Irrational anyons under an elastic membrane

Transcription

1 Irrational anyons under an elastic membrane Claudio Chamon Collaborators: Siavosh Behbahani Ami Katz Euler Symposium on Theoretical and Mathematical Physics D.I. Diakonov Memorial Symposium

2 Fractionalization in the Quantum Hall Effect: Chern-Simons perspective L = p 4π µνλ a µ ν a λ + e 2π µνλ A µ ν a λ

3 Fractionalization in the Quantum Hall Effect: Chern-Simons perspective L = p 4π µνλ a µ ν a λ + e 2π µνλ A µ ν a λ J µ = δs δa µ = e 2π µνλ ν a λ 0= δs δa µ = p 2π µνλ ν a λ + e 2π µνλ ν A λ J µ = 1 p e 2 h µνλ ν A λ

4 Fractionalization in the Quantum Hall Effect: Chern-Simons perspective J µ = 1 p e 2 h µνλ ν A λ J i = 1 p e 2 h ij E j ρ = 1 p e 2 h ẑ ( A) Q = 1 p e Φ φ 0

5 Fractionalization in the Quantum Hall Effect: Chern-Simons perspective J µ = 1 p e 2 h µνλ ν A λ J i = 1 p e 2 h ij E j ρ = 1 p e 2 h ẑ ( A) e e Q = 1 p e Φ φ 0 Laughlin (1983) φ 0 = hc e

6 Does statistical angle need to be rational? counter-examples: Wilczek, PL (1982) Jackiw and Nair, Phys. Lett. B (2000)... Fitzpatrick, Kachru, Kaplan, Katz, and Wacker, arxiv: counter-examples with realization on a lattice: Chamon, Hou, Jackiw, Mudry, Pi, and Schnyder, PL (2008) yu, Mudry, Hou, and Chamon, PB (2009)

7 Perspective Departure from Chern-Simons Theory a i = α ij j φ H = 1 2m ψ (p i α ij j φ) 2 ψ b = ij i a j = α 2 φ Wilczek, PL (1982) Fitzpatrick, Kachru, Kaplan, Katz, and Wacker, arxiv:

8 Perspective Departure from Chern-Simons Theory a i = α ij j φ H = 1 2m ψ (p i α ij j φ) 2 ψ b = ij i a j = α 2 φ 2 φ(r) = g τ ψ ψ(r) Wilczek, PL (1982) Fitzpatrick, Kachru, Kaplan, Katz, and Wacker, arxiv:

9 Perspective Departure from Chern-Simons Theory a i = α ij j φ H = 1 2m ψ (p i α ij j φ) 2 ψ b = ij i a j = α 2 φ 2 φ(r) = g τ ψ ψ(r) e Φ = α g/τ Wilczek, PL (1982) Fitzpatrick, Kachru, Kaplan, Katz, and Wacker, arxiv:

10 Perspective Departure from Chern-Simons Theory a i = α ij j φ H = 1 2m ψ (p i α ij j φ) 2 ψ b = ij i a j = α 2 φ 2 φ(r) = g τ ψ ψ(r) e Φ = α g/τ vs e = ν e Φ 0 = hc/e Wilczek, PL (1982) Fitzpatrick, Kachru, Kaplan, Katz, and Wacker, arxiv:

11 Flux attachment under an elastic, electric and magnetic membrane Behbahani, Chamon, and Katz, arxiv:1307. θstat = π + θ δ Φ = α g/τ e e Φ = α g/τ σe σm θ = 2π e τ 2 2

12 Flux attachment under an elastic, electric and magnetic membrane S= ρ 2 τ κ 2 dt d x φ ( φ) ( 2 φ) ( i iα ij j φ) ψ g φ ψ ψ + ψ i t ψ 2m Hψ = p + ea ea0 2m δ Φ = α g/τ e e Φ = α g/τ

13 Flux attachment under an elastic, electric and magnetic membrane m δ + φ(x ) X X A i (X) = d 2 X σ m ij (X X ) j (X X ) 2 +(δ + φ (X )) 2 3/2,

14 Flux attachment under an elastic, electric and magnetic membrane A = A BG + A φ A φ i (X) 2πσ m ij j φ(x) sgn(δ) A φ 0 (X) 2πσ e φ(x) sgn(δ) α = 2πσ m e sgn(δ) g =2πσ e e sgn(δ) θ = 2π 2 e 2 σ eσ m τ

15 Experimentally probing fractional statistics? V G V C V G L! I t 1 I t 2 L L V G V G Chamon, Freed, Kivelson, Sondhi, Wen, PB (1997) B = Φ /A Φ = e B = Φ 0 /A e Φ 0

16 Experimentally probing fractional statistics? V G V C V G L! I t 1 I t 2 L L V G V G Chamon, Freed, Kivelson, Sondhi, Wen, PB (1997) B = Φ /A Φ = e B = Φ 0 /A e Φ 0

17 Experimentally probing fractional statistics? V G V C V G L! I t 1 I t 2 L L V G V G Chamon, Freed, Kivelson, Sondhi, Wen, PB (1997) B = Φ /A Φ = e B = Φ 0 /A e Φ 0 Webb et al., PL (1985)

18 Experimentally probing fractional statistics? VC VG VG! It 1 L It 2 L L VG VG Chamon, Freed, Kivelson, Sondhi, Wen, PB (1997) FIG. 1. Two point-contact interferometer. Two gates are placed a distance a apart. The gate voltages are adjusted so as to bring the edges of a FQH state with filling fraction ν close together, but not pinch the constriction. In this way, quasiparticles carrying fractional charge and statistics can tunnel from one edge to the other. A magnetic flux Φ can be inserted in the region between the point-contacts that is bounded by the edge states. A central gate allows the charge in the region to be selectively depleted. The transmitted current (the Hall current IH = νe2 /h minus the tunneling current It1 + It2 ) oscillates as a function of the inserted flux, the voltage difference between the edges and the voltage of the central gate. An overall back gate on the device allows magnetic field sweeps at constant filling factor. B = Φ /A B = Φ0 /A e Φ = Φ0 e We would like to emphasize that we will always be interested in the limit where the barriers are weak, so that the constrictions are far from being pinched off, as in Figure 1. (The opposite limit, where the two point-contacts are near pinch off, is similar to tunneling through a quantum dot, except that for our geometry the central island would be larger. It can be analyzed using methods similar to those in this paper, but we will not address it here.) The chief advantage of this restriction is that we stay far from the regime near pinch off where, experimentally, poorly understood resonances arise already for a single point-contact [27].etConsequently, we expect that the only resonances Webb al., PL (1985) are the ones explicitly created by the two point-contact geometry and the resulting energy and field scales for these are set by the parameters of the device and can be chosen to lie in an observable range. (Given the lack of understanding of the near pinch-off resonances, it is hard to say theoretically whether they are absent for weak barriers; however,

19 Proposed experimental set-up to probe anyons under a membrane D

20 Proposed experimental set-up to probe anyons under a membrane membrane layer electron gas layer D 2 φ(r) = g τ ρ(r) α τ ẑ J(r) B = α e 2 φ

21 Proposed experimental set-up to probe anyons under a membrane membrane layer electron gas layer Q D 2 φ(r) = g τ ρ(r) α τ ẑ J(r) B = α e 2 φ Q = en

22 Proposed experimental set-up to probe anyons under a membrane membrane layer electron gas layer Q Φ D 2 φ(r) = g τ ρ(r) α τ ẑ J(r) B = α e 2 φ } Q = en Φ = αg τ Q e 2

23 Proposed experimental set-up to probe anyons under a membrane membrane layer electron gas layer Q Φ D 2 φ(r) = g τ ρ(r) α τ ẑ J(r) B = α e 2 φ } Q = en Φ = αg τ e Q e 2 Φ = α g/τ

24 Proposed experimental set-up to probe anyons under a membrane G(Φ total /Φ 0 ) Φ total = Φ + Φ ext Q Φ 2π Φ/Φ 0 = (αg/τ) Q/e D θ = αg 2τ = π Φ ext/φ 0 Q/e = 1 2 e 2 C Φ ext V

25 Possible experimental realization membrane layer: graphene electron gas layer: metal D Magnetic impurities spaced about 1 nm away, with each impurity having a magnetic moment of order µ B, yields a magnetic density of order σ m ev. 3µm = D nm φ max 300 nm

26 Possible experimental realization Q e θ π = 2πσ m e φ max 1 ln D / 2πσ m e φ max N e 1 D

27 Comments on anyon interaction log potential S = dt d 2 x ρ 2 φ 2 τ 2 ( φ)2 κ 2 ( 2 φ) 2 + ψ i t ψ 1 2m ( i iα ij j φ) ψ 2 g φψ ψ

28 Comments on anyon interaction log potential S = dt d 2 x ρ 2 φ 2 τ 2 ( φ)2 κ 2 ( 2 φ) 2 + ψ i t ψ 1 2m ( i iα ij j φ) ψ 2 g φψ ψ no log potential 2 φψ σ z ψ 2 φψ ψ Zeeman quadrupole

29 Comments on anyon interaction log potential S = dt d 2 x ρ 2 φ 2 τ 2 ( φ)2 κ 2 ( 2 φ) 2 + ψ i t ψ 1 2m ( i iα ij j φ) ψ 2 g φψ ψ no log potential 2 φψ σ z ψ 2 φψ ψ Zeeman quadrupole

30 Comments on anyon interaction D metal: screening of interactions!

31 Amusement: log-interaction allows the mesoscopic study of anyon stars vary θ

32 Conclusions Anyon statistics mediated by electromagnetism and phonons Statistical angle is a continuous function of membrane properties magnetic dipolar density electric charge density tension Proposed physical realization with graphene as membrane Proposed experiment that does not suffer from area problem