Fractional charge and fractional statistics in the quantum Hall effects
FFractional Charge and Fractional Statistics in the Quantum Hall Effects
D. E. Feldman
Brown Theoretical Physics Center and Department of Physics, Brown University, Providence, RI 02912, USA
Bertrand I. Halperin
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Dated: February 19, 2021)Quasiparticles with fractional charge and fractional statistics are key features of the fractionalquantum Hall effect. We discuss in detail the definitions of fractional charge and statistics andthe ways in which these properties may be observed. In addition to theoretical foundations, wereview the present status of the experiments in the area. We also discuss the notions of non-Abelianstatistics and attempts to find experimental evidence for the existence of non-Abelian quasiparticlesin certain quantum Hall systems.
CONTENTS
I. Introduction 1II. The meaning of fractional charge and fractionalstatistics 2A. Fractional charge 2B. Fractional statistics 31. Definition in terms of effective wavefunctions and effective Hamiltonian 32. Illustrative example 43. Relation to the microscopicHamiltonian 54. Non-Abelian statistics 6C. Application to quantized Hall states 61. Fractional charge 62. Fractional statistics 73. Edge modes 8III. Experimental probes of fractional charge 9A. Shot noise 91. Photo-assisted shot noise 11B. Charging spectroscopy 111. Measurements of tunneling through anantidot 112. Single-electron transistor technique 12IV. Experimental probes of fractional statistics 12A. Fabry-Perot Interferometry 131. The ideal case 132. Interferometer with a compressible bulk 153. Quasiparticle charges from Fabry-Perotexperiments 17B. Anyon collider 19V. Non-Abelian statistics 20A. Basic principles 20B. Edge modes 24C. Examples of non-Abelian statistics 251. Possible states at ν = 5 / I. INTRODUCTION
The experimental discovery in 1982, in a two-dimensional electron system, of quantized Hall plateauswith Hall conductivity σ xy = νe /h showing fractionalvalues ν = 1 / and ν = 2 / , marked the beginning of oneof the most surprising and far-reaching developments incondensed matter physics in the second half of the twen-tieth century. These fractional quantized Hall (FQH)plateaus, together with plateaus at other rational frac-tional values of ν , were understood to be manifestationsof a new type of correlated electron state, with a num-ber of peculiar properties. Continuing experimental andtheoretical efforts have revealed a wide variety of FQHstates, as well as other unusual phenomena that can oc-cur in two-dimensional electron systems in a magneticfield at low temperatures, in different materials and un-der different conditions. Indeed, experiments on thesesystems continue to produce surprises, and the field ofquantum Hall effects remains a vital area of condensedmatter research today.[1] Moreover, insights gained fromthe exploration of FQH states have also inspired predic-tions of a variety of other unusual states in other systems. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b One peculiar feature of the FQH states, which was un-derstood quite early, is that they must necessarily havewell-defined charged excitations (quasiparticles) with acharge that is a fraction of the electronic charge. Itwas also predicted that collections of these quasiparticlesshould obey fractional statistics, such that the effectivewave function for the quasiparticles would be multipliedby a complex phase factor when two quasiparticles areinterchanged, in contrast with the factor of ± obtainedon interchange of the familiar bosons or fermions.As we shall describe below, the existence of quasiparti-cles with fractional charge and statistics is essentially aninescapable logical consequence of the existence of frac-tional quantized Hall states. Thus, in one sense, the ob-servation of an FQH plateau might be considered as a di-rect demonstration of the existence in principle of quasi-particles with fractional charge and statistics. However,it is not necessarily true that isolated quasiparticles willform the lowest energy configurations when electrons areadded to or subtracted from a quantized Hall state, andit is not clear how easy it might be to prepare isolatedquasiparticles or to measure directly their charges.In a similar vein, we may ask whether it is possible tosee a direct effect of fractional statistics in an experiment,such as one where there is interference between two pos-sible paths, in which a pair of quasiparticles encircle eachother a different number of times. We shall see that thereare numerous obstacles that need to be overcome to carryout such an interference experiment in practice. Further-more, there are many complications, due particularly tosubtle effects of Coulomb interactions and to the possibleparticipation of different species of quasiparticles, whichmay complicate the interpretation of these experiments.Nevertheless, major progress has been made.In addition to quasiparticles with fractional statistics, certain FQH states have been predicted to have quasi-particles with non-Abelian statistics . In this case, theinterchange of two or more quasiparticles can give riseto a unitary transformation between orthogonal quan-tum states in a Hilbert space containing many degenerateground states. In principle, the existence of such quasi-particles should give rise to some striking experimentalmanifestations, with possible consequences for technol-ogy. However, direct demonstration of the predicted phe-nomena has, again, proved challenging.In the following section, we shall introduce precise def-initions of fractional charge and fractional statistics, andexplain why quasiparticles with these properties are pre-dicted to occur in fractional quantized Hall states. InSections III and IV, we discuss in greater detail the the-ory behind experiments designed to demonstrate most di-rectly the effects of fractional charge and fractional statis-tics, and we review the current status of these experi-ments. In Section V, we discuss in greater detail the con-cept and implications of non-Abelian statistics, and wediscuss some examples of FQH states where non-Abelianstatistics have been proposed to occur. The search fora clear manifestation of non-Abelian statistics by means of Fabry-Perot interferometry is discussed in Section VI.In Section VII, we discuss the alternate geometry of aMach-Zehnder interferometer, which has been realized inan integer quantized Hall state but not yet in an FQHstate, and we review how the combination of fractionalstatistics and fractional charge leads to a flux period con-sistent with the Byers-Yang theorem. In Section VIII, wediscuss several other experimental techniques, which re-veal aspects of FQH effect related to fractional chargeand statistics, but which would not be considered to bedirect observations of these properties. We present con-cluding remarks in Section IX.
II. THE MEANING OF FRACTIONALCHARGE AND FRACTIONAL STATISTICSA. Fractional charge
Fractional charge is relatively easy to define in amodel where the Hamiltonian H contains only short-range forces.[2] For any state | Ψ (cid:105) that is an eigenstateof H , we can define a charge density ρ Ψ ( r ) , which is thetime-independent expectation value of the charge densityoperator ρ ( r ) . If there is an energy gap ∆ E separating | Ψ (cid:105) from all other states of the Hamiltonian, then thedensity ρ Ψ may be obtained with arbitrary precision, inprinciple, by using an apparatus that measures the den-sity averaged over a time scale large compared to (cid:126) / ∆ E .If properly carried out, such a measurement will not af-fect the quantum state | Ψ (cid:105) , and the measurement may berepeated many times with the same results. In practice,the requirement for an energy gap of size ∆ E given abovemay be weakened, in that one may exclude eigenstates of H of lower energy if they result from excitations, relativeto | Ψ (cid:105) that are localized in space far from the measuringpoint r .In this paper, we shall consider typically a large systemwith a Hamiltonian of the form H = H + V (1)where H is at least approximately translationally in-variant in regions far from the system boundaries, and V is a sum of local perturbations, centered around a setof points { r j } , which will be assumed to be far fromthe boundaries. Let ρ ( r ) be the charge density in theground state of H , let Ψ be a low-lying eigenstate ofthe full Hamiltonian H , and and let δρ Ψ = ρ Ψ − ρ . Weshall say that the state | Ψ (cid:105) contains one or more localizedquasiparticles if δρ Ψ ( r ) differs substantially from zero inthe vicinity of at least one of the points r j , but is expo-nentially close to zero at points r that are far from all r j and far from the boundaries. If the point r j is well sep-arated from other regions where V is non-zero, we mayintegrate δρ Ψ ( r ) over the region containing r j where it isnon-zero and thereby obtain the excess charge q j associ-ated with the quasiparticle or quasiparticles near point r j .For an ordinary insulator, if one ignores the possibleeffects of long-range Coulomb interactions, one finds thatthe quasiparticle charge q j will necessarily be an integermultiple of the electron charge − e . For a fractional quan-tized Hall state, as we shall show below, the quasiparticlecharge can have values which are specified rational frac-tions of e .More generally, we can see that the quasiparticlecharge will be a protected quantity, at least for a sys-tem with short range interactions. Its value must remainconstant if the microscopic Hamiltonian is continuouslyvaried, as long as the bulk material retains an energy gapat the Fermi energy and the magnetic field is fixed. Be-cause the bulk material remains effectively an insulator,it cannot carry an electric current over long distances to-wards or away from the quasiparticle. Consequently, thelocalized charge, as well as the background charge densityin the bulk, must remain constant.By an extension of this reasoning, a localized quasi-particle can be moved around if we allow the localizingperturbation V to be time dependent. In particular, if weallow the center r j of an isolated localizing well to movesufficiently slowly as a function of time t , a state that isinitially in eigenstate of the Hamiltonian H at time t willbe in the corresponding eigenstate of the time-dependentHamiltonian H ( t ) at any later time. If the initial statehad a quasiparticle localized at point r j ( t ) , the state attime t will have a quasiparticle at point r j ( t ) . Clearly,the quasiparticle charge q j cannot change in this processif the quasiparticle remains isolated from all other quasi-particles throughout.It should be emphasized that the requirements that V varies only slowly and that the measurement of thecharge density takes place over a time that is slow onthe scale of the ground-state energy gap is essential forthese arguments. An instantaneous measurement of theelectronic charge in any spatial region will always yieldan integer multiple of e .In the presence of long-range Coulomb interactions,the definition of quasiparticle charge is complicated bythe induced polarization of the dielectric medium. Asa familiar example, for a localized electron embedded ina three-dimensional insulator with dielectric constant κ ,the total excess charge in the vicinity of the electron willactually be equal to − e/κ , with the remaining chargedistributed around the boundary of the sample. By con-vention, we divide the local charge into free charge andbound charge, so that the free charge associated with theelectron is said to be − e . Similarly, for a quasiparticlein a fractional quantized Hall state with Coulomb forcesembedded in a dielectric medium, we define the quasi-particle charge q j as the free charge associated with thequasiparticle, which will be equal to the local charge mul-tiplied by κ in this case. It is this free charge which willbe quantized in rational multiples of e .We remark that if an electron is injected at one placeon the surface of a three-dimensional insulator and ismoved through the bulk of the sample to another place on the surface, the total charge transferred between thetwo points will be − e , not − e/κ . This is because theimage charge on the surface of the insulator moves alongwith the electron so the total charge is transferred. Thuswe may say that the total electric current associated withan electron moving at a velocity v is given by − e v , eventhough the local charge is − e/κ . In the case of a quan-tized Hall system, the current associated with a quasi-particle moving through the bulk is more difficult to de-fine, as the system will necessarily have conducting statesalong its edges.What happens if we turn off the localizing perturbation V ? As H is supposed to be translationally invariant, alocalized quasiparticle will not, in general, be an eigen-state of the Hamiltonian. In an ordinary insulator, inthe absence of a magnetic field, the energy eigenstateswill be plane-wave-like superpositions of localized statescentered at positions throughout the sample. For quan-tized Hall states, however, it is possible to create localizedstates for a charged quasiparticle that are eigenstates ofthe Hamiltonian. Of course, these states will be highlydegenerate, due to the many possibilities for choosingthe center r j , and the localized states can be mixed byan arbitrarily small perturbation.As one example, in the presence of a strong magneticfield and a weak electrostatic potential V ( r ) that variesslowly in space, energy eigenstates will generally extendall the way along contour of constant potential, whilebeing localized in the perpendicular direction. One finds,in this case, that a quasiparticle wave packet , which isinitially localized at some point in space, will move alongthe potential contour line, with a velocity v D , given bythe classical formula, v D = E × B /B , where E is thelocal in-plane electric field and B is the perpendicularmagnetic field. B. Fractional statistics
1. Definition in terms of effective wave functions andeffective Hamiltonian
Whereas fractional charge can be easily defined for asingle isolated quasiparticle or for a collection of localizedquasiparticles, the concept of fractional statistics requiresthe consideration of two or more quasiparticles that areable to move around each other or to interchange posi-tions. If one is confined to a suitable low-energy subspace,one may hope to describe the quantum mechanical stateof such a system by an effective wave function ψ eff thatdepends only on the coordinates of the quasiparticles,rather than of all the electrons in the system. The ef-fective wave function should evolve in time according toa Schrödinger equation with some effective Hamiltonian H eff . Fractional statistics will be a characteristic of thecombination ψ eff and H eff .As was first noted by Leinaas and Myrrheim, in 1977,in two dimensions it is possible to extend the formula-tion of quantum mechanics to a situation where the wavefunction of a set of identical particles is multiplied by acomplex phase factor different from ± , provided we mayexclude from consideration points where two quasiparti-cles coincide precisely in space.[3] Specifically, one mayrequire that if one interchanges the positions of two iden-tical particles by moving their coordinates in a counter-clockwise direction along a closed contour C that encloses N C other quasiparticles of the same type, the wave func-tion should be multiplied by a phase factor e − iθ , where θ = (1 + 2 N C ) θ m , (2)where the angle θ m , defined modulo π , is a character-istic of of the type of quasiparticle in question. (We usethe index m to distinguish between different species ofquasiparticles.) If the position of a single quasiparticle ismoved along a closed loop enclosing N C other identicalquasiparticles, the wave function must be multiplied by e − iN C θ m . For cases other than θ m = 0 or π , this requiresthat the wave function be multivalued, or equivalentlythat it is defined on a multi-sheeted Riemann surface.Nevertheless, quantum mechanics can be generalized ina straightforward way to deal with this situation. In acase where θ m (cid:54) = 0 mod π , if the effective Hamiltonian H eff can be written as a local function of the positions r j and the momenta p j = − i (cid:126) ∇ j , with the possibleaddition of long-range Coulomb forces that depend onposition variables only, one says that the quasiparticlesobey fractional statistics , with statistical angle θ m . Suchquasiparticles are often referred to as anyons .[4]To describe quasiparticles with fractional statistics,however, it is not actually necessary to employ multi-valued wave functions. The multiple phase factors canbe eliminated by implementation of a unitary transfor-mation, essentially a singular gauge transformation.[3]Specifically, if ψ eff is a multivalued wave function as de-scribed above, let us define a transformed wave function ψ (cid:48) eff by ψ (cid:48) eff { r j } = ψ eff { r j } (cid:89) k 2. Illustrative example As an example to illustrate a physical consequence offractional statistics, let us consider a system containingeither one or two identical anyons with charge q m in anexternal magnetic field B . We shall assume that thereis at most a short-range interaction between the anyons.We also assume a weak circularly symmetric parabolicelectrostatic potential of the form Φ( r ) = K r , (6)with q m K > , in addition to a stronger short-range at-tractive potential that can trap a localized quasiparticlenear the origin. The case of a single charged particle ina uniform magnetic field and a weak parabolic potentialis exactly solvable. For a particle in the lowest Landaulevel, the energy eigenstates will consist of a series of cir-cular orbits with (cid:104) r (cid:105) = 2( n + 1) (cid:126) / | q m B | , n = 0 , , , ..., (7)and energy given by E = E ∗ + q m K (cid:104) r (cid:105) , (8)where E ∗ is a constant. The addition of a localized po-tential well near the origin will have negligible effect onthe energies or eigenstates for large values of n .Let us now consider a system with one quasiparticle,say quasiparticle 1, localized in the well near the origin,and the second quasiparticle sitting in a circular orbit oflarge radius. According to the Bohr-Sommerfeld rules,we should calculate the allowed radii by requiring thatthe action for the circular orbit should be equal to toan integer multiple of π . Because of the Chern-Simonsterm due to the presence of particle 1, the action forquasiparticle 2 will be shifted by an amount δS = (cid:73) a ( r ) · d r = 2 θ m . (9)The result is that (7) will be replaced by (cid:104) r (cid:105) = 2( n + 1 − σθ m /π ) (cid:126) / | q m B | , (10)where σ = sign ( q m B z ) . If θ m (cid:54) = 0 mod π , the set of al-lowed values for r , and hence for the energies for quasi-particle 2, will be different depending on whether quasi-particle 1 is present or not.The above arguments can be generalized to the casewhere one has two indistinguishable quasiparticles in or-bits that are not localized near the origin. In this case,one finds that the set of allowed energy levels will besensitive to θ m mod π . 3. Relation to the microscopic Hamiltonian To make these ideas more concrete, let us return to themicroscopic states for a system containing a given num-ber N of identical quasiparticles. Let | Ψ( { r j } ) (cid:105) be themany-electron state with quasiparticles localized at spec-ified positions ( r , .., r N ) . The set of such states, whichwe here assume to be unique except for a phase, will forman (over-complete) basis for the set of states we are in-terested in. The set of allowed positions r j may includerestrictions, such as a minimum separation between twoquasiparticles. We shall assume that any state in theHilbert space of interest can be written as a superposi-tion of basis states, in the form | Ψ (cid:105) = (cid:90) d r ...d r N ψ eff ( { r j } ) | Ψ( { r j } ) (cid:105) . (11)Once we have made a specific phase choice for the basisstates | Ψ( { r j } ) (cid:105) , we can define a Berry connection, α k ( { r j } ) = i (cid:104) Ψ( { r j } ) |∇ k | Ψ( { r j } ) (cid:105) , (12)where ∇ k is the gradient with respect to the position r k .We may now consider a situation in which the positions of two quasiparticles, labeled k and l , are interchangedby moving them around a specified contour C in a coun-terclockwise fashion, until their final positions are inter-changed from their initial positions, while the positionsof all other quasiparticles are held fixed. The Berry phasefor the process is given by θ Ckl = (cid:90) d r k · α k + (cid:90) d r l · α l , (13)where the integral is taken along the contour. Whereasthe Berry connections α j depend on the particularchoices made for the phases of the basis states, theBerry phase θ Ckl may be seen to be independent of thosechoices, up to an additive multiple of π . Thus the quan-tity e iθ Ckl is independent of the phase choice and is there-fore gauge invariant. For a system of identical anyons ofcharge q m in an external magnetic field, the value of θ Ckl should be given by θ Ckl ∼ (1 + 2 N C ) θ m + 2 πq m B z A C / (cid:126) , (14)where A C is the area enclosed by the contour C , θ m is aconstant characteristic of the type of quasiparticle underconsideration, and N C , as before, is the number of addi-tional quasiparticles enclosed. Equation (14) is supposedto be exact when the contour C is large and quasipar-ticles k and l stay far from all other quasiparticles, butthere can be corrections if these conditions are violated.Nevertheless, the implication of (14) is that with a suit-able choice of gauge, α j may be written in the form α j = q m A ( r j ) + a j ( r j ) , (15)where A is the vector potential due to the applied mag-netic field B , and a j is just the Chern-Simons vectorpotential given by (4). The discussion may be readilyextended to the case where there are several types ofquasiparticle present, in which case the first term in (14)should be replaced by ( θ m +2 (cid:80) (cid:48) m N m (cid:48) C θ mm (cid:48) ) , where N m (cid:48) C is the number of quasiparticles of type m (cid:48) enclosed bythe contour, and the definition of a j must be extended,as described in Eq. (5).Next, we must examine the time evolution of a state | Ψ (cid:105) of the form (11). It is convenient for this purpose touse path integral approach. Then the state at time t canbe related to the state at time 0 by a unitary transfor-mation of the form ψ eff ( { r j } , t ) = (cid:90) d { r (cid:48) k } K ( { r j } , { r (cid:48) k } ) ψ eff ( { r (cid:48) j } , , (16)where the kernel K is given by the sum of e − iS over allpaths connecting the initial and final configurations ofpositions, with S being the action associated with thepath. To a good approximation, we may evaluate S as S = (cid:90) dt (cid:48)(cid:48) U ( { r (cid:48)(cid:48) j } ) + (cid:88) j (cid:90) d r (cid:48)(cid:48) j · α j ( r (cid:48)(cid:48) j ) , (17)where U ( { r (cid:48)(cid:48) j } ) is the expectation value of the microscopicHamiltonian H in the basis state | Ψ { r (cid:48)(cid:48) j }(cid:105) , and the inte-gral is taken along the path from the initial to the finalconfiguration. This expression coincides with the formulafor the action of a collection of particles with charge q m subject to an applied magnetic field and a Chern-Simonsvector potential, in the presence of a position-dependentpotential energy U , in the limit where the effective massof the particle is taken to zero, i.e. , in the limit wherethe particles are all in the lowest Landau level.More generally, U should be replaced by an operatorthat may include terms that are slightly off-diagonal inthe position variables, which would lead to additionalmomentum-dependent terms in the Hamiltonian, includ-ing, perhaps, short-range momentum-dependent inter-actions between the quasiparticles. Matrix elements ofthe microscopic Hamiltonian that mix states in the low-energy subspace we are considering with states outsidethat subspace may be taken into account via perturba-tion theory as corrections to the matrix elements of U .In a similar fashion, the effects of mixing between Lan-dau levels due to interactions in a system of particleswith nonzero mass may be included in a model that isprojected onto a single Landau level by including suit-able corrections to the interactions within the Landaulevel. As long as corrections to the interaction terms re-main short-ranged in space, they can be distinguishedfrom the Chern-Simons interaction, and will not affectthe behavior of well-separated quasiparticles. Thus, thevalues of the statistical angles θ mm (cid:48) remain well-definedand unchanged.The interplay of Landau-level mixing with long-rangeforces can change the apparent values of θ mm , as dis-cussed in Refs. 5 and 6. However, the deviations decayas a power of the distance between quasiparticles. 4. Non-Abelian statistics In our previous discussions, we have assumed that ifthe locations and types of all quasiparticle are specified,there will be a unique low-energy state of the Hamil-tonian corresponding to this specification. However, avery different situation is believed to occur in some spe-cial quantized Hall states. For these states, in a situationwith N localized quasiparticles, there should be a numberof nearly-degenerate low-energy eigenstates which growsexponentially with N . The energy differences betweenthese states should fall off exponentially with the sep-aration between quasiparticles, and they are frequentlytreated as negligible in theoretical discussions.Now, if a set of quasiparticles are slowly moved aroundeach other or interchanged, in such a way that the set offinal positions for each quasiparticle type is identical tothe initial set, the final state of the system will be re-lated to the initial state by a unitary transformation inthe Hilbert space of low-energy eigenstates. Furthermore,if the braiding process is fast compared to the “exponen- tially small" energy splittings of the Hilbert space, theunitary transformation will depend on the topology ofthe braiding, but will be independent of all other detailsof the paths that are taken. For processes that involvemultiple interchanges of quasiparticles, the resulting uni-tary transformation will generally depend on the order inwhich the interchanges have been performed. Hence, thequasiparticles are said to obey “non-Abelian statistics".Because a full discussion of various types of non-Abelian statistics and the ways in which they may bemanifest in quantum Hall systems is complicated, weshall defer that discussion until later sections of this pa-per, and shall first concentrate on states with Abelianfractional statistics. C. Application to quantized Hall states 1. Fractional charge In Laughlin’s landmark 1983 paper, which describedhis trial wave function for the fractional Hall groundstate at ν = 1 / and related fractions, he also pro-posed wave functions for the elementary quasiparticles,often denoted as quasielectrons and quasiholes.[7] Theadded electric charges associated with these proposedwave functions were, indeed, fractions of an electroncharge, viz. , q c = ± e/ . Since the trial wave functionsare not exact eigenstates of the Hamiltonian for a re-alistic model with Coulomb interactions, one might betempted to question the exactness of the charge quan-tization based on them. However, Laughlin presenteda more general argument that quasiparticles with frac-tional charge must be a feature of any fractional quan-tized Hall state.Consider a two-dimensional electron system in a frac-tional quantized Hall state with filling factor ν on a largedisk of radius R . Let us puncture the disk with a holeof diameter a at the center of the disk, and let us threadan infinite solenoid with radius less than a through thehole. In two dimensions, the scattering cross section ofa barrier of radius a will vanish [8] in the limit a → ,proportional to / ln | a | . Thus, in the limit a → , thesolenoid will have no effect on electrons in the systemwhen there is no flux through the solenoid.Now, start with a situation where the system is initiallyin its ground state and there is no flux in the solenoid,and gradually increase the flux until the solenoid containsprecisely one flux quantum, pointing in the same direc-tion as the uniform magnetic field. [Note: In our discus-sions of quantum Hall systems, throughout this paper,we shall assume that the applied magnetic field B pointsalong the negative z axis, unless otherwise specified, and B = | B | > . ] The time-dependent flux will generatean azimuthal electric field, which will drive electrons intowards the origin, due to the non-vanishing Hall conduc-tance. A simple calculation shows that the total chargeaccumulated near the origin will be equal to − νe . Thisextra charge will have come from the edge of the sys-tem, where there is necessarily a reservoir of low-energyconducting states.[9] Since there is a finite energy gapin the bulk of the system, we expect, according to theadiabatic theorem, that the final state will again be anenergy eigenstate of the system. (Although, in principle,the adiabatic theorem could break down at an instantwhere the added energy of the system due to the chargeat the origin crosses the energy for adding the chargeback to a state at the edge of the system, the matrixelement for such a transfer will be exponentially small,if the radius R is very large compared to the magneticlength. In addition, for a system with circular symme-try, the matrix element will be identically zero by angularmomentum conservation.)Although the Hamiltonian with the added flux quan-tum is mathematically different from the original Hamil-tonian, we can make a gauge transformation that elim-inates the vector potential due to the solenoid, multi-plying the wave functions by a position-dependent phasefactor and restoring the Hamiltonian to its original form.Thus, the original Hamiltonian must have an eigenstatewith the same energy and charge distribution as the onewe have found for the state with an added flux quantum.Of course, we can generate a quasiparticle with charge + νe by repeating the above procedure with solenoid fluxin the opposite direction. There is no guarantee, however,that quasiparticles with charge ± νe have the lowest en-ergy or the smallest charge of any possible quasiparticlein the system. In particular if ν = p/q , where p and q are integers with no common divisor, one can alwaysconstruct a quasiparticle with charge q m = ± e/q . Since p and q have no common divisor, there will necessarilyexist integers n and n (cid:48) such that nq − n (cid:48) p = 1 . Thena combination of n (cid:48) quasiparticles of charge νe and n electrons will have total charge − e/q .These arguments do not require that e/q is necessarilythe smallest charge for a quasiparticle in the system. Forexample, the various competing models [10] proposed toexplain the even-denominator quantized Hall state ob-served at ν = 5 / have quasiparticles with charge ± e/ .Levin and Stern have argued[11], in fact, that for anyfractional quantized Hall state with even denominator q ,there must exist quasiparticles with charge q m = ± e/ q . 2. Fractional statistics The prediction that quasiparticles in fractional quan-tized Hall states should obey fractional statistics wasmade in 1984, by Halperin,[12] and slightly later, byArovas, Schreiffer and Wilczek.[13] The analysis ofHalperin was based on the behavior of effective wavefunctions for collections of quasiparticles, similar to thediscussion in Subsection (II B 1), above. By contrast theanalysis of Arovas, et al. , made use of the definition pre-sented in Subsection (II B 3), specifically, by calculatingthe Berry phase acquired on interchanging the positions of two quasiholes in the ν = 1 / state, using Laughlin’strial wave function for the quasiholes.The analysis of Ref. 12 was motivated by the fol-lowing set of observations. Laughlin’s wave functionsfor the FQH states at ν = 1 /m involve a factor of (cid:81) j 3. Edge modes In our previous discussions, we have focused on theproperties of localized quasiparticles, or collections ofquasiparticles, that are far from any edges of the sample.However, fractionally charged quasiparticles can also ex-ist in delocalized states along the edges of a sample, or ata boundary between two quantized Hall states with dif-ferent Hall conductivity. As was originally noted in Ref.9, there must be gapless modes at a boundary between agapped quantum Hall liquid and a vacuum. In the caseof integer quantized Hall states, these edge modes maybe understood as orbits for electrons at the Fermi level,which propagate along the edge in a particular direction.For FQH states, the edge modes may be similarly inter-preted as orbits for quasiparticles of various types.Though the charge on an edge will be conserved if thereare no contacts to the edge and the edge is far from allother edges of the sample, charges can tunnel betweentwo opposite edges of a sample if there is a narrow con-striction which brings them close together. When thetunneling strength is small, transfer of charge from oneedge to another can occur in discrete units which may beinterpreted as the charge of a tunneling quasiparticle. Ina geometry with two or more constrictions, there can beinterference features that one can attribute to the differ-ence in phase accumulated by a quasiparticle in tunnelingfrom one edge to the opposite edge via the possible pathsinvolving tunneling at different constrictions. For quasi-particles with fractional statistics, the accumulated phasewill be sensitive to the number of other quasiparticles en-closed by the difference in paths. Therefore, interferenceexperiments can provide a means for observing effects ofthe fractional statistics.Although fractional statistics as well as fractionalcharge may be measured, in principle, with experimentson quasiparticles far from any sample edges, as illustratedby the gedanken experiments described above, in prac-tice studies of fractional statistics have always employedinterferometers with tunneling between edges. As de-scribed below, a few experiments have succeeded in mea-suring fractional charge accumulation in localized regionsfar from any edge of the sample, but even here, the ma-jority of experiments have involved edge modes and havemeasured the charges of quasiparticles tunneling fromone edge to another across a constriction.A more detailed discussion of edge modes in FQHstates will be given below in the section on non-Abelianstatistics. III. EXPERIMENTAL PROBES OFFRACTIONAL CHARGE Fractional charge was one of the earliest predictionsof the FQH theory, but it took more than a decade todirectly observe it. Three experimental techniques havebeen implemented: shot noise [25–33], Aharonov-Bohminterferometry [34–37], and charging spectroscopy [38–44]. The bulk of our knowledge comes from shot noise ex-periments, and we start with their review. This includesa discussion of a recent experiment on photo-assisted shotnoise [32]. We then discuss two approaches to chargingspectroscopy. Because the interferometry technique usesthe same setup to probe fractional statistics and frac-tional charge, we shall defer discussion of both applica-tions until the following section. A. Shot noise Suppose that particles of charge q m tunnel througha high barrier between two conductors. The tunnelingrate from the higher to lower electrochemical potential is T q so that the average current (cid:104) I ( t ) (cid:105) = q m T q . The shotnoise technique focuses on the low-frequency fluctuationsof the current. The noise is defined as S = (cid:90) ∞−∞ dt exp( iωt ) (cid:104) I ( t ) I (0) + I (0) I ( t ) (cid:105) , (19)where we are interested in the ω → limit. The in-tegral reduces to the mean square fluctuation of thetotal transmitted charge over a long time t , S =lim t →∞ (cid:104) [∆ Q ( t )] (cid:105) /t . In the low-transmission limit, this simplifies to S = 2 q m I T , (20)where I T is the average tunneling current. The quasipar-ticle charge can be extracted, if both noise and currentare known. The derivation does not depend on any de-tails of the Hamiltonian and applies as long as T q is smalland no charges tunnel uphill from the lower to higherelectrochemical potential. The latter is true as long asthe temperature is low compared to the voltage energyscale q m V . Measurements of the current noise at a finitefrequency ω can be used to determine the quasiparticlecharge provided that ω is less than a value that is neces-sarily smaller than microscopic frequencies such as (cid:126) − times the energy gap and I/e but in practice is likelyto be limited by details such as capacitive lags on thesample or characteristics of the measuring apparatus. FIG. 1. Shot noise setup with chiral edges. The incomingcurrent from source S splits into the transmitted current I into drain D1 and the tunneling current I T into drain D2.Current fluctuations can be measured at D1 or D2. Shot noise was used with success to measure the elec-tron charge [45] as early as in 1918, but almost a centuryelapsed before it was extended to FQH quasiparticles inRefs. 25 and 26. The schematics[46–48] of the experi-mental setup are shown in Fig. 1. In the quantum Halleffect, the bulk is gapped, and charges travel along edges,which are maintained at different voltages in the settingof Fig. 1. A narrow constriction allows charge tunnelingbetween the edges. Since tunneling charges cross the bulkof the sample, they are restricted to the allowed quasi-particle charges in the bulk. Usually, but not always, thelowest quasiparticle charge dominates the limit of weaktunneling and can be extracted from shot noise. Thenoise is detected in the drain at the end of one of theedges. In the absence of the Nyquist noise, at zero tem-perature, the drain noise is the same as the noise of thetunneling current I T .In real experiments, the temperature and the frequency ω remain finite. A finite frequency does not affect theinterpretation of the data as long is /ω exceeds all othertime scales in the problem, such as the thermal scale0 (cid:126) /T and the Josephson scale (cid:126) /q m V . Experiments aretypically performed at ω ∼ MHz.The effect of a finite temperature is more complex.Access to fragile FQH states requires simultaneous lowtemperatures and voltages, and the limit of T (cid:28) q m V might not be available. Fortunately, a universal relation[49] exists among the tunneling current I T , the voltage,the temperature, and the noise: S = 2 q m I T coth q m V k B T − k B T ∂I T ∂V + 4 k B T G, (21)where G = νe /h is the quantized Hall conductance. Eq.(21) contains the non-linear tunneling conductance ∂I T ∂V and is a consequence of detailed balance [50] and fluctu-ation relations [51, 52]. It applies irrespectively of mi-croscopic details as long as tunneling is weak and theedges are chiral, that is, all edge modes propagate inthe same direction, as is the case at the filling factors ν = 1 / (2 n + 1) .The interpretation of experiments on FQH states withnon-chiral edges is complicated by hot spot formation[49]. A non-chiral edge contains a downstream chargemode that carries charge and energy and one or moreneutral modes that carry energy in the opposite upstreamdirection. When a biased charged mode arrives to agrounded terminal, Joule heat is dissipated. Some of itis carried back by neutral modes. This heat arrives tothe source and affects the thermal noise of the outgoingcurrent. This, in turn, affects the measured noise in thedrain so that Eq. (21) no longer applies. The problemcan be alleviated [49] with floating contacts along theedge. (See Fig. 2.) The contacts absorb the excess heatcarried by the neutral modes. FIG. 2. Shot noise setup useful for edges with upstream neu-tral modes. Drain D sends the current in a narrow rangeof frequencies to the amplifier that measures the voltagenoise.The rest of the current arrives to the ground, wherea hot spot h forms. Floating contacts F absorb excess heatcarried by the upstream modes emitted from the hot spot andthe tunneling contact. The first experiments [25, 26] revealed charges e/ atthe filling factor ν = 1 / . Subsequent work [27–30, 33] reported quasiparticle charges e/ at ν = 2 / , / , / and / , e/ at ν = 2 / , and e/ at ν = 3 / in agreementwith the lowest quasiparticle charges predicted at thosefilling factors. Charge values, consistent with the lowesttheoretical values, q m ∼ . e at ν = 5 / and q m ∼ . − . e at ν = 7 / were also observed [30, 31], but onlyat intermediate tunneling rates. The observed tunnelingcharge grows in the weak-tunneling limit.A challenge for Eq. (21) is the growth of the observed q m as the temperature goes down at several filling fac-tors. This does not happen at ν = 1 / , where q m staysat e/ (see, however, Ref. 31). On the other hand [28], q m reaches e/ at the lowest temperatures at ν = 2 / ,the low-temperature q m reaches e/ at ν = 2 / , and q m reaches . e/ at the lowest available temperatures at ν = 3 / . A possible explanation consists in the compe-tition of the tunneling of quasiparticles with the charge e/q and composite quasiparticles of the charge pe/q at ν = p/q . The bulk energy cost of a quasiparticle growswith its charge. Hence, the bare tunneling amplitude ofa quasiparticle through a constriction between two edgesis higher for a lower charge. This is not necessarily thecase for the renormalized tunneling amplitude that con-trols low-temperature transport. It was indeed observedin Refs. 53 and 54 that the tunneling of the charges pe/q is more relevant in the renormalization group sense thanthe tunneling of the charges e/q at low energies at thefilling factors p/q = p/ (2 p + 1) . Thus, high-temperatureand low-temperature tunneling may be dominated by dif-ferent charges, and both charges compete at intermedi-ate temperatures. Similar physics was also proposed at ν = 5 / in the presence of /f noise [55]. Recent dataraise a question about this interpretation. The tunnelingcharge is usually extracted from the autocorrelation ofthe drain current. It is also possible to extract it fromthe cross-correlation of the currents in two drains (Fig.1). It was observed [56] that the autocorrelation givesthe effective charge that grows at low T , yet the chargeextracted from the cross-correlation in the same sampleremains at its high-temperature value.The above discussion focuses on weak tunneling sinceEq. (20) holds only in that limit. Much interestingphysics is observed at intermediate transmissions (see,e.g., Ref. 49), but the Fano factor of the noise cannot beinterpreted in terms of the tunneling charge in that case,which is thus beyond the scope of the review. Theorypredicts a rather boring picture at strong quasiparticletunneling. This case is best understood in terms of thedual geometry (Fig. 3), where electrons tunnel betweentwo separate FQH liquids. The Schottky noise followsEq. (20) with the electron charge in place of q m . How-ever, a surprise was found when a dilute beam of frac-tionally charged quasiparticles impinged on a weak linkbetween two FQH liquids. The observed tunneling chargeequaled a fractional quasiparticle charge [57]. The stan-dard theoretical toolbox sheds no light on this puzzlingphenomenon [58].It should be noted that charge density in an electrostat-1 FIG. 3. The geometry with two FQH regions is dual to thegeometry from Fig. 1. ically defined constriction is lower than than in the FQHbulk. This may affect the filling factor in the constrictionand the nature of the quasiparticles whose tunneling isallowed. In particular, fractional tunneling charges wereobserved in the integer quantum Hall effect [59]. Chargefractionalization on integer edges is also possible [60] dueto purely edge physics that is beyond the scope of thisreview. 1. Photo-assisted shot noise This technique combines a dc bias V with an ac bias V ac ∼ cos( ωt ) in the geometry of Fig. 1. A charge q , emitted from the source, acquires a time-dependentphase φ ( t ) = q (cid:82) dtV ac ( t ) / (cid:126) . This can be interpreted as ashift in the energy of tunneling quasiparticles. Withoutan ac bias, the available energy is q m V . The absorptionof n quanta of the ac field shifts the energy to q m V + n (cid:126) ω .The observed shot noise is then a weighted sum of thenoises at dc voltages V + n (cid:126) ω/q m , S = ∞ (cid:88) n = −∞ w n S dc (cid:18) V + n (cid:126) ωq m (cid:19) , (22)where the weight w n reflects the probability to absorb n quanta. The low-temperature noise is singular at V = (cid:126) ω/q m . This was used [32] to verify q m = e/ at ν = 1 / and q m = e/ at ν = 2 / .In a related experiment [33], high frequency noise mea-surements at f ≈ GHz with dc bias show q = 0 . e at ν = 4 / and q = 0 . e at ν = 2 / . B. Charging spectroscopy In principle, the most direct way of measuring thecharge of a quasiparticle is to form a weak potential well,using a local external gate, which is just strong enough to bind a single quasiparticle, and to measure the changein the electric charge in the region about the well when aquasiparticle is induced to enter or leave the trap. Thiscould be done by varying the depth of the potential well,by applying voltage to a contact which varies the electro-chemical potential of the two-dimensional electron sys-tem, or by changing the magnetic field to vary the chem-ical potential of the surrounding FQH state. Alterna-tively, one could employ a potential well big enough toaccommodate many quasiparticles, and one could mea-sure the jump in charge each time a new quasiparticleenters the well.In practice, however, an absolute measurement of thelocal electric charge is difficult. It is much easier to mea-sure values of the varying gate voltage or other parame-ters where the quasiparticle enters, and to calculate thequasiparticle charge based on the spacing between succes-sive charge jumps. Even if discontinuities in the accumu-lated charge are largely smoothed out due to finite tem-perature effects or external noise, weakened sinusoidaloscillations in the accumulated charge may persist, andone could measure their period. Under proper conditions,the dominant factor determining the number of quasipar-ticles in a well will be a charging energy, which would beminimized when the accumulated charge Q is as close aspossible to a value Q ∗ that varies continuously with pa-rameters such as the gate voltage. If charges can onlyenter in units of the quasiparticle charge, q m , then thespacing ∆ Q ∗ between charge jumps will be equal to q m .Furthermore, if one can carry out the same experiment inthe FQH state and an integer quantized Hall state, andone can be confident that the geometry of the well is thesame in both cases, then the value of q m /e is given bythe ratio between the periods of oscillation as a functionof gate voltage in the FQH and integer cases. 1. Measurements of tunneling through an antidot The earliest version [38, 40] of charging spectroscopyinvolved tunneling through an antidot inside a constric-tion between two FQH edges (Fig. 4). This setting isrelated to that of interferometry, addressed in the nextsection. The electron gas is depleted inside the antidotand hence an FQH edge forms around it. Quasiparti-cles travel through the constriction by tunneling in andout of that edge. The size of the antidot is controlledby the depleting gate voltage. The technique probes howthe conductance through the antidot depends on the gatevoltage and the magnetic field.The dependence can be understood from the pic-ture of quasiparticle orbits, introduced in Section II.B.2.Changes of the magnetic field or the gate voltage resultin an orbit periodically crossing the chemical potential.When this happens, a resonance is seen in the transmis-sion through the constriction. The period in voltage cor-responds to adding or subtracting a quasiparticle fromthe antidot. The quasiparticle charge can be found from2 FIG. 4. Antidot geometry. Charge tunnels between the up-per and lower edges through the edges of an antidot (greydisk) where the charge is depleted. Purple area is occupiedby electrons in an FQH state. the observed period in the voltage and the geometric ca-pacitance. The latter can be approximately extractedfrom the nominal area of the antidot, and can be checkedwith the magnetic field periodicity of the conductance.The calibration may be further checked by comparingwith measurements in an integer quantized state.To understand the magnetic field periodicity, it is nec-essary to take into account fractional statistics as well asfractional charge. The allowed orbits are determined bythe Bohr-Sommerfeld quantization rule in terms of thephase, accumulated by an anyon on a closed orbit. Thephase has two contributions, First, there is an Aharonov-Bohm phase for a particle making a circle around theantidot. For an e/ particle in the Laughlin state at ν = 1 / , the Aharonov-Bohm phase is φ AB = 2 π Φ / ,where Φ is the magnetic flux through the antidot and Φ = hc/e is the magnetic flux quantum. In addition,each localized quasihole in the dot contributes the statis-tical phase − π/ . Since a new quasihole is created ordestroyed every time the flux changes by Φ at a fixedcharge density, the total phase accumulated on an orbitis periodic with the period of a flux quantum. Thus, inrelating the magnetic field period to the area of the anti-dot, one must take into account the fractional statisticsas well as the fractional charge.The experimental results [38] are consistent with thetheoretically predicted charge e/ at ν = 1 / . Yet, itwas argued that the fractional charge is not the only wayto understand the data [39]. One could instead startwith a picture of single-electron orbits around the an-tidot and assume that electron correlations ensure thatonly / of them are populated. Resonant transmissionwould still be observed when an orbit crosses the chem-ical potential. This predicts the same periodicity as thequasiparticle picture. Besides, electrostatic effects maynot be captured by the above single-particle picture (seethe next section).An additional drawback of this technique is that in or-der for tunneling to occur, the antidot must be physicallyclose to an edge of the sample. Thus one might questionwhether the results are necessarily reflective of the prop- erties of excitations in the bulk. One might also questionwhether the gate period obtained from a transport mea-surement necessarily reflects the period for charge occu-pancy of the antidot.It should be noted that a similar technique [43] showedexcitations of charge e/ at ν = 2 / . Very re-cently, charge e/ was reported in an antidot tunnelingexperiment[44] in graphene at ν = ± / . 2. Single-electron transistor technique Difficulties in the interpretation of the antidot data ne-cessitated a different strategy. Thus, later experimentsused a different approach [41, 42]: A single electron tran-sistor (SET), which is sensitive to variations in the localelectrostatic potential, was placed on the surface of theheterostructure that embeds the FQH liquid, or on ascanning tip just above the surface. The SET was used todetect potential jumps when a quasiparticle or quasiholeenters or leaves a local potential well created by fluctua-tions in the doping density, which were found to have aphysical scale on the order of 200 nm, large compared tothe magnetic length.The SET technique was first developed [61] for 2D het-erostructures outside the quantum Hall regime and al-lowed the spatial resolution of 100 nm. It was extendedto studies of the integer quantum Hall effect in Refs. 62and 63.In a subsequent development, fractional charges inFQH liquids were reported in Refs. 41 and 42. As was ex-plained above, the entry of new quasiparticles into a wellcan be controlled with the gate voltage, and the quasipar-ticle charge, relative to that of an electron in an integerquantized state, can be extracted from the spacing ofthe jumps as a function of the voltage, The experimentalresults [41] are consistent with charge- e/ excitations at ν = 1 / and ν = 2 / . The absolute value of the quasipar-ticle charge could also be extracted, with lesser accuracy,from the SET measurements and were consistent withthe value e/ . In the second Landau level, comparisonbetween measurements at ν = 5 / and ν = 7 / obtainedthe ratio [42] q m, / /q m, / = 3 / in agreement with thetheoretical expectation that the charges should be e/ and e/ in the two cases. For a detailed theoretical dis-cussion at ν = 5 / , see Ref. 64. IV. EXPERIMENTAL PROBES OFFRACTIONAL STATISTICS Fractional statistics were defined in terms of phases ac-cumulated by anyons exchanging their positions or run-ning around other anyons. This makes interferometry[65] the most direct probe of statistics, since that tech-nique is directly sensitive to phase differences accumu-lated by particles on different possible paths between the3same endpoints, which could depend on whether the dif-ference in paths encloses some other quasiparticles.The simplest Fabry-Perot geometry [65] is illustratedin Fig. 5. In the illustrated ideal case, the bulk of the sys-tem is in an almost perfect quantum Hall state, where theFermi level falls inside an energy gap of the pure system,but there are a small number of localized states inside thegap, due to impurities, which may become occupied orempty as the Fermi level is varied inside the gap. We havealso assumed that there is only one chiral mode at thesample boundaries, carrying quasiparticles in the direc-tion shown by the arrows, and that tunneling between op-posite edges can take place only in the two constrictions.In the weak tunneling limit, two paths connect the sourcein the lower left corner and the drain in the upper leftcorner. Their phase difference combines an Aharonov-Bohm phase in the external magnetic field with a sta-tistical phase. Thus, the technique allows probing bothfractional charge and statistics. Several other geometrieshave been proposed, with Mach-Zehnder interferometry[66] attracting particular interest. Experimental imple-mentation proved difficult for all geometries, but recentyears have brought promising results [34, 37, 67, 68] inthe Fabry-Perot approach, and we restrict ourselves tothat geometry in the current Section.Very recently, a somewhat less direct observation offractional statistics was accomplished with an anyon col-lider [69, 70], which will be discussed in Section IV.B.,below. Mach-Zehnder interferometry will be discussed inSection VII. Several other techniques yield informationabout statistics, which we briefly address in Section VIII,with the emphasis on thermal transport and tunnelingexperiments. Interferometer experiments designed to re-veal effects of non-Abelian statistics near filling factors ν = 5 / and 7/2 will be discussed in Section VI.As we shall see below, there are several major chal-lenges to the interpretation of interferometry data. Somemajor complications arise due to effects of Coulomb in-teraction. The situation also becomes more complicatedin states with more than one propagating edge mode.Most significantly, in most experiments, the region insidethe interferometer is not in the ideal quantum Hall statedescribed above, which we refer to as an incompressible state. Rather, the bulk is typically in a compressible state, where the Fermi level does not fall inside an en-ergy gap of the pure system.[71, 72] In this case, therewill be a large density of localized quasiparticles or quasi-holes present in equilibrium, and it costs relatively littleenergy to add or subtract one quasiparticle. The result,after thermal fluctuations are taken into account, is thatinterference patterns tend to fall into one of two cate-gories, which are generally described as Aharonov-Bohm (AB) and Coulomb-Dominated (CD), as, at least for inte-ger quantum Hall states, the difference between the twostates is determined by the importance of Coulomb inter-actions between charges in the bulk and charges on the in-terferometer edge, relative to an energy scale set by char-acteristics of the edge. [73, 74]. For FQH states, these labels may be somewhat of a misnomer, as one predictsin some cases that behavior of the Coulomb-Dominatedtype may be found in the compressible regime, even whenthe interaction between bulk and edge is very weak. Toemphasize this point we will sometimes use “CD-like” inplace of “CD”. It should also be emphasized that the be-havior of an FQH state in an incompressible regime isdifferent than either the Aharonov-Bohm or Coulomb-Dominated behavior in the compressible regime, as willbe discussed below. FIG. 5. Ideal Fabry-Perot interferometer. Current tunnelsbetween the lower and upper edges at two tunneling contacts.Two crosses show localized quasiparticles inside the interfer-ence loop. A. Fabry-Perot Interferometry 1. The ideal case We shall first consider the ideal case described above.We assume that the electron density drops rather sharplyto zero at the boundary, and there is just a single edgemode propagating along the boundary, as indicated inFig. 5. A quasiparticle propagating on the interferometerboundary may then be described by a Hamiltonian of theform ˆ H = ˆ H edge +[Γ exp( iφ ) ˆ T + Γ exp( iφ ) ˆ T + h.c. ] , (23)where ˆ H edge describes charge propagation on the upperand lower edges, the operators ˆ T , move a quasiparticleof charge q m from the lower edge to the upper edge atthe two constrictions, and Γ , exp( iφ , ) are the associ-ated tunneling amplitudes. We shall focus on the limitof weak tunneling, where the current between the lowerand upper edges is much less than the incoming current νe V /h , where V is the voltage difference between thelower and upper edges. The quasiparticle tunneling ratecan then be extracted from Fermi’s golden rule, p = [Γ + Γ ] r ( V, T ) + 2Γ Γ cos( φ − φ ) r ( V, T ) , (24)4where r and r depend on V and the temperature T .Hence the tunneling current between the lower and upperedges of the interferometer will be given by I t = q m [Γ + Γ ] r ( V, T )+2 q m Γ Γ cos( φ − φ ) r ( V, T ) . (25)To realize the experimental configuration illustrated inFigure 5, one can connect a current source at voltage V to the lower left corner, and attach grounded contacts tothe lower right and and upper left corners. The back-scattered current I t will then be equal to the currentflowing into the upper left contact.The phase difference θ = φ − φ = α − φ AB − φ s combines two key pieces of information: the quasiparti-cle charge q m through the Aharonov-Bohm phase φ AB = − πq m Φ /e Φ and the statistical phase φ s accumulatedby a quasiparticle on the trajectory around the anyons,trapped inside the interferometer. Here Φ = BA is thetotal magnetic flux through the area A enclosed by thepaths of the edge states between the constrictions. Weignore any additional slow dependence of the matrix ele-ments r , on the magnetic field. The constant α will beset to zero without the loss of generality.To get access to the information encoded in θ , an ex-perimentalist needs to look for oscillations in the currentas one varies the magnetic field and/or the area of theinterferometer. The area may be varied by applying avoltage V g to external gates along the sides of the de-vice. We assume that the area does not depend on themagnetic field. This assumption is not crucial for the in-terpretation of the data in the incompressible regime. Wewill lift it in the discussion of the non-ideal compressiblecase.If no quasiparticles enter or leave the interference re-gion, contours of constant θ should lie along lines in theplane of | B | and V g with slope dVdB = − AB ( ∂A/∂V g ) . (26)In the case of ν = 1 / , the spacing ∆ B between succes-sive conductivity maxima at fixed V g should equal /A ,while the spacing ∆ V g at fixed B should correspond toan area change that contains one electron. At certainvalues of the parameters, however, it may be favorablefor a quasiparticle to enter or leave a localized impu-rity state in the interferometer, at which point we wouldexpect a jump in phase by an amount equal to ± θ m ,caused by a change in the value of φ s . For a quasiparti-cle in the Laughlin state at ν = 1 / , it is predicted that θ m = 2 π/ .Behavior of this type was observed in recentexperiments[68] by Nakamura, et al., as shown in Fig. 6.It is possible, of course, to ask whether the reported phasejumps could have been caused by some effect other thanfractional statistics, such as Coulomb interactions be-tween localized quasiparticles and the conducting statesat the interferometer edges, which could cause a jump d V g ( m V ) B (T) -4-3-2-101234 d G (x10 -2 e /h) Δ𝜃 2𝜋 = −0.32Δ𝜃2𝜋 = −0.38Δ𝜃2𝜋 = −0.28Δ𝜃 2𝜋 = −0.29 FIG. 6. From Ref. 68. Conductance through the Fabry-Perotinterferometer oscillates when the magnetic field or the side-gate voltage changes. The Aharonov-Bohm-type behavior iscombined with phase jumps ∆ θ π when anyons enter the in-terferometer. Gray lines and dashed lines are guides to theeye. in the area enclosed by the interfering trajectories. (Seediscussion in Subsection IV A 2.) However, the samplein these experiments had nearby conducting planes de-signed to screen Coulomb interactions as much as possi-ble. Moreover, it would be peculiar if phase jumps causedby residual Coulomb interactions would all have the samesize for quasiparticle states localized at different impuritypositions in the sample, and that these phase jumps justhappened to be close to the value predicted by theory.The alternate possibilities should be further checked andhopefully ruled out by additional experiments, but as-suming that the interpretation is correct, the results ofRef. 68 provide as direct a demonstration as one couldimagine of fractional statistics and a measurement of thestatistical phase of quasiparticles in the ν = 1 / FQHstate.In order to prepare a sample where one could enterthe ideal incompressible regime and still see Aharonov-Bohm oscillations, the authors of Ref. 68 had to over-come major difficulties. The challenge comes from con-flicting demands on the interferometer size imposed byweak Coulomb interaction and strong phase coherence.The interaction can be suppressed in a large interfer-ometer, but phase coherence is favored by a small de-vice size. A key improvement, described in Refs. 37and 68 came from introducing ancillary wells that screenCoulomb forces in the heterostructure.It is important to note that the simple results shownin Fig. 6 were only observed over a limited range of mag-netic field. This is to be expected because outside a cer-tain range, the Fermi level will no longer be inside theenergy gap of the ideal FQH state. In that case, we canexpect that the sample would fall into the compressibleregime described above, where there will be a large num-ber of quasiparticles or quasiholes inside the interferom-eter, with only a small energy barrier to add or subtract5an additional quasiparticle[75]. 2. Interferometer with a compressible bulk We present here a brief summary of our current the-oretical expectations for the behavior of a Fabry-Perotquantum Hall interferometer in the compressible situa-tion, which will apply to the integer quantum Hall regimeas well as to FQH systems. Although these theoreticalpredictions have been confirmed in a variety of experi-ments in the integer regime, the reader should be warnedthat there has been little success so far in observing os-cillations of the predicted type in FQH states. FIG. 7. From Ref. 74. Fabry-Perot interferometer in theinteger quantum Hall regime. The channel that separates ν = 0 from ν = 1 is fully transmitted. The channel thatseparates ν = 2 from ν = 3 is fully reflected. The channelthat separates ν = 1 from ν = 2 is partially reflected in theconstrictions, where the charge density is lower than in thecenter of the device. Dotted lines show tunneling across theconstrictions, applicable to the case of weak backscattering.Arrow heads, which indicate the directions of particle propa-gation, are shown here for the case of a magnetic field pointingtowards the viewer. Consider the example from the integer quantized Hallregime, illustrated in Fig. 7. The bulk of the system is ina state with a quantized Hall conductance given by ν = 3 .The LL filling factor f near the center of the sample couldbe anywhere in the range . < f < . ; deviations fromthe ideal value of f = 3 are accommodated by a finitedensity of localized electrons in regions with f > andlocalized holes for f < . Near edges of the sample, theelectron density drops to zero, and f drops off accord-ingly. In our somewhat simplified model, we assume thatthere are quantized Hall strips with quantum numbers ν = 2 , and 0 in the edge region, with a single propa-gating edge mode separating each region. The locationsof the propagating modes should fall roughly where theelectron density is such that the local Landau-level fill-ing factor is 2.5, 1.5, or 0.5. For situation illustrated inthe figure, the density in the constriction is supposed tobe slightly less than f = 2 . The edge state separating ν = 2 and ν = 3 is totally reflected outside the constric-tion, the edge state separating ν = 0 and ν = 1 is totallytransmitted, while the edge state separating ν = 1 and ν = 2 is partially transmitted. It is this last mode thatis relevant in an interference experiment.The situation in Fig. 7 can readily be extended toFQH states. For example, if we interpret the labeledfilling factors as effective fillings for composite fermions,with two flux quanta attached to each electron, the Hallstates become quantized states with ν = 1 / , / and / . More generally, we shall assume that there is asingle partially-transmitted edge state, which separatesinner and outer regions with quantum numbers ν in and ν out , with ν in > ν out We shall assume that the tunnel-ing processes occur at one well-defined point within eachconstriction, and we shall define the interference area A I as the area enclosed by the interfering edge state betweenthese points. We also define q in and q out as the charges ofthe fundamental quasiparticles in quantized Hall stateswith ν in and ν out . For the fractional case, we shall con-fine our discussion to the situation where ν in correspondsto a Jain state in the bottom half of the lowest LL, so wemay write ν in = p ps + 1 , q in = e ps + 1 , (27)where p and s are positive integers. The values of ν out and q out are obtained by replacing p by p − in the aboveformulas. The situation in Fig. 7 corresponds to p = 2 . Next we define N L as the net number of quasiparticlesof charge q in inside the area A I . The number N L takesinto account the total excess charge in area, includingcharges in regions with ν > ν in as well as positive or neg-ative quasiparticles localized at density inhomogeneitieswithin the ν in region. Specifically, it is related to thetotal electric charge Q inside the interference area A I by Q = N L q in − A I ν in eB/ Φ (28)If we assume that the dominant tunneling processes atthe constrictions involve quasiparticles with charge q in ,then the interference phase seen by the tunneling parti-cles will be given by θ = − N L θ in +2 πBA I q in /e Φ , (29)where θ in is the statistical phase associated with thequasiparticles of charge q in , given by [16] θ in = π (cid:20) − s ps + 1 (cid:21) . (30)We remind the reader that our sign conventions assumethat the field points along the − z direction, and B = | B | is the magnitude of the field. The phase θ in would havehad the opposite sign if the magnetic field had been cho-sen to be in the positive z direction. Note that the statis-tical phase is the same for a particle and its antiparticle.A key assumption is that N L is restricted to take oninteger values (positive or negative), because the local-ized states inside the interfering edge are isolated fromthe states outside and from the edge itself. [74] This does6not mean that N L is frozen in time, only that it is con-stant on the time scale for a quasiparticle in the edgestate to move along the length of the interferometer. Weassume that on the longer laboratory time scale, chargescan hop readily from one localized state to another andthat occupations will take on an equilibrium distributiondetermined by the temperature, the magnetic field, andany voltages applied to the gates and the current con-tacts. From this point of view, the entire region insideinterfering edge state, as well as the region surroundingthe edge state, should be considered as compressible inmost cases.[71, 72]In contrast, the charge on the edge state can varyrapidly, because it is connected directly to the edge statesoutside the interferometer, and we consider here a situ-ation where the backscattering probabilities at the con-strictions are small. Thus, the edge charge is not quan-tized, and the area A I , related to it by Eq. (31) below,may be considered to be a continuous variable.We now define an energy function E ( N L , A I ) , whichdescribes the free energy of the system after all othervariables have been integrated out.[74] We assume thatthe time-average interference current measured in an ex-periment is proportional to the thermodynamic averageof Re ( e iθ ) , weighted by the factor e − E ( N L ,A I ) /T .It is convenient to introduce another variable δn L , sothat we can write δn I ≡ − ( ν in − ν out ) B ( A I − ¯ A ) / Φ , (31) δn L ≡ N L q in e − ν in B ¯ A Φ − ¯ Q, (32)where ¯ A and ¯ Q are quantities chosen such that δn L and δn I would be zero if we were to minimize E without theconstraint that N L be an integer. The values of ¯ A and ¯ Q should be smooth monotonic functions of any appliedgate voltages, with perhaps a weak smooth dependenceon B . The variable δn I describes charge fluctuations onthe interfering edge, while δn L is determined by fluctua-tions in the interior. Then for small fluctuations in thevariables, we can expand E in the form E = K L δn L + K I δn I + K IL δn L δn I , (33)where the constants K L , K I and K IL depend on the ge-ometry and are largely determined by the Coulomb in-teractions between charges. Eq. (33) contains only theeffect of long-range Coulomb forces and no contributionfrom the quasiparticle gap since the random potentialcreates an essentially continuous spectrum for anyons.At T = 0 , there will be no thermal fluctuations, andthe phase factor e iθ will exhibit jumps at discrete valuesof the parameters, where N L increases or decreases byone. At finite temperatures, fluctuations become impor-tant, and one rapidly enters a regime where one or twoFourier components are dominant in a plot of the thermal expectation value (cid:104) e iθ (cid:105) as a function of the parameters B and V g . The allowed contributions have the form [74] D m exp (cid:26) πi (cid:20) m (cid:18) B ¯ A Φ (cid:19) − ¯ Q ( q in − me ) eν in (cid:21)(cid:27) , (34)where m are integers restricted to values of form m = − ν out eq out + g ν in eq in , (35)where g is an integer. The amplitudes D m fall off expo-nentially with temperature, | D m | ∝ exp( − π T /E m ) , sotypically only the component with largest E m is visible.An explicit expression for E m in terms of the parametersof the model is given by Eqs. (20) and (27) of Ref. 74.According to those formulas, the largest value of E m oc-curs when g is the closest integer to − ∆ θ/ π , where ∆ θ = − θ in + 2 πq e ( ν in − ν out ) K IL K I = 2 π (cid:18) sq in e − q in q out K IL K I (cid:19) (36)is the jump in interferometer phase that would occur if N L is increased by one at T = 0 . The favored value of g corresponds to the Fourier component of the interferenceoscillations that is least sensitive to thermal fluctuationsin N L and A I at higher temperatures.The case g = 1 has been termed the Aharonov-Bohmor AB regime, while the case g = 0 has been termed theCoulomb-Dominated or CD regime. For integer quan-tum Hall states, where s = 0 , the AB regime occurswhen K IL /K I < / , so that the coupling between edgeand bulk is relatively weak, while the CD regime occursfor / < K IL /K I < / , where the coupling is rela-tively strong. For a fractional state of the form (27),there will again be a CD-like regime with g = 0 , andat least in principle, an AB regime with g = 1 . How-ever, the value of K IL /K I separating the two regimeswill be < / , and the AB regime may be difficult toaccess. In fact, for the Laughlin states, with p = 1 and s ≥ , one is in the CD-like regime, with g = 0 , evenfor K IL = 0 , so the traditional CD designation is actu-ally a misnomer in this case. [To reach the AB regime at ν in = 1 / , one would actually need an attractive interac-tion between the edge mode and localized charges, with − / < ( K IL /K I ) < − / .] For integer states and forJain states of the form (27), the dominant term in theAB region has m = 1 , while in the g = 0 CD-like region,it has m = 1 − ( ν in e/q in ) = 1 − p ,According to (34), if the gate voltage is held fixed,and if one can assume that ¯ A and ¯ Q are insensitive tothe magnetic field, then the oscillations in conductanceshould have a period in the magnetic field given by ∆ B =Φ / | m | A . In the AB regime, where m = 1 , the fluxperiod is Φ for all the states under consideration. Bycontrast, in the CD regime, the period depends on the7state, and it will be a submultiple of Φ for states wherethere are two or more fully transmitted edge states, suchas ν in = 3 or ν in = 3 / ,If one fixes B and varies V g , one will generally see anoscillating conductance with a period that will dependon d ¯ A/dV g and d ¯ Q/dV g . A color plot of the conductanceoscillations as a function of B and V g will lead to a seriesof parallel stripes, similar to those seen in Fig. 6. It wasargued in Ref. 74 that at least for integer quantized Hallstates, lines of equal phase should have a negative slope,similar to the stripes in Fig. 6, in the AB regime, but theyshould have a positive slope in the CD regime, providedthere is at least one fully transmitted edge mode. Fabry-Perot experiments in the integer quantized regime haveseen both types of behavior, depending on the details ofthe sample.[35, 76] Also, in certain samples, the two typesof stripes were seen to coexist, leading to a checkerboardpattern of diamond shapes in the color plot. However,the situation is more complicated in the FQH case. If wedefine the measured phase ˜ θ as arg( (cid:104) e iθ (cid:105) ) , then followingEq. (34), ∂ ˜ θ/∂B will again have the same sign as m .However, for FQH states, the sign of ∂ ˜ θ/∂V g can dependon microscopic details.For the case of ν in = 1 / and ν out = 0 , where there areno fully-transmitted edge modes, one has m = 0 in theCD-like regime, as noted above. Then the conductancewill not show oscillations as the magnetic field is varied,and stripes of equal phase will be horizontal in the colorplot. Behavior of this type was indeed observed in theexperiments reported in Ref. [68] for magnetic fields onoutside of the range shown in Fig. 6. We remark thatthis result differs from the original prediction of Ref. 75that there should be a flux period of Φ in this region, asone would expect in the AB compressible regime; how-ever, that prediction has now been corrected.Note that in the compressible domain, FQH states in ahigher Landau level will have different flux periods thanthe corresponding states in the spin-polarized lowest Lan-dau level. For example [74], for a state at ν = 7 / , whichwe assume to consist of a Laughlin liquid at ν = 1 / on top of an integer quantized Hall state with ν = 2 ,we would have ν in = 7 / and ν out = 2 , so the allowedvalues of m in (34) will be equal to − g . As at ν in = 1 / , we expect that the dominant term shouldhave g = 0 . Experimental results at ν = 7 / by Willettand collaborators[77] showed a flux period Φ / , con-sistent with predictions for the compressible regime with m = − . However, the dependence on gate voltage wasnot reported in this reference.In another experiment at ν = 7 / , An and collabora-tors [78] reported a gate period with phase jumps, ap-pearing in the form of telegraph noise, which was consis-tent, at least qualitatively, with what one might expectfor a state in the incompressible regime. However, theflux period was not reported at this filling fraction, norwas there a reported calibration of the amount of chargeentering the interferometer in one gate period.Motivated by the experiments of Ref. 43, Schreier et al. [79] have analyzed interference effects to be expectedin a geometry where there is tunneling through an anti-dot inside a constriction. In particular, they considereda situation where there are two edge states around theantidot, and they found that the system was likely to bein an AB regime for an FQH state for the same geome-try where one would observe CD behavior in the integercase. They advanced this as an explanation for the dif-ferent behaviors observed in Ref. 43 between bulk fillingfactors ν = 2 / and ν = 2 .It should be cautioned that our discussion of the Fabry-Perot interferometer ignored the possible effects of tun-neling between different edge modes along the perime-ter of the interferometer. While this has been justifiedby experiments in the integer regime in many cases, itmay be more questionable for FQH states, particularlywhen there are edge modes propagating in two directions.Inter-mode scattering may contribute to decoherence ef-fects, which may be a reason why interference oscilla-tions have proved much more difficult to observe for FQHstates than for integer states.The analyses which led to the results described above,for both the AB and CD regimes in the FQH case with acompressible bulk, certainly made use of the property offractional statistics. More generally, if one accepts thatthe interfering particles have fractional charge, then oneneeds to invoke fractional statistics to avoid flux periodswhich are integer multiples of Φ . However, the flux pe-riods predicted above for the Jain states are identical tothe ones predicted for integer states in both the AB andCD regimes, where the tunneling particles are electrons.Consequently, it might be hard to rule out the possibil-ity that the interfering particles in an experiment[80] areelectrons rather than fractionally charged quasiparticles.For this reason, observations of the predicted flux periodin either regime might not be accepted as a convincingdirect observation of fractional statistics. 3. Quasiparticle charges from Fabry-Perot experiments Measurements using the Fabry-Perot geometry can beused to measure the charges of quasiparticles in variousquantized Hall states in either the CD or AB regime. Us-ing expression (34), if the values of d ¯ A/dV g and d ¯ Q/dV g are known, one can predict the oscillation period ∆ V g when B is held fixed. In the CD regime, this period cor-responds to the addition of a charge equal to q out to theinterior of the interferometer.Importantly, although (34) was derived in the regimeof weak backscattering, the same result obtains, for agiven partially-transmitted edge state, in the regime ofstrong backscattering, where the partially transmittededge state is almost totally reflected at the constrictions,and there is only weak forward scattering. (See Fig. 8a.)In this limit, the area enclosed by the interfering edgeforms an isolated droplet of material in a quantum Hallstate with quantum number ν in , embedded in a region8with quantum number ν out . Charge can then enter orleave the droplet only in units of q out , and the total chargein the droplet must be an integral multiple of this unit.If V g is varied, periodic oscillations will occur in the am-plitude for forward tunneling through the constriction,as the quantized charges enter or leave the droplet. FIG. 8. Schematic of a constriction with (a) weak forwardscattering and (b) no scattering. Dashed line shows electrontunneling. In typical experiments, the filling factor f c in the con-strictions and the filling factor in the bulk are varied si-multaneously by changing the magnetic field, while theoverall electron density and gate voltages are held con-stant. A region of weak forward scattering should oc-cur when f c is slightly above a rational value ν c thatcorresponds to a well-established quantized Hall state.(This will obtain when the magnetic field is slightly lowerthan the value at which f c = ν c .) In this case, we have ν out = ν c , so the charges measured in the CD regime willbe that of the elementary quasiparticles in the region ofthe constriction. Figure 8a shows a case where ν c = 2 and f c ≈ . . By contrast, the constrictions shown inFig. 7 are in a regime of weak backscattering, where f c is slightly below ν c , meaning that the magnetic field isslightly higher than the value where f c = ν c . (In Fig. 7,we have ν c = 2 and f c < .) In such cases, we have ν c = ν in , so if ν c is a fraction, the charge q out measuredin the experiment will differ from the elementary chargein the constriction. Note that the partially reflected edgestates are not the same in Figs. 7 and 8a.In a well-made constriction, as parameters are varied,there will be intervals where f c is sufficiently close tosome quantized value ν c that there is neither apprecia-ble forward nor back scattering. (See Fig. 8b.) In thisregime, the measured Hall resistance of the device andthe two-terminal conductance will sit on a quantized Hallplateau, where no interference oscillations will be seen.Quasiparticle charge measurements in the CD regimeobtained in Ref. 35 were consistent with the expectedanyon charges e/ and e/ at ν = 1 / and ν = 2 / respectively. Charge ≈ e/ was reported for ν =1 / , / , / , and / in Ref. 36. A recent Coulombblockade experiment [81] reveals charges e/ at ν = 1 / and / .As mentioned above, the color plots of conductance asa function of B and V g presented in Ref. 68 showed aseries of horizontal stripes, for fields outside the range ofincompressible bulk, which is what one predicts for theCD regime when the bulk is in a compressible state onthe ν = 1 / plateau and the filling in the constrictions isless than that of the bulk. Moreover, the observed gateperiod ∆ V g is consistent with the predicted period in theCD regime, since q out = e .Quasiparticle charge can also be obtained fromAharonov-Bohm oscillations in the incompressible bulkregime shown in Fig. 6, as was done in Ref. 37. Assum-ing the interference area A is known, if one can neglectCoulomb coupling between the edge and the bulk, thecharge of the interfering particle can be extracted fromthe magnetic-field period in an interval where no local-ized quasiparticles enter or leave, by use of the equation q m A ∆ B = e Φ . Alternatively, if the dependence of A on gate voltage is known, the charge may be extractedfrom the gate period using q m B ∆ V g = e Φ / ( ∂A/∂V g ) .The authors of Ref. 37 used the second method to ex-tract the value of q m at ν = 1 / , assuming that value of ( ∂A/∂V g ) was unchanged from the value at ν = 1 , andthey obtained the value q m = 0 . e , in good agreementwith the expected value e/ . On the other hand, mea-surements of the same type at ν = 2 / obtained a resultof . e , suggesting that the tunneling charges in thatcase might be electrons rather than fractionally chargedquasiparticles.Using the model defined by Eq. (33), we can addressthe effects of the Coulomb interactions, omitted aboveand in Section IV A, on interferometer experiments inthe incompressible region. If we continue to assume thatthe background parameters ¯ A and ¯ Q are insensitive to themagnetic field, then modifications of the interference area A I are controlled by the coupling constants K IL and K I .In this case one finds that the jump in the interferometerphase on entry of a quasiparticle will be given by Eq.(36) while the magnetic field period, between jumps, willbe renormalized to ∆ B = e Φ q in ¯ A (cid:20) − K IL K I ν in ( ν in − ν out ) (cid:21) − . (37)The slope of lines of equal phase on the plane of B and V g should not be affected by a non-zero K IL . The distinctjumps predicted by (36) should be visible in the incom-pressible regime at temperatures much higher than in thecompressible regime, in so far as the energy to create aquasiparticle is typically much higher than the scales ofcharging energies, K I and K L .The claim in Refs. 37 and 68 that Coulomb couplingmay be neglected in their sample is supported by the factthat the interference stripes they observe at the integerfilling ν = 1 are consistent with what one would expect9in the incompressible regime on neglecting the correctionproportional to K IL /K I in (37), or in the Aharonov-Bohm regime if the bulk is compressible.Note that in the incompressible region, one finds onlya gradual transition between the regimes of weak andstrong Coulomb interaction, as the predictions for ∆ θ and ∆ B vary continuously as a function of K IL /K I .This is in contrast to the compressible region, where thetransition between AB and CD-like regimes is marked bysimultaneous manifestation of two distinct periodicities,rather than a single intermediate period.We conclude this section by mentioning puzzling be-havior [82, 83] observed in a geometry with a ν = 1 / channel going around a ν = 2 / island, where the trans-port data were interpreted as showing a magnetic-fieldperiod of and a period of in the interferometer chargeof e . The explanation advanced by the experimenterssupposed that the enclosed ν = 2 / region was in a com-pressible state, where e/ quasiparticles could readily en-ter or leave, so as to keep the electron-density and areafixed as the magnetic field was varied. However, accord-ing to the analysis presented above, in a compressibleregion, regardless of whether one was in the AB regimeor the CD regime, any observed flux periods should be Φ or a submultiple of it, not a period larger than Φ .(See Refs. 74, 84–86 for further discussions.)A possible resolution of the puzzle might be obtainedif the quantum dots in these experiments were actuallymeasured in a magnetic field interval where the interiorstate was essentially incompressible, as in the centralmagnetic-field region of Refs. 68 and 37. In that case, ifthe interfering qusaiparticles have charge e/ , one wouldnaturally expect to find a flux period of and a gateperiod corresponding to the addition of two electrons.It should be noted, however, that the varying gate em-ployed in these experiments was not a side gate but rathera back gate, separated from the sample by the thicknessof a sapphire substrate, which may complicate the anal-ysis. In any case, a more detailed analysis, and perhapsfurther experiments, are needed to resolve these issues. B. Anyon collider It is known that the scattering of identical fermions dif-fers from the scattering of identical bosons with the sameinteraction potential. This suggests the use of anyon col-lisions to probe fractional statistics.An anyon collider at ν = 1 / was implemented in Ref.70 following the proposal from Ref. 69. The setup is il-lustrated in Fig. 9. Charge from two sources arrives alongthe edges to two point contacts QPC1 and QPC2, wheretunneling gives rise to two dilute beams of anyons prop-agating along the edges towards cQPC. Anyons, arrivingfrom the two sides to that contact, collide. This affectsthe currents, collected in the two drains, D1 and D2. Ifthe anyons were fermions, the Pauli principle would pro-hibit the two arriving anyons from ending up on the same side of cQPC. In other words, the two fermions wouldblock each other from tunneling through cQPC. Mathe-matically, this would result in an absence of correlationsbetween the two drain currents. Bosons are known tobunch, and this would result in non-zero correlations.Laughlin anyons are intermediate in their properties be-tween bosons and fermions. Thus one might expect someintermediate form of current correlations for a Laughlinliquid. FIG. 9. Anyon Collider. The currents from sources S1 andS2 give rise to dilute beams of quasiparticles from QPC1 andQPC2 to cQPC, where anyons collide. The correlations of thecurrents in drains D1 and D2 are determined experimentally. Reference 69 made specific predictions for variousLaughlin liquids, and the experimental results, obtainedat ν = 1 / , were found to be in excellent agreement withthe theory. However, there were other ingredients in thetheory in addition to the assumption of fractional statis-tics. The theory employed a specific model of the edgeHamiltonian ˆ H edge , given by Eq. (23). The Hamilto-nian is important because a quasiparticle tunneling be-tween two edge states will leave behind trace in exci-tations along the edges, which can affect the amplitudefor tunneling of a second quasiparticle. Indeed, it is ex-pected that the form of the current correlations may bealtered if there is reconstruction at the edges of the sam-ple. The results of the anyon collider experiment, whilevery interesting, would therefore seem to be a less directmeasurement of fractional statistics than those obtainedfrom the Fabry-Perot experiments.Several other setups have been proposed theoreticallyto obtain signatures of fractional statistics from othercurrent correlations [87–92], but these have not yet beenrealized experimentally.Another type of correlation experiment, which requiresthe simultaneous presence of two identical particles andwhich depends on their mutual statistics is the Hanbury-Brown Twiss interferometer. A beautiful experiment ofthis type, demonstrating the interference between twoelectrons from independent sources injected into a quan-tum Hall edge state at integer filling, was reported inRef. 93. Results that might be expected for a similarexperiment with FQH edge states have been explored0theoretically in Refs. 87, 94, and 95. V. NON-ABELIAN STATISTICS In our previous discussion of Abelian anyons, we fo-cused on the statistical angle acquired during anyonbraiding, that is, in a process in which anyons exchangetheir positions or run full circles around other anyons.By measuring braiding phases, accumulated by variousanyon types on a circle around a localized anyon, thelocalized anyon could be identified. This was the ideabehind the interferometry technique in Section IV.A.Two other important processes which characterize thetopological behavior of anyons are fusion and splitting.In fusion, two anyons combine into a single excitation.Splitting is the reverse process. These processes will beof particular importance in our discussion of non-Abeliananyons.In the Laughlin states [7] at ν = 1 /m , fusion is triv-ial. Anyon types are fully determined by anyon charges.Combining two anyons of charges q and q produces ananyon of charge q + q . As an example, consider the ν = 1 / liquid of charge e bosons[96, 97]. The elemen-tary quasiparticles have charge e/ and are semions, thatis they have statistical angle θ m = π/ , half that of afermion. A semion is a non-local or topologically non-trivial object. This means that there is no way to cre-ate an isolated semion. Semions can only be created inpairs. Two semions fuse into a boson, which is a topologi-cally trivial object that can be created locally and cannotbe detected with interferometry. We say that it belongsto the vacuum topological sector. Similarly, adding anynumber of bosons to a semion does not affect the outcomeof an interferometry experiment and does not change thetopological sector of the excitation. If we label the vac-uum sector with 1 and the semion sector with s , we getthe following fusion rules for the particles from the twosectors: × , × s = s, s × s = 1 . (38)This is an example of Abelian statistics . More compli-cated states with Abelian statistics, such as Halperin’s nnm liquids[14], allow neutral anyonic excitations. Thus,anyons of the same electric charge may belong to differ-ent sectors a i . Still, the sectors form an Abelian groupwith the fusion rules a i × a j = a k ( i,j ) , where k is uniquelydetermined by i and j .In systems with non-Abelian statistics , we encountersituations where two given anyons can fuse into excita-tions from more than one topological sector. We shallbe particularly interested in systems with the simplesttype of non-Abelian statistics, known as Ising statistics ,which emerges in the exactly solvable Kitaev model[98]of a magnet on a hexagonal lattice and is relevant forvortices in p -wave superconductors[99] as well as FQH states at half-integer fillings[100]. Systems with Isingtopological order have three topological sectors: vacuum1, fermion ψ , and Ising anyons σ . Fusion with the vac-uum has no effect on the topological sector. The remain-ing three fusion rules are ψ × ψ = 1 , σ × ψ = σ, σ × σ = 1 + ψ. (39)The last rule means that the fusion of two Ising anyonsmay yield a boson or a fermion. If the two σ particles arefar apart, the two fusion channels cannot be distinguishedby local measurements and are present at the same localquantum numbers of the two anyons. The informationabout the fusion channel is stored globally. This servesas the foundation for the idea of topological quantumcomputing [101].As an example, consider two vortices in a spinlesstwo-dimensional superconductor with pairing of the form p x + ip y [99]. Each vortex binds a Majorana zero modedescribed by a real fermion Ψ , = Ψ † , . The tworeal fermions combine into a single complex fermion Ψ = Ψ + i Ψ that can populate a single energy level.The states with the filled and empty level differ by theirfermionic parity but cannot be distinguished locally bylooking at a single vortex. Thus, we can think of thevortices as σ -anyons and the filled and empty level as thetwo fusion channels from Eq. (39).A system with conserved fermionic parity cannot movebetween the two fusion channels, but the same physicsis present in a parity conserving system with four Isinganyons. The trivial total parity can be obtained in twolocally indistinguishable ways: anyons 1 and 2 fuse tovacuum and anyons 3 and 4 fuse to vacuum, or alterna-tively, anyons 1 and 2 fuse to fermion and anyons 3 and4 fuse to fermion. A system of n Ising anyons has n − locally indistinguishable states. A. Basic principles The theory of fusion and braiding was dubbed the al-gebraic theory of anyons in Ref. 98, which is the ap-proach we follow. In pure mathematics, fractional statis-tics correspond to modular tensor categories [102]. Thesame mathematical structures emerge in topological fieldtheory[103] and in conformal field theory[104] (CFT).This reflects the physics of the problem: topological fieldtheories capture some of the bulk physics in topologi-cal liquids; we will see below that CFT captures univer-sal aspects of the edge physics. The complete topologi-cal classification of a system with non-Abelian statisticsgoes beyond the rules stating which sectors can fuse intowhich others, but will depend also on various amplitudesassociated with fusions and braidings. For example, onefinds[98] that that there are eight distinct topological or-ders for systems obeying the fusion rules (39)1 F FF F F FIG. 10: Pentagon equation. The two-dimensional nature of the problem does not make any difference for fusion. We will follow the conventionof placing all anyons on a line. Braiding exchanges anyon positions on that line while fusing and splitting changes thenumber of the occupied sites. We also ignore all local quantum numbers of the anyons and focus solely on topologicallydistinct states. Thus, we consider just one anyon state in each topological sector. In other words, we introduce aone-dimensional Hilbert space for each anyon type.We will use diagrammatic language to speak of fusion and braiding. The key objects are the splitting operators [ ψ i ] abc and their Hermitian conjugate fusion operators, illustrated below: ca b [ ψ i ] abc ca b [ ψ † i ] abc . We use the convention that the time axis runs up. The right diagram suggests moving two anyons a and b into thesame point, where they fuse into c , but a different way of thinking is often useful. We can assume that particles do notmove and the fusion and splitting operators are just linear maps between the Hilbert space of the combined systemof the two anyons and a one-dimensional space.The most general fusion rule is a × b = (cid:88) c N cab c, (40)where the fusion multiplicities N cab show the number of independent ways to fuse anyons a and b into anyon c ; in otherwords, N cab is the dimension of the Hilbert space V abc of the states of the two anyons with the total topological charge c . All fusion multiplicities equal 1 for the Ising statistics and for any Abelian statistics. Assuming the normalization ca bψ j c ψ † k = δ jk c , (41)where j, k = 1 , . . . , N cab , we decompose the identity operator as a b = (cid:80) c,j ψ j ψ † j a ba bc . (42)2One of the anyon sectors is vacuum 1, and the fusion multiplicity with vacuum is always N aa = N a a = 1 . Also, N ab can only be 0 or 1. It is always possible to add a vacuum line to any diagram.Calculations with diagrams often involve F -moves uxa b c = F abcu u ya b c , (43)where F abcu are matrices with the indices x and y and additional numerical indices, if the fusion multiplicity exceedsone in any node of the diagram. The diagrams on the right and on the left represent two compositions of splittingoperators. A gauge freedom exists in the choice of the F -symbols and other topological data of an order. See Ref.105 for numerous examples of such data. For Abelian statistics the F -matrices are × , i. e. , just numbers. Forexample, for semions, F ssss = − and all the other F -symbols are 1 in the gauge [105] we use. For the Ising statisticswith the braiding rules (47), the following F -symbols are non-trivial: [ F σσσσ ] rs = (cid:18) / √ / √ / √ − / √ (cid:19) , (44) (cid:104) F σψσψ (cid:105) σσ = (cid:2) F ψσψσ (cid:3) σσ = − , (45)where r, s = 1 , ψ with r = s = 1 in the upper left corner of the matrix.Thinking of fusion operators as linear maps naturally leads to an infinite number of associativity relations such asthe pentagon equation (Fig. 10), which tells that the two upper F -moves in the diagram are equivalent to the threelower ones. It can be proven that any other ‘obvious’ relation follows from the pentagon equation and the hexagonequation (Fig. 11).Braiding is described by the unitary operators called R -symbols: R ab = ab ba R − ab = ba ab (46)The statistical phase, accumulated at the exchange of a and b , is unaffected by local operators acting on each of theanyons. As a consequence, lines can be moved over crossing points of other lines. For example, splitting b into twoanyons below or above the crossing point in Eq. (46) produces equivalent diagrams. For the semion topological order,the only nontrivial R -symbol describes the exchange of two semions: R ss = i . For Ising anyons with the fusion rules(39), eight topological orders are known with different braiding rules [98]. In this subsection we consider one example,where the non-trivial R -symbols depend on the fusion channel of the excitations in the following way: R σσ = e − iπ/ , R σσψ = e iπ/ ,R σψσ = R ψσσ = e − iπ/ , R ψψ = − . (47)A combination of F -moves and R -moves generates the hexagon equation (Fig. 11): the composition of the the two R -moves and one F -move in the upper part of the diagram is equivalent to the composition of the two F -moves andone R -move in the lower part of the diagram. A similar equation holds for R − -moves. The R - and F -symbols satisfythe equations in Figs. 10 and 11 and form a key part of the data that defines a topological order.Each particle a has a unique antiparticle ¯ a for which the vacuum 1 is a possible outcome of the product a × ¯ a .The antiparticle of a Laughlin anyon of charge q m carries the opposite electric charge − q m . In the Ising and semionorders, each particle is its own antiparticle. Since the fusion multiplicity with an antiparticle to the vacuum has tobe 1, the following diagram is defined up to an arbitrary phase factor κ a :3 a a ¯ a = κ a d a a , (48)where the quantum dimension d a describes the scaling of the number of states ∼ d aN of N (cid:29) anyons a . Allquantum dimensions are for Abelian statistics. For the Ising order, one quantum dimension is nontrivial: d σ = √ in agreement with n − states for n anyons. The quantum dimensions are the same for a particle and its antiparticle.A useful identity relates quantum dimensions with fusion multiplicities: (cid:88) c N cab d c = d a d b . (49)If a = ¯ a , the phase factor κ a in Eq. (48) is no longer arbitrary and is known as the Frobenius-Schur indicator.This invariant equals ± and indicates the breaking of the spin-statistics correspondence at κ a = − . [See Eq. (53)below.]. The indicator is for all excitations of the Ising liquid with the braiding rules (47), while κ s = − in thesemion order. R F RF R F FIG. 11: Hexagon equation. It proves profitable to redefine the normalization of thesplitting and fusion operators (41) in terms of quantumdimensions: (cid:113) d c d a d b ca bψ j c ψ † k = δ jk c . (50)The resulting diagrammatic technique has a nice featurethat topologically equivalent diagrams are equal. For ex-ample, for κ a = 1 , a a ¯ a = a . (51)Negative Frobenius-Schur indicators are accounted for bydecorating lines with arrows. We will ignore this com-plication since the Ising order with the R -symbols (47),which is of primary interest for this review, has triv-ial κ a . With the new normalization, a closed non-self-intersecting loop from a particle line and an antiparticleline equals the quantum dimension of the particle.Braiding properties can largely be deduced from a sin-gle number called the topological spin for each anyon4type: θ a = d a − a . (52)The topological spin is a root of unity [106]. For vacuum, θ = 1 . For semions, θ s = i . For the anyons in Isingliquids, θ ψ = − and θ σ = exp( iπ/ .Naively, the topological spin defines the statisticalphase at the exchange of a particle with its antiparti-cle, but this only holds in some cases. First of all, thestatistical phase of non-Abelian anyons depends on theirfusion channel. Second, even in the vacuum fusion chan-nel, the standard spin-statistics relation may not hold.In particular, for a = ¯ a , R aa = θ ∗ a κ a . (53)On the other hand, interferometry involves the phase φ abc accumulated by an anyon a on a full circle around b as-suming that a and b fuse to c . This phase depends onlyon the topological spins: exp( iφ abc ) = θ c θ a θ b . (54)The proof of this expression illustrates the power of thediagrammatic approach and immediately follows fromthe diagrammatic identity in Fig. 12. FIG. 12. The two lines of each diagram represent anyons a and b . The left diagram can be interpreted as a double line,representing anyon c . B. Edge modes As previously mentioned, a boundary between agapped FQH liquid and the vacuum necessarily carriesgapless modes. [9] The simplest example is the fillingfactor ν = 1 for non-interacting spinless electrons. Far from the boundary, all electrons occupy degenerate statesof the lowest Landau level at the energy (cid:126) ω C / , where ω C is the cyclotron frequency. Assume that the confin-ing potential near the edge changes slowly on the scaleof the magnetic length (cid:112) (cid:126) c/eB . Then the energy of astate localized at the distance x from the boundary is E ( x ) = (cid:126) ω C / V ( x ) , where V ( x ) is the confining po-tential. The boundary x of the occupied electron statescorresponds to E ( x = x ) = E F , where E F is the Fermienergy. Gapless excitations are localized at x ≈ x . Theexcitations are chiral, that is, they propagate only clock-wise or counterclockwise, depending on the direction ofthe magnetic field. That direction is called downstream.In the simplest free-fermion model, the Lagrangiandensity of the low-energy mode is L = iψ † ( ∂ t + v∂ x ) ψ, (55)where v is the mode velocity, and ψ is a fermionic Grass-mann field. It is often convenient to bosonize [107] theabove Lagrangian density, substituting ψ ∼ exp( iφ ) ,where − e∂ x φ/ π is the linear charge density: L B = − π ∂ x φ ( ∂ t + v∂ x ) φ. (56)A closely related chiral Luttinger liquid model [108] isoften used to describe Laughlin states at ν = 1 / (2 n + 1) : L n +1 = − νπ ∂ x φ ( ∂ t + v∂ x ) φ, (57)where the electron operator ψ ∼ exp( iφ/ν ) . Generaliza-tions of this model are broadly applied to describe edgesof Abelian FQH liquids. Besides a downstream chargemode, additional modes are generally present, whose di-rections can be both downstream and upstream [108].The additional modes are typically charge-neutral, due toeffects of impurity scattering and/or long-range Coulombinteractions.The chiral Luttinger liquid model (57) misses compli-cated physics due to the long-range Coulomb interac-tion and often fails to quantitatively describe the data[55, 109–111]. One effect overlooked by the model isedge reconstruction [112]. It was shown theoretically [71]and confirmed experimentally [113] that a realistic FQHedge in GaAs heterostructures is formed by a sequenceof compressible and incompressible stripes. Their widthsdepend on the depletion length, where the electron den-sity drops to zero near the sample boundary [71]. Thelatter is set by the gate voltage for gate-defined edgesand is determined by the physics of the localized surfacestates for the edges defined by chemical etching [114].Narrow incompressible stripes are fixed-density regionswith the filing factor between 0 and the bulk filling factor.They carry current proportional to the voltage differencebetween their edges. Incompressible stripes are sepa-rated by compressible stripes of fixed electrostatic poten-tial and coordinate-dependent charge density. Naively,5this picture suggests several co-propagating modes on theedge. Yet, general arguments based on thermal conduc-tance (Section VIII) show that each downstream modemissed by the chiral Luttinger liquid model must beaccompanied by an upstream mode. Inevitable disor-der localizes pairs of contra-propagating modes on largelengths. On the longest length scales, the only surviv-ing topologically-protected neutral modes are the onespresent even on a sharp edge without reconstruction.Despite their limitations, chiral Luttinger liquid mod-els produce deep insights about the FQH effect. One suchinsight is bulk-edge correspondence [115]. It turns outthat the bulk wave-function of a Laughlin FQH liquid at ν = 1 / (2 n +1) can be extracted from the correlation func-tion of the electron operators ψ ( x, t ) in the chiral CFT(57), where the imaginary time plays the role of the sec-ond spatial coordinate y . Moore and Read conjectured[115] that this represents a more general relation betweenconformal field theories of the edges and the ground-statewave functions in topological matter. This led them toa proposal for a non-Abelian state at half-integer fillingfactors dubbed the Pfaffian state. The Pfaffian edge the-ory contains two modes: L P f = − π ∂ x φ c ( ∂ t + v∂ x ) φ c + iψ ( ∂ t + v n ∂ x ) ψ, (58)where the Bose-mode φ c defines the charge density − e∂ x φ c / π , and ψ = ψ † is a neutral Majorana fermion.The electron operator Ψ = ψ exp(2 iφ c ) . Each Bose modehas the central charge of 1 but the central charge [104] ofthe Majorana fermion is / (roughly speaking, the cen-tral charge counts the degrees of freedom, and a Majo-rana fermion can be seen as a half of a complex fermion).This is an example of a general rule: the chiral centralcharge of the edge theory in non-Abelian liquids is non-integer. The topological order in the Pfaffian state isclosely related to the Ising order from the previous sub-section and will be reviewed below.Exceptions to bulk-edge correspondence are known[116]. Nevertheless, it remains a useful heuristic prin-ciple. For example, consider particle-hole conjugation oftopological orders [117] and its effect on the edge struc-ture. Imagine some topological order at ν = n + f ina fractionally filled Landau level of filling f on top of n filled spin-resolved Landau levels. The same order can beinterpreted as a particle-hole conjugate order of holes atthe filling factor − f on top of n + 1 filled Landau lev-els. One can also define a particle-hole conjugate orderfor electrons at the filling factor n + 1 − f . The clas-sification of the excitations and the fusion rules are thesame as in the original order. All braiding phases changetheir sign. The effect of the particle-hole transformationon the edge structure is the following. A boundary be-tween ν = n and ν = n + 1 − f , with f < / , shouldbe understood as a composition of an outer boundarybetween ν = n and ν = n + 1 and an inner boundarybetween ν = n + 1 and ν = n + 1 − f . The original state at ν = n + f would have had one or more edgemodes between ν = n and ν = n + f . The particle-holeconjugate order corresponds to the opposite propagationdirection of each of those modes plus an additional down-stream integer mode describing the boundary of ν = n and ν = n + 1 .Electron tunneling between edge channels can modifythe description of the modes and sometimes reduces theirnumber. For example, the Laughlin liquid at ν = 1 / hasa single downstream mode. According to the above pre-scription, the / edge contains two charge modes [118]:an integer downstream mode and an upstream mode.Electron tunneling due to inevitable disorder is knownto reorganize the edge into a single downstream chargemode and an upstream neutral mode [119].We finish the discussion of edge modes by observingthat the central charge in a CFT description is propor-tional to the heat conductance of a mode [22, 120, 121]. Itwas proven that a chiral mode of the central charge c hasthe thermal conductance cκ T , where κ T = T π k B / h is known as a thermal conductance quantum. The cen-tral charge is integer for every edge mode in any Abelianstate. Hence, fractional quantization of the thermal con-ductance is a sign of non-Abelian statistics. C. Examples of non-Abelian statistics It was long suspected that the filling factor[122] / hosts a non-Abelian FQH liquid [115]. Experiment hasbrought strong evidence in favor of that view [123]. Pre-dictions were made for non-Abelian orders at other fill-ing factors [124–134] of the second Landau level in GaAs.Experiment is consistent with Abelian orders on the rel-atively more robust ν = 7 / and / plateaus (for areview, see Ref. 135). At the same time, the analysis ofthe gap dependence on the filling factor [136] supportsdifferent nature for FQH states at ν = 7 / , / on theone hand and at / < ν < / on the other hand. Itmight be that all states in the latter interval are non-Abelian. Very little is known about the plateaus [136–139] that presumably exist at ν = 19 / , / and / .We will not address them and will not dwell on possiblenon-Abelian states in the first Landau level. Our focuswill be on the filling factors[122, 137] / and / . Afragile state[140] at ν = 7 / is expected to be the sameor closely related to the state at ν = 5 / .The existing theoretical pictures at ν = 5 / and / were influenced by CFT ideas[115, 141]. Thus, we startwith a brief summary of the CFT approach to the frac-tional statistics. A reader who is not familiar with CFTwill be able to follow the bulk of the discussion in thissection. The starting point is an edge theory, which isa combination of chiral CFTs of perhaps opposite chi-ralities (downstream and upstream). Anyons correspondto the products of primary fields from each chiral CFT.One such product is postulated to describe electrons. Allother allowed anyons must have single-valued operator6product expansions with the electron operator. The ra-tionale for this requirement comes from two considera-tions. First, electrons are in the vacuum topological sec-tor and hence braid trivially with all excitations. Second,wave functions of systems of anyons are identified withconformal blocks of the CFT. Trivial braiding impliessingle-valued conformal blocks. The topological spin ofeach anyon a is determined [98] by its conformal weights ( h a , ¯ h a ) in the CFT: θ a = exp(2 πi [ h a − ¯ h a ]) , (59)where h a comes from the counterclockwise holomorphicpart of the edge theory, and ¯ h a from the clockwise anti-holomorphic modes. Below we identify the holomorphicdirection with the downstream direction of the chargemode. 1. Possible states at ν = 5 / In our discussion of the proposed ν = 5 / and ν = 12 / orders we will ignore the two filled spin-resolved Landaulevels. We will think of electrons in the partially-filledlevel in terms of composite fermions that combine anelectron and two flux quanta [142]. Composite fermionsmove in zero effective magnetic field. Thus, one mightexpect that they form a gapless Fermi-liquid-like state.Gapless states are indeed observed [142] at ν = 1 / and ν = 3 / in GaAs. The gap at ν = 5 / can be explainedby Cooper pairing [22] of composite fermions. However,multiple ways exist to build a Cooper pair. In an isotropicsystem, one can have pairing in various angular momen-tum channels l . In an anisotropic system, where l is nota good quantum number, we can instead talk about thewinding number of the phase of the order parameter asthe fermion momentum moves around the Fermi surface.In general, for a spinless single-component Fermi surface,only odd values of l are allowed. However, in the presenceof electron-electron interaction it is possible for a Fermisystem to spontaneously divide itself into several compo-nents with independent Fermi surfaces, and in that casepairing with even values of l is allowed. For example, ithas been proposed that for a wide quantum well at to- tal filling ν = 1 / , electrons might organize themselvesinto two parallel sheets with 1/4 filling in each.[143] Thebulk-edge correspondence gives a convenient principle forclassifying the various states.For a half-filled Landau level, the charge mode is de-scribed by the Lagrangian density (57) with ν = 1 / .Operators that create and annihilate an electron chargeare proportional to Φ ± = exp( ± iφ ) . One can checkthat operators Φ ± are bosonic and must be multipliedby a neutral fermion to produce a legitimate electronoperator. This means that the edge theory should con-tain one or more gapless Fermi modes. Since a complexfermion is a combination of two Majorana fermions, wecan assume without loss of generality that all fermionmodes are Majorana. We can also assume that all Majo-rana modes are co-propagating since contra-propagatingmodes can be gapped out by electron tunneling betweenedge modes. The net number C of the Majorana modesis often referred to as a Chern number, because it has theform of a Chern index in the analysis presented in Ref. 98.The Chern number is positive for downstream modes andnegative for upstream modes. The Lagrangian density is L P f = − π ∂ x φ c ( ∂ t + v∂ x ) φ c + | C | (cid:88) k =1 iψ k ( ∂ t + v n sign C∂ x ) ψ k . (60)There is no a priori reason for the velocities of the Ma-jorana fermions to be the same, but edge disorder makesthem equal in the long-scale limit [144–146].No Majorana modes are present at C = 0 . In thatspecial case[14, 147, 148], known as the K = 8 state,electrons are gapped on the edge. For subtleties in the113 state at C = − , see Ref. 149. Subtleties that emergeat C = − are addressed in Ref. 146.The states with even Chern numbers are Abelian andthe states with odd Chern numbers are non-Abelian. Thetopological order depends only on C mod 16 . Thus, thereare 8 Abelian and 8 non-Abelian possibilities known to-gether as the 16-fold way [10, 98]. Orders with a largeChern number are seen as unlikely, and the bulk of re-search has focused on the orders listed in Table I. Thestates with the Chern numbers C and − ( C + 2) are re-lated by the particle-hole conjugation. The PH-Pfaffianorder at C = − is unique in being its own conjugate[150, 151]. C − − − name anti-Pfaffian PH-Pfaffian K = 8 Pfaffian SU (2) Ref. 144 and 145 149 145, 150, and 151 (see 152 and 153 for related surface states) 14, 147, and 148 115 14 154 and 155TABLE I. Proposed topological orders at half-integer filling factors. The Chern number C is the difference between the numberof forward- and backward-propagating Majorana modes on a sample edge. It appears that multiple orders of the 16-fold way are realized in nature. Numerical work has brought a prepon-7derance of evidence in favor of the non-Abelian Pfaffianand anti-Pfaffian liquids at ν = 5 / in GaAs withoutimpurities [156–158]; see, e.g. , Refs. 159–162. Experi-ment appears consistent with a different non-Abelian PH-Pfaffian liquid [123, 151]. Some data were interpreted interms of the Abelian 113 and 331 states [163–165]. Re-cent theoretical work suggests a complicated phase dia-gram in realistic disordered systems in which all topolog-ical orders with − ≤ C ≤ are present [166–168] (seealso Refs. 169–172 for the role of Landau level mixing).On the other hand, some theoretical proposals question[173–175] the existence of an energy gap at ν = 5 / .Half-integer FQH plateaus have also been found inseveral systems beyond single-layer GaAs. The SU (2) order was predicted in graphene [176, 177]. The 331order[14] is believed[178–180] to be present in GaAsbilayer[143, 181] at the filling factor / . Recent exper-iments on single-layer graphene have demonstrated theexistence of gapped QH states in the N = 3 Landaulevel, corresponding to ν = C = 3 orto its particle-hole conjugate with C = − .[177] BesidesGaAs and graphene[177, 182–186], half-integer plateaushave been observed[187–189] in ZnO and WSe .We finish the discussion of the 16-fold way by describ-ing the quasiparticle types, fusion rules, and topologicalspins for each order [10]. (See Table I.) All non-Abelianorders possess excitations with three topological charges:1, ψ , and σ . 1 and ψ carry half-integer electrical charges ne/ . σ carriers charge e/ ne/ . The fusion rules forthe topological charges are given by Eq. (39). Electricalcharges of the excitations, of course, add up in fusion.The topological spins of the excitations are determinedby their topological charge t and their electrical charge ne/ as θ ( t,n ) = θ t exp( iπn / , (61)where θ = 1 , θ ψ = − , and θ σ = exp( iπC/ .The K = 8 state is effectively a Laughlin-like liquidof charge e bosons at Landau-level filling 1/8, whichgives rise to Abelian anyons labeled by their electricalcharges ne/ , with the topological spins exp( iπn / . Inthe remaining Abelian states, there are four topologicalcharges , ψ, σ and µ . The electrical charges of - and ψ -excitations are ne/ , while σ and µ carry electricalcharges e/ ne/ . The fusion rules depend on the parityof C/ . For odd C/ , σ × σ = µ × µ = ψ and σ × µ = 1 .For even C/ , σ × σ = µ × µ = 1 and σ × µ = ψ . Inall cases, ψ × ψ = 1 , σ × ψ = µ , and µ × ψ = σ . Thetopological spins are given by Eq. (61) with θ = 1 , θ ψ = − , θ σ = θ µ = exp( iπC/ .A key difference between Abelian and non-Abelianstates is the existence of one charge- e/ particle σ forthe non-Abelian orders and two charge- e/ particles σ and µ for the Abelian orders. This leads to subtleties inthe interpretation of experiment since two quasiparticletypes in Abelian states may have the same experimen-tal consequences as two fusion channels for non-Abelian anyons [190].Evidence exists for a ν = 1 / plateau in wide GaAsquantum wells [191–193]. The 16-fold way was extendedto that filling factor in Ref. 194. 2. Read-Rezayi states The thermal conductance of a Majorana mode is de-termined by its central charge c = 1 / . A more generalclass of CFTs is known with c = (2 k − / ( k + 2) , where k is an arbitrary positive integer. The case k = 2 re-duces to the Ising CFT, while the CFTs with k > areknown as parafermion theories [104]. They were used byRead and Rezayi to generate a family of FQH states[141]at the filling factors k/ ( k + 2) . Anyon types [195] aredistinguished by their electrical charge and their topo-logical charge, which comes from the list of the primaryfields in the parafermion CFT. There are k ( k + 1) / pri-mary fields Φ jm with j = 0 , / , . . . , k/ , ( j − m ) ∈ Z .Two identifications are made: ( j, m ) ≡ ( j, m + k ) and ( j, m ) ≡ ( k − j, m + k ) . This allows choosing j > and − j < m ≤ j . The topological spin of Φ jm is θ jm = exp (cid:18) πi (cid:20) j ( j + 1) k + 2 − m k (cid:21)(cid:19) . (62)The fusion channels are given by Φ jm × Φ j (cid:48) m (cid:48) = min( j + j (cid:48) ,k − j − j (cid:48) ) (cid:88) j (cid:48)(cid:48) = | j − j (cid:48) | Φ j (cid:48)(cid:48) m + m (cid:48) . (63)Electrons carry the topological charge Φ k/ − k/ . The topo-logical spin of an anyon of electrical charge se is theproduct of the neutral contribution (62) and exp( πi [ k +2] s /k ) . The allowed combinations of the topological andelectrical charges make the braiding phase (54) with anelectron φ aea × e trivial. The lowest quasiparticle charge is e/ ( k + 2) .The state at k = 4 corresponds to the observed fill-ing factor / / . There is some numericalevidence for a Read-Rezayi state at that filling factor[124], but experiment suggests that it hosts a Laughlin-like state (see Ref. 135 for a review). No plateau hasbeen seen[139] at ν = 13 / / , which wouldcorrespond to k = 3 . A plateau is known [137] at theparticle-hole conjugate filling factor / . Apparently,Landau level mixing effects[131, 132] are responsible forthe difference between ν = 12 / and ν = 13 / . Numericssuggests a non-Abelian state[126–128] at ν = 12 / thatis the particle-hole conjugate of the k = 3 Read-Rezayistate. Such state is interesting from the point of view ofquantum computing since it allows universal topologicalcomputation[24], impossible with the topological ordersof the 16-fold way.8Note that a generalization of the Read-Rezayi statesapplies[141] to the filling factors ν = k/ ( M k + 2) with anodd M . A negative-flux version [196] of the states wasproposed at ν = k/ (3 k − , k > .Another non-Abelian candidate at ν = 12 / is aBonderson-Slingerland state[129, 130], whose fractionalstatistics is closely related with that in the Ising topolog-ical order. VI. FABRY-PEROT INTERFEROMETRY WITHNON-ABELIAN QUASIPARTICLESA. The even-odd effect The theory of Fabry-Perot interferometry for non-Abelian anyons has attracted much attention. [197–206].Fabry-Perot interferometry exhibits particularly inter-esting behavior for non-Abelian states, because the in-terference picture can depend on the fusion channel ofthe anyons traveling through the interferometer and theanyons trapped inside the device. Let the trapped topo-logical charge be b , the topological charge of the tunnel-ing anyon be a , and assume that a and b fuse to c . Thenthe tunneling current through the interferometer can becomputed from Eq. (25) with the statistical phase (54)in the cosine. When multiple fusion channels c exist forgiven a and b , the contributions of each fusion channelshould be added with the weight [207] p cab = N cab d c d a d b , (64)where N cab and d x are fusion multiplicities and quantumdimensions. The weights in Eq. (64) reflect the fact thatthere is no correlation between the incoming quasiparti-cle and the particles inside the interferometer, and wemay simply add probabilities, because different fusionoutcomes are always orthogonal. One can check usingEq. (49) that the weights add up to one.For Ising anyons, multiple fusion channels lead to an even-odd effect [199, 200]. Suppose that the leading con-tribution to the current through the interferometer comesfrom e/ quasiparticles with the topological charge σ (39). Consider two possibilities for the trapped topo-logical charge t : (i) t = 1 or ψ ; or (ii) t = σ . Since thetraveling anyon has topological charge σ , there will be aunique fusion channel for it and the trapped topologicalcharge in case (i). The theory from Section IV applieswith the statistical phase determined by the topologicaland electrical charges inside the interferometer. In case(ii) two equally likely fusion channels exist according toEq. (39). They correspond to the statistical phases (54)that differ by π . Hence, the two fusion channels interferedestructively with each other and no dependence of thecurrent through the interferometer on the magnetic fieldcan be seen[199, 200]. After a sufficiently strong change in the magnetic field,a new anyon of topological charge σ is expected to enterthe bulk of the interferometer. This leads to the switch-ing between regimes (i) and (ii). The name “even-oddeffect” reflects that (i) corresponds to an even numberof trapped quasiparticles and (ii) corresponds to an oddnumber.The even-odd effect is present in all non-Abelian statesof the 16-fold way[10] and its absence in an experimentwould prove Abelian statistics. The opposite is not nec-essarily true [190]. Indeed, the existence of two charge- e/ anyons in Abelian states of the 16-fold way maymimic the two fusion channels of the e/ -particles in non-Abelian topological orders[190].The above physical picture assumes that the trappedtopological charge does not fluctuate randomly on thelaboratory time scale. If there are a non-zero even num-ber of e/ particles inside the interferometer, one shoulddistinguish between the cases where they exist in thetopological sector 1 or ψ . The interferometer will exhibitthe same field periodicity in either case, but the phase ofthe signal will differ by π between the two cases. Overthe laboratory time scale necessary to accumulate datain an experiment, it is possible that neutral fermions ψ can tunnel between the edges of the device and one ormore localized states in the bulk, and thereby changethe topological sector of the interior. In this case, themean occupations during the measurement should be athermal equilibrium distribution, determined by the en-ergy differences between states in the different sectors. Ifone or more of the trapped e/ particles is far from theothers and far from the boundaries, then the energy dif-ference between the 1 and ψ sectors will be smaller than k B T , and the two sectors will have equal probability inequilibrium. In this case, the e/ signal would be lost foreven occupation numbers as well as for odd. Effects offrequent ψ -tunneling were investigated in Refs. 208–212.The discussion above also ignores the tunneling ofanyons of charge e/ between the edges at the constric-tions in the interferometer. Such anyons carry topologicalcharges 1 and ψ and do not exhibit an even-odd effect. Acontribution to the tunneling current from e/ particlesshould exhibit a periodicity with respect to the magneticfield that is two times shorter than for charges e/ , andit should be present regardless of the number of enclosed e/ particles. In addition, if there is a significant contri-bution to the signal from e/ particles that wind twicearound the interferometer, that contribution should be-have similarly to that of e/ particles. B. Experimental investigations In a series of papers dating back to 2007, Willett andcollaborators have reported observations of even-odd al-ternation in carefully prepared GaAs samples at fillingfactors 5/2 and 7/2, in a Fabry-Perot geometry. (SeeRef. 213 and references therein.) In the most recent ofthese papers, they reported extensive measurements on9eleven different samples, including analyses of the oscil-latory dependences on magnetic field and gate voltages.The interpretation of these experiments assumes thatthe interferometer is in a compressible Aharonov-Bohmregime, where the periods are strongly affected by under-lying filled Landau levels. It assumes, further, that whenthere is an even number of e/ particles enclosed by theinterferometer path, the system is consistently in one ofthe two possible topological sectors, 1 or ψ , and that itreturns to the same sector when two more quasiparticlesare added. At ν = 5 / , ten e/ quasiparticles will leavethe interferometer as the flux Φ = B ¯ A is increased by Φ , so that the parity will switch from even to odd andback five times in this interval. As it turns out, if onetakes into account the Abelian phase acquired when an e/ particle encircles an even number of e/ particles in afixed topological sector but ignores the even-odd switch-ing that turns the interference on and off, one would pre-dict an interference with a flux period of Φ . When thissignal is modulated by the rapid switching with period Φ / , the power spectrum is predicted to have promi-nent peaks at frequencies / Φ , / Φ and / Φ . A sim-ilar analysis at ν = 7 / predicts that interference peaksdue to the circulation of e/ quasiparticles should occurthere at frequencies . / Φ , . / Φ and . / Φ .In addition to the signal from e/ particles, one shouldexpect contributions from other processes with differentflux periods, as well as aperiodic features due to disorder,etc. These contributions tend to obscure the underlyingperiodicities in the raw data and lead to complicatingfeatures in the Fourier transform. Nevertheless, it ap-pears that strong peaks were observed at the predictedpositions at ν = 5 / , and to a lesser extent at ν = 7 / .These results give support for the occurrence of even-oddalternations, consistent with the existence of non-AbelianIsing anyons. (The observations do not distinguish be-tween the Pfaffian, Anti-Pfaffian, or PH-Pfaffian states.).There are, however, some aspects of the experimentswhich are not well understood. The interference areasneeded to fit the data were very small, typically of order0.25 µ m , while the lithographic areas were squares rang-ing from 2.5 to 5.7 µ m on a side. Moreover, since the in-terference area should presumably connect the openingsin the defining gates on two ends of the interferometer,the width in the perpendicular direction must be lessthan 0.1 µ m. It is not clear what are the physical mech-anisms that give rise to this unusual geometry. Theremay also be questions about the extent to which it is ap-propriate to talk about the existence of a quantized Hallstate in a region of these dimensions.Nevertheless, the small interferometer area appears tobe reproducible, as it is seen in many samples, and per-sists at a variety of filling factors. The samples used inthese experiments include a number of special features,including carefully designed screening layers, which maybe important for understanding the resulting geometry.It should also be noted that in contrast to the procedures most commonly employed in quantum Hall interferencemeasurements, the samples in these experiments were il-luminated before measurement.Experiments by An et al. [78] have measured phaseslips and telegraph noise at ν = 5 / analogous to thosethey found at ν = 7 / , which were discussed in SectionIV.A.2, above. They find a distribution of phase-slipsizes with several peaks, including a prominent broadpeak centered at ∆ θ ≈ π/ , which they attribute tosimultaneous entry of an e/ quasiparticle into the in-terferometer region and tunneling of a Majorana fermion( ψ -particle) between the edge of the system and the lo-cation of an e/ particle in the interior. However, theirdata is less extensive than that of Willett et al., and itseems difficult to rule out alternative explanations fortheir results. VII. MACH-ZEHNDER INTERFEROMETRY A different type of interference geometry, which hasalso been realized in quantum Hall states, is Mach-Zehnder geometry [214, 215], Fig. 13. It is natural toask whether this geometry can lead to a demonstrationof fractional statistics. We shall see that the geometryis also of theoretical interest since it appears in a gen-eral explanation why fractional charge entails fractionalstatistics [214]. FIG. 13. The current from source S splits between two drainsD1 and D2. Drain D2 is inside the interference loop. FQHliquid is shaded. At first sight, nothing changes compared to the Fabry-Perot case, at least, in the incompressible limit with weakbulk-edge interactions. The tunneling rate is still givenby Eq. (24) and the current seems to be the same asabove. But, the magnetic-flux periodicity e Φ /q m > Φ of the so-computed current would conflict with the rig-orous Byers-Yang theorem [216]. Indeed, the Mach-Zehnder interferometer has a hole. A change of the fluxthrough the hole by one quantum should be invisible tothe electrons from which the system is made. Hence, themagnetic flux period cannot possibly exceed Φ . φ s contributing to θ = φ − φ in Eq.(24) changes after each tunneling event, and the tunnel-ing rate changes accordingly. (The contradiction withthe Byers-Yang theorem would be unavoidable if frac-tional charges could have Bose or Fermi statistics.) νe νe νee − νee − νe FIG. 14 Transitions between the states of a Mach-Zehnderinterferometer in a Laughlin liquid at T = 0 . The precise expression for the current depends on thedetails of statistics [10, 151, 194, 214, 215, 217–221] andsimplifies greatly for the Laughlin states at zero tem-perature [214]. Fig. 14 illustrates possible transitionsbetween topological charges of the drain at the Laugh-lin filling factors ν = 1 /m . At zero temperature, chargeonly goes from the higher chemical potential to the lowerchemical potential and hence all transitions are onlypossible in the direction of the arrows. The transitionrate p n along the arrow connecting the trapped charges nνe mod e and ( n + 1) νe mod e is given by Eq. (24)with the statistical phase that depends on n . The av-erage time a transition takes is t n = 1 /p n . The totaltime for one full circle in the diagram Fig. 14 is thus ¯ t = (cid:80) p − k =0 /p n . Since a charge me/m = e is transmittedin that sequence of tunneling events, the total current I = e/ ¯ t is the harmonic average of the currents (25) atall possible values of the statistical phase φ s . If the fluxthrough the hole is increased by Φ , this only has theeffect of shifting the transition times t n to t n +1 , so thenet current is unchanged. ( − e/ , σ ) (0 , ψ ) p ( π ) / , p (0) / e/ , σ ) p ( π ) p (0) ( e/ , p ( − π/ / e/ , ψ ) p ( π/ / p ( − π/ p ( π/ FIG. 15: Transitions between the states of a Mach-Zehnderinterferometer in a PH-Pfaffian liquid at T = 0 . Non-Abelian statistics results in more complicated be-havior due to multiple fusion channels for non-Abeliananyons. Fig. 15 illustrates possible transitions amongdrain states for a PH-Pfaffian liquid, in which charge- e/ quasiparticles tunnel at the QPCs. Each vertex islabeled by the trapped electric charge mod e and thetrapped topological charge in the drain. The transitionrates are defined in terms of p ( θ ) = [Γ +Γ ] r ( V, T )+2Γ Γ cos( φ AB + θ + α ) r ( V, T ) , (65)where the statistical phase θ can be , π, or ± π/ , Γ and Γ are the tunneling amplitudes at the two QPCs, φ AB is the Aharonov-Bohm phase, and r , and α havethe same origin as in Eq. (24). The transition rates equal p ( θ ) / whenever two fusion channels are available. It isapparent from the figure that the system can return tothe initial state in multiple ways. One finds, as in theAbelian case, that the current is unchanged if the fluxthrough the hole is increased by Φ .Shot noise in the weak tunneling limit yields the moststriking signature of statistics in Mach-Zehnder interfer-ometry [207]. According to Eq. (20) one would naivelythink that the noise does not contain any new informa-tion compared to the current. This is indeed the casein the Fabry-Perot geometry. In Mach-Zehnder interfer-ometry, however, Eq. (20) does not hold since tunnel-ing events are not independent, as their probability isaffected by the topological charge, accumulated in thedrain. The noise exhibits particularly interesting behav-ior in non-Abelian states [10]. For example, it can evendiverge at some values of the magnetic flux [151]. Indeed,consider Fig. 15 and suppose that Γ ≈ Γ , r ≈ r in Eq. (65). Changing the magnetic field allows tun-ing φ AB so that p (0) ≈ . Consider an interferometerin the initial state ( − e/ , σ ) . The rate p ( π ) / of theprocess ( − e/ , σ ) → (0 , ψ ) is much faster than the rate p (0) / of the process ( − e/ , σ ) → (0 , . Hence, before1the interferometer enters the (0 , state, it will evolvethrough multiple loops ( − e/ , σ ) → (0 , ψ ) → ( e/ , σ ) → ( e/ , t ) → ( − e/ , σ ) , where t = 1 or ψ . The aver-age charge q t , transmitted during those loops, is large: q t (cid:29) e . Eventually, the drain reaches the (0 , state.The transition rate p (0) out of that state is small, andthe interferometer will be stuck in the (0 , state for along time t ∼ /p (0) . At some point, a quasiparticlewill tunnel through the device, and it will rapidly reachthe ( − e/ , σ ) state again. One observes the alternationof periods of high current and periods of no transport,when the drain is stuck in the (0 , state. This implieshigh noise.On the other hand, possible tunneling between inter-ferometer edges and localized states in the bulk does nothave much effect on the current and hence does not leadto telegraph noise as long as the tunneling events are sep-arated by longer time intervals than the tunneling eventsat the QPCs.Calculations tend to be rather involved in the theoryof Mach-Zehnder interferometers even in the lowest orderperturbation theory. Yet, curiously, in some cases, theBethe ansatz allows an exact solution for the current andnoise in a model of a Mach-Zehnder interferometer [217,219, 222, 223].It is instructive to reconsider Fabry-Perot interferome-try in light of the Byers-Yang theorem in a geometry witha hole in the center of the interferometer [224]. If the fluxthrough the hole is increased by Φ on a sufficiently shorttime scale, the charge in the hole will increase by νe . ForFQH states, this will alter the statistical phase, which, inaddition to possible effects of the Coulomb interaction,will generally change the transmission of the interferom-eter. This does not contradict the Byers-Yang theorem,however, because the theorem only applies in equilib-rium. Equilibrium is established on a long time scale byrelatively rare tunneling events between the edge of theinterferometer and the inner edge around the hole. Suchtunneling also leads to telegraphic noise [225]. A relatedphenomenon of switching noise in a Fabry-Perot interfer-ometer with a fluctuating number of trapped anyons wasproposed [226] as a probe of statistics.Mach-Zehnder interferometry has not yet been imple-mented for FQH states, despite its success in the inte-ger quantum Hall regime. A recent reference [227] shedslight on that challenge by measuring the dependence ofthe interference visibility on the filling factor. The visi-bility of the interference in the outer ν = 1 channel di-minishes as the bulk filling factor decreases towards 1.This is accompanied by signatures of edge reconstruction, i.e. , the emergence of topologically unprotected pairs ofcontra-propagating edge modes. It has long been recog-nized that quantum Hall edges exhibit complicated spa-cial structure. Progress in interferometry will likely de-pend on a deeper understanding of edge states.Various other geometries have been considered in theliterature. Ref. 228 considers a “wormhole” geometry inwhich a tunneling contact creates a shortcut along a chi- ral FQH edge. Long tunneling contacts were proposed asprobes of neutral modes in Ref. 229. Ref. 230 introducesa modification of the Fabry-Perot setup that reveals aneffect of topological vacuum bubbles. Ref. 231 reports anexperimental realization of a version of a Mach-Zehnderinterferometer in which a single edge is split into twoconducting channels that provide two interfering paths. VIII. OTHER TECHNIQUES Several other approaches can give information abouttopological order. While that evidence may be indirect,it has importance because of the challenges faced by in-terferometry. In this section we focus on four methods:thermal conductance experiments [123, 232, 233], detect-ing upstream neutral modes [234], thermoelectric trans-port [235, 236], and tunneling into the edge [109, 237].Thermal conductance is particularly useful as a probeof non-Abelian statistics. Tunneling seems an enticinglystraightforward probe of topological order. The actualinformation it gives turns out rather limited due to thecomplex physics of real edges. The complications, un-covered in tunneling experiments, are likely relevant forother probes, including interferometry. A. Thermal transport The quantization of thermal conductance has longbeen recognized in non-interacting 1D systems [238].Quantum Hall liquids are unique in that their thermalconductance remains quantized even for strong interac-tions [22, 120, 121]. Consider first an Abelian FQH sys-tem with chiral edges such that all edge modes propagatein the same downstream direction, clockwise or counter-clockwise, depending on the direction of the magneticfield. Since the bulk is gapped, heat is only carried bythe edge at the lowest temperatures. A chiral edge, em-anating from a source at the temperature T , remains inthermal equilibrium at that temperature. The local ther-mal current along the edge in any point x depends on thetemperature and the details of the Hamiltonian of a localsubsystem around point x . At the same time, the heatcurrent must be the same in all points of the edge sinceenergy cannot accumulate on any portion of the edge ina steady state. This implies that the heat current J h ( T ) depends only on the temperature and is not sensitive tomicroscopic details such as the mode velocities and inter-mode interactions. As a consequence, the thermal cur-rent on an edge with n chiral modes reduces to the sumof n thermal currents in the simplest chiral systems withharmonic Lagrangians of the form, equivalent to (56): L = (cid:126) π (cid:90) dx [ ∂ t φ∂φ x − v ( ∂ x φ ) ] (66)2with an arbitrary edge velocity v . An easy calculationyields the quantized thermal conductance for an FQHbar with two edges emanating from two terminals at thetemperatures T and T + ∆ T : κ = lim ∆ T → J h ( T + ∆ T ) − J h ( T )∆ T = nκ T, (67)where κ = π k B / h .Many quantum Hall states possess topologically pro-tected upstream neutral modes that travel in the di-rection, opposite to that of the charge mode. In par-ticular, Jain’s states at ν = [ p + 1] / [2 p + 1] have onedownstream mode and p upstream modes [108]. The ef-fect of the upstream modes on the thermal conductancedepends on the edge length L in comparison with theequilibration length ξ on which the energy exchange be-tween the upstream and downstream modes is significant[135, 232, 239]. If L (cid:28) ξ , the thermal conductances of the n u upstream modes and the n d downstream modes addup: κ = ( n u + n d ) κ T . This can be understood by observ-ing that ( n u + n d ) noninteracting modes emanate fromeach of the two terminals, maintained at different temper-atures (Fig. 16). A long edge reaches thermal equilibriumso that κ = | n u − n d | κ T . The absolute value sign arisesbecause heat can only flow from the hotter terminal tothe colder terminal. For n u (cid:54) = n d , thermal equilibriumat the temperature of the majority modes is establishedon the length scale ∼ ξ . At n u = n d the approach to theequilibrium is slow [232] and the thermal conductance κ ∼ ξ/L . At low T , the equilibration length is predictedto diverge as a power of the temperature [239]. FIG. 16. One downstream and two upstream modes are shownon each edge. The modes, emanating from the left terminal,have the temperature T . The modes, emanating from theright terminal, have the temperature T + ∆ T . Thermal transport in non-Abelian liquids is qualita-tively similar to the Abelian case. The integer num-bers n u and n d should be substituted with the combinedcentral charges of the upstream and downstream modes[22, 121]. Those central charges are not integer in gen-eral. In particular, a Majorana edge mode contributes κ T / to the thermal conductance. This can be under-stood by interpreting a real Majorana fermion mode ashalf of a complex Dirac fermion mode that can be presenton Abelian edges and carries the central charge c D = 1 .Indeed, a complex fermion Ψ D can be represented as thecombination Ψ D = Ψ + i Ψ of two real fermions with Ψ = Ψ † and Ψ = Ψ † .Since the FQH effect is observed at low tempera-tures, the relevant heat currents are low and challengingto measure. An ingenious approach was introduced inRef. 240 in an experiment in the integer quantum Halleffect. The current I = GV enters the central float-ing contact (Fig. 17) from a biased source. The cur-rents I/N leave the contact along N arms. The dis-sipated Joule heat Q = [ GV − N G ( V /N ) ] / raisesthe temperature T m of the central floating contact andis carried away along the edges of the n arms, so that Q = N [ T m κ ( T m ) − T κ ( T )] / , where T is the tem-perature of the cold contacts. κ can be found after T m is determined from the current noise. A possiblephonon contribution to the heat escaping the centralfloating terminal can be eliminated with a subtractiontrick[240]. The success of the experiment depends onhow fast charge leaves the central floating contact. Fora short dwell time, full equilibration cannot be achievedand the thermal conductance cannot be measured cor-rectly [135, 241, 242].Our discussion so far has ignored heat losses from theedge to the bulk by phonons or other possible processes,which can contribute at finite temperatures. Such pro-cesses do not appear to be a major issue in current ex-periments. For a theoretical discussion of bulk losses, seeRef. 243. FIG. 17. The current from source S partitions in the centralfloating contact into N = 4 currents along the N = 4 arms ofthe device. Using an adaption of the above geometry, Banerjee et al. measured the thermal conductance at several frac-tionally quantized states in GaAs, finding the results[232] κ = κ T at ν = 1 / and / , and κ = 2 κ T at ν = 4 / ,consistent with theory. The thermal conductance at ν = 2 / remained relatively far from the equilibratedvalue as expected, since there is one upstream mode3and one downstream mode a that filling factor. A re-cent experiment on graphene[233] measured κ ≈ κ T at ν = 4 / , in agreement with theory.The second Landau-level filling factors / , / , and / in GaAs were explored in a different sample from theone used for the states of the first Landau level [123]. Theobserved κ = 2 . κ T at ν = 7 / is consistent with theLaughlin topological order: two units of thermal conduc-tance come from two integer edge modes and one moreunit comes from one fractional edge channel. The ob-served thermal conductance was . κ T at ν = 8 / .The topological order at ν = 8 / is expected to be thesame as at the filling factor / . The predicted equilib-rium thermal conductance is κ theor = 2 κ T for an infiniteedge. Indeed, the edge contains two downstream integeredge channels, and one downstream and one upstreamfractional channels. The difference between the theoreti-cal and experimental thermal conductances is similar tothe case of ν = 2 / . This can be understood by observingthat two of the downstream channels interact only weaklywith the remaining downstream and upstream channels[135]. We first observe that the overall charge mode ismuch faster than the rest of the modes in the secondLandau level [135]. Thus, its excitations leave the sys-tem before they can exchange energy with the rest of theedge channels on a realistic finite edge. Besides, the in-teger spin mode is only weakly coupled with the othermodes [135]. Thus, the thermal conductance containsthree independent contributions: one quantum from thecharge mode, one quantum from the spin mode, and thecontribution of the remaining downstream and upstreammodes. The latter contribution is subject to strong finite-size corrections just like at ν = 2 / .The observed thermal conductance at ν = 5 / is (2 . ± . κ T at higher temperatures and growsrapidly at low temperatures. Both properties are consis-tent with the non-Abelian PH-Pfaffian order [135, 239],but the interpretation of the data is still debated [135,175, 244–247].To finish this section, we note that Ref. 248 proposesan experiment with shot noise induced by a temperaturegradient in a quantum point contact. B. Upstream modes Thermal conductance experiments cannot distinguisha state with n downstream modes and no upstreammodes from a state with n + m downstream modes and m upstream modes, under conditions where energy is equi-librated between different modes on an edge. Thus, itis helpful to supplement thermal transport experimentswith a tool for detecting upstream modes. Several setupshave been used for that purpose. Fig. 18 illustrates anearly theoretical proposal [249]. Upstream neutral modescarry no current but they can carry energy. Charge tun-neling from source S at QPC1 induces Joule heat thatis carried upstream to QPC2. A thermoelectric effect generates excess current noise in drain D and reveals thepresence of upstream neutral modes. The role of QPC1can also be played by a hot spot [250] at an ohmic con-tact. Much of the early theoretical work [249] was focusedon the states of the 16-fold way at ν = 5 / . See Ref.251 for the application of the setup[249] to Read-Rezayistates. FIG. 18. Charge tunnels into the edge from source S atQPC1. The upstream neutral mode (dashed line) carriersenergy to QPC2. Non-equilibrium noise is generated in drainD at QPC2. Experimental probes of upstream neutral modes arewell established now. Topologically protected upstreammodes were observed at the filling factors / and / inthe first Landau level [234, 252] in agreement with theory.No evidence of an upstream mode was seen[253] at ν =7 / in agreement with a Laughlin order at that fillingfactor. An upstream mode was found at ν = 8 / , as itshould be for a particle-hole conjugate state of the / liquid. An upstream mode has also been detected[234,253] at ν = 5 / in agreement with the anti-Pfaffian andPH-Pfaffian models.The above experiments deal with relatively long edgesof several tens of microns. At the scale of microns, evi-dence of upstream modes was seen[254, 255] at ν = 1 / and ν = 4 / even though no topologically protected up-stream mode is expected at those filling factors. Thiscan be understood as an example of edge reconstruc-tion [112]. The reconstructed upstream modes do notsurvive on longer edges since inter-channel tunneling lo-calizes them. Topologically protected modes are alwaysdelocalized. The dependence of the upstream noise onthe length of the edge is addressed theoretically in Refs.256 and 257 and experimentally in Refs. 258 and 259.A very recent experimental development consists inprobing upstream modes on interfaces of different fillingfactors [260]. Several other approaches have been pro-posed theoretically for the detection of neutral modes.Much attention has focused on Coulomb blockade physics[261–263], which is closely related to the interferometryideas addressed above. Other proposals include the useof the Kondo effect in a quantum dot, coupled to a quan-tum Hall edge [264], and a proposed experiment to use4a quantum dot transport to distinguish FQH states fromtheir particle-hole conjugates [265]. Neutral modes couldbe detected with momentum resolved tunneling into theedges [266–268], but this technique requires very weakdisorder. The same limitation applies to a proposal [269]to probe topological order by measuring the Hall viscos-ity [270]. C. Thermoelectric transport In the Seebeck effect, a gradient of the electric po-tential builds in response to a thermal gradient. Thestrength of the effect is measured by the Seebeck coeffi-cient Q = −∇ Q/ ∇ T . In a uniform FQH system, underconditions where energy is equilibrated more rapidly thanmomentum is transferred to impurities, the Seebeck coef-ficient should reflect [235] the entropy per charge carrier: Q = − S/ ( eN e ) , (68)where S and N e are the entropy density and the elec-tron density. For a non-Abelian FQH state with a smallnumber of well-separated localized quasiparticles, the en-tropy at low temperatures should be determined [235] bythe number K of states at the fixed positions of quasi-particles. The latter number depends on the number ofthe quasiparticles N q and their quantum dimension d , K ∼ d N q . The number N q is controlled by the mag-netic field. Thus, a measurement of the Seebeck coef-ficient Q ∼ N q log d reveals the quantum dimension ofnon-Abelian anyons.The existing experimental data [236, 271] are limited.Qualitative agreement with the theory for non-Abelianstates of the 16-fold way was reported[236] at ν = 5 / ,but more work is needed before the data are understood.Related theoretical ideas are explored in the papers272–274. See Ref. 275 for a proposal of a thermoelectricprobe of neutral edge modes.The thermoelectric technique differs profoundly fromall the approaches addressed in the previous sections ex-cept the single-electron-transistor probe of anyon charges(Section III.B). Indeed, all those proposed and imple-mented probes of fractional charge and statistics involveedge physics. On the other hand, thermoelectric trans-port occurs in the bulk. Thus, this technique should beinsensitive to the complications of edge physics (see thenext subsection). We note that it has also been suggestedto use a scanning tunneling microscope for a bulk probeof anyon statistics [276]. Another proposed bulk probeinvolves Raman scattering [277]. D. Tunneling It was predicted long ago that the tunneling conduc-tance through a weak link of two FQH liquids follows a universal power dependence G t ∼ T g e − , where g e de-pends only on the topological order [108]. A similar be-havior, G t ∼ T g q − with a universal q q , was predictedfor weak quasiparticle tunneling between two edges of anFQH system [108]. These predictions were based on thechiral Luttinger liquid model.Early results [278] on electron tunneling at ν = 1 / were consistent with the theoretical expectations for g e .Yet, at other filling factors a puzzling dependence g e ∼ /ν was observed [279]. This does not agree with thetheory [280, 281]. Note that edge reconstruction was pre-dicted to occur in experimental samples [282, 283]. SeeRef. 237 for a review.Later experiments focused on quasiparticle tunneling.The observed g q is typically greater than the predictions[109]. Three mechanisms beyond the chiral Luttingerliquid model were introduced to explain the discrepancy:edge reconstruction [110], long-range Coulomb forces be-tween segments of the edge [111], and /f noise and dis-sipation [55]. It is possible that a combination of mecha-nisms is at play. Thus, tunneling experiments only yieldan upper bound on g q and provide limited informationabout topological order [151]. This probably explainsthe difficulties in the interpretation [163–165, 284] of thequasiparticle tunneling experiments at ν = 5 / . Differ-ent ideal theoretical g q are predicted for different statesof the 16-fold way. The observed g q has also differed indifferent experiments [163–165, 284]. Data from differ-ent samples and even from the same sample at differentgate voltages were interpreted in terms of several differ-ent states of the 16-fold way. However, the tunnelingexponent g q was found to change continuously with thegate voltage at the gates that form the tunneling con-tact. The observed values were consistent with an upperbound on the ideal theoretical value for the Pfaffian andPH-Pfaffian orders [151].Tunneling data were used to extract both the tunnelingexponent g q and the quasiparticle charge [163–165, 284]at ν = 5 / from a fit to a theoretical I − V curve.The confidence intervals are elongated ovals in the g q -charge plane and hence the uncertainty in both quanti-ties is high. At the same time, the quantized quasiparticlecharge is known independently. Fitting for g q at a fixedcharge reduces error bars.Note, finally, that tunneling noise was proposed as an-other probe of non-Abelian statistics [285]. IX. CONCLUDING REMARKS Quantum mechanics textbooks usually state that onlytwo types of quantum statistics are possible: Fermi andBose. The argument goes as follows. For two indis-tinguishable particles, there is no way to tell the con-figuration with particle 1 in point r and particle 2 inpoint r from the configuration with particle 1 in point r and particle 2 in point r . Thus, the probabilitiesof the two configurations P ( r , r ) = | ψ ( r , r ) | and5 P ( r , r ) = | ψ ( r , r ) | must equal. Hence, the particleexchange generates a phase change in the wave-function: ψ ( r , r ) = θψ ( r , r ) , where | θ | = 1 . After two particleexchanges, one finds ψ ( r , r ) = θ ψ ( r , r ) (69)so one must have θ = ± . The plus sign describes bosonsand the minus sign describes fermions.The argument might look convincing but it containsmultiple loopholes. First, it may not be necessary forthe wave-function to be single-valued, as is implicitly as-sumed in Eq. (69). Alternatively, the wave function doesnot have to depend just on the positions of the particlesbut may depend on how the system reached a particu-lar configuration. In other words, a single-valued wavefunction may be defined not on the configuration spacebut on the Riemann surface whose points are equiva-lency classes of trajectories in the configuration space.Besides, θ does not have to be a number but may be aunitary operator, if the Hilbert space associated with afixed set of particle positions is multidimensional. Thislast loophole opens the particularly interesting possibilityof non-Abelian statistics.The loopholes have some surprising consequences[286,287] in 3D, but it is in 2D where things become trulyexciting, as systems with anyons, particles with fractionalstatistics or non-Abelian statistics, are mathematicallypossible.But, physics is an experimental science, and the the-ory of anyons is only relevant, if anyons exist in nature.Fortunately, observation of the fractional quantized Halleffect makes their existence an almost mathematical cer-tainty. Indeed, fractional quantization of the Hall con-ductance in appropriate systems is well established ex-perimentally. As explained in Section II, such fractionalquantization of the Hall conductance in an insulator nec-essarily entails the existence of fractional charges, andfractional charges entail fractional statistics.Yet, general arguments do not tell us everything wemight want to know about the particular anyons thatmight occur in a given quantum Hall system. The quan-tum number ν obtained from a measurement of the Hallconductance sets constraints on the possible charges andstatistics of the elementary quasiparticles hosted by theFQH state, but it does not completely determine them.Moreover, general arguments do not tell us whether indi-vidual anyons, or small collections of them, will be man-ifest in any practical experiment.For a long time, our knowledge about fractional chargeand statistics was derived in a rather unsatisfactory way.First, theoretical predictions were made based on as-sumptions about the nature of the ground state in anobserved FQH state. Second, numerics on small idealizedsystems would verify some of the theoretical predictions,most importantly, the form of the ground-state wavefunction. Third, some experimental data would showagreement with some aspects of numerics, such as the spin polarization. This would be interpreted as a proofof the theoretical picture. Such evidence is inevitablyindirect and not always reliable. For example, there re-main persistent discrepancies between calculated energygaps and the activation gaps measured in experiments.Although these discrepancies have been attributed to ef-fects of disorder, theoretical attempts to understand theprecise manner in which impurities affect the measure-ments have only been partially successful.[288]The last decade of the twentieth century saw a break-through in the detection of fractional charges. Theshot noise technique proved particularly fruitful (Sec-tion III.A). A clear direct evidence of fractional statisticshad to wait until very recently. While promising inter-ferometric results for fractional statistics in FQH statesat ν = 1 / and ν = 2 / were published more thana decade ago,[82, 83], interpretation of those data hasproved challenging. Similarly, though promising interfer-ometry results[67] concerning non-Abelian statistics werepublished some ten years ago at ν = 5 / , there have beenquestions about the interpretation of those data, partic-ularly because of the very small interferometer area in-ferred from the experiments.In 2020, a clear direct observation of the anyonic sta-tistical phase in interferometry at ν = 1 / has finallyarrived[68]. Another achievement of 2020 is the imple-mentation of an anyon collider[70] at ν = 1 / . Althoughthe relation of these experiments to fractional statisticsmay not be direct, the experiments do probe effects of col-lisions between pairs of diluted anyons, where fractionalstatistics is an essential ingredient. Results presented in2019 of improved interferometer experiments at ν = 5 / and 7/2, using a large number of samples, have confirmedthe previous measurements on this system, and give ad-ditional support to the existence of particles with Ising-type non-Abelian statistics in these states. Our under-standing of interferometer experiments has increased aswe have seen that one should distinguish measurementswhere the central region is in an incompressible state,with at most a few localized quasiparticles, and the moreusual situation, where there are many quasiparticles inthe system, which can enter and leave on a laboratorytime scale as parameters such as the magnetic field andgate voltages are varied.Probing potentially non-Abelian states on fragileplateaus of the second Landau level is certainly chal-lenging. Yet, the distinction between non-Abelian andAbelian statistics is more dramatic than the distinc-tion of Abelian fractional statistics from the Fermi andBose statistics. This opens a way for probes that woulddemonstrate the existence of non-Abelian statistics eventhough they would not allow distinguishing Abeliananyons from fermions. One such probe is thermal con-ductance (Section VIII.A). Remarkable evidence of non-Abelian statistics at ν = 5 / came from a thermal con-ductance experiment[123] in 2018The main focus of the experimental work on anyonicstatistics has been on the simplest Abelian and non-6Abelian filling factors / and / . We eagerly awaitextension of the recent experimental breakthroughs toother filling factors. As this review shows, there is nolack of theoretical proposals to detect fractional statis-tics, and the ball is in the experimentalists’ court. Yet,there is much work for theory too, since the interpretationof the data is often challenging. Major puzzles surroundkey probes, such as Fabry-Perot interferometry. For ex-ample, it has been found that Fabry-Perot interferometryexhibits an enigmatic pairing effect at certain integer fill-ing factors [289, 290]. Until that effect is understood, itis hard to be confident in the interpretation of FQH data.Almost all probes that have been proposed or imple-mented are based on edge physics. This is not surprising,since edges dominate transport and it is easier to accessand manipulate the edges than the bulk. Yet, fractionalcharge and statistics are defined in the bulk. The successof edge probes hinges on the bulk-edge correspondencehypothesis (see Section V.B). It is noteworthy that mea-surements of fractional charge in puddles far from theedge of a sample have been successfully carried out usingsingle electron transistors as charge sensors.[41, 42]. It would be highly desirable to also implement bulk probesof fractional statistics that would not rely on bulk-edgecorrespondence. 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