1. Proposed experimental probes of non-abelian anyons Ady Stern (Weizmann) with: N.R. Cooper, D.E. Feldman, Eytan Grosfeld, Y. Gefen, B.I. Halperin, Roni Ilan, A. Kitaev, K.T. Law, B. Rosenow, S. Simon

2. Outline: 1.Non-abelian anyons in quantum Hall states – what they are, why they are interesting, how they may be useful for topological quantum computation. 2.How do you identify a non-abelian quantum Hall state when you see one ?

3. Introductory pedagogical Comprehensive More precise and relaxed presentations:

4. The quantized Hall effect and unconventional quantum statistics

5. zero longitudinal resistivity - no dissipation, bulk energy gap current flows mostly along the edges of the sample quantized Hall resistivity The quantum Hall effect  is an integer, or q even or a fraction with q odd, I B

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

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

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

9. Topological quantum computation (Kitaev 1997-2003) Subspace of dimension 2 N, 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 Up to a global phase, the unitary transformation depends only on the topology of the trajectory 1 1 3 2 2 3

10. The goal: experimentally identifying non-abelian quantum Hall states The way: the defining characteristics of the most prominent candidate, the =5/2 Moore-Read state, are 1.Energy gap. 2.Ground state degeneracy exponential in the number N of quasi-particles, 2 N/2. 3.Edge structure – a charged mode and a Majorana fermion mode 4.Unitary transformation applied within the ground state subspace when quasi-particles are braided.

11. In this talk: 1.Proposed experiments to probe ground state degeneracy – thermodynamics 2.Proposed experiments to probe edge and bulk braiding by electronic transport – Interferometry, linear and non-linear Coulomb blockade, Noise

12. Probing the degeneracy of the ground state (Cooper & Stern, 2008 Yang & Halperin, 2008)

13. Measuring the entropy of quasi-particles in the bulk The density of quasi-particles is Zero temperature entropy is then To isolate the electronic contribution from other contributions:

14. Leading to (~1.4) (~12pA/mK)

15. Probing quasi-particle braiding - interferometers

16. When a vortex i encircles a vortex j, the ground state is multiplied by the operator  i  j Nayak and Wilczek Ivanov A localized Majorana operator. All  ’ s anti-commute, and  2 =1. Essential information on the Moore-Read state: Each quasi-particle carries a single Majorana mode The application of the Majorana operators takes one ground state to another within the subspace of degenerate ground states

17. The interference term depends on the number and quantum state of the quasi-particles in the loop. even oddeven Number of q.p.’s in the interference loop, Interference term Brattelli diagram Interferometers:

18. Odd number of localized vortices: vortex a around vortex 1 -    a 1 a The interference term vanishes:

19. Even number of localized vortices: vortex a around vortex 1 and vortex 2 -    a    a      1 a 2 The interference term is multiplied by a phase: Two possible values, mutually shifted by 

20. Interference in the =5/2 non-abelian quantum Hall state: The Fabry-Perot interferometer S1S1 D1D1 D2D2

21.  Gate Voltage, V MG (mV) Magnetic Field (or voltage on anti-dot) cell area The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.

22. Coulomb blockade vs interference (Stern, Halperin 2006, Stern, Rosenow, Ilan, Halperin, 2009 Bonderson, Shtengel, Nayak 2009)

23. Interferometer (lowest order) Quantum dot For non-interacting electrons – transition from one limit to another via Bohr-Sommerfeld interference of multiply reflected trajectories. Can we think in a Bohr-Sommerfeld way on the transition when anyons, abelian or not, are involved? Yes, we can (BO, 2008) (One) difficulty – several types of quasi-particles may tunnel

24. Thermodynamics is easier than transport. Calculate the thermally averaged number of electrons on a closed dot. Better still, look at Poisson summation The simplest case, =1. Energy is determined by the number of electrons Partition function

25. Sum over electron number. Thermal suppression of high energy configurations Sum over windings. Thermal suppression of high winding number. An Aharonov-Bohm phase proportional to the winding number. At high T, only zero and one windings remain

26. The energy of the dot is made of A charging energy An energy of the neutral mode. The spectrum is determined by the number and state of the bulk quasi-particles. The neutral mode partition function χ depends on n qp and their state. Poisson summation is modular invariance And now for the Moore-Read state (Cappelli et al, 2009)

27. The components of the vector correspond to the different possible states of the bulk quasi-particles, one state for an odd n qp ( “  ” ), and two states for an even n qp ( “ 1 ” and “ψ”). A different thermal suppression factor for each component. The modular S matrix. S ab encodes the outcome of a quasi-particle of type a going around one of type b

28. Low T High T

29. Probing excited states at the edge – non linear transport in the Coulomb blockade regime (Ilan, Rosenow, Stern, 2010)

30. A nu=5/2 quantum Hall system =2 Goldman ’ s group, 80 ’ s

31. Non-linear transport in the Coulomb blockade regime: dI/dV at finite voltage – a resonance for each many-body state that may be excited by the tunneling event. dI/dV V sd

32. Energy spectrum of the neutral mode on the edge Single fermion: For an odd number of q.p. ’ s E n =0,1,2,3, …. For an even number of q.p. ’ s E n = ½, 3/2, 5/2, … Many fermions: For an odd number of q.p. ’ s Integers only For an even number of q.p. ’ s Both integers and half integers (except 1!) The number of peaks in the differential conductance varies with the number of quasi-particles on the edge.

33. Current-voltage characteristics (Ilan, Rosenow, AS 2010) Source- drain voltage Magnetic field Even number Odd number

34. Interference in the =5/2 non-abelian quantum Hall state: Mach-Zehnder interferometer

35. The Mach-Zehnder interferometer: (Feldman, Gefen, Kitaev, Law, Stern, PRB2007) S D1D1 D2D2

36. Compare: S D1D1 D2D2 S1S1 D1D1 D2D2 M-Z F-P 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)

37. Number of q.p.’s in the interference loop                 Interference term 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 four)

38. 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. If all rates are equal, current flows in “ bunches ” of one quasi-particle each – Fano factor of 1/4.

39. Summary: Interference magnitude depends on the parity of the number of quasi-particles Phase depends on the eigenvalue of Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers Mach-Zehnder: Temperature dependence of the chemical potential and the magnetization reflect the ground state entropy Coulomb blockade I-V characteristics may measure the spectrum of the edge Majorana mode

40. even oddeven Number of q.p.’s in the interference loop, Interference term S1S1 D1D1 D2D2 For a Fabry-Perot interferometer, the state of the bulk determines the interference term. Das-Sarma-Freedman-Nayak qubit The interference phases are mutually shifted by 

41. even oddeven Number of q.p.’s in the interference loop, Interference term The sum of two interference phases, mutually shifted by  The area period goes down by a factor of two. S1S1 D1D1 D2D2

42. Gate Voltage, V MG (mV) Magnetic Field (or voltage on anti-dot) cell area Ideally, The magnetic field Quasi-particles number The gate voltage Area

43. Are we getting there? (Willett et al. 2008)

44. From electrons at n=5/2 to non-abelian quasi-particles: 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 Step I: Read and Green (2000) to: spin polarized composite fermions at zero (average) magnetic field GM87 R89 ZHK89 LF90 HLR93 KZ93

45. B e-e- B 1/2 = 2n s  0 B (c) 2020 CF (b) B Electrons in a magnetic field B Composite particles in a magnetic field Mean field (Hartree) approximation H  = E 

46. 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 and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles. Step III: fermions at zero magnetic field pair into Cooper pairs Spin polarization requires pairing of odd angular momentum a p-wave super-conductor of composite fermions

48. 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 and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles. Step III: fermions at zero magnetic field pair into Cooper pairs Spin polarization requires pairing of odd angular momentum a p-wave super-conductor of composite fermions For a single vortex – there is a zero energy mode at the vortex ’ core Kopnin, Salomaa (1991), Volovik (1999)

49. g(r) is a localized function in the vortex core A zero energy solution is a spinor A localized Majorana operator. A subspace of degenerate ground states, with the  ’ s operating in that subspace. In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator  i  j Nayak and Wilczek (1996) Ivanov (2001) All  ’ s anti-commute, and  2 =1.

50. 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 =5/2

51. Closing the island into a quantum dot:  Interference involving multiple scatterings, Coulomb blockade cell area Current (a.u.)

52. is very different from But, so, interference of even number of windings always survives.  Equal spacing between peaks for odd number of localized vortices Alternate spacing between peaks for even number of localized vortices

53.  n is – a crucial quantity. How do we know it ’ s time independent? What is ? By the fluctuation-dissipation theorem, C – capacitance t 0 – relaxation time = C/G G – longitudinal conductance Best route – make sure charging energy >> Temperature A subtle question – the charging energy of what ??

54. And what if n is is time dependent? A simple way to probe exotic statistics: A new source of current noise. For Abelian states (  ): For the  state: Chamon et al. (1997) G = G 0 (n is odd) G 0 [1 ±  cos(  + n is /4)] (n is even)

55. time G GG compared to shot noise bigger when t 0 is long enough close in spirit to 1/f noise, but unique to FQHE states.

56. When multiple reflections are taken into account, the average conductance and the noise, satisfy and A signature of the =5/2 state (For abelian Laughlin states – the power is ) A “ cousin ” of a similar scaling law for the Mach-Zehnder case (Law, Feldman and Gefen, 2005)

57. For a lattice, expect a tight-binding Hamiltonian Analogy to the Hofstadter problem. The phases of the t ij ’ s determine the flux in each plaquette Finally, a lattice of vortices When vortices get close to one another, degeneracy is lifted by tunneling.

58. Since the tunneling matrix elements must be imaginary. The question – the distribution of + and - For a square lattice: Corresponds to half a flux quantum per plaquette. A unique case in the Hofstadter problem – no breaking of time reversal symmetry.

59. Spectrum – Dirac: E k A mechanism for dissipation, without a motion of the charged vortices is varied by varying density Exponential dependence on density

60. Protection from decoherence: The ground state subspace is separated from the rest of the spectrum by an energy gap Operations within this subspace are topological But: In present schemes, the read-out involves interference of two quasi-particle trajectories (subject to decoherence). In real life, disorder introduces unintentional quasi-particles. The ground state subspace is then not fully accounted for. A theoretical challenge! (Kitaev, 1997-2003)

61. Summary 1.A proposed interference experiment to address the non-abelian nature of the quasi-particles, insensitive to localized quasi-particles. 2.A proposed “ thermodynamic ” experiment to address the non-abelian nature of the quasi-particles, insensitive to localized quasi-particles. 3. Current noise probes unconventional quantum statistics.

62. Closing the island into a quantum dot:  Coulomb blockade ! Transport thermodynamics For a conventional super-conductor, spacing alternates between charging energy E c (add an even electron) charging energy E c + superconductor gap   (add an odd electron) The spacing between conductance peaks translates to the energy cost of adding an electron.

63. Reason: consider a compact geometry (sphere). By Dirac’s quantization, the number of flux quanta (h/e) is quantized to an integer, the number of vortices (h/2e) is quantized to an even integer In a non-compact closed geomtry, the edge “completes” the pairing But this super-conductor is anything but conventional … For the p-wave super-conductor at hand, crucial dependence on the number of bulk localized quasi-particles, n is a gapless (E=0) edge mode if n is is odd corresponds to  =0 a gapfull (E≠0) edge mode if n is is even corresponds to  ≠0 The gap diminishes with the size of the dot ∝ 1/L

64. So what about peak spacings? When n is is odd, peak spacing is “ unaware ” of  peaks are equally spaced When n is is even, peak spacing is “ aware ” of  periodicity is doubled B cell area odd even No interference B cell area odd even Interference pattern Coulomb peaks

65. From electrons at =5/2 to a lattice of non-abelian quasi-particles in four steps: A half filled Landau level on top of Two filled Landau levels Step II: From a half filled Landau level of electrons to composite fermions at zero magnetic field - the Chern-Simons transformation Step I: Read and Green (2000)

66. The original Hamiltonian: Schroedinger eq. H  Define a new wave function: The Chern-Simons transformation describes electrons (fermions) describes composite fermions The effect on the Hamiltonian:

67. |t left + t right | 2 for an even number of localized quasi-particles |t right | 2 + |t left | 2 for an odd number of localized quasi-particles  The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.

68. The new magnetic field: B nsns e-e- (a) B 1/2 = 2n s  0 B (c) 2020 CF (b) B Electrons in a magnetic field Composite particles in a magnetic field Mean field (Hartree) approximation

69. Spin polarized composite fermions at zero (average) magnetic field 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 an Abrikosov lattice of vortices in a p-wave super-conductor Look for a ground state degeneracy in this lattice

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

71. 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)

72. g(r) is a localized function in the vortex core A zero energy solution is a spinor A localized Majorana operator. A subspace of degenerate ground states, with the  ’ s operating in that subspace. In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator  i  j Nayak and Wilczek Ivanov All  ’ s anti-commute, and  2 =1.

73. backscattering = |t left +t right | 2  Interference experiment: Stern and Halperin (2005) Following Das Sarma et al (2005) interference pattern is observed by varying the cell ’ s area

74. vortex a around vortex 1 -    a vortex a around vortex 1 and vortex 2 -    a    a      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 1 a 2

75. 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 interference is dephased

76. |t left + t right | 2 for an even number of localized quasi-particles |t right | 2 + |t left | 2 for an odd number of localized quasi-particles  the number of quasi-particles on the dot may be tuned by a gate insensitive to localized pinned charges

77. cell area even odd even occupation of anti-dot interference no interference interference Localized quasi-particles shift the red lines up/down

78. B e-e- B 1/2 = 2n s  0 B (c) 2020 CF (b) B Electrons in a magnetic field B Composite particles in a magnetic field Mean field (Hartree) approximation H  = E 

79. A yet simpler version:  B cell area equi-phase lines odd even No interference

80. For a lattice, expect a tight-binding Hamiltonian Analogy to the Hofstadter problem. The phases of the t ij ’ s determine the flux in each plaquette And now to a lattice of quasi-particles. When vortices get close to one another, degeneracy is lifted by tunneling.

81. Since the tunneling matrix elements must be imaginary. The question – the distribution of + and - For a square lattice: Corresponds to half a flux quantum per plaquette. A unique case in the Hofstadter problem – no breaking of time reversal symmetry.

82. Spectrum – Dirac: E k What happens when an electric field E(q,  ) is applied? is varied by varying density

83. E k Given a perturbation the rate of energy absorption is Distinguish between two different problems – 1.Hofstadter problem – electrons on a lattice 2.Present problem – Majorana modes on a lattice

84. E k For both problems the rate of energy absorption is The difference between the two problems is in the matrix elements for the electrons for the Majorana modes

85. So the real part of the conductivity is for the electrons for the Majorana modes The reason – due the particle-hole symmetry of the Majorana mode, it does not carry any current at q=0.

86. From the conductivity of the Majorana modes to the electronic response The conductivity of the p-wave super-conductor of composite fermions, in the presence of the lattice of vortices From composite fermions to electrons

87. Summary 1.A proposed interference experiment to address the non-abelian nature of the quasi-particles. 2. Transport properties of an array of non-abelian quasi-particles.

88. localized function in the direction perpendicular to the  =0 line

90. When vortex i encircles vortex i+1, the unitary transformation operating on the ground state is How does the zero energy state at the i ’ s vortex “know” that it is encircled by another vortex? No tunneling takes place? Unitary transformations: A more physical picture?

91. 2N localized intra-vortex states, each may be filled (“1”) or empty (“0”) Notation: means 1 st, 3 rd, 5 th vortices filled, 2 nd, 4 th vortices empty. and all possible combinations with odd numbers of filled states Product states are not ground states: The emerging picture – two essential ingredients: Full entanglement: Ground states are fully entangled super-positions of all possible combinations with even numbers of filled states

92. Phase accumulation depend on occupation When a vortex traverses a closed trajectory, the system’s wave-function accumulates a phase that is N – the number of fluid particles encircled by the trajectory ( | 0000 + |1100 ) ( | 0000 - |1100 ) Halperin Arovas, Schrieffer, Wilczek

93. Vortex 2 encircling vortex 3 A “+” changing into a “  ” Vortex 2 and vortex 3 interchanging positions Permutations of vortices change relative phases in the superposition Four vortices:

94. A vortex going around a loop generates a unitary transformation in the ground state subspace 1 2 3 4 A vortex going around the same loop twice does not generate any transformation 1 2 3 4

95. The Landau filling range of 2< <4 Unconventional fractional quantum Hall states: 1.Even denominator states are observed 2.Observed series does not follow the rule. 3.In transitions between different plateaus, is non-monotonous as opposed to (Pan et al., PRL, 2004) Focus on =5/2

96. The effect of the zero energy states on interference Dephasing, even at zero temperature No dephasing (phase changes of )

97. More systematically: what are the ground states? The goal: ground states ….. that, as the vortices move, evolve without being mixed. position of vortices The condition:

98. To answer that, we need to define a (partial) single particle basis, near each vortex find the wave function describing the occupation of these states is a purely zero energy state is a purely non-zero energy state = defines a localized function correlates its occupation with that of There is an operator Y for each vortex. How does the wave function near each vortex look?

99. We may continue the process defines a localized function correlates its occupation with that of This generates a set of orthogonal vortex states near each vortex (the process must end when states from different vortices start overlapping). The requirements for j=1..k determine the occupations of the states near each vortex.

100. The functions are solutions of the Bogolubov de-Gennes eqs. Ground state should be annihilated by all ‘s For uniform super-conductors

101. The simplest model – take a free Hamiltonian with a potential part only To get a localized mode of zero energy, we need a localized region of . A vortex is a closed curve of  =0 with a phase winding of 2  in the order parameter 

102. The phase winding is turned into a boundary condition A change of sign Spinor is localized function in the direction perpendicular to the  =0 line The phase  depends on the direction of the  =0 line. It changes by  around the square. A vortex is associated with a localized Majorana operator.

103. For a lattice, expect a tight-binding Hamiltonian Questions – 1.What are the t ij ? 2.How do we calculate electronic response functions from the spinors ’ Hamiltonian? Analogy to the Hofstadter problem. The phases of the t ij ’ s determine the flux in each plaquette

104. Two close vortices: Solve along this line to get the tunneling matrix element We find t ij =i A different case – the line going through the tunneling region changes the sign of the tunneling matrix element.

105. These requirements are satisfied for a given vortex by either one of two wave functions: or: The occupation of all vortex states is particle-hole symmetric. Still, two states per vortex, altogether and not We took care of the operators For the last state, we should take care of the operator which creates and annihilates  quasi-particles 

106. Doing that, we get ground states that entangle states of different vortices (example for two vortices): For 2N vortices, ground states are super-positions of states of the form 1 st vortex 2 nd vortex 2N ’ th vortex The operator creates a particle at the state near the vortex. When the vortex is encircled by another

107. Open questions: 1.Experimental tests of non-abelian states 2.The expected QH series in the second Landau level 3.The nature of the transition between QH states in the second Landau level 4. Linear response functions in the second Landau level 5.Physical picture of the clustered parafermionic states 6.Exotic directions – quantum computing, BEC ’ s

108. One fermion mode, two possible states, two pi-shifted interference patterns Several comments on the Das Sarma, Freedman and Nayak proposed experiment n=0 n=1

109. Comment number 1: The measurement of the interference pattern initializes the state of the fermion mode initial core state after measurement current has been flown through the system A measurement of the interference pattern implies The system is now either at the     =i or at the     =-i state.

110. At low temperature, as we saw odd n qp ( “  ” )even n qp ( “ 1 ” and “ψ”). At high temperature, the charge part will thermally suppress all but zero and one windings, and the neutral part will thermally suppress all but the “ 1 ” channel (uninteresting) and the “  ” channel. The latter is what one sees in lowest order interference (Stern et al, Bonderson et al, 2006). High temperature Coulomb blockade gives the same information as lowest order interference.

111. For both cases the interference pattern is shifted by  by the transition of one quasi-particle through the gates.

112. Summary The geometric phase accumulated by a moving vortex Entanglement between the occupation of states near different vortices Non-abelian statistics

113. The positional entropy of the quasi-particles If all positions are equivalent, other than hard core constraint – positional entropy is ∝ n log(n), but - Interaction and disorder lead to the localization of the quasi- particles – essential for the observation of the QHE – and to the suppression of their entropy. (~1.4) (~12pA/mK)

114. The positional entropy of the quasi-particles depends on their spectrum: qhe gap ground state excitations non-abelians temperature localized states Phonons of a quasi-particles Wigner crystal

115. 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

116. Physical picture of the effect on magnetization: The magnetization is determined by the current on the edge, which is proportional to the number of particles on the edge. The edge and the bulk are two systems at thermal equilibrium. As the temperature is varied, the free energy per particle in the two sub-systems varies, and the two subsystems exchange particles, hence changing the magnetization of the sample.

117. S D1D1 D2D2 M-Z S1S1 D1D1 D2D2 F-P Interferometers: (Stern & Halperin) (Bonderson, Shtengel, Kitaev) (Feldman&Kitaev) (Law, Feldman, Gefen, Kitaev Stern)

118. Shot noise as a way to measure charge: p 1-p I2I2 Binomial distribution For p<<1, current noise is S=2eI 2 coin tossing S D1D1 D2D2