10 ( po

Recently an experiment [174, 175] investigated the presence (or absence) s k

8 e of such upstreamc neutral modes, that is, neutral modes propagating in the

pea 6 en t t r opposite directione 4 of the electric current. The detection was made by looking en ff i c d a 2 at the eﬀects ofj the neutral mode on the shot noise at a quantum point contact

ad 0 (both with and without0 a40 tunneling 80 120 charge 160 mode): 200 240 the (additional) 280 320 impinging neutral quasiparticles fragmentschange into in s chargedide gate v quasiparticlesoltage Vs (mV) and increases the

Supplementary I41nformation Figure S4.

26 current noisewas charged. at this point,Excess asnoise shownwas inobserv Fig. 2.13.ed with Aan theoreticalapproximate discussionquadratic of theincrease with In. This proves, right from the experimentstart, isthe givenprese inn [176],ce of an whereupstream the outcomesneutral mo arede. ratherWhile the interpretedincrease of asthe thenoise was the smallest among the fractional result ofstate heatb transportedeing tested, it byw theas in neutralrelative modesterms tothe thehighest pointsince contact,the actual causingcurren a t that was carried by the fractional state temperaturewas the differencesmallest between(as 4/5 of thethe twocurren edgest is carried and thusby the influencingfirst two, lo thewer currentlying, integer Landau levels). Similarly, the excess noise due to current arriving from source #1, when charged, was strongly affected by In; with an apparent increase noise across the point contact. of the quasiparticles temperature while the quasiparticle charge dropped (Fig. 7c and 7d). Again, like in the νb =2/3 The measurements were performed at different filling fractions: at ν = 1, ∗ state, this temperature increase cannot account for the charge drop [28]. The charge dropped with In from e =0.75e to 1/3 and 2∗/5 no counter-propagating neutral mode was observed, as expected by e =0.32e at In=10nA. A similar evolution of the charge, but as function of temperature, had been reported recently theory; on[28]. theThe contrarynon-linear at νtransmission= 2/3, 3/5also andc 5hanged,/2 suchalb aeit modeby was very detected.small amount (Fig. 7b).

100 ô = 0nA 5 99b ò ) 25 n = ôu = 0 z 2 ôò = 4nA H

/ 98 2 ) A 20 M=0.77 %

30 x 97 - (

on 96 i ì 15 ss 33 i . 5mK

5 95 m (

e 10 s n

i 94 r t no

5 93 ss

total 10.4mV e 20.8mV 92 xc

e 0 91

90 0 1 2 3 4 5 6 7 8 9 10 -4 -3 -2 -1 0 1 2 3 4 neutral current, ôò (nA) source current,ôu (nA)

12 0.9 ôò = 0nA

) c d

10 ) 0.8 z ôò = 4nA

H / Figure 2.13: Signature of an upstream neutral mode at ν = 5/2: excess f noise at 2 o

A s

8 t 0.7 i a point contact as30 a function of the impinging neutral current In. (From [174].) - un

10 n ì i

6 ( 0.6

33 . 5 Of particular interest for us are the results for ν = 5/2, being ange indication ( r

e 4 0.5

h 1 s.d. s i of non-Abelian braiding statistics [126]: for example the “Antipfaﬃan”c state al- 10.4mV e no

v i ways has upstream neutral2 modes, the “Pfaﬃan” only in the case oft 0 a.4 wide edge ss c e 2mK e 20.7mV ff

(but not for a sharpxc one), while the (Abelian) “K = 8” and “331” states does e

e 0 0.3 not have them. However from this experiment it is not yet possible to determine 1/4 -2 which state is actually-4 realized:-3 -2 for this-1 to0 be achieved,1 2 an3 understanding4 0 of1 the2 3 4 5 6 7 8 9 10 edge proﬁle and more quantitativesou measurementsrce current,ôu (nA) are needed. neutral current, ôò (nA)

Figure 7: Testing for the existence of the neutral mode at νb =5/2. The 2DEG used for these measurements was embedded ..and other transportin 30nm experimentswide quantum well, which was doped on both sides, buried pproximately 160nm below the surface of the 11 −2 6 2 heterostructure. The carrier density was 3.10×10 cm and the low temperature dark mobility was >30×10 cm /Vs. (a) Many other theoreticalExcess noise studiesas function investigatedof injecting theIn signaturesprovides direct ofevidence the “Pfaffian”of an upstream andneutral mode. (b) The dependence of the non-linear conductance as a function of Is on the presence of In. The relative change in the transmission is very small, ν ν of the other proposedmoun statesting to m inuch (notless than yet realized)1%. Since 80% experimentsof the total curren involvingt flows in transportthe two underlying edge channels ( b =1 and b =2), the effective transmission is about 77%. (c) The dependence of the excess noise as a function of Is on the presence of In. ∗ ∗ measurements at (d)ν =The 5/dep2,endence in differentof the quasiparticle setups andcharg geometries,e on In [extracted all aimingfrom (c)]. atThe thecharge drops from e ∼0.75e to e ∼0.32e. identification of the underlying state. These proposals involve: interferometry in a Mach-Zehnder [177, 178] (see also [36]) and in an Aharonov-Bohm geometry [179]; signatures in the shot noise at a single quantum pointDiscussion contact [180] or in a Mach-Zehnder interferometer [181]; voltage and noise measurements in a “wormhole” geometry [182]; a dissipative transport caused by tunneling of Ma- jorana fermionsPresenting betweenthe first localizedevidence quasiparticlesof the existence [183];of Coulombneutral mo blockadedes in the exper-fractional states νb =2/3, νb =3/5 and νb =5/2, iments [169,184–187]using a ubiquitous and interferenceQPC constr noiseiction measurementsserving as a detector, [188] in athe Fabry-Pérotfollowing findings can be summarized: (i) A flux of geometry;neutral transportquasiparticles, measurementsemitted infrom a geometrya biased ohmic with threecontact edgesdoes connectednot carry current or shot noise. Moreover, a neutral mode impinging on a macroscopic ohmic contact, does not increase its temperature by a measurable amount. (ii) An by two quantum point contacts [189]; the coupling between a quantum dot and impinging flux of neutral quasiparticles on a QPC constriction having a finite transmission t, results in excess shot the edgenoise. of a quantumThe excess Hallnoise bulk,is ap bothproximately at ν = 5prop/2 [190–192];ortional to transportt(1 − t) and acrossto the voltage of the injecting contact. The a point contactupstream separatingenergy flux, twoin linethe junctionsodd denominator [193]; tunnelingfractions, resonancesseems to be acrosscorrelated with the ratio between the number of a singleupstream long-contactand regiondownstream [194];mo momentum-resolveddes. (iii) Having a neutral tunnelingmode intoimpinging the edgeon a QPC constriction, while a charge mode is simultaneously being partitioned, alters dramatically the noise and the deduced partitioned quasiparticle charge. The charge drops inversely in proportion to the injecting voltage. (iv) In the same experiment, the temperature of the simultaneously partitioned quasipar42 ticles increases with increasing the injecting voltage. However, the temperature increase is too small to account for the observed drop in charge. The mechanism responsible for modifying the tunneling cross-section of the quasiparticles in the QPC constriction is not currently understood. (v) Assuming a temperature −2 ∼ dependent energy decay of T , the typical length scale is 100µm at 25mK for νb =2/3. (vi) Observing an upstream

7 of a ν = 5/2 quantum Hall liquid (from outside of it) [195, 196]; detection of the neutral edge modes through thermal eﬀects in the transport across a point contact [197] or signatures in the scattering at a lower-density constriction [198].

Bulk physics To investigate the nature of the fractional quantum Hall eﬀect at ν = 5/2, besides these many proposals related to edge and transport physics, also a few experimental tests involving bulk properties has been proposed. The key point for these experiments is the exponential ground state degeneracy related to non- Abelian braiding statistics: in a system with Nq localized and well separated non-Abelian quasiparticles the degeneracy D grows as dNq (up to a prefactor), where d is the quantum dimension of the quasiparticles and depends on the particular non-Abelian state. 8 For example, for the quasiholes in the “Pfaﬃan” state the quantum dimension is d = √2, as seen in section 2.6. This degeneracy leads to a ground state entropy

Sd = kB log D = kBNq log d + O(1), (2.17) where kB is the Boltzmann constant. That is, in a non-Abelian state, each quasiparticle carries an entropy kB log d; on the other hand for Abelian quasiparticles there is no degeneracy and thus no such contribution to the entropy. If it were possible to measure it, it would clearly signal the non-Abelian nature of the experimentally realized ν = 5/2 state. Some methods for this investigation has been proposed, basing on the measurements of thermopower [199], electro- chemical potential or orbital magnetization [200] and adiabatic cooling [201]: for the first some (still inconclusive) results has been published, while the others still wait for an experimental realization. In a thermopower experiment, the sample is submitted to a temperature gradient T and the voltage gradient E = V generated as a response by the system∇ is then measured. The thermopower−∇ is defined as the ratio between them: Q = V/ T . Yang and Halperin [199] showed that the thermopower Q directly measures−∇ ∇ the entropy per electron and could thus allow to access the non-Abelian entropy (2.17) (if it is present), in the (low) temperature interval where it dominates over the other entropy contributions. The number of quasiparticles localized in the sample can be for example varied by changing the external magnetic field around the value that corresponds to the ν = 5/2 fractional quantum Hall state (and keeping constant the electron density in the sample). The results of a first experimental attempt has been published [202], but the measured entropy is larger than the predicted value, as shown in Fig. 2.14. Furthermore, the slope of the measured entropy (red dots in the figure) as function of the external magnetic field seems to be larger than the expectation for the “Pfaffian” state (represented with a dashed line). The exact value of this slope could be important, because it depends on the quantum dimension of the

8The quantum dimension d comes directly from the conformal ﬁeld theory describing the system. For a quasiparticle of the type a, the quantum dimension is the biggest eingenvalue c P c of the matrix (Na)b that encodes the fusion rules with a: a × b = c(Na)bc. Fusing M times the quasiparticle a with itself, the degeneracy can be expressed as the matrix multiplication P a b1 b2 bM−2 M (Na) (Na) (Na) ... (Na) , that grows asymptotically as d [32]. For a direct bi b1 b2 b3 a connection with the 5/2 state see for example the review [35].

43 7

12 12 clear minimum versus magnetic ﬁeld at ν = 5/2. The 10 10 magnetic ﬁeld location of the thermopower minimum co- 8 8 incides, within experimental uncertainty, with that of

6 6 the longitudinal resistance Rxx, which is also shown in

¡ ¢ ¢ the ﬁgure. Taken together, Figs. 7 and 8 convincingly 4 4 R demonstrate that the incompressibility of the ν = 5/2 xx

( FQHE state is detectable in the thermopower of the 2D Ω

2 2 ) electron system. -S (µV/K) The incompressible ground state of a 2DES at ν = 5/2 is currently believed to be well approximated by the Moore-Read, or Pfaffian, wavefunction6. This state, which may be viewed as a BCS condensate of p-wave paired composite fermions, has come under intense 5 10 15 20 scrutiny recently owing to the expected non-abelian ex- -1 -1 Temperature (K ) change statistics of its quasiparticle excitations. Unlike conventional abelian FQHE states (e.g. at ν = 1/3), multiple pairwise interchanges (braidings) of localized non- non-AbelianF quasiparticles,IG. 7: (color o thusnline) indicatingArrhenius whichplots o off t theherm statesopowe isr realized.S The (red circles, left axis) and longitudinal resistance Rxx (blue abelian quasiparticles at ν = 5/2 generate a large Hilbert sources of thesqu excessares, rig thermopowerht axis) at ν = could5/2. beDas desorderhed line fit effectss give e and,nergy particularly,space of degenerate ground states. This Hilbert space, the relativelyga highps of temperature∆ ≈ 370 mK atand which450 m theK f measurementor the thermop hasower beenand performed.which is topologically protected from local disturbances Thus betterr experimentalesistance, respe datactively are. still needed to support or confute thethat exis-might otherwise lead to decoherence, has been sug- tence of the non-Abelian entropy. gested as an ideal venue for the storage and processing of quantum information32. 5 40 The anomalous ground state degeneracy arising from non-abelian quasiparticle statistics is anticipated to have observable consequences in certain thermal transport and 4 30 thermodynamic measurements7,33,34. Most relevant here is the prediction7 of Yang and Halperin (YH) that the R

3 xx excess entropy arising from the ground state degener- (

20 Ω acy of a collection of localized non-abelian quasiparticles ) is expected, under certain conditions, to dominate the -S (µV/K) 2 thermopower of the 2DES near filling factor ν = 5/2. 10 YH find that under ideal circumstances the low (but not 1 too low7) temperature thermopower of a 2DES very near ν = 5/2 is temperature independent and proportional to 0 0 |1 − B/B0|, where B0 is the magnetic field correspond- 4.70 4.75 4.80 4.85 4.90 ing to ν = 5/2. (A deviation of the magnetic field from B (T) B0 is necessary, in the ideal case, to produce quasiparticles in the ground state of the system. In real samples FIG. 8: (color online) Thermopower vs. magnetic field (red quasiparticles are doubtless present even at B = B0 and Figure 2.14:c Throughircles) alon thermopowerg with Rxx vs. measuredmagnetic fi entropyeld (blue (redcurv dots),e) abou ast functionT = 0 of owing to density inhomogeneities and other forms ≈ the externalν magnetic= 5/2 at field,T 82 inm comparisonK. The dash withed lin thee re predictionpresents the forthe ther- “Pfaffian”of disorder.) mopower of Eq. (3) for B0 = 4.80 T. non-Abelian entropy (dashed line). (The blue curve represents the longitudinalThe dashed lines in Fig. 8 represent the quantitative prediction7 of YH for the thermopower of the 2DES near resistance Rxx; from [202].) ν = 5/2. Clearly, this prediction underestimates the ex- Cooper andin m Sternesa 2 [200]) at ν proposed= 5/2. alternativeIn spite of waysthe so tome probewhat thelim- lowp temper-erimentally observed thermopower. While it is not yet ature entropy:ited by da simplyta set, usingit is cle twoar f Maxwellrom the relations,figure that theyboth showedthe thatpossib thele to unambiguously identify the sources of the ex- thermopower and the longitudinal resistance are consis- cess thermopower we observe, certain possibilities come non-Abelian entropy (2.17) can lead to measurable contributions in the tem- tent with simple thermal activation (i.e. both scale as to mind. First, as YH stress, their calculation is for an perature dependence∼ e−∆/2T of) f magnetizationor the roughly o densityne orde andr of electrochemicalmagnitude varia- potentialideal, (atdisorder-free 2DES. Our sample, while extraordi- fixed magnetiction fieldof e andach delectronata set. density).From the Atslop lowes o enoughf the da temperaturesshed lines n thisarily de-pure, is certainly not disorder-free. Indeed, the pendence wouldin th bee fi linear,gure, w withe find a slope∆ ≈ 370 dependingmK and on450 them quantumK for the dimensionvery exdistence of Hall plateaus demonstrates this. Sec- and with a signther dependingmopower a whethernd resist thereivity d areata localized, respectiv quasielectronsely. These (νond > 5, /t2)he YH prediction applies only within a somewhat or quasiholesva (luνe

~~44 cooling. On the other hand for an Abelian state Sd is absent: in order to keep Sn constant, the increase in the number of quasiparticles is counterbalanced by an increase of temperature.~~

...in conclusion In conclusion, these first experimental results agree with the elementary quasiparticle excitations being e/4-charged and are not inconsistent with a fractional quantum Hall state at± filling factor ν = 5/2 having non-Abelian braiding statistics. In particular, the “Antipfaffian” state seems to be a better candidate than the “Pfaffian”, but more results are needed for a clear answer.

~~2.13 The “Pfaﬃan” (or others) at ν = 5/2? Nu- merics.~~

As we have seen, the experimental tests have not yet given a clear answer concerning the exact nature of the quantum Hall state at ν = 5/2. Luckily, it turned out that numerical investigations in systems containing only few particles can give some useful information about this issue. However, also the results from this approach are not fully conclusive, mostly because of finite size or geometry effects, the definitive “smoking gun” being only an experimental evidence. We review here some of the numerical works, with electron systems in different geometries and with different interactions.

ν = 5/2 with the “Pfaffian”? As discussed in section 2.5, the first numerical investigations, with exact diagonalizations in the spherical [60] and torus [69] geometry, gave the first hints that the quantum Hall system at ν = 5/2 could indeed realize the “Pfaffian” state. Further numerical studies, mostly of them based on exact diagonalization, supported these results, at least in some region of some parameter space around or in the vicinity of the experimentally realized quantum Hall state. In a series of works Wójs(partially in collaboration with Quinn) [203–206] investigated the exact ground state at ν = 5/2 in the sphere geometry for Coulomb interaction and for modifications of it (by slightly changing the lowest order Haldane pseudopotentials), consistently finding for all examined system sizes (Nel 16) a gap and a sizeable overlap with the “Pfaffian” state around the Coulomb≤ point. Furthermore, around this point the three-body correlations for the ground state show the caracteristic behaviour of the “Pfaffian” state (cf. section 2.4): the avoidance of the three-body state with the smallest relative angular momentum L = 3~. In the planar geometry Xin Wan et al. [207,208] investigated small systems (with up to 12 electrons) in the ν = 5/2 state confined in a disk by a tunable confinement potential, thus allowing to study both the bulk and the edge of the system. They considered an electron interaction interpolating between the three-body interaction that has the “Pfaffian” as ground state and the Coulomb interaction, identifying the phase in which the system is, by looking at the total angular momentum of the ground state (obtained by exact diagonalization). They found that the “Pfaffian” phase extends until the Coulomb limit, however

45 only in a small range of strength of the confinement potential, with a strong dependence on it: for both a weaker and a stronger confinement, the system is in a stripe phase; for an even weaker confinement eventually in the “Antipfaffian” phase [see Fig. 2.15a)]. 5 8 a) b) 1.5

~~(iv) Mgs = 146 1 +e/2 quasihole (stripe? anti-Pfaffian?) stripe (Mgs = 138) (Mgs = 126) 1 ) (iii) Mgs = 136 (stripe) B B / ε l 2~~

~~d / l 0.1 +e/4 quasihole (ii) M = 126 (Pfaffian) gs W (e (M = 132) 0.5 gs~~

~~(i) Mgs = 121 (stripe)~~

~~Pfaffian (Mgs = 126) 0 0.01 0.0001 0.001 0.01 0.1 1 0.001 0.01 0.1 λ λ~~

FIG. 2: (color online). Total angular momentum of theFIG. global 4: (color ground online). stateGround as a functi stateon angular of the mixing momentum parameter as aλ function and of trapping potential and three-body interaction strength. background charge distanced for 12 electrons in 22 orbitalsWe have with 12 the electrons mixed Hamiltonian in 22 orbitals, [Eq. with (1)]. theThemixed ground Hamiltonian state in region [Eq. (1)] and the Gaussian tip potential [Eq. (14)].λ andW M (ii) has the same gs = 126 asFigure the Moore-R 2.15:ead (or Phase Pfaffian)characterize diagrams wave thefunction. three- obtainedbody The interaction ground bystate and looking in regionstip potential (i) at an thed strength (iii) ground are respectively. state The angular background charge is fixed atd=0.7l B believed to be stripe phases. In region (iv), the groundabove state the is a electron candidate lay forer the (the so-calle groundd anti-Pfaffian state hasM stategs = (see 126, Sec. same VII as the Moore-Read state in the absence of the Gaussian for detail). momentum Mgs for systems in the disk geometry. The parameter λ describes potential). For large enoughλ, as the tip potential strengthW increases, states withM gs = 132 orM gs = 138, believed the interaction interpolation via V = (1 λ)V + λV (V is the Coulomb, V to contain a +e/4 quasihole or a +−e/2 quasihole,C become3b thCe global ground state. For small3b λ, another ground state with the “Pfaffian” interaction).Mgs = 126 (which a) is a System stripe state without with occupation quasiholes;pattern|00000111111111111000 the parameter d00is�), the separates these two quasihole states. momentum isM gs. In our approach the ground state angular momentum is a result that comes out of the calculation, rather than a parameter fixeddistance a priori based (from on the the property two-dimensionalof the state that one electron is interested system) in. Therefore of, wethe can background charge quantitatively analyze the stabilitythat of generates the ground state. the systemA. confinement Edge spectrum potential.of a Hamiltonian b) System with mixed with electron-e a localizedlectron interaction Figure 2 is a phase diagram that shows the total angular momentum of the global ground stateM gs for 12 electrons in 22 orbitals with the Hamiltonian in Eq. (1). We vary the mixing parameterλ and the background ch arge distance quasihole at theIn center this subsection, of the we disk; demonstrate the parameter a clear separationW ofdescribed the fermionic the and strength bosonic modes for the Moore-Read state, d. The Moore-Read state for 12 electrons hasM MR =N(2N−3)/2 = 126. In the smallλ limit, the ground state of the pinningand potential, try to obtain the their backroundvelocities for chargeλ=0.5. We distance will thenis try fixed to e xtend at d the= results 0.7` to. the pure Coulomb case in the aroundd=0.6-0.7 persist to haveM gs = 126. To be precise, the ground state is stable for 0.51≤d≤0.76 for the pure 0 Coulomb caseλ=0. 26 On the(Results other hand, for the 12rangenext electrons extends subsection. asλ in increases, 22We orbitals;begin since by three-b recalling fromody the [208].) interaction procedure favors to extract the edge mode dispersion in the simpler Laughlin case Moore-Read state. The two regions withM gs = 121at andν=1 136/3, surround whereing there the is Moore-Read only o ne gro bosonicund state branch are believed of edge t mode.o Then, we apply a similar analysis to the Pfaffian be stripe phases. They can be represented by two stringscase, where of 0 and we 1’s have|M gs a fermionic= 121�=|10000011111111111000 branch of edge mode00 in� additionand to a bosonic one. Of course, unlike the Laughlin |Mgs = 136�=|11000000011111111110They00�, were respectively. alsocase, Thehere able 0 we an to needd 1’s localize areto re thely occupatioon a severalsinglen reasonablenumberse/4 charged ofassumptions, single-ele quasiholectron which can atbe the justified center post priori. angular momentum eigenstates (smaller angular momentumIn an earlier orbitals work,to the29 we left). studied Alternativ the energyely, one spectrum can understand of the electron system atν=1/3, trying to identify the single such a string as the Slater determinantof the disk, of the corresponding usingbosonic a branch smooth single-ele predictectron pinningd angular by the potential.chiral momentu Luttingem eigenstates Alsor liquid in labeledtheory. this1 The case basic they idea foundis that the low-lying excitations of the by 1. At this system size, numericalthat, groundin an st intermediateatesquantum have an overlap Hall range system of about aoftν 30-40%=1 the/3 system can with be the describedcorresponding confinement by a Slater-branch potential of single-boson and ofedge the states with angular momentum determinant states in their rangequasihole of stability. pinningFor veryl(l = smallstrength, 1, 2,d(d 3,≈0 ...).1l the andB),lowest eneM gsrgycan� lyingb jump(l). Therefore, to state 110 for hasλ< we0. can01,the which la angularbel each low-energy momentum stateof by a set of (bosonic) occupation is believed to be a finite-size artifact. On the othernumbers hand, there{n(l) is}, a whose region total with angular ground stat m omentumeM gs = 146, is which the authors already showed in Fig.a 1be/ of4 Ref. charged 26. Toward Moore-Read the pure Coulom quasihole,b case, this ground even stat ine theis stable Coulomb over a range limit. For a weaker ofd twice as large as that for the Moore-Read state. We speculate this is related to the so-calledM=anti-PfaffianM 0 + ΔM= stateM 0 + n(l)l, (15) 17,18 pinning, the system is in the “Pfaffian” state (without quasiholes),� and for a discussed very recently. We will discuss this state in greater detail in Sec. VII. l In the absence of three-bodystronger interactions, pinning, the overlap after between a the wide ground intermediate state wave function stripeand the phase, Moore-Read one finds a localized 2 wave function, |�Ψgs(Mgs = 126)|ΨMR�| , is aboutand 0.5 for energy the Coulomb interaction (it jumps up to 0.7 when we tune 26 “double quasihole”, with charge e/2 [see Fig. 2.15b)]. theV 1 pseudopotential ). While this is quite substantial considering the already quite large size of Hilbert subspace, it is significantly smaller comparedIn with the the sameLaughlin work, state at Xinν=1/3 Wan at comparableet al. also system studiedsize.E= CombinedE the0 + Δ edgeE with=E thstates0 e+ � n of(l)� theb(l), system, (16) l narrow window ofd within whichfinding,M gs = 126 as in expected, the pure Coulo amb (neutral) interaction case,fermionic these suggest branch that the of Moo excitationsre-Read in addition to state may be quite fragile when system parametersrespectively, are varied. This where is consistentM andE withare earlie totalr numerical angular work mom onentum the and energy of the corresponding ground state. In 7 0 0 torus and the experimentalthe observation (charged) that the bosonicEq. FQH (16) state we mode. atν assumed=5/2 Remarkably, disappearsthe interactions in a tilted the betwee velocitiesmagneticn the excitations field of for the are fermionic negligible, which and turns out to be an excellent modest tilting angle, even thoughbosonic the state edge is believed modesapproximation. to arebe spin found pol Beingarized. to edge be We excitations, verynote the different, ph suchase boundaries states which can in be thecould independen betly a sourceverified by of calculating the squared matrix smallλ limit persist in the sma ll negative-λ regime (−0.02<λ< 0). This suggests the M oore-Read–like† ground2 state with pure Coulomb interaction is stable against a smallelements attractiveT[{n(l) th}]=ree-body |�ψ {n interaction,(l)}(N+1)| wc hich3N+Δ mayM |ψ arise,0(N)�| e.g.,numerically due in the microscopic model, and comparing them decoherence for the interference experiments described in29,37 section 2.12, which to Landau level mixing. with the predictions of the chiral Luttinger liquid theory. Note thatM 0(N+1)−M 0(N)=3N is the difference in involve e/4 chargedtotal angular quasiholes momenta propagatingbetween theN- and at the(N +edge 1)-electron of the ground sample. st ates. LookingAs shown in Ref. 29, even in the presence at the dependenceof background of the edge confining statespotential, on the the ansatz presence of Eqs. (respectively (15) and (16) can thebe used absence) to unambiguously identify the bosonic mode energies� b(l), given that edge excitations are not significantly mixed with bulk excitations. The calculation of of the e/4 chargedT[{n(l) quasihole}], while not atnecessary, the center d oes ensure of the us the system, correct identification they found of thes thate excitations the as edge states. quasihole changesEncouraged the spectrum by the succ of theess of fermionic identifying the mode, edge mode signaling dispersion its and non-Abelianeven predicting the energies of edge excita- nature. These resultstions in the give Laughlin a strong case, we support apply the to same the analysis realization to the Moore-Re of thead “Pfaffian” state. The complication is that, in addition state, however they cannot be considered as conclusive, because they all strongly depend on the strength of the confinement potential, signaling that probably the

46 investigated systems were still to small to accomodate and adeguately describe both the bulk and the edge of the quantum Hall system. At the same time, Peterson et al. [209, 210] investigated the dependence of some quantum Hall state on the ﬁnite width of the system (in the direction perpendicular to the (quasi) two-dimensional electron layer). This ﬁnite width leads to a softening of the Coulomb interaction, particularly modifying its lowest order Haldane pseudopotentials (describing the short range strength of the interaction). 15

~~w [l] w [l] w [l] 0 0.25 0.5 0.75 1 1.25 1.5 1.75 0 0.25 0.5 0.75 1 1.25 1.5 1.75 0 0.25 0.5 0.75 1 1.25 1.5 1.75 1 1 0.24 0.98 0.99 SQ 0.21 SQ 0.96 FH 0.98 FH ZDS ZDS 0.18 0.94 0.97~~

~~0.92 0.96 0.15 ( d ´) 〉 ( d ´) 〉 ( d ´) 〉~~

~~1/2 0.9 5/2 0.95 9/2~~

~~|Ψ |Ψ SQ |Ψ 0.12 Pf Pf FH Pf 〈Ψ 0.88 〈Ψ 0.94 ZDS 〈Ψ 0.09 0.86 0.93 0.06 0.84 0.92~~

~~0.03 0.82 0.91~~

0.8 0.9 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 d´ [l] d´ [l] d´ [l] (a) LLL (b) SLL (c) TLL Figure 2.16: Overlaps of the “Pfaffian” state with the exact ground state (for FIG. 9: (Color online) Samea system as Figs. of N 7= and 8 electrons 8 except in for the fractional sphere geometry), filling 1/2. as function Hence, of the therelevant system overlaps are between the thickness d0. (The three curves� correspond to three different models for the exact ground state wavefunction at some thicknessd (|Ψν (d)�) and the Pfaffian wavefunction (|ΨP f �). The particular system hasN = 8 electrons atl=6finite.5. thickness; from [210].) For ν = 5/2 in the sphere geometry (with up to 12 electrons), they found that a finite system thickness enhances thecreased overlap between to the theoptimal “Pfaffian”value state should give more a stable 1 and the ground state obtained by exact diagonalization, reaching a maximum 5/2 FQHE! This is further elaborated (and reinforced) (of almost unity) at about 5 magnetic lengths (see Fig. 2.16). They interpreted by studying the ground state topological degeneracy on 0.99 this result as a stabilization of the “Pfaffian” state, thanks to the finite width. They also investigated the effects of an in-planea torus, magnetic a defining field, which hallm effectivelyark of non-Abelian states in the reduce the width of the system, thus reducingnext the section. overlaps and eventually de- 0.98 stroying the quantum Hall effect, as experimentally observed (and discussed in ( d ´) 〉 section 2.2). 5/2

|Ψ In our work [211] we will confirm (withB. bigger Threefold systems) this topo enhancementlogical degeneracy signature of Pf SQ 0.97 〈Ψ Choi,of the et al. overlap, discussing it in a more general framework;the we Pfaffian will also statesee that on the torus Pan,at ν et= al. 5/2 the increase of the layer thickness leads to a reduction of the gap in Dean, et al. the quantum Hall system, in general not helpingRecall to its (cf. stabilization. Sec. III A) that in the spherical geometry, 0.96 Peterson et al. [209, 210] also investigated the torus geometry (with up to the signature of an incompressible FQH state is the ex- 16 electrons), looking for the six-fold degeneracy characteristic of the “Pfaffian” istence of a rotationally symmetric uniform state with 0.95 0 2 4 6 8 10 total angular momentumL = 0 with a finite excit ation d´ [l] 47 gap to higher energy states. A given FQH state generally has a “shift” in the equation relating the number of elec- FIG. 10: Same as Fig. 9 (middle panel, SLL) except only tronsN to the total flux 2Q through the finite sph ere. � the SQ potential is shown. The experimentald of Refs. 15 The MR Pfaffian wavefunction is written on the sphere and 16 are shown as a asterisk and solid square, respectively. as N l . The particular system has = 8 electrons at =6 5. Also 1 show is the result for a very recent experiment (Ref. 85) by Ψ = Pf (u v −u v )2, (12) P f �u v −u v � i j j i Dean, et al. i j j i i

−iφj /2 where the spinor coordinates areu j = cos(θj /2)e andv = sin(θ /2)eiφj /2 with (θ, φ) being the coordinates tively. Using the standard formula for magnetic length j j on the surface of a sphere. The Pfaffian symbol above in GaAs/AlGaAs quantum wells (l≈ 25nm/ B[T]) we � corresponds to plot in Fig. 10 the value ofd these particular� experiments correspond to on our overlap plot at 1/2 in the Pf (A ) = � A ...A − , (13) SLL (ν=5/2) for the SQ potential . From this compar- ij σ σ(1)σ(2) σ(N 1)σ(N) �σ ison we see that the experimental systems are not optimized to observe the strongest possible Pfaffian state at whereσ are permutations of t he N particle indices. It ν=5/2. Somewhat increasin g the value of the quan- is found that the wavefunction in Eq. 12 requires a flux tum well width so thatd �/l (or equivalently d/l) is in- 2Q=2N− 3. While this corresp onds to filling factor state (see section 2.9). Because of the degeneracy with the “Antipfaffian” in the torus geometry one should actually observe two such sets of six-fold degenerated states, corresponding to a symmetric and an antisymmetric “Pfaffian”- “Antipfaffian” combination. They claim to see these two sets of (almost) degenerated states, and that the finite system width enhances their visibility; however it is not always clear how to discern them (from the rest of the spectrum) and the results strongly depend on the aspect ratio of the system.9 In a subsequent work, Papić et al. [212] investigated the behaviour of the system at ν = 1/2. For zero width the system is in a (compressible) Fermi- liquid-like phase [213], but increasing the system width a phase transition to a quantum Hall state could occur: indeed Papić et al. showed that the overlaps of the exact ground states with the “Pfaffian” state sharply increase at a critical value of the system width. However, the corresponding quantum Hall gaps are very small and in our general framework such a phase transition seams unlikely to occur. We also note that recent numerical investigations for systems with an odd number of electrons studied (for Coulomb interaction) the energy gap eventually related to the (neutral) Majorana fermion ψ [214, 215]. If the “Pfaffian” model is realized, in these systems an unpaired neutral fermion ψ is indeed present, as mentioned in section 2.6. The size of this “neutral fermion gap” is found to be of the same order of magnitude of the gap obtained from ( e/4) charged excitations [43, 46, 47]. ±

Entanglement entropy (and spectrum) Besides the “standard” exact diagonalization techniques, recently the quantum Hall systems were also investigated within the framework of the so-called “topological entanglement entropy” [216, 217] and “entanglement spectrum” [218], with results consistent with the Moore-Read state. The topological entanglement entropy is obtained by spatially dividing the quantum Hall system in two parts and tracing out the degrees of freedom of one of these; for a topological ordered system, the von Neumann entropy of the so obtained reduced density matrix ρA (measuring the entanglement of the two parts) has then form 1 S = αL γ + O , (2.18) − L where L is the length of the boundary between the two parts and γ is the topological entanglement entropy. The latter is a direct measure of the total 2 quantum dimension of the system, γ = a da (where the sum is over all the sectors) [216,217], thus helping to discriminate in which state the system is: for pP the Moore-Read state γMR = √8. Numerical investigations of the entanglement entropy for both the “Pfaffian” wavefunction [219] and the exact ground state of realistic systems at ν = 5/2 [220, 221] gave indeed results compatible with this value. Li and Haldane [218] pointed out that the spectrum (called “entanglement spectrum”) of the fictitious Hamiltonian H, ρ = exp( H), extracted from A − 9We also did calculations in the torus geometry, looking for these six-fold degenerated states, however finding no consistent picture, the results strongly depending on the system size, the aspect ratio and the electron interaction.

48 the reduced density matrix ρA (described above) of a fractional quantum Hall ground state contains more information than the entanglement entropy: the low- lying “entanglement spectrum” (corresponding to high probability levels) can be used as a “fingerprint” allowing to identify the corresponding conformal field theory. Indeed it provides information about the counting of the edge states (as function of angular momentum) at the boundary between the two half systems; a counting consistent with the underlying conformal field theory [222, 223]. 3

ξ ξ ξ 10 10 10 8 8 8 6 10 6 10 6 10 8 8 8 4 6 4 6 4 6 2 4 2 4 2 4 2 (a)P [0|0] 2 (b)P [0|1] 2 (c)P [1|1] 56 58 60 62 64 56 58 60 62 64 56 58 60 62 64 0 A 0 A 0 A 40 45 50 55 60 65 Lz 40 45 50 55 60 65 Lz 40 45 50 55 60 65 Lz

FIG. 2: The low-lying entanglement spectra of theN e = 16 andN orb = 30 ground state of the Coulomb interaction projected into the second Landau level (there are levels beyond the regions shown here, but they are not of interest to us). The insets Figureshow the low-lying 2.17: par Low-lyingts of the spectra “entanglement of the Moore-Read state, spectra” for comparison (as function[see Figure (1)]. of Note the that angular the structure mo- of the low-lying spectrum is essentially identical to that of the ideal Moore-Read state. mentum component Lz) extracted from the exact ν = 5/2 Coulomb ground state for Nel = 16 electrons on the sphere, for three different ways of partition- ingversa. the However, orbitals. the electron In the density insets anywhe there oncorresponding the system, this results rule expla forins the the counting “Pfaffian” only for state. small sphere must remain constant, which can be achieved if ΔL; for large ΔL, the finite size limits the maximal an- (Fromthe quasihole [218].) excitations inA andB are correlated (en- gular momentum that can be carried by an individual tangled). This gives the empirical rules of counting the quasihole. Therefore the number of levels at large ΔL levels. Take the spectrum in Figure (1(a)) as an example. in a finite system will be smaller than the number ex- The partitioningThe low-lyingP [0|0] results “entanglement in the root co nfiguration spectrum”pected for in the an infinite exact system. Coulomb Not only ground is this empirical state at110011001100110ν = 5/2 on (for the northern small hemisphere systems (re ingion theA), sphererule consistent geometry) with all our has numerical indeed calcula thetion, same but and it corresponds to the single “level” at the highest pos- it also explains whyP [0|0] andP[0|1] have essentially A A A (counting)sible value ofL z structure=L z,max = 64. as We that measure for theL thez by “Pfaffian”identical low-lying wavefunction structures. This (see is because Fig. the 2.17), (semi- A A A infinite) configuration “··· 1100110” is essentially equiva- signalingits deviation from thatL z,max they, i.e. share ΔL:=L z,max the− sameL z , which nature [49, 218, 221, 224]. In the Coulomb has the physical meaning of being the totalz-angular lent to “···11001100” (with an ext ra “0” attached to the casemomentum there carried are by additionalthe quasiholes. generic At ΔL = higher-lying 1, the right). levels, We expect however thatP [0|0] separated andP [0|1] become from exactly the low-lyinglevels correspond “topologic to edge excitations part” upon by a the so-calledΔL=0 “entanglementidentical in the thermo gap”,dynamic that limit. remains finite root configuration. There is exactly one edge mode For completeness, we list the root configurations asso- in this the case, thermodynamic represented by the MR limit. root Theconfiguration same investigationciated with the first (also few low-lying in the levels Coulomb in Figure case) (1(c)). 110011001100101. was made [225] for exact states containing quasiholes,L finding similar results and The number of ΔL = 2 levels can be counted in exactly Δ =0: 110011001 10011001 alsothe same observing way. There ar aesector three of them, change of whic (thath the root is a changeΔL in=1: the edge 110011001 states100110001 counting) as aconfigurations quasihole are is taken through the system cut. 110011001100101010 1100110011001001 ΔL=2: 110011001 1001100001 1100110011001010010 ν = 5/2 without1100110011000110 the “Pfaffian”? 1100110010101010 1100110011001001100 1100110010101010100 Onwhile the for Δ otherL = 3, the hand, five root some configurations numerical are results were also published [226–228], casting some doubts11001100110010001 on the relevance of the “Pfaffian”Figure (2) at showsν = the 5/ sp2.ectra of the system of the same size as in Figure (1), i.e.,N e = 16 andN orb = 30, but for On one11001100110001010 side, T˝oke et al. [226, 228] calculatedthe ground state the of energy the Coulomb spectra interaction forprojected small electron systems11001100101001100 in the sphere geometryinto containing the second Landa(non-localized)u level, obtained two by di orrect four diag- 11001100101010010 onalization. Interestingly, the low-lying levels have the quasiholes for both the second Landau levelsame Coulomb counting structu interactionre as the correspondin and theg three- Moore- 11001010101010100 body interaction having the “Pfaffian” asRead ground case. We state. identify The these obtained low-lying lev spectraels as the The counting for the levels at small ΔL forP[0|1] and “CFT” part of the spectrum, in contrast to the other wereP [1|1] can interpreted be obtained simi aslarly. qualitatively different,generic in particular, non-CFT levels because that are of expected the apparent for generic lackingFor an infinite of the system (degenerate) in the thermodynamic “Pfaffian” limit, the quasiholemany-body band states. inAtr theelatively Coulomb small ΔL (up case, to a and limit above idea gives an empirical counting rule of the number which grows with the size of the system), the CFT lev- thusof levels making at any ΔL, in i.e., principle it is the number impossible of independent anels adiabatic are separated continuity from the generic between levels by a theclear two gap, interactionquasihole excitations limits. upon the semi-infinite root config- which we define as the distance from the average of the urationIn uniquely this context, defined by inthe apartitioning. more recent For a finite work WójsCFT levelset toal. the[131] bottom showed of the generic that levels. the inclusion of the three-body force generated by Landau level mixing to the Coulomb interaction provides energy spectra that are similar to the spectra in the “Pfaf- fian” case, with an increase of the overlaps between the corresponding states,

49 collective mode for Pf QHs in this figure as well as 0.2 6 Pfaffian (c) HPf (d) HBN (e) H1 attributed to the closer proximity of this 0.0 0.0 0246810 01230.00 0.05 0.10 APf, leading to a stronger interference with κ L δV1 ics. This, however, is a finite size effect relevant in the thermodynamic limit where FIG. 2 (color online). Spectra of 16 particles at flux 2Q ¼ the APf ground state would be chosen. 2N À 3 ¼ 29 for (a) VCoulomb, (b) HBN, and (c) HPf (black as The importance of finite quantum well well as colored dots). The energy expectation values of the Pf stressed previously [13]. To assess the wave functions are also shown for VCoulomb and HBN in panels (a) for a system with finite thickness, we and (b) (dashes) except when they are so high that they fall 2 approximate model in which we modify outside the frame. The energies are given in units of e = ( is ing infinite square quantum well confinement) the magnetic length) in panels (a) and (b) and in units of W3 in V W (c). The numbers near the ground state (red [medium gray]) and scale and by the ratio VCoulomb the low energy neutral excitations (blue [dark gray]) indicate the VCoulombðm ¼ 1;w¼ 0Þ (which is 0.91 squared overlaps with the corresponding Pf eigenstates. Panels optimal values of shift downward with (d) and (e) show the evolution of the overlaps of the exact states ness, as seen in Figs. 1(e) and 5(e)–5(g) of HBN and H1 with the Pf states as a function of and V1, the absence of LL mixing, the Pf QH and respectively; results for only alternate L are shown for clarity. not improve as rapidly with increasing the Pf ground state. (shown as dashes) are in qualitative and semiquantitative The assumption of full spin polarization agreement with the actual band. The somewhat worse justified. The 5=2 FQHE has been seen agreement at the smallest L in the exciton branch in high field of B ¼ 10 T [26], where the state Fig. 2 and at the largest L in Fig. 3 suggests that the Pf be fully spin polarized, but also at very model is less accurate for short distance physics, because There is numerical evidence that the the Pf QH and the Pf QP in the exciton branch are at their fully polarized even at low fields [10,22, closest separation at the smallest angular momentum (L ¼ scattering experiments suggest a lack of 4) and the two QHs are nearest at the largest L in Fig. 3. tion at somewhat elevated temperatures These comparisons demonstrate that, with LL mixing, the directly probe the 5=2 FQHE state, and Pf physics emerges for quasiparticles and quasiholes. [30] in terms of a polarized ground state

~~-6.42 -6.65 )~~

~~-6.42 λ -6.28 / -62 .44 ) E (e λ 07 / 2 64~~

~~-6.44 51 0.3 e 2 618 -6.46 0.8 72 7 0. 0. 25 E ( 43 19 70 0.3 853 -6.70 0.1 0.~~

~~0.4 (b) H 281~~

~~0. BN 403 851 0.8 (a) V 878 Coulomb 0. 0. -6.32 0. (a) V κ -6.48 Coulomb -6.46 0. =2 5 5 2 B(T) 0.8 1.0 2.0 1.0 L=2 L=2 6 0.8 L=2 0.8 4 3 2~~

~~2 6 W p 3 0.6 0 0.6 ap E/~~

~~E/W 8 0.4 8 0.4 overla overl 1.0 0.2 0.2 0 4 2QHs (c) HPf 2QPs (d) HBN (e) H1 (c) HPf 0.0 0.0 0.0 0246810 01230.00 0.05 0.10 0246810 0123 κ L δV1 L~~

FIG. 3 (color online). Same as in Fig. 2 but for two Pf quasi- FIG. 4 (color online). Same as in Fig. Figureholes 2.18: Comparisonat 2Q ¼ 2N ofÀ the2 ¼ energy30. spectra as function of the angular mo-quasiparticles at 2Q ¼ 2N À 4 ¼ 28. mentum L for 2 (non-localized) quasiholes in systems of 16 electrons interacting via (a) Coulomb interaction, (b) Coulomb interaction with inclusion of096802-3 Landau level mixing and (c) the three-body interaction that has the “Pfaffian” as ground state. Panels (d) and (e): squared overlaps of the quasihole states with the corresponding “Pfaffian” quasihole states in (c), as function of Landau level mixing strength κ and modification of the first Haldane pseudopotential δv1. (The blue numbers in (a) and (b) are the corresponding squared overlaps; from [131].) as shown in Fig. 2.18 (a), (b) and (c). On the other side, T˝oke et al. [227] also tried to localize quasiholes, using weak δ functions as pinning potentials. In the pure “Pfaffian” case they could localize only e/2 charged “double quasiholes” and not the e/4 charged elementary quasiholes, since the former have a zero in the electron density, thus giving a zero contribution to the localization energy, while the latter only have a local minimum of the density, giving a finite localization energy. In the Coulomb case they were not able to localize quasiholes. Exactly these two points are the main topics of our work: we will show how to localize and manipulate e/4 charged quasiholes (also performing a braiding of quasiholes in the Coulomb case) and we will obtain an adiabatic continuity between the pure Coulomb and the three-body interaction limit, both for the ν = 5/2 ground state and for systems containing localized quasiholes [211,229], signaling that the “Pfaffian” model is indeed relevant for the description of the experimentally realized state at ν = 5/2 We believe that the differences in the energy spectra for the systems without localization of quasiholes (Fig. 2.18) are a finite size effect, with the inclusion of (Landau level mixing) three-body forces helping to get larger overlaps for the quasiholes states in the small systems considered, but without driving a phase transition into the “Pfaffian” phase; indeed our results suggest that the pure Coulomb limit is already in this phase. Figures 2.18 (d) and (e) show that an en-

50 hancement of the overlaps similar to that obtained by Landau level mixing, can be also obtained by merely modifying the first (two-body) Haldane pseudopotential v1 from its Coulomb value, indicating that the three-body interaction is not an essential ingredient for the “Pfaffian” phase (although such an interaction is actually present in real samples and would help to select between the “Pfaffian” and the “Antipfaffian”, as seen in section 2.10).

~~51 52 Chapter 3~~

~~ν = 5/2: our numerical investigations~~

~~3.1 Introduction~~

With our numerical investigations we aim to strengthen the evidence that the experimentally realized fractional quantum Hall state at ν = 5/2 is indeed in the “Pfaffian” phase proposed by Moore and Read [51]. On one side we confirm and reinforce previous work already pointing in this direction [60,69,86,107,203–210] (also “improving” it, by looking at bigger system sizes and by studying the system in a more general framework) and on the other side we directly confute the results of other work [226–228] that put in question the relevance of the “Pfaffian” state for the description of the quantum Hall effect at ν = 5/2 (as discussed in section 2.13). We consider systems with a small number (Nel 20) of electrons on the surface of a sphere [6–8, 230]. We choose this geometry≤ because on the sphere the finite size effects turn out to be less important than in other geometries (disk and torus), also because of the absence of a boundary and of other “geometry aspects” (like the shape and aspect ratio of a torus). The only geometry parameter of the sphere is its radius, which is fully determined by the system size, once the perpendicular magnetic field at the surface is fixed to a constant. We assume that the electrons in the considered half filled Landau level are fully spin polarized. For ν = 5/2 the lower-lying Landau level is full (with electrons of both spins) and we treat it as inert: in our calculations we consider only the (spin polarized) electrons in the half filled second Landau level. This electrons in the second Landau level interact with each other via the corresponding Coulomb interaction (or modifications of it): we implement this by taking the matrix elements of the interaction with the second Landau level wavefunctions [231]. We take (indirectly) into account the finite thickness of the quantum Hall system by varying the first Haldane pseudopotentials that parametrize the two- body electron interaction [8]. On the other hand we do not consider Landau level mixing effects, which should play an important role in selecting between the “Pfaffian” and the “Antipfaffian” (as discussed in the paragraph “Antipfaffian” of section 2.10). Our calculations on the (edgeless) sphere, with particle-hole

53 symmetric two-body interactions, are not able to distinguish between them. (For simplicity in the following pages we always only speak of the “Pfaffian”.) For the ground state of a spin-polarized half filled second Landau level, in the sphere geometry, the number of electrons Nel and the number of magnetic flux quanta Nφ are related through [60]

N = 2N 3, (3.1) φ el − that is, the “shift” has the value S = 3 (the same as for the “Pfaﬃan” state). We insert Nqh e/4-charged quasiholes into the system by adding Nqh/2 supplementary ﬂux quanta:

N (N ,N ) = 2N 3 + N /2. (3.2) φ el qh el − qh We perform exact diagonalizations in the Hilbert space for various system sizes using a (block) Lanczos algorithm [232] and this way ﬁnding the low-energy spectra (and states) of the investigated systems. Starting from an initial (normalized) vector, the Lanczos algorithm iteratively constructs an orthonormal basis in which the original matrix takes a tridiagonal form; the obtained tridiagonal matrix can then be easily diagonalized. In the “block version” of the Lanczos algorithm one starts from various vectors, iterating for each of these and keeping the generated “block-basis” orthogonal to each other; this method allows us to determine degenerate (or almost degenerate) energy states. In most of our calculations, we use as starting vectors states that are already “near” to the exact eigenstates: for example as we slightly modify the interaction, we use as starting point the eigenstates of the preceding calculation, speeding up the convergence. In the following two sections we present the publications reporting our results. In section 3.2 we investigate the ground state at ν = 5/2, while in section 3.3 we insert (and localize) e/4 charged quasiholes into the system.

~~54 3.2 The ground state~~

In the following 4 pages we attach the work published in Phys. Rev. Lett. 104, 076803 (2010). Summarizing, we show that the ground state for Coulomb interaction is adiabatically connected with the Moore-Read state: interpolating between the two limits, the ground state is protected by a large gap, with no sign of phase transition for all examined system sizes. We also modify the (two-body) electron interaction, by varying the Haldane pseudopotentials v1 and v3 (keeping all others at their Coulomb value), drawing a phase diagram in the (v1, v3)-plane. (For example the finite thickness of the quantum Hall sample causes a variation of these pseudopotentials.) We find that in the (v1, v3)-plane the quantum Hall energy gap and the overlap of the exact ground state with the “Pfaffian” state form two hills, whose positions and extents coincide: the energy gap is large there where the overlap is large. We interpret this as a sign that the fractional quantum Hall (ground) state at ν = 5/2 is indeed in the “Pfaffian” phase. During these investigations we also look at the system at ν = 1/2, finding that it is in an compressible phase, but near to a phase transition to the quantum Hall (gapped) phase. The variation of the system thickness could lead to a phase transition into the quantum Hall phase, however from our calculations this seams unlikely to happen.

~~55 The Fractional Quantum Hall State at ν = 5/2 and the Moore-Read Pfaan~~

M. Storni1, R. H. Morf1 and S. Das Sarma2 1Condensed Matter Theory, Paul Scherrer Institute, CH-5232 Villigen, Switzerland and 2Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

Using exact diagonalization we show that the spin-polarized Coulomb ground state at 5 is ν = 2 adiabatically connected with the Moore-Read wave function for systems with up to 18 electrons on the surface of a sphere. The ground state is protected by a large gap for all system sizes studied.

~~Furthermore, varying the Haldane pseudopotentials v1 and v3, keeping all others at their value for the Coulomb interaction, energy gap and overlap between ground- and Moore-Read state form~~

hills whose positions and extent in the (v1, v3)-plane coincide. We conclude that the physics of the Coulomb ground state at 5 is captured by the Moore-Read state. Such an adiabatic connection ν = 2 is not found at 1 , unless the width of the interface wave function or Landau level mixing eects ν = 2 are large enough. Yet, a Moore-Read-phase at 1 appears unlikely in the thermodynamic limit. ν = 2

~~PACS numbers: 73.43.Cd 71.10.Pm~~

One of the most intriguing strongly correlated ishing Zeeman energy. Furthermore, the GS for Nel = 8 electronic states discovered in nature is the even- electrons was found to have substantial overlap with the denominator fractional quantum Hall eect (FQHE) at MR state although that state is the exact ground state of the Landau level lling factor ν = 5/2 = 2 + 1/2 [1], i.e., an unphysical short-range three-body interaction Hamil- at the half-lled second orbital Landau level (LL) of a 2D tonian. Subsequent theoretical [9, 10] and experimental electron system. The 5/2 FQHE cannot be understood [11] studies yielded results consistent with these ideas. within the canonical hierarchical (Laughlin) theory, since These ndings led Das Sarma et al. [12] to propose the odd-denominator rule is a necessity to preserve the the use of the ν = 5/2 FQH state for the realization Pauli principle. A particularly interesting proposal by of non-Abelian topological qubits which, they argued, Moore and Read (MR) [2] extending Laughlin's ideas to would permit fault tolerant and robust quantum compu- quantum Hall states at half lling is the Pfaan wave tation. Their proposal prompted a great surge of activity function (wf), characterized by quasiparticle excitations [3] to further elucidate the nature of the 5/2 FQHE both obeying non-Abelian braiding statistics [3]. theoretically [1316] and experimentally [1719]. How- The rst numerical study of this wf was carried out by ever, whether the FQHE at ν = 5/2 observed in exper- Greiter et al. [4] who considered it as a candidate for iments has the properties of the non-Abelian MR state the observed FQHE at both ν = 1/2 and 5/2. Their cal- remains an open problem, especially since the relevance culations done for systems on the sphere with Nel 10 of the Pfaan state at ν = 5/2 has been questioned by electrons did not allow a determination of the excitation≤ [20]: in their exact diagonalization studies of quasiholes gap and the dierence between ν = 1/2 and 5/2 was (qhs), they only observed qhs with charge e/2, while the not explored in any detail. A rst hint at possible adi- qhs in the MR state are predicted to have charge e/4 [21]. abatic continuity (AC) between the MR state and the In this Letter we provide theoretical evidence, using ground state (GS) of a two-body model interaction was state of the art exact diagonalization, that the MR wf and mentioned briey in a subsequent paper by Wen [5], but the spin-polarized ν = 5/2 FQH state belong to the same limited to a single system size Nel = 10. universality class [22]. Following the pioneering work by Shortly after its discovery, the ν = 5/2 state was Haldane and Rezayi [23] who established that the GS at studied in a tilted magnetic eld [6]. Examining the ν = 1/3 is in the universality class of the 1/3-Laughlin temperature dependence of the longitudinal resistivity state, we adiabatically change the electron interaction by

ρxx exp (∆/2kBT ), the activation gap ∆ was found interpolating between the three-body interaction V3b, for to decrease≈ − with increasing tilt angle and the Hall plateau which the Pfaan wf is the unique GS, and the Coulomb disappeared beyond some critical tilt angle. These re- interaction VC and follow the evolution of GS and energy sults suggested that the quantized state is at most par- spectrum by exact numerical diagonalization. tially spin-polarized until at some critical tilt angle the For all even system sizes examined (Nel 18) we ob- increasing Zeeman energy produces a phase transition to serve AC of the GS and no indication of a decrease≤ of the a gapless polarized state [7]. gap for interactions interpolating between VC and V3b, This scenario was challenged by one of us [8]: exact thus implying AC between the spin-polarized 5/2 state diagonalization results for small systems on a sphere for and the MR state in the thermodynamic limit. spin-unpolarized and fully polarized states at ν = 5/2 A related study is mentioned in a recent paper by have shown that the GS is spin polarized even for van- Möller and Simon [16]. They report that in systems of Figure 1. Nel = 16, Nφ = 29: low-lying energy spectra (the lowest L = 0 state is the reference) and overlaps G1 Gx and h | i G0 Gx as function of the interaction parameter x. Figure (a) ν = 5/2, (b) ν = 1/2, (c) ν = 1/2 for nite width w/`0 = 2.46. h | i In (b) overlap curves are shown as dashed lines for x values for which the state Gx is not the GS (color online). | i

12, 14 and 16 electrons they see no gap closing when in- a half lled LL the particle number Nel and the number terpolating the interaction between V3b and a particular of ux quanta Nφ are related by Nφ = 2Nel S. Here, type of two-body interaction near the Coulomb interac- the shift S is a topological quantum number− [27] and tion, but supposedly in the weak pairing phase [16]. depends on the particular FQH state: for the MR state Contrary to our present work, no details are given and S = 3. We consider particle interactions of the form, the dierence between ν = 1/2 and 5/2 is not discussed. In addition, we systematically vary the two-body in- V = (1 x)V2b + xV3b, (1) teraction, by using the Haldane pseudopotentials deter- − mining the pairwise interaction among the electrons, and with 0 x 1, interpolating between a generic 2-body potential≤ ≤ and the 3-body interaction for which construct a phase diagram which elucidates the dierence V2b V3b the MR wf is an exact GS [5]: between ν = 1/2 and ν = 5/2 and allows a discussion of the inuence of experimental parameters and Landau N level mixing on the nature of the state. In this phase di- A el V = S ∆ δ(i j)∆2 δ(i k) , (2) agram the region that corresponds to the gapped phase 3b N 5 ijk j − k − el i 1/2. By projection of V3b on a LL the singularities of should be a few magnetic lengths wide) the ν = 1/2 the δ functions are regularized. The 2-body interaction state can become adiabatically connected to the MR wf V2b can be written as for small systems. Yet for increasing system size, the gap decreases in a way that it is doubtful that the Moore- NΦ (3) Read state can occur in single-layer systems at ν = 1/2. V2b = vm Pm(ij), m=0 1 i 0, and thus van- largest systems (Nel 16). Remarkably, these two hills ishing overlap with the Pfaan and no AC to the MR are congruent in position≤ for a given system size, while state. The situation changes when the nite width of the their extent and shape show only little system size de- wave function in the direction perpendicular to the 2D pendence. The two hills of gap and overlap thus belong electron system are taken into account. Our results of together and are a manifestation of the MR phase; be-

Fig. 1(c) reveal that a small gap opens down to x = 0 low the hills, for smaller v3, we nd a compressible phase. and the overlaps are comparable in size or even larger We also note that, if we plot the gaps and overlaps than in the second LL. Thus the nite width induces as functions of y1 = v1/v5 and y3 = v3/v5, the result- adiabatic connection between MR wf and Coulomb GS ing plots for ν = 1/2 and ν = 5/2 are quite similar, in the lowest LL. In analogy, we also look at the eect of the dierences being of the same magnitude as those due a nite width in the second LL: In agreement with [14], to nite size eects. This results from the fact that the we obtain a decrease of the Coulomb gap, together with higher order vm's change only little when going from the an increase of the overlap between MR wf and Coulomb lowest to the second LL. In Fig. 2(c) we summarize our

GS. We note LL mixing eects can be accounted for by results for both LL in the (y1, y3)-plane: the gap con- an eective width in the range , depending tour plot shows the incompressible region, in addition 1 . w/`0 . 6 on the cyclotron energy (and electron density) [10, 28]. the black line marks the top of the overlap ridge; the To study in detail the nite width eect and the dif- shaded (blue) area is the compressible region. ference between rst and second LL we vary the 2-body Now looking at the nite width y3(y1) trajectories we interaction V2b by changing the pseudopotentials v1 and can view the above results in a new light: for ν = 5/2 v3 [Eq. (3)] and keeping all other vm at their Coulomb (blue curve) the Coulomb point is on the safe side of the values (in a given LL). The values of vi encode the de- MR gap ridge, with a consistent gap and a high overlap pendence of the interaction on sample characteristics, like with the MR wf; increasing the thickness of the system the width of the 2D layer and the electron density. the overlap grows somewhat, as the nite width trajec- In Fig. 2(a) we plot the gap as a function of coul tory approaches the crest of the ridge, while the gap de- v1/v1 and coul for particles in the second LL, where creases. For (red curve) the situation is very v3/v3 16 ν = 1/2 coul are the Coulomb values of the pseudopotentials. dierent: the Coulomb point is on the other side of the vi In Fig. 2(b) we do the same for the overlap of the GS MR ridge, near the line of the phase transition, for some

58 system sizes in the gapped region, for others already in d'Ambrumenil, J. Fröhlich and B.I. Halperin and the compressible phase. We thus conclude that the MR support by the Swiss National Science Foundation and phase is so close to the compressible domain that a def- the Institute for Theoretical Physics at ETH, Zurich. inite prediction of its existence in the thermodynamic limit is not possible and only experiment can answer. Indeed, the gaps calculated for ν = 1/2 for nite width are small [Fig. 2(d)] and show a marked, although non- [1] R. Willett, et al., Phys. Rev. Lett. 59, 1776 (1987). monotonic, decrease with increasing system size (N el ≤ [2] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991). 18) while the layer width at which the gap opens increases [3] C. Nayak, et al., Rev. Mod. Phys. 80, 1083 (2008). with system size. It is unlikely that a gap survives in the [4] M. Greiter, X.-G. Wen, and F. Wilczek, Phys. Rev. Lett. thermodynamic limit for any layer or quantum well width 66, 3205 (1991); Nucl.Phys. B 374, 567 (1992). supporting a single-layer system [29]. This proves the im- [5] X.-G. Wen, Phys. Rev. Lett. 70, 355 (1993). portance of careful studies of the system size dependence [6] J. P. Eisenstein, et al., Phys. Rev. Lett. 61, 997 (1988); for valid conclusions about the existence of FQH states. J. P. Eisenstein, et al., Surf. Sci. 229, 31 (1990). As a test of our methods in discriminating the MR [7] For this reason, it was questionable whether the results of [4] had any relevance for the observed ν = 5/2 state. phase from Abelian FQH phases we studied the sys- [8] R. H. Morf, Phys. Rev. Lett. 80, 1505 (1998). tem with Nel = 12 and Nφ = 2Nel 3 = 21, which [9] E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 84, is aliased with the hierarchical 2/5 −state of 10 holes, 4685 (2000). 5 [30]. Indeed, our results for the [10] R. Morf and N. d'Ambrumenil, Phys. Rev. B 68, 113309 Nφ = 2 Nholes 4 = 21 second LL show− AC between the Coulomb GS and the (2003). MR state. However, in the lowest LL as the interaction [11] W. Pan, et al., Sol. St. Comm. 119, 641 (2001). is varied from pure three-body to Coulomb, the gap in- [12] S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. 94, 166802 (2005). creases linearly while the overlap of the MR wf with the [13] Xin Wan, Kun Yang, and E. H. Rezayi, Phys. Rev. Lett. GS decreases strongly and its largest overlap is with a 97, 256804 (2006); Xin Wan, Zi-Xiang Hu, E. H. Rezayi, high-lying L = 0 state (∆E 0.128): we have entered Kun Yang, Phys. Rev. B 77, 165316 (2008). the Abelian hierarchical phase.≈ The phase transition, as [14] M. R. Peterson, Th. Jolicoeur, and S. Das Sarma, Phys. the interaction is varied, from the non-Abelian MR phase Rev. Lett. 101, 016807 (2008); Phys. Rev. B 78, 155308 to the Abelian hierarchy phase is signalled by a signi- (2008). cant and sharp decrease of the overlap between the GS [15] A. E. Feiguin, et al., Phys. Rev. Lett. 100, 166803 (2008); A. E. Feiguin, et al., Phys. Rev. B 79, 115322 (2009). and the MR state. To identify the universality class of [16] G. Moller and S.H. Simon, Phys. Rev. B 77, 075319 a FQH state, AC is thus only a necessary condition, one (2008). must also study the overlap between GS and prototype [17] I. P. Radu, et al., Science 320, 899 (2008). FQH state as well as its system size dependence. [18] M. Dolev, et al., Nature 452, 829 (2008). Finally, we address the choice of the shift S = 3: three [19] C. R. Dean, et al., Phys. Rev. Lett. 100, 146803 (2008); 101, 186806 (2008). important features characterize states at S = 3: (i) the GS at 5/2 has angular momentum for all even [20] C. Töke and J. K. Jain, Phys. Rev. Lett. 96, 246805 L = 0 (2006); C. Töke, N. Regnault, and J. K. Jain, Phys. Rev. system sizes N 20 explored by us; (ii) the excitation el ≤ Lett. 98, 036806 (2007); Sol. St. Comm. 144, 504(2007). gap shows a smooth size dependence as expected for a [21] In contrast to [20], our calculations show well separated FQH state [8, 26]; (iii) low energy states at S0 = 3 can qhs with charge e/4, R.H. Morf et al., to be published. 6 be consistently identied as states with Nqp = 2(S0 3) [22] With universality class we mean the class of all states quasiparticles of charge e/4 nucleated in the underlying± − that are adiabatically connected to a prototypal state. FQH state with S = 3,± while the GS has small angular [23] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983); F. D. M. Haldane and E. H. Rezayi, ibid. 54, 237 (1985). momentum 0 indicating that quasiparticles L = O(Nel) [24] Song He, S. Das Sarma, X. C. Xie, Phys. Rev. B 47, 4394 with charge e/4 are well separated. ± (1993). We have shown that the polarized GS for Coulomb [25] Y. W. Suen, et al., Phys. Rev. Lett. 72, 3405 (1994). interaction at ν = 5/2 is adiabatically connected to the [26] R. H. Morf, N. d'Ambrumenil, and S. Das Sarma, Phys. Moore-Read state for all sizes studied. If the gap does Rev. B 66, 075408 (2002). not close in the thermodynamic limit - we have not seen [27] X.-G. Wen and A. Zee, Phys. Rev. Lett. 69, 953 (1992). [28] J. Nuebler et al., Phys. Rev. B 81, 035316 (2010). any sign that it will - the polarized GS at ν = 5/2 has the characteristics of the MR state. The same may happen [29] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993). in the lowest LL at ν = 1/2: While nite width and LL [30] N. d'Ambrumenil and R. Morf, Phys. Rev. B 40, 6108 mixing eects may help establish a Moore-Read phase, its (1989). realization in the thermodynamic limit remains doubtful. We acknowledge fruitful discussions with N.

~~59 3.3 Quasihole excitations~~

In the following 9 pages we attach the work published in Phys. Rev. B 83, 195306 (2011). Summarizing, we look at systems at ν = 5/2 containing a small number of localized e/4 charged quasiholes. We first show that it is indeed possible to localize such quasiholes on the surface of the sphere, using δ-function pinning potentials. Using a smooth pinning potential we also show that it is possible to reduce the radius of the quasihole to a minimum of about three magnetic lengths. We then perform the adiabatic connection investigations for systems containing two and four localized quasiholes. For two quasiholes, the lowest energy state evolves adiabatically between the “Pfaffian” and the Coulomb limit, without mixing with higher lying energy states. For four quasiholes we find that the lowest two states in the “Pfaffian” limit, corresponding to the degenerate Moore-Read doublet, remain the lowest-energy states even for pure Coulomb interaction. We conclude that the adiabatic continuity holds also for systems containing quasiholes. Finally, in the Coulomb limit (only slightly modifying the first Haldane pseudopotential), we perform quasiholes braidings in systems containing four quasiholes, by keeping two of them fixed and exchanging the positions of the other two, by stepwise changing the location of their pinning potentials. We find that under such a braiding, the system goes from one of the states of the Moore-Read doublet to the other: a sign of their non-Abelian braiding statistics.

~~60 Localized quasiholes and the Majorana fermion in fractional quantum Hall state at ν = 5/2 via direct diagonalization~~

~~M. Storni and R. H. Morf Condensed Matter Theory, Paul Scherrer Institute, CH-5232 Villigen, Switzerland~~

Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at ν = 5/2 for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge e/4 quasihole can be easily localized by means of a δ-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the ground state, while for four quasiholes, the lowest two energy states exhibit adiabatic continuity. This implies the existence of a Majorana fermion for pure Coulomb interaction. We also present preliminary results in the Coulomb limit for braiding in systems containing four quasiholes, with up to 14 electrons, diagonalizing in the full spin-polarized sector of the second Landau-level Hilbert space.

~~PACS numbers: 73.43.Cd 71.10.Pm~~

I. INTRODUCTION In spite of the AC of the GS, the question whether the elementary charged excitations preserve their non- The possibility of realizing non-Abelian braiding Abelian properties in going from the MR to the Coulomb statistics1 in a condensed-matter system has generated limit still needs to be examined: This is a very important a great deal of interest both for theoretical and experi- issue, because braiding of non-Abelian quasiparticles has mental studies. One such system that may be closest to been proposed for topological quantum computation.4,11 experimental realization is the fractional quantum Hall At ν = 5/2, one expects that QHs can have either charge with the possibility of having non-Abelian braiding (FQH) state at lling fraction ν = 5/2, rst observed by e/4 2 statistics or with Abelian fractional statistics. In- Willett et al. Whether the experimentally observed 5/2 e/2 state indeed has charged excitations with non-Abelian deed, the results by T®ke et al.12 have cast doubt on the braiding statistics is still unknown, altough the discovery existence of localized QHs with charge e/4. of a neutral current in experiment may make us hopeful.3 Here, we study fully spin-polarized systems with local- For a review, we refer the reader to Nayak et al.4 and to ized QHs in the spherical geometry. This polarization Stern.5 choice is motivated by theoretical investigations concern- In an earlier work6 we presented numerical evidence ing this issue: One of us showed that the GS of the disorder-free FQH state at is spin polarized that the FQH ground state (GS) at ν = 5/2 for Coulomb ν = 5/2 interaction, when spin polarized, is in the same univer- even for vanishing Zeeman energy;13 recent theoretical sality class as the non-Abelian Moore-Read (MR) Pfaf- work14,15 conrmed this result. However, from the exper- an state.7 Calculating the energy spectrum of the few imental point of view, the situation is less clear. Trans- lowest-energy states by exact diagonalization for electron port experiments with variable electron density16,17 or in interactions interpolating between the Coulomb interac- a tilted magnetic eld at high density18 have been interpreted in controversial ways: Varying the electron den- tion VC and the three-body interaction V3b, for which the MR state is the unique GS, we found for all examined sity by more than a factor 2,16 the excitation gap varies system sizes adiabatic continuity (AC) of the GS and no smoothly with no break in slope or discontinuity, indicating that neither GS nor excitations change their character sign of a decrease of the gap between the two limits V C while the Zeeman energy varies by an amount that ex- and V3b. We concluded that AC can be expected even in the thermodynamic limit. In addition, we drew a phase ceeds the measured energy gap by a factor of 5 10. That diagram in the two-body interaction space (in the vicin- an unpolarized state would survive under this− condition ity of the Coulomb interaction), showing that the FQH appears unlikely. Also, in high-density samples, the dis- gapped phase coincides with the MR phase. appearence of the gapped phase in a tilted magnetic eld Previous theoretical work810 gave indications for AC cannot be explained by a Zeeman energy term with a GaAs bulk factor.18 Nevertheless, arguments have been for the spin-polarized ν = 5/2 GS between the MR and g the Coulomb limit. In the disk geometry Xin Wan et al.9 presented that these experimental results might be con- were also able to localize a single quasihole (QH) at the sistent with a spin-unpolarized 5/2 state.19 Also rst center of the disk showing that, in a certain range of QH results from direct spin polarization measurements, us- pinning and system connement potential strength, the ing optical methods,20,21 appear to be consistent with a lowest-lying state belongs to the total angular momentum spin-unpolarized state. As the experimental samples are likely to be inhomogeneous and consist of compressible value that corresponds to a MR e/4 charged QH, for regions separated by percolating incompressible laments electron interactions interpolating between V3b and VC . at ν = 5/2,22 it is not clear that the optical data relate ν = 5/2 we study a half lled second LL, while the two solely to the laments or to the compressible regions.21 lower-lying lled levels are considered as inert. For such The separate determination of the polarization state of a system the GS is at shift S = 3 (as obtained by exact the incompressible parts has not yet been possible. In diagonalizations13), the same value as for the MR state.32 addition, charged impurities may also induce skyrmions We can insert NQH e/4-charged QHs into the system by 21,23 1 in the incompressible domains, which will decrease adding NQH supplementary ux quanta, i.e., the total spin polarization. We also note that, once ther- 2 22 mally assisted tunneling is taken into account, the gap Nφ(N,NQH ) = 2N 3 + NQH /2. (1) obtained from analyzing the dissipative conductance can − 6 be as consistent with expectations for a spin-polarized In the spirit of our previous work we consider interactions of the form system at ν = 5/2 (Refs. 17, 24 and 25) as for the hierarchy state at . ν = 4/9 (1) (0) (2) In our investigations at rst we show that charge V = (1 x)V + xV + Vpin(x), e/4 − C 3b QHs can be localized by a -function potential and, vary- δ where , interpolating between the Coulomb ing the form of the localization potential away from the 0 x < 1 interaction≤ and the three-body interaction , whose simple -function, we also show that there is an optimal VC V3b δ exact GS is the MR wave function,8 localizing potential that minimizes the density oscillations around the QHs. We also nd that in systems with A N two QHs, in the Coulomb case, it is sucient to have a V = S ∆ δ(i j)∆2 δ(i k) , (3) single -function: This localizes a single QH in the GS, 3b N 5 ijk j − k − δ i 1/2). mix with the higher lying ones. For systems containing The superscripts (0) and (1) in Eq. (2) indicate that four QHs, the two lowest-lying states forming a degener- (1) the Coulomb energy V is evaluated in the n = 1 LL, ate doublet in the limit26,27 remain the two lowest- C V3b as required for the experimentally realized ν = 5/2 state, energy states even in the limit of pure Coulomb inter- while the three-body interaction (0) is evaluated in the action . Thus, the Majorana fermion associated with V3b VC LL, since the MR state is the GS of in the the doublet28 appears to survive in the Coulomb limit. n = 0 V3b lowest LL. To better study the MR limit we then repeat the inves- The pinning term localizes the QHs at xed tigations for four QHs using a dierent pinning potential: Vpin(x) positions on the surface of the sphere. In our rst Also with this approach we obtain AC for the lowest-lying calculations (in sections III and IV A) we consider a doublet between the two interaction limits, conrming parametrization of the pinning potential of the form the above results. Finally we present some preliminary results of braiding with QHs, showing that exchanging V (x) = (1 x) (1 x) (1) + x (0) , (4) the position of two QHs, in the presence of two others pin − · − Vpin Vpin xed QHs, the system goes from one of the MR doublet where the pinning function is still to be chosen, states to the other, showing non-Abelian braiding statis- pin such that it localizes exactlyV an e/4-charged QH. This tics. Furthermore, we investigate the fusion of two QHs, couples to the electron density, interpolating like getting an estimate for the MR doublet splitting.29 Vpin(x) the electron interaction between LL n = 1 and n = 0 with weights (1 x) and x, respectively. The overall factor makes− the localization term vanish in the MR II. THE SYSTEM (1 x) limit as− x 1, just like the Coulomb term, both helping to separate→ the energies of states with charge e/4 QHs. In the following we consider fully spin-polarized elec- However, in the MR limit ( ), the -charged 30 x 1 e/2 tron systems on the surface of a sphere: By exact diag- double QH has a zero in the electron→ density and thus onalization we obtain their low-lying states and energy zero pinning energy, while the e/4-charged QH shows a spectra. In the spherical geometry for a half lled Lan- nite local minimum of the density and thus nite pin- dau level (LL) the number of electrons and the number N ning energy. To separate e/4-charged QHs requires ei- of ux quanta Nφ are related by Nφ = 2N S, where ther an additional repulsive interaction between QHs or the shift S (Ref. 31) is a topological quantum− number a special pinning potential coupling to the monopole and that depends on the particular FQH state. quadrupole moment of the QHs (cf. problems discussed For our investigations of the FQH at lling factor in T®ke et al.12). To avoid this diculty, in this rst series of calculations we never reach exactly the MR point,

~~62 1.25 1.25 2 quasiholes at poles: N=20 density a) b) 2 quasiholes at poles: N=18 missing charge 1 1~~

~~0.75 0.75 8~~

~~density weak b-pinning ] 0 l d) missing charge weak -pinning c) ¡ 0.02 b / density strong -pinning 6 2 b 0.5 0.5 missing charge strong b-pinning -1/4 4~~

~~0.25 density 0.25 2 0.01 energy of L eigenstates pinning potential [arbitrary units] 0 -function pinning~~

~~excitation energy [e b 0 1 2 3 4 5 6~~

~~density (missing charge) smooth pinning 0 0 great circle distance [l0] 0 0 2 4 6 8 10 angular momentum -0.25 -0.25~~

~~0 2 4 6 8 10 12 14 0 2 4 6 8 10 12~~

~~great circle distance [l0] great circle distance [l0]~~

Figure 1. (Color online) Polar angle dependence of electron density and missing charge for ν = 5/2 systems with two QHs localized at the poles of the Haldane sphere, for Coulomb interaction. In (a) the QHs are localized by a δ-function potential pin [setting x = 0 in Eq. (4)], for two dierent pinning strengths; in (b) by a smooth pinning potential, whose radial dependenceV is shown in (c). (d) Energy spectrum of a system with two QHs, without pinning, as function of the total angular momentum L, and comparison with the Coulomb energy for δ function and smooth pinning [cf. the text for more details; results are for N = 20,Nφ = 38 in (a) and for N = 18,Nφ = 34 in (b),(c) and (d)]. but approach it closely enough to identify the lowest- shape, depicted in Fig. 1(c), we obtain an electron den- energy states. In Section IV B we will then use a dierent sity without oscillations, as illustrated in Fig. 1(b): The localization potential33 to circumvent this problem and smooth potential allows to properly separate the two QHs study the full AC up to the MR point. and to reduce their radius to a minimum of 3`0. 2 ∼ All energies are measured in the usual units e /ε`0, How is this possible? Figure 1(d) shows a comparison where is the magnetic length, and taking of the interaction energies, not including the energy due `0 = ~c/eB the energy of the lowest state as reference. to the pinning forces, for δ-function pinning (upper line) p and for smooth pinning (lower line). The circles depict the spectrum of a system with two free QHs, i.e., without III. LOCALIZATION OF QUASIHOLES pinning force, as function of the total angular momentum L. We see that the use of a smooth pinning potential We start by studying the QH localization in the pure allows to obtain a smaller interaction energy, admixing only the lowest-lying L eigenstates, while the δ-function Coulomb limit, taking for pin a δ-function pinning potential: Choosing a suitableV pinning strength we see that potential admixes also higher-lying states, giving rise to the large density oscillations. We also note that a better it is actually possible to obtain separated e/4-charged QHs, in contrast to the results of T®ke et al.12 Figure localization, with a pinning potential that further lowers 1(a) shows the polar angle dependence of the density (up- the interaction energy, is unlikely to be possible, because per curves) and of the integrated missing charge (lower a minimal number of low-energy L eigenstates is needed curves) for two dierent pinning strengths, for a system to obtain the desired QH localization. We note that the energy of two-QHs systems is aected containing two QHs with localizing δ functions, one at the north and the other at the south pole of the sphere. by how well the density oscillations t with the separa- We observe that the two QHs are localized at the poles. tion distance of the QHs, such that the energy will show However they are large in extent and are accompanied oscillations as function of the QH separation. The en- by signicant and slowly decaying density oscillations (cf. ergy oscillations observed in the variational calculation 17 of the Majorana fermion energy29 may to some extent be Nuebler et al. ). The missing charge e/4 is reached (for caused by such commensuration eects. An optimized the stronger pinning potential) at R 4`0 and at the equator as required by symmetry. ≈ localization potential might help to reduce them, hope- As the system is compressible at the position of a QH, fully allowing a more accurate calculation of the coher- the shape of the pinning potential will inuence its struc- ence length associated with the Majorana fermion. ture. We therefore study how the pinning potential can In the following, we use δ-function pinning potentials. be tuned to minimize the size of the QH and to suppress We rst study what happens if only a single δ-function the density oscillations around it. With the choice of is inserted in a system of N = 16 electrons with two

~~63 0.04 0.03 a) 0.035 weak b-pinning at N- and S-pole strong b-pinning at N- and S-pole ] 0 l~~

~~0.03 ¡ / 2 ] 0 l ¡~~

~~/ 0.02~~

~~2 0.025~~

~~0.02~~

~~0.015~~

~~total energy [e 0.01 0.01~~

~~0.005 excitation energy [e~~

~~0 0 1 2 3 4 5 6 7 8 9 0 0.035 b) weakangular b-pinning momentum at S-pole Lz 0 0.2 0.4 0.6 0.8 1 strong b-pinning at S-pole interaction parameter x 0.03 ]~~

0 Figure 3. (Color online) Low-lying energy spectrum as func- l ¡ / 2 0.025 tion of the interaction parameter x for a system of N = 12 electrons with two QHs localized at the poles by δ-function 0.02 pinning potentials; the lowest-energy state is the reference.

~~0.015 total energy [e 0.01 Coulomb interaction, this double QH state has quite large interaction energy and, except perhaps in the limit 0.005 of very strong pinning, is unlikely to become the GS.~~

~~0 0 1 2 3 4 5 6 7 8 9 IV. ADIABATIC CONTINUITY angular momentum Lz~~

Figure 2. (Color online) (a) Energy spectrum of a two-QH A. Quasiholes localization with δ-function pinning system as a function of the angular momentum component Lz when δ functions are inserted at both poles (for two dierent We now turn to the study of AC for systems containing pinning strengths, ); the lowest-energy state N = 16,Nφ = 30 localized QHs. We begin with two QHs, pinned at the is the reference. (b) The same for a single function at the δ poles by functions, and we vary the particle interaction south pole. δ as in Eq. (2) from x = 0 (the Coulomb case) to near x = 1 (the MR limit). Figure 3 shows the evolution QHs, and whether they are still separated or if a sin- of the excitation energies as function of the parameter for electrons and ux units; other gle e/2-charged double quasihole is formed. Figure x N = 12 Nφ = 22 2(a) shows the energy spectrum as function of the an- system sizes give similar results. The horizontal line at zero energy represents the GS energy: Clearly none of the gular momentum component Lz when two δ functions are inserted at the poles, for weak and strong pinning - lines for the excited states come close to it and there is no cf. Fig. 1(a). In both cases the lowest-energy state has mixing between them. Thus there is no phase transition and describes two separated QHs localized at the in going from the Coulomb case to the MR limit, exactly Lz = 0 as observed for systems without QHs.6 poles. Figure 2(b) shows the same, but for a single δ function at the south pole. The lowest energy state has Next we investigate a system of N = 14 electrons and ux quanta that contains lo- also , indicating that the QHs are still at max- Nφ = 27 NQH = 4 Lz = 0 34 imal separation, which minimizes their Coulomb repul- calized QHs. This is particularly interesting because in 35 sion: The QH pinned at the south pole repels the other the MR limit the GS is a degenerate doublet, which 28 to the north pole of the sphere. For increasing pinning is associated with a Majorana fermion and with the non-Abelian braiding statistics of the QHs:26,27 For any strength some states with L = 0 are reduced in energy. z 6 four-QH conguration there are two linearly independent With strong enough pinning, a state with Lz = 0, in which QHs would cease to be maximally separated,6 may wave functions that describe it; braiding QHs around become the lowest energy state. Of particular interest is each other induces a linear transformation in the degen- the state at which corresponds to a dou- erate subspace of the doublet. Does this doublet survive Lz = N/2 = 8 in the Coulomb limit? ble QH with charge e/2 localized at the south pole, but no charge deciency at the north pole. At , for Figures 4(a) and 4(b) show results for four QHs local- ν = 5/2 ized at the corners of a square on a great circle and at

~~64 1 groundstate (a) square (b) tetrahedron first excitation (d) tetrahedron: Coulomb interaction 0.9 (c) 0.015 0.015 0.015 overlap~~

~~] ] 0.8 ] 0 0 l l 0 l ¡ ¡ / / ¡ / 2 2 2 0.7 0 0.2 0.4 0.6 0.8 1 0.01 0.01 0.01 interaction parameter x~~

~~Pfaffian Pfaffian~~

~~0.005 0.005 excitation energy [e 0.005 excitation energy [e excitation energy [e~~

~~0 0 0~~

~~0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 1 1.5 2 2.5 interaction parameter x interaction parameter x strength of pinning potential~~

Figure 4. (Color online) Low-lying energy spectrum as function of the interaction parameter x for systems with four QHs localized with δ-function pinning potentials: (a) At the corners of a square on a great circle and (b) at the vertices of a tetrahedron; the curves labeled Pfaan give the variational estimate of the doublet splitting obtained using the GS doublet for the MR limit. (c) Overlaps of the two doublet-states for Coulomb interaction with the corresponding states for interaction parameter x. (d) Low-lying energy spectrum for Coulomb interaction, as function of the pinning strength, for four QHs in tetrahedral position. (The lowest-energy state is the reference; N = 14, Nφ = 27.) the vertices of a tetrahedron, respectively. In both cases we see that the two lowest-energy states close to the MR interaction parameter x limit, i.e., the GS at energy 0 and the rst excited state, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 remain the lowest-energy states over the entire parame- 0.03 (a) tetrahedron, N=12 ter interval, even at the Coulomb point x = 0. During the whole evolution there is neither crossing nor mixing 0.02

with higher-lying states, indicating the absence of a phase ] 0 l ε / transition. Note that in the MR case, at x = 1, all the 2 considered states have zero energy, because the pinning 0.01 potential Vpin(x) vanishes at this point. In the tetrahedral case, right in the Coulomb limit, 0.00 a higher-energy state comes down in energy and nearly -0.01 reaches the upper state of the doublet. But Fig. 4(d) excitation energy [e shows that this is actually not a problem: The Coulomb spectrum is plotted as function of the strength of the -0.02 pinning potential and it is evident that the near degeneracy is easily lifted by increasing the pinning strength. -0.03 We also note that the two states of the doublet are not (b) tetrahedron, N=14 degenerate in the presence of a Coulomb interaction but 0.025 we expect the splitting to vanish in the thermodynamic ] 0 l

~~29 ε / 0.020~~

limit. 2 The curves denoted by Pfaan in Figs. 4(a) and 4(b) show the variational results for the splitting of the dou- 0.015 blet, computed using the states at x = 0.95 (compare 29 with Baraban et al. ): They overestimate the splitting 0.010

in the Coulomb limit, in the square geometry by a factor excitation energy [e of 2.5 and in the tetrahedral geometry by 70%. 0.005 Finally, Fig. 4(c) shows the evolution of∼ the overlaps ψ (0) ψ (x) of the two states ψ (x)(i = 0, 1) of the 0.000 h i | i i i doublet as the interaction varies from Coulomb VC at 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 interaction parameter x x = 0 to the vicinity of the three-body limit V3b at x = 0.95. We see that there is no sign of an abrupt drop of the overlaps which could signal a phase transition, and Figure 5. (Color online) Low-lying energy spectrum as func- the values reached near the MR limit are reasonably large tion of the interaction parameter x for systems with four QHs localized at the vertices of a tetrahedron using the pinning for a system of this size (N = 14,Nφ = 27). potential in Eq. (5): (a) N = 12,Nφ = 23, q = e/4, (b) All these results provide evidence in favor of adiabatic − N = 14,Nφ = 27, q = 0.19e (the lowest-energy state in the continuity from the MR to the Coulomb limit for the two MR limit is the reference).−

65 Figure 6. (Color online) Particle density on the sphere surface for the two states of the MR doublet: (a),(b) in the MR limit and (d),(e) in the Coulomb case; (c) and (f) show the density dierence between the two states (N = 14, Nφ = 27). lowest-energy states in systems with four localized QHs, was rst used by Prodan and Haldane36 in their inves- suggesting that the non-Abelian doublet, and the associ- tigation of the (non-Abelian) braiding properties of MR ated Majorana fermion, can be expected in the limit of QHs, although projecting the pinning potential on the pure Coulomb interaction. zero energy space of their Hamiltonian. Figure 5 shows the results for the low-lying energy spectrum, as the interaction is varied from the Coulomb B. Quasiholes localization with STM tip pinning (x = 0) to the MR limit (x = 1), for systems with four QHs localized at the corners of a tetrahedron by four In the AC investigations described so far, the pinning such pinning potentials. The results in Fig. 5(a) are for electrons and a localizing charge , while potential Vpin(x) [in Eq. (4)] was not able to localize QHs N = 12 q = e/4 in the MR limit (as discussed in section II), but we ap- in Fig. 5(b) for N = 14 electrons34 and q−= 0.19e: proached this limit near enough to recognize the lowest- These charges were chosen in order to obtain that− the lying states. One can pose the question if the permanent two states of the MR doublet are exactly the lowest-lying presence of the Coulomb interaction (that vanish as 1 x states in the MR limit. in the MR limit x 1) could inuence our results,− se- In both cases we see that (once q is xed) the two lecting exactly the→ right states from the zero energy lowest energy states remain the lowest over the whole states at the MR point. To strenghten our evidences and parameter interval (for N = 12 with a level crossing). to show a complete AC between Coulomb and MR limit, The third energy level, corresponding to an excitation in we repeat here the calculations for systems containing the pinning potential, is exactly threefold degenerate (for symmetry reasons) and can thus be easily distinguished four QHs, using a dierent pinning potential Vpin(x) that allows to reach both limits: from the two lowest states forming the MR doublet. We note that this doublet has a nite energy splitting, (1) (0) (5) not only for pure Coulomb interaction, but also in the Vpin(x) = (1 x)Vq + xVq , − MR limit: This is caused by the pinning potential (5) where Vq, interpolated between the n = 1 and the n = 0 that mixes in states with nonzero energy. We also nd LL like the electron interaction, is the Coulomb potential that the energy separation between the upper state of the of a pointlike object, e.g., a STM tip, with a (negative) doublet and the third (threefold degenerate) state can be charge q, positioned on the surface of the FQH sample.33 small, particularly in the MR limit, and depends on the This potential repels the electrons from its center and, charge q used in the pinning potential. For example for if the charge q is chosen in an appropriate way, namely, N = 14 electrons, using a localizing charge q = e/4, in an interval near e/4, it can only localize e/4-charged the pinning potential is too strong and at the MR− point QHs, thus avoiding− the problem of the localization of the threefold degenerate state is slightly lower in energy e/2-charged double QHs in the MR limit. This idea

66 than the upper state of the doublet; to obtain the right braiding angle ϕ [degrees] 0 30 60 90 120 150 180 order we chose a weaker potential, with q = 0.19e. -4.504 The AC between the Coulomb and the MR− limit for the (a) N=12 lowest energy doublet shown in Fig. 5 thus supports the -4.506 evidences of the previous section in favor of a survival of the Majorana fermion till the Coulomb point. -4.508 ] 0 In Fig. 6 we also show the particle density on the sur- l ε / face of the sphere for the two states of the MR doublet in 2 -4.510 the two limiting cases: (a), (b) for the MR and (d), (e) for the Coulomb limit. The two states of the doublet are -4.512 energy [e indeed very similar, as also shown in (c) and (f), where -4.514 the density dierences are plotted. Note that the density dierences in the MR case are approximately twice as -4.516 large as the dierences in the Coulomb limit.

~~-4.518 (b) N=14 V. QUASIHOLES BRAIDING AND FUSION -5.223~~

~~In this last section we show some preliminary results -5.225~~

of QH braiding and fusion in systems containing four lo- ] 0 l ε / calized QHs. We perform this investigation for electron 2 -5.227 interactions near to the Coulomb limit, but slightly in- 30 -5.229 creasing v1, the Haldane pseudopotential that describes the interaction between two particles having a relative energy [e 6 -5.231 angular momentum 1~. In our previous work we showed that, under such interaction modications, the system remains in the MR phase, even improving the overlap be- -5.233 tween the exact GS and the Pfaan wave function and -5.235 simultaneously with an enhancement of the gap. For the 0 30 60 90 120 150 180 QH localization we use the pinning potential introduced braiding angle ϕ [degrees] in the previous section [Eq. (5)], setting , that is x = 0 Figure 7. (Color online) Braiding of two QHs (in the pres- evaluating it fully in the n = 1 LL, and chosing a suitable ence of two other xed QHs): low-lying energy spectrum as value for the localizing charge , as explained below. q function of the braiding angle ϕ, dened through Eq. (6). Investigating QH braiding in systems with four QHs, The electron interaction is near to the Coulomb limit (with a the non-Abelian nature of the MR doublet states should slightly increased Haldane pseudopotential v1) and the QHs become manifest: Interchanging the positions of two QHs are localized by the pinning potential of Eq. (5), setting x = 0. (a) , , Coul, (b) , by stepwise changing the location of their pinning po- N = 12 q = 0.1715e v1 = 1.05v1 N = 14 , − Coul, where Coul is the Coulomb tentials (and keeping the other two QHs xed), the two q = 0.19e v1 = 1.075v1 v1 states of the MR doublet should transform into each value− of the rst Haldane pseudopotential. other, in particular, switching their position in the energy spectrum. For this to happen, during the QHs interchange process, an odd number of level crossings be- and 2 around the vertical axis through the poles, tween them is needed. In the following calculations, we (6) engineer an exact energy degeneracy for the two states of ϕ1(ϕ) = 90◦ + ϕ, ϕ2(ϕ) = (270◦ + ϕ) mod 360◦, the doublet at the midpoint of the QHs interchange pro- keeping the polar angle constant, until cess, by choosing suited values for the localizing charge θ1,2 = 54.736◦ q they exchange their original positions: . and the rst Haldane pseudotential . Then, performing 0◦ ϕ 180◦ v1 This is a -periodic process; we call the≤ braiding≤ the interchange, we investigate if this is indeed a crossing 180◦ ϕ angle. point and whether it is the only one or if others arise. The polar angles and are chosen in such a We chose as starting (and ending) conguration four θ1,2 θ3,4 way that at the midpoint of the rotation process, that QHs at the corners of a rectangle: Two of them on the is at braiding angle , the four QHs are in a upper half-sphere, at the same polar angle ϕ = 90◦ θ1,2 θ1 = tetrahedral conguration. The localizing charge and and at the opposite azimuthal angles≡ q θ2 = 54.736◦ ϕ1 = Haldane pseudopotential (for the whole braiding pro- ; the other two on the lower half-sphere at v1 90◦, ϕ2 = 270◦ cess) were previously chosen, such that at this point the polar angle and azimuthal θ3,4 θ3 = θ4 = 125.264◦ the two states of the MR doublet are degenerate in en- angles and≡ . We keep QHs and ϕ3 = 90◦ ϕ4 = 270◦ 3 ergy: , Coul for and xed at their locations and, by stepwise changing the q = 0.1715e v1 = 1.05v1 N = 12 4 ,− Coul for electrons, where positions of the pinning potentials, we rotate both QH q = 0.19e v1 = 1.075v1 N = 14 1 Coul− is the Coulomb value of the pseudopotential . v1 v1

~~67 -5.220 These are reasonable values for q and v1: The localized QHs have charge e/4 and the electron interaction is -5.225 modied such that the system is still in the MR phase. -5.230~~

~~Figure 7 shows the evolution of the low lying energy -5.235 spectrum as function of the braiding angle for (a)~~

~~ϕ ] -5.240 0 l ε~~

and (b) electrons. In both cases we / N = 12 N = 14 2 observe that the two states of the doublet remain the -5.245 lowest-energy states during the whole braiding process, -5.250 never mixing with higher-lying states or with each other energy [e -5.255 (except at the degeneracy point). Indeed the overlaps -5.260 between the corresponding states of the doublet for suc- -5.265 cessive braiding steps are consistently high: In the range -5.270 - for ( steps) and - for 0.94 0.97 N = 12 5◦ 0.97 0.99 N = 14 -5.275 (3◦ steps). 10 20 30 40 50 From the plots it is evident that the degeneracy point polar angle θ1,2 [degrees] at ϕ = 90◦ is indeed also a crossing point for the two doublet states: If one follows the lines with a continuous Figure 8. (Color online) Fusion of two QHs at the north pole (in the presence of two other xed QHs): Low-lying energy derivative, at the degeneracy point one goes from the spectrum for electrons as function of , lower to the upper state of the doublet (and vice versa). N = 14 θ1,2 θ1 = θ2 the polar angle of both the moving QH 1 and 2.≡ The elec- No other crossing or mixing point arises and thus the QH tron interaction is near to the Coulomb limit (with a slightly braiding causes an interchange in the MR doublet states: increased Haldane pseudopotential Coul) and the v1 = 1.075v1 If one starts at ϕ = 0◦ in the lower-energy state, one ends QHs are localized by the pinning potential of Eq. (5), setting and ( Coul is the Coulomb value of the rst at ϕ = 180◦ in the upper-energy state, as expected for x = 0 q = 0.19e v1 non-Abelian braiding statistics. Two such braidings are Haldane pseudopotential).− needed to come back to the initial state. We note that we can get an exact degeneracy of the MR doublet only when the four QHs are in the tetra- possible with a study of the monodromy matrix, as was hedral conguration. Slightly modifying the geometry done by Prodan and Haldane36); nevertheless, the inter- of the conguration, a small gap opens; however, if the change of the MR doublet states should survive also in speed of the braiding is suciently high, the states of the this limit. MR doublet would still interchange at this point. This Unfortunately we can get results for QH braiding only situation is analogous to that studied by Thouless and for small system sizes34 and from them we cannot extract Gefen,37 concerning the crossing between the lowest-lying useful information about the dependence of the energy states as function of the magnetic ux, for the quantum splitting when approaching the thermodynamic limit. Hall eect at fractional llings, showing that these cross- The average splitting of the MR doublet for N = 14 ings are essential in order to get a fractional charge. On is slightly smaller than for N = 12, but this is mainly the other hand if the QHs are moved in a strictly adia- caused by the dierent shape of the energy oscillations batic way, the system always remains in the lowest-energy as function of the braiding angle ϕ in the two cases: For state, without crossing. We also wish to emphasize that N = 14 the oscillations of the MR doublet states are al- the doublet degeneracy is very dierent from the three- most in phase, while for N = 12 they are out of phase, fold degeneracy of the rst excited state: The former thus giving a larger average splitting. is obtained only by ne tuning of the parameters q and Finally we investigate the fusion of two QHs by bringing the pinning potentials that localize them close to each v1, the latter is a purely geometric degeneracy, resulting from the symmetric QH conguration and independent other, in the presence of two others xed QHs. We start from interaction and localization parameters. from the tetrahedral conguration described above and The fact that the MR doublet states are not degener- we let fuse the two QHs in the upper half-sphere, by taking them to the north pole, that is, by shrinking the polar ate (except by tuning at ϕ = 90◦) in our small system diagonalizations helps us to follow their evolution during angle θ1,2 θ1 = θ2 from 54.736◦ to 5◦ at xed azimuthal the braiding (through the evolution of the energy lev- angles. Figure≡ 8 shows the low-lying energy spectrum as els) and thus to recognize the crossing point which leads function of the polar angle θ1,2 during this process, for to the interchange of states. However we expect that in N = 14 electrons. We see that, after that the degeneracy the thermodynamic limit the two states become exactly is lifted, the evolution of the two lowest states is similar, degenerate during the whole braiding process, when the with a relatively constant splitting between them. This QHs are suciently far apart from each other: On one splitting (or at least a part of it) could come from the in- side this would make the ne tuning of the parameters q trinsic splitting of the MR doublet as two QHs are taken together (its order of magnitude is the same as that ob- and v1 unnecessary, but on the other side it would make it impossible, using our present method, to follow the tained by Baraban et al.29). However, at approximately evolution of the states under braiding (however, it is still θ1,2 = 37◦, the upper state of the MR doublet comes very

68 near to the next higher-lying state and strongly mixes Coulomb interaction. Forces breaking particle-hole sym- 38 39,40 with it (but without crossing), possibly losing a part of metry like V3b are actually present due to LL mixing its character. Thus it is not fully clear what happens at and will favor either the Pfaan7 or the Antipfaan.41,42 the MR doublet, and further investigations are needed.

ACKNOWLEDGMENTS VI. CONCLUSION We acknowledge fruitful discussions with Nicholas Our results provide evidence in favor of adiabatic conti- d'Ambrumenil, Jürg Fröhlich, Duncan Haldane, and nuity from the Moore-Read to the Coulomb limit for the Sankar Das Sarma, as well as the support by the Swiss two lowest-energy states in systems with four localized National Science Foundation and by the Institute for QHs and thus the non-Abelian doublet and associated Theoretical Physics at ETH, Zurich. Majorana fermion can be expected in the limit of pure

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~~69 Conclusions~~

~~Summarizing, in our exact diagonalization studies we have shown:~~

Localization of spatially separated e/4 charged quasiholes in the ν = 5/2 • state. Adiabatic continuity beetween the pure Coulomb limit and the “Pfaﬃan” • model for the ground state at ν = 5/2 and for states containing few localized e/4 charged quasiholes. Coincidence of the quantum Hall gapped phase with the region of high • overlap between the exact ground state and the “Pfaﬃan” state in a phase diagram obtained by varying the Haldane pseudopotentials v1 and v3 describing the two-body interaction.

~~Signals of non-Abelian braiding statistics as two quasiholes exchange their • position (in the presence of other two ﬁxed quasiholes).~~

We believe that these results are a strong indication that the “Pfaffian” (or “Antipfaffian”) model is indeed relevant for the description of the experimentally observed fractional quantum Hall effect at ν = 5/2, if spin polarized. Experiments on two-dimensional electron systems (in GaAs-based samples) investigating this issue are in progress. Recently it has been pointed out [233, 234] that the Moore-Read state could also be realized in half filled Landau levels of bilayer graphene systems. These systems offer the advantage that the effective electron interaction can be tuned over a wider range, eventually allowing to obtain a more stable state, with a larger gap and a larger overlap with the “Pfaffian” state (that is reaching the ridge of the hill in our phase diagram). In any case we hope that an irrefutable experimental proof of non-Abelian braiding statistics in a fractional quantum Hall system will soon be available.

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~~86 Curriculum Vitae~~

~~Personal data Name: Maurizio Storni Born: April 28, 1976 in Lugano (TI) Nationality: Swiss~~

Education 1991-1995: Liceo Lugano 2 (Liceo Scientifico): graduated with Attestato di MaturitàCantonale (June 1995) 1995-2000: Swiss Federal Institute of Technology (ETH) Zürich: graduated with Diploma in Physics (November 2000) 2003-2011: Swiss Federal Institute of Technology (ETH), Zürich and Paul Scherrer Institute (PSI), Villigen: doctoral studies

Publications M. Storni, R. H. Morf, and S. Das Sarma, The Fractional Quantum Hall State at ν = 5/2 and the Moore-Read Pfaﬃan, Phys. Rev. Lett. 104, 076803 (2010) M. Storni and R. H. Morf, Localized quasiholes and the Majorana fermion in fractional quantum Hall state at ν = 5/2 via direct diagonalization, Phys. Rev. B 83, 195306 (2011)