Theoretical Considerations and Experimental Probes of the =5/2 Fractional Quantized Hall State by Bertrand I. Halperin, Harvard University talk given at the Rutgers Statistical Mechanics Meeting, May 11, 2008 in honor of Edouard Brézin and Giorgio Parisi

E. Brézin and the quantum Hall effect [2]. * From the 2nd paragraph:

=5/2 Quantized Hall State In 1987, Willett et al. discovered a Fractional Quantized Hall plateau at Landau-level filling fraction =5/2, the first even- denominator QHE state observed in a single-layer system. The nature of this state is still under debate.

Moore-Read “Pfaffian” State Moore and Read (1991) proposed a novel trial wave function, involving a Pfaffian, as a model for a quantized Hall state in a half-filled Landau level. Further clarified by Greiter, Wen and Wilczek; suggested as an explanation for the quantized Hall plateau at = 5/2 = 2 +1/2. Elementary charged excitations, which have charge  e/4, obey “non-abelian statistics”. State is related mathematically to a superconductor of spinless fermions, with p x +ip y pairing. (Described as a “p x +ip y superconductor of composite fermions.”)

Non-abelian statistics for Moore-Read 5/2 state Consider a system containing 2N localized quasiparticles, far from each other and far from boundaries. Then there exist M=2 N-1 orthogonal degenerate ground states, which cannot be distinguished from each other by any local measurement. Moving various quasiparticles around each other and returning them to their original positions, or interchanging quasiparticles, can lead to a nontrivial unitary transformation of the ground states, which depends on the order in which the winding is performed. ( Unitary matrix depends on the topology of the braiding of the world lines of the quasiparticles. Matrices form a representation of the braid group). If two quasiparticles come close together, degeneracy is broken; but energy splittings fall off exponentially with separation.

Non-abelian quasiparticles may be useful for “topological quantum computation”. [ Kitaev, quant-ph/ ; Freedman, Larson, Wang, Commun. Math Phys (2002); Bonesteel, et al PRL (2005). ] Manipulation of qubits would be carried out by moving quasiparticles around each other, not bringing them close together. Advantage: exponentially long decoherence times if quasiparticles are sufficiently far apart. Caveats: 1. Current materials are very far from this regime. 2. Moore-Read state is actually not rich enough for general topological quantum computation. However, this may be possible with some other states in the second Landau level. Topological quantum computation

What is the evidence that the =5/2 Quantized Hall State is indeed of the Moore-Read type ? Evidence comes primarily from numerical calculations on finite systems.(Morf & collaborators, 2002, 2003; Das Sarma et al. 2004; Rezayi and Haldane, 2000). Using parameters appropriate to the experimental situation, find a spin-polarized ground state, which seems to have an energy gap, and which has good overlap with Pfaffian wave function. But evidence is not overwhelming, and is certainly open to question. Existing experiments do not provide clear evidence on nature of state. Recent improvements in quality of materials, and new experimental techniques,give hope of resolving these questions.

What are the theoretical alternatives? Anti-Pfaffian state: Topologically distinct from the Pfaffian state, but has similar properties, e/4 quasiparticles, non- abelian statistics. (Would be equally interesting.) Other kinds of paired states, including tightly bound pairs in a fully spin-polarized system, or partially polarized or unpolarized systems. Would have e/4 quasiparticles but not non-abelian statistics. (Not so interesting.) Other kinds of quantized Hall states we haven’t thought of?

Anti-Pfaffian State The Pfaffian (Pf) state is not symmetric under particle-hole conjugation. The anti-Pfaffian (APf) is its particle-hole conjugate. Pf and APf have been shown to be topologically distinct. (Rezayi and Haldane, 2000; Levin, Halperin, and Rosenow, 2007; Lee, Ryu, Nayak and Fisher, ) If you vary the parameters in a system so that the ground state changes from Pf to APf, there must be a phase transition separating the two phases. If the parameters of a system vary in space, so that one region is Pf and one is APf, there must be a boundary separating them, with gapless low-energy excitations. (Both states have an energy gap in the bulk.)

Boundary between a Pf or APf =5/2 state with a =2 Integer Quantized Hall state (filled Landau level) The simplest boundary between a Pf =5/2 state and a =2 state should have two low-energy chiral modes: a bosonic phonon mode and a neutral Majorana fermion mode, traveling in the same direction. The edge has a thermal Hall conductance with K = 1 +1/2 = 3/2. The boundary between an APf =5/2 state and a =2 state has a different structure, and has K=-1/2. The thermal Hall conductance is a topological invariant, cannot be altered by disorder or boundary reconstruction. Q =  T K   2 k B T/3h

Edges of a Pf or APf state =2 =3 =2 =3 Pf APf K= 3/2 K= -1/2 Thermal Conductance K: 3/2 = 1/ /2 = -1/

Nonlinear electrical resistance Experimentally, it is difficult to measure the thermal Hall conductance. However, the different boundary structures of Pf and APf with, say, a vacuum or a simple =2 state should lead to different Luttinger-liquid-type properties, which should give rise to different forms of non-linear electrical resistance at a narrow constriction, which has been studied experimentally. Recent experimental results seem to favor APf. (Marcus lab) The complications expected in a real system have not been completely sorted out.

Competition between the Pfaffian and Anti-Pfaffian State If one has only two-body interactions, and one ignores Landau- level mixing, the Hamiltonian of the half-full Landau level is particle-hole symmetric. Since the Pf and APf states are particle -hole conjugates of each other, they must have identical energies in this model. Degeneracy can be broken by inter-Landau level mixing, effects of impurities, sample boundaries, and deviations from half filling. Sample boundaries may be particularly important in a narrow constriction. Pf and APf may coexist, with a boundary between them. Note: The Pf and APf trial wave functions are exact ground states of models with three-body interactions, which break particle-hole symmetry explicitly.

Proposed experiments to look for non-abelian statistics, or at least test whether =5/2 state is Pf, APf or something else. The most direct demonstration of non-abelian statistics wold require the ability to move one quasiparticle around another in a controlled way. Possible in principle, but we are far from being able to accomplish this technologically. We seek other experiments to examine the =5/2 state to see if it is of the Moore-Read type.

Actual recent experiments *Measurements of the non-linear resistance of a narrow constriction at =5/2 can give important information about the state. Interpretation seems to be complicated *Measurements of the quasiparticle charge. Moore-Read quasiparticles have charge e/4. Recent measurements of shot noise from a quantum point contact at =5/2 support this result. (Heiblum group). Quasiparticles with charge e/4 are necessary, but not sufficient: could also result from other states without non-abelian statistics.

Proposed experiments (1). *Interference-type experiments directly related to non- abelian statistics.

Proposed Interference Experiments at =5/2 Discussed by: Ady Stern and B. I. Halperin (PRL 2006) Other theoretical papers discussing interference experiments with non-abelian quasiparticles include: Das Sarma, Freedman and Nayak, (PRL 2005) Bonderson, Kitaev, and Shtengel, (PRL 2006) Fradkin et al., Nucl Phys B 1998 Bonderson, Shtengel and Slingerland, cond-mat/ : Discuss consequences for Read-Rezayi parafermion states, possibly applicable to =12/5.

II 12 Side Gate = 5/ t1t1 t2t2 Fix gate voltage at point contacts. Vary area A by varying voltage on side gate. Measure resistance V 12 /I. Expect oscillations in the resistance as a function of A + = quasihole

If =5//2 state is non-abelian Pfaffian or Anti-Pfaffian state: the period of resistance oscillations should depend on whether the number of localized quasiholes encircled by the path is even or odd.

II 12 = 5/ t1t1 t2t2 Weak back-scattering: V 12  t 1 + t 2 e  i   2, with  =  A B/4  0, only if the qh number is even. If interference path contains an odd number of localized quasiholes, quasiparticle path tunneling at point t 2 changes the state of enclosed zero-energy modes, and cannot interfere with path tunneling at t 1.

Will these experiments actually work and show non-abelian statstics? We don’t know. Real systems can be pretty complicated.

Acknowledgments Co-authors: Ady Stern, Bernd Rosenow, Michael Levin, Steve Simon, Chetan Nayak, Ivalo Dimov. Discussions with experimentalists, too numerous to name. Financial support: NSF, Microsoft Corporation, US-Israel Binational Science Foundation.

Proposed experiments (2). : *Measurements of spin polarization. Moore-Read has complete polarization in second Landau Level. Measurement should be possible, but difficult at very low temperatures. (Would be a consistency check, because some of the alternatives to Moore- Read are not fully polarized, but not a definitive test.)

Existence of a quantized Hall state at =5/2

If central region contains an odd number of localized quasiparticles, this interference term is absent. Then leading interference term varies as Re [t 1 * t 2 e 2  i  ] 2. (Period corresponds to an area containing two flux quanta, rather than four.)

Zero-energy modes Specifically, in a p x +ip y superconductor, an isolated vortex, at point R i, has a zero energy mode, with Majorana fermion operator  i : ( from solution of the Bogoliubov-de Gennes equations)  i =  i †,  i 2 = 1, {  i,  j } = 2  ij To form ordinary fermion creation or annihilation operator: need pair of vortices: e.g. c 12 = (    i     c 12 † = (    i     obey usual fermion commutations rules    c 12 † c 12 has eigenvalues = 0, 1. [N 12,N 34 ] = 0, etc. Constraint : Number of occupied pairs = N electrons (mod 2). -> 2N vortices gives 2 N-1 independent states

Explicit relation between Majorana operator and electron operators  i =  dr [ u(r)  (r) + v(r)  † (r) ] with v(r) = u*(r), localized near vortex. If vortices are far apart, so there is no overlap between the wave functions of their zero-energy states, then these states must have precisely zero energy. This relates to the fact that solutions of the BdG equations must occur in pairs with E 1 =-E 2.

Braiding properties of vortices Vortices at points R 1 R 2 R 3 R 4.

Braiding properties Vortices at points R 1 R 2 R 3 R 4 Move vortex 2 around vortex 3. Gives unitary transformation   2  3. Changes N 12 -> (1-N 12 ), N 34 -> (1-N 34 ).

Braiding properties Vortices at points R 1 R 2 R 3 R 4 Move vortex 2 around 3 and 4. Gives unitary transformation ~  2  4  2  3 =  3  4 : leaves N 12 and N 34 unchanged. Since vortices are indistinguishable, get other unitary transformations by simply interchanging positions of two vortices. Order of interchanges matter: The unitary transformations do not commute.