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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 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 ? More precise and relaxed presentations: Introductory pedagogical Comprehensive The quantized Hall effect and unconventional quantum statistics The quantum Hall effect • zero longitudinal resistivity - no dissipation, bulk energy gap current flows mostly along the edges of the sample B I • quantized Hall resistivity xy 1 h e2 is an integer, or a fraction p q or q even with q odd, Extending the notion of quantum statistics Electrons A ground state: Laughlin quasiparticles (r1,.....................rN ; R1,.., R4 ) Energy gap Adiabatically interchange the position of two excitations ei More interestingly, non-abelian statistics (Moore and Read, 91) 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 2 N -degenerate. Interchange of quasi-particles shifts between ground states. 2 N ground states g .s. 1 R1 , R2 ... g .s. 2 R1 , R2 ... ….. R1 , R2 ...position of quasi-particles g .s. 2 R1 , R2 ... N Permutations between quasi-particles positions unitary transformations in the ground state subspace Up to a global phase, the unitary transformation depends only on the topology of the trajectory 1 2 2 3 3 1 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 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. 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 Probing the degeneracy of the ground state (Cooper & Stern, 2008 Yang & Halperin, 2008) Measuring the entropy of quasi-particles in the bulk The density of quasi-particles is 4 n n5 / 2 4 n Zero temperature entropy is then 4n 5 B 2 0 5 B log 2 2 0 To isolate the electronic contribution from other contributions: s n T ; s m B T Leading to 2 log 2 sgn 5 / 2 T (~1.4) m 5 log 2 sgn 5 / 2 T 0 (~12pA/mK) Probing quasi-particle braiding - interferometers 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 A localized Majorana operator g i g i . All g’s anti-commute, and g2=1. When a vortex i encircles a vortex j, the ground state is multiplied by the operator gigj g.s. g ig j g.s. Nayak and Wilczek Ivanov Interferometers: The interference term depends on the number and quantum state of the quasi-particles in the loop. Interference term even Brattelli diagram odd even Number of q.p.’s in the interference loop, Odd number of localized vortices: vortex a around vortex 1 - g1ga a left 1 left core states right g ag 1 core states The interference term vanishes: right core states g ag 1 core states * left right Even number of localized vortices: vortex a around vortex 1 and vortex 2 a left g1gag2ga ~ g2g1 - 1 2 left core states right g 2g 1 core states The interference term is multiplied by a phase: right core states g 2g 1 core states * left Two possible values, mutually shifted by p right Interference in the =5/2 non-abelian quantum Hall state: The Fabry-Perot interferometer D2 S1 D1 The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field. Gate Voltage, VMG (mV) 5/2 cell area Magnetic Field (or voltage on anti-dot) Coulomb blockade vs interference (Stern, Halperin 2006, Stern, Rosenow, Ilan, Halperin, 2009 Bonderson, Shtengel, Nayak 2009) 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 Thermodynamics is easier than transport. Calculate the thermally averaged number of electrons on a closed dot. Better still, look at N A The simplest case, =1. Energy is determined by the number of electrons E Ec N Nbg A 2 Partition function Ec ( N N bg ) 2 Z exp N T Poisson summation Ec ( N N bg ( A)) 2 T Z ~ dN exp 2piNs ~ exp (ps) 2 2piN bg ( A) s T s s Ec Ec 2 exp ( N N ) bg T N • Sum over electron number. • Thermal suppression of high energy configurations ~ T 2 s exp E (ps) 2piN bg ( A)s c • 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 And now for the Moore-Read state The energy of the dot is made of • A charging energy • An energy of the neutral mode. The spectrum is determined Ec N Nbg A 2 by the number and state of the bulk quasi-particles. Ec Z exp ( N N bg ( A)) 2 N (T ) T N The neutral mode partition function χ depends on nqp and their state. Poisson summation is modular invariance v / L n T T S vn / L (Cappelli et al, 2009) T vn / L S T vn / L (T ) The components of the vector correspond to the different possible states of the bulk quasi-particles, one state for an odd nqp (“s”), and two states for an even nqp (“1” and “ψ”). A different thermal suppression factor for each component. S The modular S matrix. Sab encodes the outcome of a quasi-particle of type a going around one of type b Low T High T Probing excited states at the edge – non linear transport in the Coulomb blockade regime (Ilan, Rosenow, Stern, 2010) A nu=5/2 quantum Hall system =2 Goldman’s group, 80’s 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 Vsd Energy spectrum of the neutral mode on the edge Single fermion: For an odd number of q.p.’s En=0,1,2,3,…. For an even number of q.p.’s En= 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. Current-voltage characteristics En (n 1 / 2) Even number En n Odd number (Ilan, Rosenow, AS 2010) Sourcedrain voltage Magnetic field Interference in the =5/2 non-abelian quantum Hall state: Mach-Zehnder interferometer The Mach-Zehnder interferometer: (Feldman, Gefen, Kitaev, Law, Stern, PRB2007) S D2 D1 Compare: M-Z F-P S D2 D1 D2 S1 D1 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) Interference term G2 G1/2 G1 G2/2 G4 G3 G3/2 G4/2 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 four) If all rates are equal, current flows in “bunches” of one quasi-particle each – Fano factor of 1/4. 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. Summary: Temperature dependence of the chemical potential and the magnetization reflect the ground state entropy Interference magnitude depends on the parity of the number of quasi-particles Phase depends on the eigenvalue of 2 log 2 sgn 5 / 2 T m 5 log 2 sgn 5 / 2 T 0 g 2g 1 Coulomb blockade I-V characteristics may measure the spectrum of the edge Majorana mode Mach-Zehnder: Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers D2 For a Fabry-Perot interferometer, the state of the bulk determines the interference term. S1 D1 Interference term even odd even Das-Sarma-Freedman-Nayak qubit The interference phases are mutually shifted by p. g () g m e m 0 2pim p / 2 Number of q.p.’s in the interference loop, Interference term even odd Number of q.p.’s in the interference loop, even The sum of two interference phases, mutually shifted by p. g () g m e m 0 4pim D2 S1 The area period goes down by a factor of two. D1 Ideally, Quasi-particles number The gate voltage Area Gate Voltage, VMG (mV) The magnetic field cell area Magnetic Field (or voltage on anti-dot) Are we getting there? (Willett et al. 2008) From electrons at n=5/2 to non-abelian quasi-particles: Read and Green (2000) Step I: A half filled Landau level on top of two filled Landau levels 5 1 2 2 2 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 B Electrons in a magnetic field B e- H=E (b) 20 B Composite particles in a magnetic field B 2 0n( r ) CF ({ri }) ({ri })e i2 i j arg( ri rj ) Mean field (Hartree) approximation (c) B B1/2 = 2ns0 B B 20 n 0 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 H H 0 dr( r ) h.c. 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. Shot noise (A2/Hz) 5 x 10 -29 4 e/2 3 2 1 0 -1 -2 e/4 -1 0 1 Impinging current (nA) 2 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 H H 0 dr( r ) h.c. 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. For a single vortex – there is a zero energy mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999) A zero energy solution is a spinor g i dr g ( r Ri ) ( r ) g * ( r Ri ) ( r ) g(r) is a localized function in the vortex core A localized Majorana operator g i g i . All g’s anti-commute, and g2=1. A subspace of degenerate ground states, with the g’s operating in that subspace. In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator gigj g.s. g ig j g.s. Nayak and Wilczek (1996) Ivanov (2001) 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 Closing the island into a quantum dot: 5/2 -9.0 -7.5 -6.0 -4.5 -3.0 Current (a.u.) 15 Interference involving multiple scatterings, Coulomb blockade 10 5 0 0 50 100 150 cell area 200 5/2 g 2g 1 But, is very different from g 2g 1 2 g ag 1 g ag 1 1 2 so, interference of even number of windings always survives. Equal spacing between peaks forfor oddeven number of localized vortices Alternate spacing between peaks number of localized vortices nis – a crucial quantity. How do we know it’s time independent? What is 5/2 nis (t )nis (0) ? e Qis nis 4 By the fluctuation-dissipation theorem, t Qis (t )Qis (0) 2CTe t0 C – capacitance t0 – relaxation time = C/G G – longitudinal conductance Best route – make sure charging energy >> Temperature A subtle question – the charging energy of what ?? And what if nis is time dependent? A simple way to probe exotic statistics: nis (t ) Gnis (t ) I t A new source of current noise. For Abelian states (1/3): 2pnis G(nis ) G0 1 cos q Chamon et al. (1997) For the 5/2 state: G = G0 (nis odd) G0[1 ± cos( + nis/4)] (nis even) G dG time nis (t )nis (0) e t t0 compared to shot noise dI 2 0 dG 2V 2t0 e*GV bigger when t0 is long enough close in spirit to 1/f noise, but unique to FQHE states. When multiple reflections are taken into account, the average conductance and the noise, satisfy I 2 0 1 cos 4 I B I B I1 cos 4 and A signature of the =5/2 state I1 IB , 1 IB IB 3 (For abelian Laughlin states – the power is 1 1 ) A “cousin” of a similar scaling law for the Mach-Zehnder case (Law, Feldman and Gefen, 2005) Finally, a lattice of vortices When vortices get close to one another, degeneracy is lifted by tunneling. For a lattice, expect a tight-binding Hamiltonian H tijg ig j h.c. i, j Analogy to the Hofstadter problem. tij t e iij t e i i The phases of the tij’s determine the flux in each plaquette j Adl Since g i g i the tunneling matrix elements must be imaginary. H it ij g ig j i, j The question – the distribution of + and For a square lattice: Corresponds to half a flux quantum per i i i i plaquette. A unique case in the Hofstadter problem – no breaking of time reversal symmetry. Spectrum – Dirac: E (k ) v0k v0 ta v0 is varied by varying E density k A mechanism for dissipation, without a motion of the charged vortices s e q 0, 0 ~ q2 2 Exponential dependence on density Protection from decoherence: (Kitaev, 1997-2003) • 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! 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. Closing the island into a quantum dot: Coulomb blockade ! 5/2 Transport thermodynamics The spacing between conductance peaks translates to the energy cost of adding an electron. For a conventional super-conductor, spacing alternates between charging energy Ec (add an even electron) charging energy Ec + superconductor gap (add an odd electron) 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, nis a gapless (E=0) edge mode if nis is odd a gapfull (E≠0) edge mode if nis is even corresponds to =0 corresponds to ≠0 The gap diminishes with the size of the dot ∝ 1/L 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 So what about peak spacings? When nis is odd, peak spacing is “unaware” of peaks are equally spaced When nis is even, peak spacing is “aware” of periodicity is doubled B Interference pattern Coulomb peaks even even No interference odd odd even even No interference cell area From electrons at =5/2 to a lattice of non-abelian quasi-particles in four steps: Read and Green (2000) Step I: A half filled Landau level on top of Two filled Landau levels 5 1 2 2 2 Step II: From a half filled Landau level of electrons to composite fermions at zero magnetic field - the Chern-Simons transformation The Chern-Simons transformation • The original Hamiltonian: 1 H 2m P i A( r ) i 2 1 2 i, j e2 | ri rj | • Schroedinger eq. H E • Define a new wave function: ({ri }) ({ri })e i2 i j arg( ri rj ) ({ri }) describes electrons (fermions) ({ri }) describes composite fermions The effect on the Hamiltonian: Pi A(r ) Pi A(r )a(r ) r arg r 5/2 |tleft + tright|2 for an even number of localized quasi-particles |tright|2 + |tleft|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. The new magnetic field: ~ a( r ) 0 ( r ) (a) e- B ns A B 20 (b) B Electrons in a magnetic field CF Composite particles in a magnetic field A a B 2 0n( r ) Mean field (Hartree) approximation (c) B B1/2 = 2ns0 B A a B 2 0 n 0 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 B A a B 20 n 1 2 0 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 Dealing with Abrikosov lattice of vortices in a p-wave super-conductor First, a single vortex – focus on the mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999) H H0 r r drdr ' ( R, r ) ( R ) ( R ) h.c. 2 2 ( R, r ) R f (r ) A quadratic Hamiltonian – may be diagonalized (Bogolubov transformation) H E0 E g E g E E g E dr u (r ) (r ) v(r ) (r ) BCS-quasi-particle annihilation operator Ground state degeneracy requires zero energy modes The functions u( r ), v ( r ) are solutions of the Bogolubov de-Gennes eqs. g dr u(r)(r) v(r) (r) E Ground state should be annihilated by all For uniform super-conductors ik r u(r) v(r) e g E ‘s const. g.s. 1 g k ck ck vac k For a single vortex – there is a zero energy mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999) A zero energy solution is a spinor g i dr g ( r Ri ) ( r ) g * ( r Ri ) ( r ) g(r) is a localized function in the vortex core A localized Majorana operator g i g i . All g’s anti-commute, and g2=1. A subspace of degenerate ground states, with the g’s operating in that subspace. In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator gigj g.s. g ig j g.s. Nayak and Wilczek Ivanov Interference experiment: Stern and Halperin (2005) Following Das Sarma et al (2005) 5/2 backscattering = |tleft+tright|2 interference pattern is observed by varying the cell’s area vortex a around vortex 1 - g1ga vortex a around vortex 1 and vortex 2 a left 1 g1gag2ga ~ g2g1 - 2 right 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 left right core states After encircling left core states right g 2g 1 core states for an even number of localized vortices only the localized vortices are affected (a limited subspace) left core states right g ag 1 core states for an odd number of localized vortices every passing vortex acts on a different subspace interference is dephased 5/2 |tleft + tright|2 for an even number of localized quasi-particles |tright|2 + |tleft|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 occupation of anti-dot interference no interference interference cell area Localized quasi-particles shift the red lines up/down B Electrons in a magnetic field B e- H=E (b) 20 B Composite particles in a magnetic field B 2 0n( r ) CF ({ri }) ({ri })e i2 i j arg( ri rj ) Mean field (Hartree) approximation (c) B B1/2 = 2ns0 B B 20 n 0 A yet simpler version: equi-phase lines 5/2 B even odd No interference even No interference cell area And now to a lattice of quasi-particles. When vortices get close to one another, degeneracy is lifted by tunneling. For a lattice, expect a tight-binding Hamiltonian H tij g i g j h.c. i, j Analogy to the Hofstadter problem. tij t e iij t e i i The phases of the tij’s determine the flux in each plaquette j Adl Since g i g i the tunneling matrix elements must be imaginary. H it ij g ig j i, j The question – the distribution of + and For a square lattice: Corresponds to half a flux quantum per i i i i plaquette. A unique case in the Hofstadter problem – no breaking of time reversal symmetry. Spectrum – Dirac: E (k ) v0k v0 ta v0 is varied by varying E density k What happens when an electric field E(q,) is applied? Given a perturbation E J Acost the rate of energy absorption is 2p i J A f f 2 d f i Distinguish between two different problems – 1. Hofstadter problem – electrons on a lattice 2. Present problem – Majorana modes on a lattice k For both problems the rate of energy absorption E 2 is Re sE 2p i J A f f 2 d f i electric field E ~ iA dos ( ) ~ k v0 2 The difference between the two problems is in the matrix elements i J f ev0 if ev0 t 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. So the real part of the conductivity is e2 8h e2 8h t for the electrons 2 for the Majorana modes 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 2 n e2 qa i 2 m 8 v0 q 1 s CF q, 0 0 n e2 qa i m 8 From composite fermions to electrons s e q 0, ~ q 2 2 2 v q 1 0 s s 1 e 1 CF h 0 2 2 e 2 0 s e q, v0q 0 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. ei localized function in the i direction perpendicular to the e =0 line Unitary transformations: When vortex i encircles vortex i+1, the unitary transformation operating on the ground state is g ig i 1 dr dr ' wi (r )ei (r ) wi * (r )e i (r ) i i wi 1 (r ' )eii1 (r ' ) wi 1 (r ' )e ii1 (r ' ) * • No tunneling takes place? • How does the zero energy state at the i’s vortex “know” that it is encircled by another vortex? A more physical picture? The emerging picture – two essential ingredients: • 2N localized intra-vortex states, each may be filled (“1”) or empty (“0”) • Notation: 1 0 0 1 1... means 1st, 3rd, 5th vortices filled, 2nd, 4th vortices empty. •Full entanglement: Ground states are fully entangled super-positions of all possible combinations with even numbers of filled states ei 0000 0000 ei 0011 0011 ei1100 1100 ei1010 1010 ei 0110 0110 ei 0101 0101 ei1001 1001 ei1111 1111 and all possible combinations with odd numbers of filled states ei 0001 0001 ei 0010 0010 ei 0100 0100 ei1000 1000 ei 0111 0111 ei1011 1011 ei1101 1101 ei1110 1110 Product states are not ground states: i 2 i 3 i 4 i 21 2 2 2 e * e 10 ie 01 10 ie 01 • Phase accumulation depend on occupation When a vortex traverses a closed trajectory, the system’s wave-function accumulates a phase that is 2p N Halperin Arovas, Schrieffer, Wilczek N – the number of fluid particles encircled by the trajectory (|0000 + |1100 ) (|0000 - |1100 ) Permutations of vortices change relative phases in the superposition Four vortices: Vortex 2 encircling vortex 3 ei 0000 0000 ei 0011 0011 ei1100 1100 ei1010 1010 ei 0110 0110 ei 0101 0101 ei1001 1001 ei1111 1111 Vortex 2 and vortex 3 interchanging positions ei 0000 0000 ei 0011 0011 ei1100 1100 ei1010 1010 ei 0110 0110 ei 0101 0101 ei1001 1001 ei1111 1111 A “+” changing into a “” A vortex going around a loop generates a unitary transformation in the ground state subspace 2 3 4 1 A vortex going around the same loop twice does not generate any transformation 2 3 1 4 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 p 2 p 1 5 7 19 , , 2 2 8 rule. 3. In transitions between different plateaus, xy is non-monotonous as opposed to (Pan et al., PRL, 2004) Focus on =5/2 The effect of the zero energy states on interference Dephasing, even at zero temperature No dephasing (phase changes of 4p ) More systematically: what are the 2 N ground states? The goal: 2 ground states g.s. 1 R1 , R2 ... g .s. 2 R1 , R2 ... N ….. R1 , R2 ... position of vortices g .s. 2 R1 , R2 ... N that, as the vortices move, evolve without being mixed. The condition: g.s. k g.s. n 0 Ri for kn How does the wave function near each vortex look? 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 w0 * is a purely zero energy state w0 w0 * is a purely non-zero energy state = w 0 w0 iw1 Y CE GE w*0 iw*1 E C E GE C *E GE E defines a localized function w1 correlates its occupation with that of w0 There is an operator Y for each vortex. We may continue the process w1 * w1 w1 * w1 C E (2) E GE h.c. C (2) E GE h.c. E w iw2 ( 2) Y ( 2 ) CE GE *1 * w iw 1 2 E defines a localized function w2 correlates its occupation with that of w0 , w1 This generates a set of orthogonal vortex states w0 , w1, w2 ,...wl near each vortex (the process must end when states from different vortices start overlapping). The requirements Y j g.s. 0 for j=1..k determine the occupati of the states w0 ..wk near each vortex. The functions u( r ), v ( r ) are solutions of the Bogolubov de-Gennes eqs. g dr u(r)(r) v(r) (r) E Ground state should be annihilated by all For uniform super-conductors ik r u(r) v(r) e g E ‘s const. g.s. 1 g k ck ck k vac The simplest model – take a free Hamiltonian with a potential part only h( r ) ( r ) To get a localized mode of zero energy, we need a localized region of 0. A vortex is a closed curve of =0 with a phase winding of 2p in the order parameter . The phase winding is turned into a boundary condition A change of sign ei localized function in the Spinor g i is i direction perpendicular to the e =0 line The phase depends on the direction of the =0 line. It changes by p around the square. A vortex is associated with a localized Majorana operator. For a lattice, expect a tight-binding Hamiltonian H tij g i g j h.c. i, j Analogy to the Hofstadter problem. tij t e iij t e i i j Adl The phases of the tij’s determine the flux in each plaquette Questions – 1. What are the tij? 2. How do we calculate electronic response functions from the spinors’ Hamiltonian? Two close vortices: Solve along this line to get the tunneling matrix element We find tij=i A different case – the line going through the tunneling region changes the sign of the tunneling matrix element. i i i i These requirements are satisfied for a given vortex by either one of two wave functions: p 1 c0 1 c1 1 c2 ... 1 c.. 1 c... vacuum or: m 1 c0 1 c1 1 c2 ... 1 c.. 1 c... vacuum The occupation of all vortex states is particle-hole symmetric. Still, two states per vortex, altogether 2 2 N We took care of the operators Y ( j) and not 2 N w j 1 iw j w* j 1 iw* j For the last state, we should take care of the operator which creates and annihilates E0 quasi-particles. wk * w k Doing that, we get ground states that entangle states of different vortices (example for two vortices): g.s.1 p p m m env1 p m m p env2 g.s.2 p p m m env2 p m m p env1 For 2N vortices, ground states are super-positions of states of the form 1 c (1) i i 1st vortex 1 c ( 2) i i 2nd vortex ... 1 c ( 2 N ) i vacuum i 2N’th vortex The ci operator creates a particle at the state wi (r ) near the vortex. When the vortex is encircled by another ci ci 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 Several comments on the Das Sarma, Freedman and Nayak proposed experiment One fermion mode, two possible states, two pi-shifted interference patterns n=0 n=1 Comment number 1: The measurement of the interference pattern initializes the state of the fermion mode initial core state g 1g 2 i g 1g 2 i after measurement current has been flown through the system g 1g 2 i current 1 g 1g 2 i current 2 A measurement of the interference pattern implies current 1 current 2 0 The system is now either at the g1g2=i or at the g1g2=-i state. At low temperature, as we saw N A odd nqp (“s”) even nqp (“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 “s” 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. g1 g2 For both cases the interference pattern is shifted by p by the transition of one quasi-particle through the gates. Summary Entanglement between the occupation of states near different vortices The geometric phase accumulated by a moving vortex Non-abelian statistics 2 log 2 sgn 5 / 2 T (~1.4) m 5 log 2 sgn 5 / 2 T 0 (~12pA/mK) 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 quasiparticles – essential for the observation of the QHE – and to the suppression of their entropy. The positional entropy of the quasi-particles depends on their spectrum: excitations qhe gap localized states Phonons of a quasi-particles Wigner crystal temperature ground state non-abelians Shot noise as a way to measure charge: D1 1-p S coin tossing p Binomial distribution For p<<1, current noise is S=2eI2 I22 D