Non- Abelian Quantum Hall States: An overview + experimental consequences Ady Stern (Weizmann)

1. Non- Abelian Quantum Hall States: An overview + experimental consequences Ady Stern (Weizmann)

2. Outline: • 1. What are non-abelian quantum Hall states? Why? Where? • 2. Understanding them by • Trial wave functions • Chern-Simons theories • Composite fermion theory • 3. Experimental implications

3. or a fraction with qodd, The quantum Hall effect • zero longitudinal resistivity - no dissipation, bulk energy gap current flows mostly along the edges of the sample • quantized Hall resistivity n is an integer, or q even

4. Extending the notion of quantum statistics Laughlin quasi-particles Electrons A ground state: Energy gap Adiabatically interchange the position of two excitations

5. More interestingly, non-abelian statistics(Moore and Read, 91) In a non-abelian quantum Hall state, quasi-particles obey non-abelian statistics, meaning that with 2N quasi-particles at fixed positions, the ground state is -degenerate. Interchange of quasi-particles shifts between ground states. For n=5/2 (main example)

6. ground states position of quasi-particles ….. Permutations between quasi-particles positions unitary transformations in the ground state subspace

7. 1 2 3 2 1 3 Up to a global phase, the unitary transformation depends only on the topology of the trajectory Topological quantum computation (Kitaev 1997-2003) • Subspace of dimension 2N,separated by an energy gap from the continuum of excited states. • Unitary transformations within this subspace are defined by the topology of braiding trajectories • All local operators do not couple between ground states • – immunity to errors

8. Immunity to perturbations: • Diagonalized Hamiltonian: • Energy of all ground states is set to zero • E is a general name for a positive energy of the excites states • Lowest value of E is the energy gap Perturbation: No matrix element that connects ground states Shaded part is protected. Virtual transitions introduce exponentially small splitting

9. Non-abelian states through wave functions • General comments on wave functions in the quantum Hall effect: • Surprisingly, first-quantized wave functions are a useful concept to construct. • The wave function for one full Landau level may be constructed exactly, and it inspires an approximate wave function for the n=1/3, 1/5 fractions, the Laughlin wave function.

10. Simplest many-electron problem – a full Landau level The wave function is a Slater determinant, Non-abelian states through wave functions We consider many electrons in a magnetic field, with filling fraction < 1, and weak interaction between electrons. So, we look for a wave function that’s made solely of lowest Landau level single particle w.f.’s. A single electron in the lowest Landau level (symmetric gauge):

11. From now on, we do not talk about the Gaussian factor. The rest is: • a polynomial in the zi’s – purely lowest Landau level • the maximal power for z1 is (N-1). It determines the area, and through the area the filling factor. • the polynomial in z1has (N-1) zeroes, which are on the other (N-1) electrons. Each zero is of order one. • anti-symmetric to the exchange of any two electrons

12. Laughlin wave function for other filling factors • a polynomial in the zi’s – purely lowest Landau level • maximal power for z1 is 3(N-1). It determines the filling factor to be 1/3. • The polynomial in z1has 3(N-1) zeroes, which are all on the other (N-1) electrons. Each zero is of order three. This efficient use of the zeroes is the reason to the great success of Laughlin’s wave function in minimizing the interaction energy. • anti-symmetric to the exchange of any two electrons

13. If it is so successful, why not using it for other values of n (say, 1/2)? • a polynomial in the zi’s – purely lowest Landau level - GOOD • maximal power for z1 is 2(N-1). Fixes filling factor to 1/2 - GOOD • Making efficient use of the zeroes - GOOD • The problem: • For n=1/2 the wave function is not anti-symmetric to the exchange of any two electrons • For other n’s it is not even single valued. Can the problem be fixed?

14. The role of the function F: to fix the symmetry (or single valuedness) of the wave function, without changing the filling factor. We need For example, for n=1/2 we need odd a and any finite b.

15. Theessential minimum: CFTs in this context are a device to generate functions: A CFT has a set of fields: with conformal dimensions hi and a set of fusion rules: where the d’s are determined by the conformal dimensions How shall we find a function that satisfies this? “With a little help from my friends” (L-M, 1967). Parafermionic Conformal Field Theories (CFTs) (Fateev and Zamolochikov, the Yellow Book, etc.)

16. A fusion rule determines the singular behavior of a correlator when two of its arguments get close together The function F will be a correlator of fields y1, that represents the electrons. We know what behavior we need for the wave function, so we need to find a CFT that has this behavior. Example: n=1/2 We need

17. We ask around and find the Ising CFT. The fields it has are 1, y, s. The first fusion rule is The function F will be The fusion rule gives us the short distance behavior that we need. In fact, the correlator may be exactly evaluated and shown to be identical to the BCS wave function for a p-wave super-conductor (the Moore-Read Pfaffian wave function).

18. But there are two more fusion rules, which involve the s. If y represents the electron, what is s? It is the quasi-particle(the vortex in the p-wave superconductor). Its non-abelian nature is revealed in the fusion of two s’s The F functions for two quasi-holes will be • Two functions per each pair of s’s • wave functions are single valued w.r.t. the electronic coordinates, but not w.r.t. the quasi-particle coordinates.

19. Other CFTs describe clustering of the electrons to condensates of groups of 3,4,5… with quasi-particles being sort-of vortices in these condensates. • Quasi-particles satisfy non-abelian statistics whose details are presently partially worked out. • So far, we looked at the CFTs as working in the two dimensional world of (x,y) – we looked only at ground state wave functions. • In fact, conclusions may be drawn also about dynamics, at the place where it exists – the edge.

20. The quantum Hall effect a 2+1D Chern-Simons theory + 2+1D Chern-Simons theory gauge invariance From electrons to non-abelian quasi-particles: the QFT way x,y 1+1D WZW model on the edge edge, t The spectrum and Fock space of the edge

21. One non-abelian quantum Hall state, that of nu=5/2, may be understood by Composite Fermion theory, following four steps: Step I: A half filled Landau level on top of two filled Landau levels Step II: the Chern-Simons transformation from:electrons at a half filled Landau level to: spin polarized composite fermions at zero (average) magnetic field GM87 R89 ZHK89 LF90 HLR93 KZ93

22. (c) B 20 (b) CF B B1/2 = 2ns0 B Electrons in a magnetic fieldB e- H y = E y Composite particles in a magnetic field Mean field (Hartree) approximation

23. Step III: fermions at zero magnetic field pair into Cooper pairs Spin polarization requires pairing of odd angular momentum a p-wave super-conductor Read and Green (2000) Step IV: introducing quasi-particles into the super-conductor - shifting the filling factor away from 5/2 The super-conductor is subject to a magnetic field The super-conductor is subject to a magnetic field and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles. Spin polarized composite fermions at zero (average) magnetic field

25. A quadratic Hamiltonian – may be diagonalized (Bogolubov transformation) Ground state degeneracy requires zero energy modes BCS-quasi-particle annihilation operator Dealing with vortices in a p-wave super-conductor First, a single vortex – focus on the mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999)

26. The functions are solutions of the Bogolubov de-Gennes eqs. Ground state should be annihilated by all ‘s For uniform super-conductors For a single vortex – there is a zero energy mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999)

27. A zero energy solution is a spinor g(r) is a localized function in the vortex core All g’s anti-commute, and g2=1. A localized Majorana operator . To connect between ground states one needs an even number of different g’s, which are located far away from one another. A Perturbation of the form will never translate into two different g’s and therefore will never connect one ground state to another. Immunity to perturbation

28. Furthermore, the zero energy solutions are “protected”: the BDG spectrum is even with respect to zero energy. A perturbation must connect two zero energy states for their energy to be shifted.

29. Interference magnitude depends on the parity of the number of quasi-particles Phase depends on the eigenvalue of Experimental implications: Bunching of the Coulomb peaks to groups of n and k-n – A signature of the Zk states Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers Mach-Zehnder:

30. A Fabry-Perot interferometer: Stern and Halperin (2005) Bonderson, Shtengel, Kitaev (2005) Following Das Sarma et al (2005) Chamon et al (1996) n=5/2 backscattering = |tleft+tright|2 interference pattern is observed by varying the cell’s area

31. Current (a.u.) Followed by an extension to a closed dot n=5/2 cell area Integer quantum Hall effect (adapted from Neder et al., 2006) The prediction for the n=5/2 non-abelian state (weak backscattering limit) cell area Gate Voltage, VMG (mV) Magnetic Field (the number of quasi-particles in the bulk)

32. 2 The effect of the core states on the interference of backscattering amplitudes depends crucially on the parity of the number of localized states. Before encircling vortex a around vortex 1 - g1ga vortex a around vortex 1 and vortex 2 - g1gag2ga~ g2g1 1 a

33. After encircling for an even number of localized vortices only the localized vortices are affected (a limited subspace) for an odd number of localized vortices every passing vortex acts on a different subspace

34. Interference term: for an even number of localized vortices only the localized vortices are affected Interference is seen for an odd number of localized vortices every passing vortex acts on a different subspace interference is dephased

35. n=5/2 Gate Voltage, VMG (mV) Magnetic Field (or voltage on anti-dot) The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field. cell area

36. n=5/2 When interference is seen: Interference term Interference magnitude depends on the parity of the number of quasi-particles Phase depends on the eigenvalue of

37. D1 S M-Z D2 F-P D2 D1 S1 Interferometers: Main difference: the interior edge is/is not part of interference loop For the M-Z geometry every tunnelling quasi-particle advances the system along the Brattelli diagram (Feldman, Gefen, Law PRB2006)

38. G4 G2 G3/2 G1/2 G1 G4/2 G2/2 G3 Interference term Number of q.p.’s in the interference loop • The system propagates along the diagram, with transition rates assigned to each bond. • The rates have an interference term that • depends on the flux • depends on the bond (with periodicity of 4)

39. I1 1-p a well-designed coin p I2 • The probability p, always <<1, varies according to the outcome of the tossing. It depends on flux and on the number of quasi-particles that have already tunneled. • Consider two extremes (two different values of the flux): • If all rates are equal, there is just one value of p, and the usual binomial story applies – Fano factor of 1/4. • But:

40. The other extreme – some of the bonds are “broken” Charge flows in “bursts” of many quasi-particles. The maximum expectation value is around 12 quasi-particles per burst – Fano factor of about three.

41. Effective charge span the range from 1/4 to about three. The dependence of the effective charge on flux is a consequence of unconventional statistics. Charge larger than one is due to the Brattelli diagram having more than one “floor”, which is due to the non-abelian statistics In summary, flux dependence of the effective charge in a Mach-Zehnder interferometer may demonstrate non-abelian statistics at n=5/2

42. Summary: Non-abelian quantum Hall states are a very exciting theoretical possibility, which requires much more research for a complete understanding. Even more so, it requires experimental testing.