Ψ(r 1, r 2 ) = ±Ψ(r 2, r 1 ).

Transcription

1 Anyons, fractional charges, and topological order in a weakly interacting system M. Franz University of British Columbia franz@physics.ubc.ca February 16, 2007 In collaboration with: C. Weeks, G. Rosenberg, B. Seradjeh

2 ANYONS 1 Particle statistics In 3 space dimensions indistinguishable particles can be bosons or fermions, Ψ(r 1, r 2 ) = ±Ψ(r 2, r 1 ).

3 ANYONS 1 Particle statistics In 3 space dimensions indistinguishable particles can be bosons or fermions, Ψ(r 1, r 2 ) = ±Ψ(r 2, r 1 ). In 2 space dimensions we can have exotic particles called anyons, so that Ψ(r 1, r 2 ) = e iθ Ψ(r 2, r 1 ), θ 0, π. SLIDES CREATED USING FoilTEX & PP 4

4 ANYONS 2 Anyons and quantum computation Annals of Physics 303 (2003) Fault-tolerant quantum computation by anyons A.Yu. Kitaev * L.D. Landau Institute for Theoretical Physics, , Kosygina St. 2, Germany Received 20 May 2002 Abstract Quantum computation can be performed in a fault-tolerant way by braiding non-abelian anyons. A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature. Ó 2002 Elsevier Science (USA). All rights reserved. 1. Introduction A quantum computer can provide fast solution for certain computational problems (e.g., factoring and discrete logarithm [1]) which require exponential time on an ordinary computer. Physical realization of a quantum computer is a big challenge for scientists. One important problem is decoherence and systematic errors in unitary transformations which occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to ShorÕs discovery of fault-tolerant quantum computation [2], with subsequent improvements [3 6]. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision d. Unfortunately, the threshold value of d is rather small; 1 it is very difficult to achieve this precision. Needless to say, classical computation can be also performed faulttolerantly. However, it is rarely done in practice because classical gates are reliable enough. Why is it possible? Let us try to understand the easiest thing why classical * Present address: Caltech , Pasadena, CA 91125, USA. addresses: kitaev@itp.ac.ru, kitaev@cs.caltech.edu. 1 Actually, the threshold is not known. Estimates vary from 1/300 [7] to 10 6 [4] /02/$ - see front matter Ó 2002 Elsevier Science (USA). All rights reserved. PII: S (02)

5 ANYONS 2 Anyons and quantum computation Annals of Physics 303 (2003) Fault-tolerant quantum computation by anyons A.Yu. Kitaev * L.D. Landau Institute for Theoretical Physics, , Kosygina St. 2, Germany Received 20 May 2002 Abstract Quantum computation can be performed in a fault-tolerant way by braiding non-abelian anyons. A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature. Ó 2002 Elsevier Science (USA). All rights reserved. 1. Introduction A quantum computer can provide fast solution for certain computational problems (e.g., factoring and discrete logarithm [1]) which require exponential time on an ordinary computer. Physical realization of a quantum computer is a big challenge for scientists. One important problem is decoherence and systematic errors in unitary transformations which occur in real quantum systems. From the purely theoretical point of view, this problem has been solved due to ShorÕs discovery of fault-tolerant quantum computation [2], with subsequent improvements [3 6]. An arbitrary quantum circuit can be simulated using imperfect gates, provided these gates are close to the ideal ones up to a constant precision d. Unfortunately, the threshold value of d is rather small; 1 it is very difficult to achieve this precision. Needless to say, classical computation can be also performed faulttolerantly. However, it is rarely done in practice because classical gates are reliable enough. Why is it possible? Let us try to understand the easiest thing why classical * Present address: Caltech , Pasadena, CA 91125, USA. addresses: kitaev@itp.ac.ru, FIG. 4 kitaev@cs.caltech.edu. (color online). 1 Actually, the threshold is not known. Estimates vary from 1/300 [7] to 10 6 [4]. (a) An injection weave for which the product of elementary braiding matrices, also shown, approximates the identity to a distance of 1: This weave injects a quasiparticle (or any q-spin 1 object) into the target qubit without Bonesteel et al., PRL 95, (2005). changing any of the underlying q-spin quantum numbers. (b) A weaving pattern which approximates a NOT gate to a distance of 8: (c) A controlled-not gate constructed using the weaves shown in (a) and (b) to inject the control pair into the target qubit, perform a NOT operation on the injected target qubit, and then eject the control pair from the target qubit back into the control qubit. The distance between the gate produced by this braid acting on the computational two-qubit space and an exact controlled-not gate is 1: and 1: when the total q-spin of the six quasiparticles is 0 and 1, respectively. Again, the weaves shown in (a) and (b) can be made as accurate as necessary using the Solovay-Kitaev theorem, thereby improving the controlled-not /02/$ - see front matter Ó 2002 Elsevier Science (USA). All rights reserved. PII: S (02) SLIDES CREATED USING FoilTEX & PP 4

6 ANYONS 3 Anyons in FQH fluids Abelian anyons are known to occur as excitations of the fractional quantum Hall fluids described by Laughlin wavefunctions Ψ({z i }) = i<j(z i z j ) m e P i z i 2 /4, with m odd integer. These have exchange phase θ = π m and charge q = ± e m.

7 ANYONS 3 Anyons in FQH fluids Abelian anyons are known to occur as excitations of the fractional quantum Hall fluids described by Laughlin wavefunctions Ψ({z i }) = i<j(z i z j ) m e P i z i 2 /4, with m odd integer. These have exchange phase θ = π m and charge q = ± e m. SLIDES CREATED USING FoilTEX & PP 4

8 ANYONS 4 Fractional charges e 3 have been observed by resonant tunneling experiments in the ν = 1 3 FQH state [Goldman and Su, 1995].

9 ANYONS 4 Fractional charges e 3 have been observed by resonant tunneling experiments in the ν = 1 3 FQH state [Goldman and Su, 1995]. Fractional statistics have been observed only recently in an Aharonov-Bohm type quasiparticle interferometer [Camino, Zhou and Goldman, 2005].

10 ANYONS 4 Fractional charges e 3 have been observed by resonant tunneling experiments in the ν = 1 3 FQH state [Goldman and Su, 1995]. Fractional statistics have been observed only recently in an Aharonov-Bohm type quasiparticle interferometer [Camino, Zhou and Goldman, 2005]. Non-abelian anyons occur in the so called Moore-Read Pfaffian state which may be realized in the ν = 5 2 FQH state. Experimentally as yet unconfirmed. SLIDES CREATED USING FoilTEX & PP 4

11 ANYONS 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems.

12 ANYONS 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems. Strongly correlated = { many-body wavefunction Ψ cannot be written as a single Slater determinant of the constituent electron single-particle states.

13 ANYONS 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems. Strongly correlated = { many-body wavefunction Ψ cannot be written as a single Slater determinant of the constituent electron single-particle states. Question: Are fractional statistics and fractional charges inextricably linked to strong correlations?

14 ANYONS 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems. Strongly correlated = { many-body wavefunction Ψ cannot be written as a single Slater determinant of the constituent electron single-particle states. Question: Are fractional statistics and fractional charges inextricably linked to strong correlations? Answer: Not necessarily. SLIDES CREATED USING FoilTEX & PP 4

15 ANYONS 6 Fractional charges in polyacetylene [Su, Schrieffer and Heeger, 1979]

16 ANYONS 6 Fractional charges in polyacetylene [Su, Schrieffer and Heeger, 1979] band structure

17 ANYONS 6 Fractional charges in polyacetylene [Su, Schrieffer and Heeger, 1979] bound state at domain wall band structure SLIDES CREATED USING FoilTEX & PP 4

18 ANYONS 7 For spinless electrons the charge associated with a domain wall between two degenerate ground states is δq = ± e 2

19 ANYONS 7 For spinless electrons the charge associated with a domain wall between two degenerate ground states is δq = ± e 2 e

20 ANYONS 7 For spinless electrons the charge associated with a domain wall between two degenerate ground states is δq = ± e 2 e

21 ANYONS 7 For spinless electrons the charge associated with a domain wall between two degenerate ground states is δq = ± e 2 e e/2 e/2 SLIDES CREATED USING FoilTEX & PP 4

22 ANYONS 8 In 3 space dimensions Jackiw and Rebbi argued that 1 2 bound to the monopole in the Yang-Mills gauge field: of a fermion can be

23 ANYONS 8 In 3 space dimensions Jackiw and Rebbi argued that 1 2 bound to the monopole in the Yang-Mills gauge field: of a fermion can be In the above cases charge fractionalization occurs as a result of fermions coupling to a soliton configuration of a background (scalar or gauge) field. Interactions play no significant role, systems can be regarded as weakly correlated.

24 ANYONS 8 In 3 space dimensions Jackiw and Rebbi argued that 1 2 bound to the monopole in the Yang-Mills gauge field: of a fermion can be In the above cases charge fractionalization occurs as a result of fermions coupling to a soliton configuration of a background (scalar or gauge) field. Interactions play no significant role, systems can be regarded as weakly correlated. In d = 1, 3 particle statistics is trivial (fermions or bosons): Need a two-dimensional example! SLIDES CREATED USING FoilTEX & PP 4

25 ANYONS 9 Anyon as a charge-flux composite A simple toy model based on the Aharonov-Bohm effect.

26 ANYONS 9 Anyon as a charge-flux composite A simple toy model based on the Aharonov-Bohm effect. A charge q encircling flux Φ acquires Aharonov-Bohm phase ( q ) ( ) Φ 2π e Φ 0

27 ANYONS 9 Anyon as a charge-flux composite A simple toy model based on the Aharonov-Bohm effect. A charge q encircling flux Φ acquires Aharonov-Bohm phase ( q ) ( ) Φ 2π e Φ 0 A charge-flux composite (q, Φ) encircling another (q, Φ) acquires ( q ) ( ) Φ 4π e Φ 0 SLIDES CREATED USING FoilTEX & PP 4

28 ANYONS 10 Since the encircling operation can be thought of as two exchanges, the (q, Φ) composite particles have exchange phase θ = 2π ( q e ) ( Φ Φ 0 ).

29 ANYONS 10 Since the encircling operation can be thought of as two exchanges, the (q, Φ) composite particles have exchange phase ( q ) ( ) Φ θ = 2π. e Φ 0 For fractional charge q and flux Φ these composite particles could be anyons.

30 ANYONS 10 Since the encircling operation can be thought of as two exchanges, the (q, Φ) composite particles have exchange phase ( q ) ( ) Φ θ = 2π. e Φ 0 For fractional charge q and flux Φ these composite particles could be anyons. Anyon fusion: ( nq, n! ) Fusing together n identical particles with statistical phase θ 0 results in a particle with statistical phase θ = n 2 θ 0. n*( q,!) SLIDES CREATED USING FoilTEX & PP 4

31 ANYONS 11 A new device: Superconductor-semiconductor heterostructure. type!ii SC 2DEG

32 ANYONS 11 A new device: Superconductor-semiconductor heterostructure. type!ii SC 2DEG B %&'(!))*+,!"#$

33 ANYONS 11 A new device: Superconductor-semiconductor heterostructure. type!ii SC Periodic array of pinning sites for vortices 2DEG vacancy B interstitial %&'(!))*+,!"#$ [top view]

34 ANYONS 11 A new device: Superconductor-semiconductor heterostructure. type!ii SC Periodic array of pinning sites for vortices 2DEG vacancy B interstitial %&'(!))*+,!"#$ [top view] Superconductor quantizes magnetic flux in the units of Φ 0 /2 = hc/2e. Vortices preferentially occupy the pinning sites. Vacancies and interstitials in the vortex lattice produce localized flux surplus or deficit ±Φ 0 /2. SLIDES CREATED USING FoilTEX & PP 4

35 ANYONS B Magnetic field profile with one vacancy and one interstitial, based on a simple London model with penetration depth λ = a intervortex spacing. SLIDES CREATED USING FoilTEX & PP 4

36 ANYONS 13 Three principal claims At 2DEG filling fraction ν = 1 bound to each defect there is charge +e/2 for interstitial and e/2 for vacancy.

37 ANYONS 13 Three principal claims At 2DEG filling fraction ν = 1 bound to each defect there is charge +e/2 for interstitial and e/2 for vacancy. These (±e/2, ±Φ 0 /2) charge-flux composites behave as anyons with exchange phase θ = π/4.

38 ANYONS 13 Three principal claims At 2DEG filling fraction ν = 1 bound to each defect there is charge +e/2 for interstitial and e/2 for vacancy. These (±e/2, ±Φ 0 /2) charge-flux composites behave as anyons with exchange phase θ = π/4. At 2DEG filling fraction ν = 5 2 bound to each defect should be a quasiparticle of the Moore-Read pfaffian state with non- Abelian statistics. SLIDES CREATED USING FoilTEX & PP 4

39 ANYONS 14 Fractional charges: Simple general argument Consider the effect on 2DEG of adiabatically adding or removing vortex in a perfect flux lattice.

40 ANYONS 14 Fractional charges: Simple general argument Consider the effect on 2DEG of adiabatically adding or removing vortex in a perfect flux lattice. 2DEG E According to the Faraday s law E = 1 c H t, Φ j time-dependent flux produces electric field. The field, in turn, gives rise to Hall current j = σ xy (ẑ E), with σ xy = e 2 /h at ν = 1.

41 ANYONS 14 Fractional charges: Simple general argument Consider the effect on 2DEG of adiabatically adding or removing vortex in a perfect flux lattice. 2DEG E According to the Faraday s law E = 1 c H t, Φ j time-dependent flux produces electric field. The field, in turn, gives rise to Hall current j = σ xy (ẑ E), with σ xy = e 2 /h at ν = 1. Integrate: δq = e2 hc t2 t 1 dt ( ) dφ dt ( ) δφ = e Φ 0 SLIDES CREATED USING FoilTEX & PP 4

42 ANYONS 15 Statistical angle Naive counting would assign the (e/2, Φ 0 /2) object statistical phase θ 0 = 2π( ) = π/2.

43 ANYONS 15 Statistical angle Naive counting would assign the (e/2, Φ 0 /2) object statistical phase θ 0 = 2π( ) = π/2. This is, however, incorrect because it ignores the intrinsic statistical phase of the electron.

44 ANYONS 15 Statistical angle Naive counting would assign the (e/2, Φ 0 /2) object statistical phase θ 0 = 2π( ) = π/2. This is, however, incorrect because it ignores the intrinsic statistical phase of the electron. Consider fusing two interstitials, (e/2, Φ 0 /2) + (e/2, Φ }{{} 0 /2) (e, Φ }{{} 0 ). }{{} θ 0 θ 0 π

45 ANYONS 15 Statistical angle Naive counting would assign the (e/2, Φ 0 /2) object statistical phase θ 0 = 2π( ) = π/2. This is, however, incorrect because it ignores the intrinsic statistical phase of the electron. Consider fusing two interstitials, (e/2, Φ 0 /2) + (e/2, Φ }{{} 0 /2) (e, Φ }{{} 0 ) }{{} θ 0 θ 0 π According to the fusion rule θ = n 2 θ 0 with n = 2 and θ = π we have. θ 0 = π 4. SLIDES CREATED USING FoilTEX & PP 4

46 ANYONS 16 Non-Abelian physics at ν = 5 2 The Moore-Read pfaffian state can be thought of as superconducting condensate of composite fermion pairs with charge 2e.

47 ANYONS 16 Non-Abelian physics at ν = 5 2 The Moore-Read pfaffian state can be thought of as superconducting condensate of composite fermion pairs with charge 2e. Thus, the natural size of flux quantum is hc/2e, just like in a superconductor.

48 ANYONS 16 Non-Abelian physics at ν = 5 2 The Moore-Read pfaffian state can be thought of as superconducting condensate of composite fermion pairs with charge 2e. Thus, the natural size of flux quantum is hc/2e, just like in a superconductor. Vacancy or interstitial then should bind a quasiparticle of the pfaffian state. These are known to exhibit non-abelian exchange statistics. SLIDES CREATED USING FoilTEX & PP 4

49 ANYONS 17 Practical issues Vortices in superconductors can be created, imaged, and manipulated by a suite of techniques developed to study cuprates and other complex oxides.

50 ANYONS 17 Practical issues Vortices in superconductors can be created, imaged, and manipulated by a suite of techniques developed to study cuprates and other complex oxides. Moler Lab, Stanford SLIDES CREATED USING FoilTEX & PP 4

51 ANYONS 18 Simple continuum model Consider 2-dimensional electron Hamiltonian, H = 1 ( p e ) ( 2 1 2m e c A, A = 2 B 0 + ηφ ) 0 2πr 2 (r ẑ), with m e the electron mass, p the momentum operator in the x-y plane.

52 ANYONS 18 Simple continuum model Consider 2-dimensional electron Hamiltonian, H = 1 ( p e ) ( 2 1 2m e c A, A = 2 B 0 + ηφ ) 0 2πr 2 (r ẑ), with m e the electron mass, p the momentum operator in the x-y plane. The total field seen by an electron is B(r) = A = ẑb 0 + ẑηφ 0 δ(r). The δ-function serves as a crude representation of the half-flux removed by the vacancy.

53 ANYONS 18 Simple continuum model Consider 2-dimensional electron Hamiltonian, H = 1 ( p e ) ( 2 1 2m e c A, A = 2 B 0 + ηφ ) 0 2πr 2 (r ẑ), with m e the electron mass, p the momentum operator in the x-y plane. The total field seen by an electron is B(r) = A = ẑb 0 + ẑηφ 0 δ(r). The δ-function serves as a crude representation of the half-flux removed by the vacancy. This simple model contains the essence of the physics and is exactly soluble. SLIDES CREATED USING FoilTEX & PP 4

54 ANYONS 19 The single particle eigenstates ψ km (r) are labeled by the principal quantum number k = 0, 1, 2,... and an integer angular momentum m. The spectrum reads ɛ km = 2 hω 1 c [2k m + η (m + η)], " km where ω c = eb 0 /m e c is the cyclotron frequency.!

55 ANYONS 19 The single particle eigenstates ψ km (r) are labeled by the principal quantum number k = 0, 1, 2,... and an integer angular momentum m. The spectrum reads ɛ km = 2 hω 1 c [2k m + η (m + η)], " km where ω c = eb 0 /m e c is the cyclotron frequency.! The eigenstates ψ 0m in the lowest Landau level are ψ 0m (z) = A m z η z m e z 2 /4, where z = (x + iy)/l B, and l B = hc/eb is the magnetic length. SLIDES CREATED USING FoilTEX & PP 4

56 ANYONS 20 If we fill the lowest Landau level by spin-polarized electrons then the many-body wavefunction can be constructed as a Slater determinant of ψ 0m (z i ), where z i is a complex coordinate of the i-th electron, Ψ({z i }) = N η z i η i z j )e i<j(z P i z i 2 /4. i

57 ANYONS 20 If we fill the lowest Landau level by spin-polarized electrons then the many-body wavefunction can be constructed as a Slater determinant of ψ 0m (z i ), where z i is a complex coordinate of the i-th electron, Ψ({z i }) = N η z i η i z j )e i<j(z P i z i 2 /4. i The charge density is given as ρ(r) = Ψ 0 ˆρ Ψ 0. For a droplet composed of N electrons occupying N lowest angular momentum states we obtain ρ(r) = e N 1 m=0 ψ 0m (r) 2.

58 ANYONS 20 If we fill the lowest Landau level by spin-polarized electrons then the many-body wavefunction can be constructed as a Slater determinant of ψ 0m (z i ), where z i is a complex coordinate of the i-th electron, Ψ({z i }) = N η z i η i z j )e i<j(z P i z i 2 /4. i The charge density is given as ρ(r) = Ψ 0 ˆρ Ψ 0. For a droplet composed of N electrons occupying N lowest angular momentum states we obtain N = 100, η = 0 ρ(r) = e N 1 m=0 ψ 0m (r) 2.

59 ANYONS 20 If we fill the lowest Landau level by spin-polarized electrons then the many-body wavefunction can be constructed as a Slater determinant of ψ 0m (z i ), where z i is a complex coordinate of the i-th electron, Ψ({z i }) = N η z i η i z j )e i<j(z P i z i 2 /4. i The charge density is given as ρ(r) = Ψ 0 ˆρ Ψ 0. For a droplet composed of N electrons occupying N lowest angular momentum states we obtain N = 100, η = 0 ρ(r) = e N 1 m=0 ψ 0m (r) 2. N = 100, η = 1/2 SLIDES CREATED USING FoilTEX & PP 4

60 ANYONS 21 Look at the charge distribution more closely: !(r) r/l B

61 ANYONS 21 Look at the charge distribution more closely:!(r) r/l B!"(r) !Q(r) r/l B r/l B Inset shows the accumulated charge deficit δq(r) = 2π r 0 r dr δρ(r ) in units of e.

62 ANYONS 21 Look at the charge distribution more closely:!(r) r/l B!"(r) !Q(r) r/l B r/l B Inset shows the accumulated charge deficit δq(r) = 2π r 0 r dr δρ(r ) in units of e. This confirms our first claim that vacancy binds fractional charge e/2. SLIDES CREATED USING FoilTEX & PP 4

63 ANYONS 22 Particle statistics The statistical angle θ can be computed from the present model by evaluating the Berry phase when we adiabatically carry one vacancy around another [Arovas, Schrieffer and Wilczek, 1984]: γ(c) = i C dw Ψ w w Ψ w.

64 ANYONS 22 Particle statistics The statistical angle θ can be computed from the present model by evaluating the Berry phase when we adiabatically carry one vacancy around another [Arovas, Schrieffer and Wilczek, 1984]: γ(c) = i C dw Ψ w w Ψ w. The first vacancy is at w while the second remains at the origin.

65 ANYONS 22 Particle statistics The statistical angle θ can be computed from the present model by evaluating the Berry phase when we adiabatically carry one vacancy around another [Arovas, Schrieffer and Wilczek, 1984]: γ(c) = i C dw Ψ w w Ψ w. The first vacancy is at w while the second remains at the origin. For two vacancies, located at w a and w b the many-body wavefunction reads Ψ wa w b = N wa w b (z i w a ) 1/2 (z i w b ) 1/2 i z j )e i<j(z P i z i 2 /4. (1) i In the above we have performed a gauge trasformation into the string gauge in which all the phase information is explicit in the wavefunction. SLIDES CREATED USING FoilTEX & PP 4

66 ANYONS 23 We now take w a = w and w b = 0 and compute the above Berry phase along a closed contour C that encloses the origin. This gives γ(c) = π ( Φ 1 ). Φ 0 2

67 ANYONS 23 We now take w a = w and w b = 0 and compute the above Berry phase along a closed contour C that encloses the origin. This gives γ(c) = π ( Φ 1 ). Φ 0 2 The first term reflects the Aharonov-Bohm phase that a charge e/2 particle acquires on encircling flux Φ due to background magnetic field B 0.

68 ANYONS 23 We now take w a = w and w b = 0 and compute the above Berry phase along a closed contour C that encloses the origin. This gives γ(c) = π ( Φ 1 ). Φ 0 2 The first term reflects the Aharonov-Bohm phase that a charge e/2 particle acquires on encircling flux Φ due to background magnetic field B 0. The second represents twice the statistical phase of the vacancy, θ = π/4, confirming our earlier heuristic result. SLIDES CREATED USING FoilTEX & PP 4

69 ANYONS 24 Lattice model When the effective Zeeman coupling in 2DEG is sufficiently strong, then, in addition to the periodic vector potential, the electrons also feel periodic scalar potential. This effect becomes important in diluted magnetic semiconductors, such as Ga 1 x Mn x As, where the effective gyro-magnetic ration can be of order 100.

70 ANYONS 24 Lattice model When the effective Zeeman coupling in 2DEG is sufficiently strong, then, in addition to the periodic vector potential, the electrons also feel periodic scalar potential. This effect becomes important in diluted magnetic semiconductors, such as Ga 1 x Mn x As, where the effective gyro-magnetic ration can be of order 100. As argued by Berciu et al. [Nature, 2005] this situation is described by an effective tight-binding model H = ij (t ij e iθ ij c j c i + h.c.) + i µ i c i c i, with Peierls phase factors θ ij = 2π Φ 0 j i A dl corresponding to magnetic field of 1 2 Φ 0 per plaquette. SLIDES CREATED USING FoilTEX & PP 4

71 ANYONS 25 In uniform field and µ i = µ = const H is easily diagonalized with the energy spectrum E k = µ±2t cos 2 k x + cos 2 k y + 4γ 2 sin 2 k x sin 2 k y, and γ = t /t.

72 ANYONS 25 In uniform field and µ i = µ = const H is easily diagonalized with the energy spectrum E k = µ±2t cos 2 k x + cos 2 k y + 4γ 2 sin 2 k x sin 2 k y, and γ = t /t. Filling the lower band with spin polarized electrons corresponds to the previous case of the filled lowest Landau level.

73 ANYONS 25 In uniform field and µ i = µ = const H is easily diagonalized with the energy spectrum E k = µ±2t cos 2 k x + cos 2 k y + 4γ 2 sin 2 k x sin 2 k y, and γ = t /t. Filling the lower band with spin polarized electrons corresponds to the previous case of the filled lowest Landau level. We model vacancy/interstitial by removing/adding 1 2 Φ 0 to a selected plaquette.

74 ANYONS 25 In uniform field and µ i = µ = const H is easily diagonalized with the energy spectrum E k = µ±2t cos 2 k x + cos 2 k y + 4γ 2 sin 2 k x sin 2 k y, and γ = t /t. Filling the lower band with spin polarized electrons corresponds to the previous case of the filled lowest Landau level. We model vacancy/interstitial by removing/adding 1 2 Φ 0 to a selected plaquette. We have diagonalized the above Hamiltonian numericaly for system sizes up to and various configurations of fluxes and µ i s. In all cases we find that vacancy/interstitial binds charge ±e/2. SLIDES CREATED USING FoilTEX & PP 4

75 ANYONS 26 Sample result for a single interstitial and random µ i with µ i = 0.05t.

76 ANYONS 26 Sample result for a single interstitial and random µ i with µ i = 0.05t. ρ(r) with interstitial

77 ANYONS 26 Sample result for a single interstitial and random µ i with µ i = 0.05t. ρ(r) without interstitial ρ(r) with interstitial

78 ANYONS 26 Sample result for a single interstitial and random µ i with µ i = 0.05t. ρ(r) without interstitial ρ(r) with interstitial induced charge δρ(r)

79 ANYONS 26 Sample result for a single interstitial and random µ i with µ i = 0.05t. ρ(r) without interstitial ρ(r) with interstitial induced charge δρ(r) The induced charge integrates to e/2 to within machine accuracy. SLIDES CREATED USING FoilTEX & PP 4

80 ANYONS 27 Conclusions Unusual phenomena, such as charge and statistics fractionalization, can occur when two weakly interacting (and well understood) systems are brought to close proximity.

81 ANYONS 27 Conclusions Unusual phenomena, such as charge and statistics fractionalization, can occur when two weakly interacting (and well understood) systems are brought to close proximity. We showed, by general argument and detailed computation, that vacancy in a vortex lattice binds e/2 charge in the adjacent 2DEG at unit filling.

82 ANYONS 27 Conclusions Unusual phenomena, such as charge and statistics fractionalization, can occur when two weakly interacting (and well understood) systems are brought to close proximity. We showed, by general argument and detailed computation, that vacancy in a vortex lattice binds e/2 charge in the adjacent 2DEG at unit filling. Such a composite object behaves as abelian anyon with θ = π/4 and can be manipulated by existing experimental techniques.

83 ANYONS 27 Conclusions Unusual phenomena, such as charge and statistics fractionalization, can occur when two weakly interacting (and well understood) systems are brought to close proximity. We showed, by general argument and detailed computation, that vacancy in a vortex lattice binds e/2 charge in the adjacent 2DEG at unit filling. Such a composite object behaves as abelian anyon with θ = π/4 and can be manipulated by existing experimental techniques. These superconductor-semiconductor heterostructures can be (and have been) built in a lab and could allow for controlled creation and manipulation of the fractional particles.

84 ANYONS 27 Conclusions Unusual phenomena, such as charge and statistics fractionalization, can occur when two weakly interacting (and well understood) systems are brought to close proximity. We showed, by general argument and detailed computation, that vacancy in a vortex lattice binds e/2 charge in the adjacent 2DEG at unit filling. Such a composite object behaves as abelian anyon with θ = π/4 and can be manipulated by existing experimental techniques. These superconductor-semiconductor heterostructures can be (and have been) built in a lab and could allow for controlled creation and manipulation of the fractional particles. At ν = 5 2 vacancy should bind a Moore-Read non-abelion which may be used to implement topologically protected fault tolerant quantum gates. SLIDES CREATED USING FoilTEX & PP 4