Ψ({z i }) = i<j(z i z j ) m e P i z i 2 /4, q = ± e m.

Transcription

1 Fractionalization of charge and statistics in graphene and related structures M. Franz University of British Columbia January 5, 2008 In collaboration with: C. Weeks, G. Rosenberg, B. Seradjeh

2 FRACTIONALIZATION 1 Fractionalization in many-body systems...occurs when low lying excitations carry charge that is a fraction of the electron charge e. e/3 e/3

3 FRACTIONALIZATION 1 Fractionalization in many-body systems...occurs when low lying excitations carry charge that is a fraction of the electron charge e. e/3 e/3 More is different! SLIDES CREATED USING FoilTEX & PP 4

4 FRACTIONALIZATION 2 Fractional quantum Hall fluids

5 FRACTIONALIZATION 2 Fractional quantum Hall fluids Fractional charges are known to occur as excitations of the FQH 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 charge q = ± e m.

6 FRACTIONALIZATION 2 Fractional quantum Hall fluids Fractional charges are known to occur as excitations of the FQH 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 charge q = ± e m. They also exhibit fractional exchange phase θ = π m. SLIDES CREATED USING FoilTEX & PP 4

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

8 FRACTIONALIZATION 3 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

9 FRACTIONALIZATION 4 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)

10 FRACTIONALIZATION 4 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

11 FRACTIONALIZATION 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems with broken time reversal symmetry.

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

13 FRACTIONALIZATION 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems with broken time reversal symmetry. 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 and broken time-reversal?

14 FRACTIONALIZATION 5 Anyons, fractionalization and strong correlations Anyons and charge fractionalization typically occur in strongly correlated electron systems with broken time reversal symmetry. 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 and broken time-reversal? Answer: Not necessarily. SLIDES CREATED USING FoilTEX & PP 4

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

16 FRACTIONALIZATION 6 Fractional charges in polyacetylene [Su, Schrieffer and Heeger, 1979] Theorists model of polyacetylene: t t H = i t i (c i+1 c i + h.c.)

17 FRACTIONALIZATION 6 Fractional charges in polyacetylene [Su, Schrieffer and Heeger, 1979] Theorists model of polyacetylene: t t EHkL k H = i t i (c i+1 c i + h.c.) -1-2 band structure: 1D metal SLIDES CREATED USING FoilTEX & PP 4

18 FRACTIONALIZATION 7 Peierls transition into a dimerized state: t!" t +" Shift every other ion by a small amount to the right; modify hopping amplitudes for electrons.

19 FRACTIONALIZATION 7 Peierls transition into a dimerized state: EHkL 2 t!" t +" k Shift every other ion by a small amount to the right; modify hopping amplitudes for electrons band structure: insulator

20 FRACTIONALIZATION 7 Peierls transition into a dimerized state: EHkL 2 t!" t +" k Shift every other ion by a small amount to the right; modify hopping amplitudes for electrons band structure: insulator Peierls transition is driven by electron kinetic energy gain upon opening of a gap at half filling. In 1d systems it is always advantageous to dimerize SLIDES CREATED USING FoilTEX & PP 4

21 FRACTIONALIZATION 8 Domain-wall solitons: (Boundaries between degenerate ground states of the dimerized chain)

22 FRACTIONALIZATION 8 Domain-wall solitons: (Boundaries between degenerate ground states of the dimerized chain) Su, Schrieffer and Heeger discovered that there exists a zero-energy bound state associated with a domain wall. Moreover, for spinless electrons, the charge associated with a domain wall is δq = ± e 2 bound state at domain wall SLIDES CREATED USING FoilTEX & PP 4

23 FRACTIONALIZATION 9 A simple argument for fractional charge at the domain wall [Goldstone & Wilczek, 1981]

24 FRACTIONALIZATION 9 A simple argument for fractional charge at the domain wall [Goldstone & Wilczek, 1981] e

25 FRACTIONALIZATION 9 A simple argument for fractional charge at the domain wall [Goldstone & Wilczek, 1981] e

26 FRACTIONALIZATION 9 A simple argument for fractional charge at the domain wall [Goldstone & Wilczek, 1981] e

27 FRACTIONALIZATION 9 A simple argument for fractional charge at the domain wall [Goldstone & Wilczek, 1981] e e/2 e/2 SLIDES CREATED USING FoilTEX & PP 4

28 FRACTIONALIZATION 10 Fractionalization of charge in polyacetylene is a concrete physical realization of a general field-theoretical concept discovered in 1976 by Jackiw and Rebbi.

29 FRACTIONALIZATION 10 Fractionalization of charge in polyacetylene is a concrete physical realization of a general field-theoretical concept discovered in 1976 by Jackiw and Rebbi. This, in turn is a manifestation of the Atiyah-Singer index theorem (1968).

30 FRACTIONALIZATION 10 Fractionalization of charge in polyacetylene is a concrete physical realization of a general field-theoretical concept discovered in 1976 by Jackiw and Rebbi. This, in turn is a manifestation of the Atiyah-Singer index theorem (1968). In the above cases charge fractionalization occurs as a result of fermions coupling to a soliton configuration of a background field. Interactions play no significant role, systems can be regarded as weakly correlated and time reversal remains unbroken.

31 FRACTIONALIZATION 10 Fractionalization of charge in polyacetylene is a concrete physical realization of a general field-theoretical concept discovered in 1976 by Jackiw and Rebbi. This, in turn is a manifestation of the Atiyah-Singer index theorem (1968). In the above cases charge fractionalization occurs as a result of fermions coupling to a soliton configuration of a background field. Interactions play no significant role, systems can be regarded as weakly correlated and time reversal remains unbroken. In d = 1, 3 particle statistics is trivial (fermions or bosons): Need a two-dimensional example! SLIDES CREATED USING FoilTEX & PP 4

32 FRACTIONALIZATION 11 Fractionalization in 2d time-reversal invariant systems [Hou, Chamon & Mudry (2007); Seradjeh, Weeks & Franz (2007)] (a) (b) (c) (d) π π π π π π π π π π π π π π π π π π Special dimerization patterns in honeycomb (graphene) lattice and square lattice with half-flux magnetic field per plaquette. SLIDES CREATED USING FoilTEX & PP 4

33 FRACTIONALIZATION 12 Follow the example of polyacetylene:

34 FRACTIONALIZATION 12 Follow the example of polyacetylene: Take electrons on a lattice and consider the band structure.

35 FRACTIONALIZATION 12 Follow the example of polyacetylene: Take electrons on a lattice and consider the band structure. Deform the lattice (dimerize) to open up a gap in the spectrum.

36 FRACTIONALIZATION 12 Follow the example of polyacetylene: Take electrons on a lattice and consider the band structure. Deform the lattice (dimerize) to open up a gap in the spectrum. Consider a topological defect in the dimerization pattern and look for zero-energy bound states. SLIDES CREATED USING FoilTEX & PP 4

37 FRACTIONALIZATION 13 Electrons hopping on a 2d square lattice t t H = ij t ij (c i c j + h.c.)

38 FRACTIONALIZATION 13 Electrons hopping on a 2d square lattice E(k) t t H = ij t ij (c i c j + h.c.) k y k x band structure: 2D metal SLIDES CREATED USING FoilTEX & PP 4

39 FRACTIONALIZATION 14 Electrons hopping on a dimerized square lattice t t H = ij t ij (c i c j + h.c.)

40 FRACTIONALIZATION 14 Electrons hopping on a dimerized square lattice E(k) t t H = ij t ij (c i c j + h.c.) k y k x band structure: still a metal! SLIDES CREATED USING FoilTEX & PP 4

41 FRACTIONALIZATION 15 Electrons hopping on a square lattice in transverse magnetic field!!!!!!!!!!!! t t!!!! H = ij t ij (e iθ ij c i c j + h.c.) with a Peierls factor θ ij = 2π Φ 0 j i dr A such that J θ ij = π.

42 FRACTIONALIZATION 15 Electrons hopping on a square lattice in transverse magnetic field!!!!!!!!!!!! t t!!!! H = ij t ij (e iθ ij c i c j + h.c.) with a Peierls factor θ ij = 2π Φ 0 j i dr A such that J θ ij = π. band structure: a semi-metal with Dirac points SLIDES CREATED USING FoilTEX & PP 4

43 FRACTIONALIZATION 16 Electrons hopping on a dimerized square lattice in transverse magnetic field!!!!!!!!!!!! t!!!! t H = ij t ij (e iθ ij c i c j + h.c.)

44 FRACTIONALIZATION 16 Electrons hopping on a dimerized square lattice in transverse magnetic field!!!!!!!!!!!! t!!!! t H = ij t ij (e iθ ij c i c j + h.c.) band structure: insulator with massive Dirac points SLIDES CREATED USING FoilTEX & PP 4

45 FRACTIONALIZATION 17 Follow the example of polyacetylene checklist:

46 FRACTIONALIZATION 17 Follow the example of polyacetylene checklist: Take electrons on a lattice and consider the band structure.

47 FRACTIONALIZATION 17 Follow the example of polyacetylene checklist: Take electrons on a lattice and consider the band structure. Deform the lattice (dimerize) to open up a gap in the spectrum.

48 FRACTIONALIZATION 17 Follow the example of polyacetylene checklist: Take electrons on a lattice and consider the band structure. Deform the lattice (dimerize) to open up a gap in the spectrum. Consider a topological defect in the dimerization pattern and look for zero-energy bound states. SLIDES CREATED USING FoilTEX & PP 4

49 FRACTIONALIZATION 18 We have 4 distinct domains: SLIDES CREATED USING FoilTEX & PP 4

50 FRACTIONALIZATION 19 Bringing together all 4 domains results in a topological defect vortex in the dimerization pattern: SLIDES CREATED USING FoilTEX & PP 4

51 FRACTIONALIZATION 20 Does the vortex produce a zero-energy bound state? The effective continuum theory is given by a Dirac Lagrangian L = ψ ( i + me iχγ 5 ) ψ with twisted mass m.

52 FRACTIONALIZATION 20 Does the vortex produce a zero-energy bound state? The effective continuum theory is given by a Dirac Lagrangian L = ψ ( i + me iχγ 5 ) ψ with twisted mass m. The associated Hamiltonian indeed possesses a zero energy bound state in the presence of a vortex in χ. Its existence is protected by an index theorem for Dirac Hamiltonians. SLIDES CREATED USING FoilTEX & PP 4

53 FRACTIONALIZATION 21 Is there a fractional charge associated with the vortex?

54 FRACTIONALIZATION 21 Is there a fractional charge associated with the vortex? One can give a simple electron counting argument as in polyacetylene: in the limit of strong dimerization each red bond represents one electron. Thus, there is e/2 charge per site except the central site, which is either empty or occupied by 1 electron. SLIDES CREATED USING FoilTEX & PP 4

55 FRACTIONALIZATION 22 Numerical evidence for the fractional charge

56 FRACTIONALIZATION 22 Numerical evidence for the fractional charge Exact numerical diagonalization of our model Hamiltonian H = ij t ij (e iθ ij c i c j + h.c.) gives convincing evidence for ±e/2 fractional charge bound to the vortex. Charge density in the presence of a vortex. SLIDES CREATED USING FoilTEX & PP 4

57 FRACTIONALIZATION 23 Numerical evidence for the zero modes

58 FRACTIONALIZATION 23 Numerical evidence for the zero modes Numerical diagonalization also confirms the existence of the zeroenergy bound states associated with vortices. It furthermore confirms the prediction (consistent with the index theorem) that that the bound states remain degenerate for two vortices, but split in the case of a vortex-antivortex pair. Charge density in the presence two vortices and v-a pair. SLIDES CREATED USING FoilTEX & PP 4

59 FRACTIONALIZATION 24 Relevance to graphene The same pattern of fractionalization occurs on the honeycomb lattice with Kekule distortion. (a) (b) (c) (d) π π π π π π π π π π π π π π π π π π [Hou, Chamon & Mudry (2007)] SLIDES CREATED USING FoilTEX & PP 4

60 FRACTIONALIZATION 25 Possible experimental realization The dimerization patterns discussed above can arise as mean-field instabilities of interacting spinless fermions hopping on a square or a honeycomb lattice. Consider a Hubbard-like Hamiltonian H = t ij (e iθ ij c i c j + h.c.) +V 1 n i n j + V 2 ij ij n i n j on a square lattice with π-flux or a honeycomb lattice with V 1, V 2 > 0, i.e. repulsive interactions. SLIDES CREATED USING FoilTEX & PP 4

61 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: 3 2 QAH V 2 /t Kekule 1 SM V /t 1 2 CDW 3 [honeycomb lattice]

62 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: 3 2 QAH SM: gapless Dirac semimetal V 2 /t Kekule 1 SM V /t 1 2 CDW 3 [honeycomb lattice]

63 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: 3 SM: gapless Dirac semimetal 2 QAH CDW: band insulator with n A n B V 2 /t Kekule 1 SM V /t 1 2 CDW 3 [honeycomb lattice]

64 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: 3 SM: gapless Dirac semimetal 2 QAH CDW: band insulator with n A n B V 2 /t Kekule 1 SM Kekule: dimerized phase with fractional excitations V /t 1 2 CDW 3 [honeycomb lattice]

65 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: V 2 /t 3 2 QAH Kekule SM: gapless Dirac semimetal CDW: band insulator with n A n B SM 1 V /t [honeycomb lattice] 1 CDW 2 3 Kekule: dimerized phase with fractional excitations QAH: Quantum Anomalous Hall (or Haldane ) T -breaking phase with c i c j 0 for n.n.n.

66 FRACTIONALIZATION 26 The standard Hartree-Fock decoupling of the interaction terms followed by a self-consistent evaluation of the resulting mean-field order parameters leads to the following phase diagram: V 2 /t 3 2 QAH Kekule SM: gapless Dirac semimetal CDW: band insulator with n A n B SM 1 V /t [honeycomb lattice] 1 CDW 2 3 Kekule: dimerized phase with fractional excitations QAH: Quantum Anomalous Hall (or Haldane ) T -breaking phase with c i c j 0 for n.n.n. Similar phase diagram for square lattice with π-flux. SLIDES CREATED USING FoilTEX & PP 4

67 FRACTIONALIZATION 27 Fractional Statistics [B. Seradjeh and M. Franz, arxiv: ]

68 FRACTIONALIZATION 27 Fractional Statistics [B. Seradjeh and M. Franz, arxiv: ] For the subsequent considerations it will be useful to modify the Lagrangian L = ψ ( i / A/ + γ 5 B/ + me iχγ 5 ) ψ. to include electromagnetic gauge field A µ and the chiral gauge field B µ.

69 FRACTIONALIZATION 27 Fractional Statistics [B. Seradjeh and M. Franz, arxiv: ] For the subsequent considerations it will be useful to modify the Lagrangian L = ψ ( i / A/ + γ 5 B/ + me iχγ 5 ) ψ. to include electromagnetic gauge field A µ and the chiral gauge field B µ. We note that the coupling between vortices and fermions can be written as m( ψ + e iχ ψ + ψ e iχ ψ + ) where ψ ± = 1 2 (1 ± γ 5)ψ are the chiral components of the Dirac fermion. SLIDES CREATED USING FoilTEX & PP 4

70 FRACTIONALIZATION 28 We now perform a singular gauge transformation ψ ± e ±iχ ± ψ ±, where we have defined an (as yet) arbitrary partitioning χ = χ + + χ of the vortex phase, such that upon encircling a vortex the spinor field remains single-valued.

71 FRACTIONALIZATION 28 We now perform a singular gauge transformation ψ ± e ±iχ ± ψ ±, where we have defined an (as yet) arbitrary partitioning χ = χ + + χ of the vortex phase, such that upon encircling a vortex the spinor field remains single-valued. Under this transformation the Lagrangian becomes with shifted gauge fields L = ψ (i / a/ + γ 5 b/ + m) ψ, a µ = A µ µ(χ + χ ), b µ = B µ µ(χ + + χ ). SLIDES CREATED USING FoilTEX & PP 4

72 FRACTIONALIZATION 29 The effective action Integrating out the Dirac fermions to one-loop order yields L eff = π 12 m ( a)2 + m 2π b2 + sgn(m 3) a ( b). 2π where m 3 is a Pauli-Villars regularization mass.

73 FRACTIONALIZATION 29 The effective action Integrating out the Dirac fermions to one-loop order yields L eff = π 12 m ( a)2 + m 2π b2 + sgn(m 3) a ( b). 2π where m 3 is a Pauli-Villars regularization mass. The last term is a topological mixed Chern-Simons term which implies fractional statistics for vortices. SLIDES CREATED USING FoilTEX & PP 4

74 FRACTIONALIZATION 30 It is possible to rewrite the topological term in a more familiar form by introducing two auxiliary gauge fields A ± mediating the statistical interaction between vortices, L CS = 1 πκ (σa σ A σ + j σ A σ ) σ=± where j ± = 1 2π ( χ ±) are currents associated with vortices in two partitions and κ = 2sgn(m 3 ).

75 FRACTIONALIZATION 30 It is possible to rewrite the topological term in a more familiar form by introducing two auxiliary gauge fields A ± mediating the statistical interaction between vortices, L CS = 1 πκ (σa σ A σ + j σ A σ ) σ=± where j ± = 1 2π ( χ ±) are currents associated with vortices in two partitions and κ = 2sgn(m 3 ). This is a doubled Chern-Simons theory which implies two species of semions (particles with statistical phase π 2 ). SLIDES CREATED USING FoilTEX & PP 4

76 FRACTIONALIZATION 31 Outlook and open questions Efforts in the recent years firmly establish possibility of charge fractionalization in weakly correlated, time-reversal invariant 2d systems.

77 FRACTIONALIZATION 31 Outlook and open questions Efforts in the recent years firmly establish possibility of charge fractionalization in weakly correlated, time-reversal invariant 2d systems. Can we have other fractions such as ±e/3 in a time-reversal invariant system? e/3 e/3

78 FRACTIONALIZATION 31 Outlook and open questions Efforts in the recent years firmly establish possibility of charge fractionalization in weakly correlated, time-reversal invariant 2d systems. Can we have other fractions such as ±e/3 in a time-reversal invariant system? e/3 e/3 What would be the exchange statistics of suchy fractional particles? SLIDES CREATED USING FoilTEX & PP 4