Topology in FQHE: effective magnetic field, fractional charge, and fractional braiding statistics

Transcription

1 Topology in FQHE: effective magnetic field, fractional charge, and fractional braiding statistics

2 Topology in FQHE Topology enters the physics of the FQHE through the formation of composite fermions. The composite fermion is a topological particle, because one of its constituents, the vortex, is topological. All liquids of the composite fermions are therefore topological quantum liquids. The topological nature of the CF/FQHE liquid manifests most dramatically through the effective magnetic field for composite fermions. This talk will consider two other consequences of the topology: fractional local charge and fractional braiding statistics. Abelian braiding statistics (Gun Sang Jeon, Kenneth Graham, JKJ) Non-Abelian braiding statistics (Csaba Toke, Nicolas Regnault, JKJ)

3 braiding statistics Leinaas, Myrheim (1977), Wilczek (1982) Imagine particles that acquire non-trivial, path independent phases when they go around one another. Half a loop of a particle around another is equivalent to an exchange, which produces a phase factor said to obey fractional braiding statistics. e iπθ. Such particles are Fractional statistics can be defined only in two dimensions, for particles that have an infinite hard core repulsion. It is a topological concept. Anyons can be modeled as fermions (or bosons) with flux lines bound to them.

4 Just because one can define fractional braiding statistics does not mean that it exists in nature. No elementary particles in nature have this property. However, there is no principle that precludes certain emergent quasiparticles from possessing fractional braiding statistics. FQHE is currently a prime candidate for its physical realization.

5 Plan Pre-composite fermion view Fractional statistics in the CF theory; its significance Nonabelian fractional statistics Fractional statistics has had two lives. The CF theory showed that it is not needed for the understanding of any experimental fact so far. But it later resolved some problems to put the idea on a firmer conceptual footing.

6 1/3 plateau R H = h 1 3 e2

7 Excitations ( ν = 1 ) m Laughlin, 1983 Ψ ground = j<k (z j z k ) m exp[ i z i 2 /4] Ψ quasihole η = j (z j η)ψ ground Quasihole at 1/m is modeled as a vortex. Ψ quasielectron η = exp j z j 2 /4 j ( 2 ) η (z i z k ) m z j i<k Quasielectron looks nothing like an electron. The name quasiparticle is more appropriate.

8 Fractional charge The charge excess or deficiency associated with a quasiparticle or a quasihole is precisely e/m. This can be confirmed by a direct integration of the charge density. Morf and Halperin

9 Berry phase At any given instant, let n(r) be the eigenstate of Ĥ(R): Ĥ(R) n(r) = E n (R) n(r). (A6.1) For the time-dependent Schrödinger equation, Ĥ(R(t)) Ψ(t) = i h t Ψ(t), (A6.2) we try the solution Ψ(t) = e iγ n(t) e (i/ h) t 0 dt E n (R(t )) n(r(t)), (A6.3) where the second phase factor on the right hand side is the familiar dynamical phase factor. A substitution into the Schrödinger equation yields γ n (t) = i n(r(t)) R n(r(t)) Ṙ. (A6.4) For a closed loop C it reduces to γ n = i n(r(t)) R n(r(t)) dr. C (A6.5)

10 Berry phase of a vortex Take a vortex adiabatically in a closed loop. The Berry phase for a vortex simply counts the number of electrons enclosed in the loop: vortex Berry phase for a closed loop Ψ η = N R (z j η)ψ η = Re iθ j γ = dt Ψ η i d dt Ψ η = dθ Ψ η i d dθ Ψ η = dθ( i) dη Ψ η dθ j = ( i)dη = 2π d 2 rρ η (r) r<r = 2πN enc. 1 z j η Ψ η d 2 r 1 z η Ψ η ˆρ(r) Ψ η

11 The charge of a vortex Arovas, Schriefffer, Wilczek, 1984 Equate the Berry phase to the Aharonov Bohm phase of a particle of charge e* Φ vortex = 2πN enc This implies: = 2πρA = 2πe BA hc e = νe (ν = ρhc/eb)

12 Braiding statistics In FQHE, the Berry phase also gets a contribution from the ordinary Aharonov Bohm effect, which must be subtracted: The braiding statistics can be evaluated with the knowledge of the many body wave function for the quasiparticles. θ = C dθ 2π Ψ η,η i ddθ Ψη,η Ψ η,η Ψ η,η C dθ 2π Ψ η i d dθ Ψη Ψ η Ψ η

13 Braiding statistics for the vortices The Berry phase for a vortex: Halperin, 1984 Arovas, Schriefffer, Wilczek, 1984 Φ vortex = 2πN enc The braiding statistics thus depends on the change in the number of electrons upon the insertion of an additional vortex inside the area enclosed by the loop: Φ = 2πθ = 2π N enc = 2πν The 1/m quasihole happens to be a vortex, thus obeys fractional braiding statistics. This is closely related to fractional charge.

14 What does fractional statistics do for us?

15 More fractions

16 Hierarchy scenario Haldane, Halperin, 1984 New FQHE states are produced when the fractionally charged, fractional statistics quasiparticles / quasiholes of a parent state form Laughlin-like daughter states. f = m ± p 2 ± 1 1 p 3 ± 1 1 p 4 ± 1 p 5 Taking the simplest values the hierarchy family tree

17 Pre-composite fermion view I am convinced that fractional statistics exists in nature, causes high Tc superconductivity, and has the potential to change the way we think about physics in a major way.... it will be known to future generations of scientists as a defining moment, the birth of one of Thomas Kuhn s paradigms of science. ----R.B. Laughlin, fractional statistics has measurable consequences, namely the values of the subsidiary fractions 2/5 and 2/7 and their daughters in the fractional quantum Hall hierarchy.... the occurrence of these fractions proves the existence of fractional statistics R.B. Laughlin, 1999 (Nobel lecture)

18 Inconsistencies with experiment Inconsistent with the observation of such a large number of fractions, because the new states are expected to be much weaker than the parent ones. No natural explanation for the observed fractions. Many fractions at deep levels are observed but those at earlier levels are not. Fractions appear in sequences rather than a family tree. Suggests a different underlying structure

19 Conceptual problems The quasiparticles are not ideal, point-like anyons, but have a finite size of ~10 magnetic lengths. The fractional braiding statistics is well defined only when they are not overlapping. A large number of quasiparticles are required to produce a new state. For example, N/2 quasiparticles are required to produce 2/5 from 1/3. The quasiparticles are so strongly overlapping at such high densities that the notion of fractional statistics is no longer meaningful. The concept of fractional statistics loses validity before producing a new fraction. One must repeat this process eight more time to get to the experimentally observed fraction 10/21. If the first link if broken, other fractions remain unborn. No microscopic theory. No quantitative confirmation.

20 Braiding statistics of the Laughlin quasiparticle Kjonsberg and Myrheim (1998): Braiding statistics of quasiholes at 1/3 for N=20, 50, 75, 100, 200. Kjonsberg and Myrheim (1998): Braiding statistics of quasiparticles at 1/3 for N=20, 50, 75, 100, 200. Laughlin s wave function is used. Fractional charge is necessary but not sufficient for fractional braiding statistics.

21 Composite fermion theory

22 The CF theory in a nutshell Electrons transform into composite fermions by capturing 2p quantized vortices. Composite fermions experience a much reduced effective magnetic field. The complex problem of strongly correlated electrons thus maps into a relatively simpler problem of weakly interacting fermions at an effective magnetic field. ( B = B 2pρφ 0 φ 0 = hc ) e ν ν = 2pν ± 1 Ψ ν = P LLL Φ ν (z j z k ) 2p j<k

23 Statistics The particle statistics is not a matter of fine detail. It has dramatic, testable consequences, both in condensed matter physics and quantum chemistry. Fermionic nature of electrons is crucial for explaining atomic physics (periodic table), Fermi sea, superconductivity. Bosonic nature of He-4 atoms is responsible for the phenomenon of superfluidity.

24 Statistics of composite fermions The fermionic statistics of composite fermions is firmly established and has many measurable consequences. Much of our understanding is based on that fact. FQHE: composite fermions fill CF-Landau levels prominent sequences of fractions Effective magnetic field CF Fermi sea Microscopic wave functions Excitations; energy level counting etc.

25 Quasiparticles and quasiholes in the CF theory A quasiparticle is a composite fermion in an otherwise empty level, and a quasihole is a missing composite fermion in an otherwise full level.

26 CF quasiholes at 1/3, 2/5, 3/7 CF quasiparticles at 1/3, 2/5, 3/7

27 CF quasiparticle / CF quasihole are essentially exact. overlaps ρ(r)/ρ ν=1/3 N=8, 2Q=20 Exact CF ρ(r)/ρ ν=1/3 N=8, 2Q=22 Exact CF CF-Quasiparticle CF-Quasihole ρ(r)/ρ ν=2/5 N=9, 2Q=18 r/l Exact CF ρ(r)/ρ ν=2/5 N=9, 2Q=19 r/l Exact CF CF-Quasiparticle CF-Quasihole r/l r/l

28 Dev, Kamilla, Jain (CF energies) He, Xie, Zhang (exact diagonalization) L L 0.05% accuracy with no adjustable parameters

29 N=8, 2Q = 19 N=8, 2Q = N=12, 2Q= E [e 2 /l] N=8, 2Q = 17 N=8, 2Q = L L L

30 It is thus established that the quasiparticles are excited composite fermions, and quasiholes are missing composite fermions. This gives us a unified description of the quasiparticles and quasiholes of all FQHE states, as well as of states containing many quasiparticles or quasiholes. (The quasihole of 1/m happens to be a vortex. But that is an exception. Other quasiparticles or quasiholes are not vortices, but have a tremendously complicated structure in terms of electrons.)

31 The phenomenon of the FQHE and many related experimental observations in the lowest Landau level can be understood, at an impressive level of qualitative and quantitative detail, as a consequence of composite fermions, without any consideration of fractional statistics. No low energy states are missed by the CF theory. There does not seem to be any need, nor any room, for fractional braiding statistics. The end of fractional statistics?

32 Resurgence of fractional braiding statistics If fractional braiding statistics is to be real, it must be a part of the CF theory. We will now see that the CF quasiparticles obey fractional braiding statistics. In fact, the CF theory provides the simplest derivation of this braiding property. Fractional braiding statistics is not an additional concept, but a simple and direct consequence of the formation of composite fermions.

33 Whence fractionalization? The charge and statistics depend on the choice of the reference state, i.e. the vacuum. Composite fermions are charge -e fermions relative to the vacuum containing no electrons. All physics can be described in terms of charge -e particles. However, if we take a nearby FQHE state as the vacuum, the CF quasiparticles have fractional local charge and fractional braiding statistics.

34 First: fractional local charge

35 Local charge $"# %;<=>' $" %&'( )) )*+, ()*+ "# *+),-./01.&20345,6./06745)1.0&).2)$( &796745)1.0&).2)$ " " #" $" $#" :" :#" &(l 0 '"( '"' '" "& "% "$ )>?@%+ -./ *647890:234:;89/524*/26/(,< /98976*;=:;89/524*/26/( "# " <" '" '<" (" *,l &"' 0 &"& &" "% (>?@$* The charge excess or deficiency relative to a FQHE state (taken as the vacuum ) is called the local charge. The local charge of a quasiparticle or a quasihole can be determined directly from its density profile. e = e d 2 r[ρ(r) ρ] "$,-./012341)53678/01239:78.413).15.;+# 87865):<9:78.413).15.; "# " =" &" &=" '" )+l 0

36 Alternatively: The local charge of a CF-quasiparticle is equal to the charge of an electron + the charge of 2p vortices (which produce a correlation hole around it). It has a precise fractional value. Fractional charge (Of course, there is no fractionalization of electrons.) For ν = n 2pn + 1, e = 1 + 2p n 2pn + 1 = 1 2pn + 1

37 CF-quasiparticles and CF-quasiholes obey fractional braiding statistics. The phase associated with a complete loop of a CFquasiparticle is given by: Berry phase difference: Fractional statistics Φ = 2π BA φ 0 Φ = 2π2p N enc = 2π θ = 2p 2pn + 1 Does the heuristic expectation hold up in a microscopic calculation? + 2π2pN enc (gives B*) 2p 2pn + 1 2πθ The fractional statistics is thus simply a statement about the change in the effective magnetic field due to order one changes in the density.

38 CF quasiparticle at 1/3 0 n he quasiparticle at ν = 1/3 n+lis then given by (b) z1 z2... Ψ 1qp CF = P (z j z k ) 2 z 1 z 2... j<k..... z1 N 2 z2 N = N 2 N i=1 [ 6 k (z i z k ) 1 j (z Ψ GS, j z i ) The form of the wave function is different from Laughlin s, and more complicated.

39 Comparing the two trial wave functions n N Non-composite n+l fermion approaches (b) n 0 n+l N 3 Table The overlap of the exact wave function for the quasiparticle excitation, obtained from diagonalization of the Coulomb Hamiltonian, with the quasiparticle trial wave functions of Eq. (12.30) and Eq. (12.37). The disk geometry with symmetric gauge is used. D is the dimension of the Fock space in the lowest Landau level. Source: Dev and Jain [121]. N D CF Laughlin Jeon and Jain, 2003 Kasner and Apel, 1993 Girlich and Hellmund, 1994 Bonesteel and Melik-Alaverdian, 1998

40

41 CF quasiparticle Laughlin Kjonsberg and Leinaas (1999): Braiding statistics of CF quasiparticles at 1/3 is meaningful.

42 Text The CF-quasiparticles obey well defined braiding statistics provided they are separated by >10 magnetic lengths. Jeon, Graham, Jain, PRL 2003, PRB 2004

43 FQHE quasiparticles are not ideal but fuzzy anyons.

44 Can fractional statistics be measured? Abelian braiding statistics is probably valid, but a rather subtle property of the long distance correlations in the wave function. So far, it has not been observed experimentally. Its observation is complicated by the fact that the quasiparticles are nonideal anyons: (i) In a many body state of quasiparticles, they overlap significantly, thus invalidating the notion of braiding statistics. (ii) In a two quasiparticle scattering experiment, the quasiparticles must be kept far apart, to ensure that they do not overlap. However, the statistics then provides but a small contribution to the full Berry phase. The measurement of fractional statistics remains an interesting challenge.

45 What good is fractional statistics? Usually, it simplifies the problem to work with a few excitations rather than all particles. That is the Landau Fermi liquid theory philosophy. However, for FQHE, this approach takes us from almost free composite fermions to strongly interacting, fractional statistics quasiparticles. (Anyons cannot be taken as weakly interacting; by definition, they have a long range gauge interaction.) Fractional statistics is neither able to produce new FQHE states, or explain the energy level quantitatively. While interesting, fractional statistics is not known to be essential for any aspect of the FQHE so far. The fractional statistics can be derived from composite fermions, but the reverse is not true. Composite fermions are thus more fundamental.

46 Logical order Even though fractional braiding statistics is (perhaps) valid, it does not play a central role in the explanation of the FQHE.

47 Nonabelian braiding statistics

48 Proposal I: The quasiparticles of paired CF state at 5/2 obey non-abelian braiding statistics. Moore and Read, 1991 Proposal II: This property can be used for topological quantum computation. Das Sarma, Nayak, Stern, Halperin, etc., 2006

49 5/2 FQHE Willett et al. Pan et al. Xia et al.

50 Pairing of composite fermions A CF Fermi sea is seen at 1/2, but a FQHE is seen at 5/2. An attractive proposal is that the 5/2 CF Fermi sea is unstable to a BCS-like p-wave pairing of composite fermions, which opens a gap (Greiter, Wen, Wilczek; Moore and Read). Pairing from purely repulsive interactions

51 Pfaffian wave function ( ) [ The paired state is described by a Pfaffian wave function (Moore and Read): ( ) 1 Ψ Pf 1/2 = Pf (z i z j ) 2 exp z i z j i<j Pf M ij = A(M 12 M 34 M N 1,N ) ds for Pfaffian. Compare with: [ 1 4 ] z k 2 Without the Pfaffian factor, the wav Ψ BCS = A[φ 0 (r 1, r 2 )φ 0 (r 3, r 4 ) φ 0 (r N 1, r N )] ( ) k

52 The Pfaffian wave function is the exact zero-energy ground state for a model three-body interaction that penalizes states in which three particles occupy the smallest angular momentum state. H Pf = V i<j<k δ (2) ( r i r j ) δ (2) (r i r k ) (bosons) The exact solutions are also known for the n- quasihole states of this Hamiltonian, which also have Pfaffian structure (and zero eigenenergy).

53 Pfaffian quasiholes vortex: (z j η)ψ Pf 1/2 = j j (z j η)pf ( 1 z i z j ) Φ 2 1 (charge 1/2) 2 quasiholes: Ψ Pf 1/2 (η, η ) = Pf (M ij ) Φ 2 1 M ij = (z i η)(z j η ) + (i j) (z i z j ) (charge 1/4 for each) 2n quasiholes: M ij = n α=1 (z i η α )(z j η α+n ) + (i j) (z i z j ) The wave function for 2n quasiholes is not fully specified by their positions. There are 2 n 1 linearly independent wave functions for a given set of positions. explic

54 N=10 two quasiholes L = 1, 3, 5 respectively four quasiholes This model predicts zero energy states a L = 0 2, 1 0, 2 4, 3 1, 4 4, 5 2, 6 3, 7 1, 8 2, 9 0, 10 1

55 The origin of non-abelian braiding statistics The initial 2n-quasihole wave function is a linear superposition of several linearly independent wave functions. After a braiding operation, the wave function ends up, in general, in a different linear superposition, which is related to the initial one by a Berry matrix. Consecutive braidings do not commute. Hence, non- Abelian braiding statistics.

56 Non-Abelian statistics is on solid grounds for a model interaction. Question: How well does it represent the solutions of the Coulomb interaction?

57 Pfaffian ground state comparison with the exact Coulomb 5/2 state: N overlap Scarola, Jain, Rezayi, 2002 Overlaps not conclusive but decent.

58 Testing the Pfaffian quasihole wave functions N=10 2 quasiholes L = 1, 3, 5 respectively N=10 4 quasiholes This model predicts zero energy states a L = 0 2, 1 0, 2 4, 3 1, 4 4, 5 2, 6 3, 7 1, 8 2, 9 0, 10 1 No one-to-one correspondence he superscript between denotes the solutions the degeneracy) of the model and the Coulomb interactions. Lack of adiabatic connection.

59 Testing the Pfaffian quasihole wave functions N=10 2 quasiholes * * * L = 1, 3, 5 respectively N=10 4 quasiholes * * * * * * * * * ** * * This model predicts zero energy states a L = 0 2, 1 0, 2 4, 3 1, 4 4, 5 2, 6 3, 7 1, 8 2, 9 0, 10 1 No one-to-one correspondence he superscript between denotes the solutions the degeneracy) of the model and the Coulomb interactions. Lack of adiabatic connection.

60 D E 2 E$19& (3) (3) C PfQH sector, for the.possibility that par- Ψ 0.6 as O which = Ψallows Φ The wave function 2 qh 0at or2 qh 2 qh $191 TABLE I: Overlaps between different the PfQH basiscombinations for two quasiticle braidings can produce linear bital angular momentum LCoulomb refers tointeraction the two quasihole eigen$1 holes and the lowest energy states for in of PfQH states, hence nonabelian statistics. (3) C Φ2 qh is two quasiholes state of H with quantum number Lz = L, and $89< the second Landau level at Q=2N-2. The overlaps are defined D the E E 2 energy state for the Coulomb interaction with the $89: L(3) lowest (3) C FIG. 1: Cha $8 $8 as O = Ψsame. The wave function Ψ at or 2 qh 2 qh Φ 2 qh quantum numbers. We note that for two quasiholes, the electrons at 2 (3) N = bital angular momentum L refers quasihole zero energy states of Hto the aretwo singly degenerate. C eigenthe presence (3) - quantum number Lz0.61 state ofnh= 10with = L, -and - Φ2 qh is The solid and the lowest the -Coulomb N =energy state - for interaction with the 1: sl thefig. ground same quantum the N = 14 numbers (3)We - note 0.13 that - for 0.39two- quasiholes, 0.27 respectively. for Ψpf 2 qh zero energy N states L =of0 H 2 are3 singly 4 degenerate (obtained by two delta between 0.54 quasiholes quasi the(d) - quasihole in-the four TABLE I: Overlaps the0.47 PfQH basis for two show th pole. 10 lowest 0.67 energy in - panel The holes and the states0.47 for Coulomb interaction split quasiholi (dotted the second Landau level at Q=2N-2. The overlaps are defined N L = 0D E E (3) (3) 2N 2. C as O = Ψ Φ. The wave function Ψ at or qh 2 qh 2 qh /3 (obtained bital0.67 angular momentum L0.21 refers to the two quasihole eigen the quasi (3) C state of HTABLE with quantum number Lz = L, Φ2 qh is -./.4 quasioverlaps between theand PfQH basis II: for four integrated the lowestholes energyand state for the Coulomb interaction with the the lowest energy states for Coulomb interaction in will exhib same quantum numbers. We note that for two quasiholes, the -./.3defined the second Landau level at Q=2N-1. The overlaps are (3) E 2 zero energy states of singly degenerate. PHN Dare(3) for four quasic TABLE II:asOverlaps the PfQH basis O = between Ψ Φ -./02 into 4 qh,j /N. We have take 4 qh,i i,j holes and the lowest energy for Coulomb interaction in We have For two quasiholes, quasihole bandstates[6] analogous toeach account the no N states degenerate value, Toke,for Regnault, and L Jain, /01 the Landau level atinq=2n-1. The overlapsspectrum are defined tion pote thesecond PfQH band is seen the exact Coulomb D E i and j refer to the corresponding degeneracy index. ThePoverlaps are notchigh, and 2 sometimes very low. (3) N as O = Φ Ψ /N. We have take into # # Overlaps

61 Particle hole symmetry &'()*+,-( #./ &'()*+,-( # 0 1./ 0 1 % $ # " % $ # p-h symmetric antisymmetric conjugate combination combination O(L = 0) O(L = 2) O(L = 3) O(L = 4) O(L = 5) O(L = 6) O(L = 8) " " # $ % 2 3" 3# 3$ 4 " # $ % 2 3" 3# 3$ 4 8 electrons; four quasiholes

62 Effect of disorder We ask how quasiholes and quasiparticles are localized by disorder. For this purpose, we study the two quasihole / quasiparticle state in the presence of one and two delta function impurities. For three-body interaction, a single delta function binds both quasiholes, producing a vortex, but might produce two charge 1/4 quasiholes for the Coulomb interaction.

63 two quasiholes in the presence of delta function impurities two delta functions: Pfaffian w.f. two delta functions: V3 single delta function: Coulomb two delta functions: Coulomb $ $ $ $ $196 $19& $1 $89: $897 $196 $19& $1 $89: $897 $896 $89& $19% $19& $191 $1 $89? 23 /#$ &'( "#$ %# &'( *+1$),-./ *#$ &'( *+$&$),-./0 )#$ &'( 451& /#$ &'( "#$ %# &'( *+1$),-./ *#$ &'( *+$&$),-./0 )#$ &'( $89; $19% $896 $19& $89% $191 $89& $1 $891 $89? $8 $89: $8 $89; $1 $19; $& $&9; $% $8 $89; $1 $19; $& $&9; $% " >/)# " >/)# # <=, # <=, # <=, #<=, $89; $896 $89% $89& $891 $8 $89; $896 $89% $89& $891 $8 $89; $896 $89% $89& $891 $8

64 two quasiparticles in the presence of delta function impurities two delta functions: V3 single delta function: Coulomb two delta functions: Coulomb $ $ $ $,3< $,35 $,3& $, $,34 $,35 $,3% $,3& $,3, $, $239 $,34 $,35 $,3% $,3& $,3, $, $239 "#$" %# &'( *+,$-./0" )#$" &'( *+$&$-./0"1 *#$" &'( :;,& :;,5 :;,< $2 $234 $, $,34 $& $&34 $% 6"-# # 78. # 78. #78. $2 23, 23& 23% $2 23, 23& 23% $2 23, 23& 23% "#$" %# &'( *+,$-./0" )#$" &'( *+$&$-./0"1 *#$" &'( $2 $234 $, $,34 $& $&34 $% 6"-# Toke, Regnault, and Jain, 2006

65 Overlap between the two-quasihole Pfaffian wave function and the exact wave function with two delta function impurities N Ψ Pf 2 qh Ψ (3) 2 qh 2 Ψ Pf 2 qh Ψ (C) 2 qh [0.9831(1)*] 0.326(2) 10 0 [0.862(3)*] 0 [0.045(3)*] 12 0 [0.959(8)*] 0.19(2)

66 Understanding 5/2 FQHE without the Pfaffian wave function Toke and Jain, 2006 The Pfaffian wave function does not contain any adjustable parameters. It does not produce the CF Fermi sea as one limit. We begin with noninteracting composite fermions and ask if the residual interaction between composite fermions opens up a gap.

67 CF diagonalization α Z Z

68 Lowest Landau level )*) +,- "./ 0 1 (% (# 2 3 4&" 2 3 4&# 2 3 4&$ 2 3 4" (# (5 )*) +,- "./ 0 1 (" (& " # $ % & ' " # $ % & &" ' " # $ % & &" ' " # $ % & &" &# '

69 Second Landau level )*) +,- "./ 0 1 ($ (# (" (&" (& 2 3 4&$ 2 3 4" )*) +,- "./ 0 1 (% ($ (# (" 2 3 4&" 2 3 4&# " # $ % & ' " # $ % & &" ' " # $ % & &" ' " # $ % & &" &# ' *Residual interactions between composite fermions open a gap at 5/2. The size of the gap (~0.02) is consistent with Morf s estimates from exact diagonalization studies. *Nonabelian statistics does not appear naturally in this picture.

70 BCS wave function for composite fermions Moller and Simon, 2007 Ψ CF BCS = P LLL Ψ BCS (z j z k ) 2 j<k Ψ BCS = Pf[g( r j r k )] g( r j r k ) = k g k φ k (z j )φ k (z k ) The parameters g_k are determined variationally.

71 The earlier Pfaffian wave function can be well approximated by this form, indicating that it belongs to a more general class of wave functions. The CF-BCS wave function also reduces to the CF Fermi sea in one limit. The adiabatic connectivity of the excitations has not yet been established, however.

72 Conclusions The FQHE state is a candidate for the realization of abelian fractional statistics. CF quasiparticles exhibit (theoretically) well-defined fractional braiding statistics in the low-density limit, which may have measurable consequences. The validity of the Pfaffian model and the notion of non- Abelian braiding statistics for quasiholes and quasiparticles at 5/2 remain unconfirmed for the actual Coulomb state. Further confirmation will be needed before we can assert that 5/2 state = Pfaffian / noabelian state