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Braiding fluxes in Pauli Hamiltonians Anyons for anyone J. Avron O. Kenneth Department of Physics, Technion Montreal, 2014 Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 1 / 34 Outline Motivation 1 Non abelian anyons Aharonov

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SLIDE 1

Braiding fluxes in Pauli Hamiltonians

Anyons for anyone

  • J. Avron
  • O. Kenneth

Department of Physics, Technion

Montreal, 2014

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 1 / 34

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SLIDE 2

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 2 / 34

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SLIDE 3

Motivation Non abelian anyons

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 3 / 34

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SLIDE 4

Motivation Non abelian anyons

Gates

Unitary: |ψ → U|ψ U n − qubits = ⇒ dim(H) = 2n Universal single qubit gates: 1 √ 2 1 1 1 −1

  • =

H , 1 eiπ/4

  • =

eiπ/4 Universal two qubits:

  • Z

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 4 / 34

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SLIDE 5

Motivation Non abelian anyons

Anyons and quantum computing

Desiderata

Fault tolerance Gap H = Protected subspace Topological quantum computing—non-abelain anyons

Lindner & Stern, Science Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 5 / 34

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SLIDE 6

Motivation Non abelian anyons

Non abelian anyons

Theory and experiment

Theory Localized modes of interacting) fermions or spins Theoretical realization Anyons in FQHE Majoranas: electron/ √ 2 Experiment Fractional charges in FQHE, Evidence for Majorana

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 6 / 34

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SLIDE 7

Motivation Aharonov Casher

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 7 / 34

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SLIDE 8

Motivation Aharonov Casher

Aharonov Casher

Topological Zero modes

Geometric setting: ΦT Pauli equation: spin 1/2, g = 2

  • (−i∇ − A) · σ

2 ≥ 0, ΦT = 1 2π

  • B dx ∧ dy

Zero modes: Zero modes Continuum

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 8 / 34

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SLIDE 9

Motivation Aharonov Casher

Aharnonov Casher

holomorphy

Decoupling in 2-D: (−i∇ − A) · σ = −2i

  • ∂z − iAz

¯ ∂z − i ¯ Az

  • Zero modes:
  • (−i∇ − A) · σ
  • (ψ, 0)t = 0, =

⇒ ( ¯ ∂z − i ¯ A)ψ = 0

  • 1−st order pde

Holomorphy: ψ(z, ¯ z) ∈ Ker( ¯ ∂z − i ¯ A) ∋ P(z)

  • holomorphic

ψ(z, ¯ z)

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 9 / 34

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SLIDE 10

Motivation Aharonov Casher

Aharonov and Casher

Index

Poissons’ equation–source B ∂z ¯ ∂z

  • ∆

log ψ0 = i∂z ¯ A

  • B

Polynomial decay: ψ0 = exp(∆−1B) − →

z→∞ |z|−ΦT ,

∆−1 = 1 2π log z Aharonov-Casher Index theorem: Number of zero modes D = ⌈ΦT⌉ − 1

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 10 / 34

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SLIDE 11

Motivation Aharonov Casher

Confined and free zero modes

Φa > 1 vs Φa < 1

Two types of Charge-Flux composite Φa > 1 Φb = Φb′ = 3/4

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 11 / 34

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SLIDE 12

Braiding fluxes

Braiding fluxes

Gates from braiding fluxes

curvature Φa Φb What gates can you make by braiding fluxons? Catch 22: Holonomy without curvature! Φa ∈ R; Think of 1/2 < Φa < 1 No gap protection

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 12 / 34

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SLIDE 13

Braiding fluxes

Adiabatic evolution

AB-Anyons

Adiabatic evolution for moving fluxes

Gapless Gauge issues Defrosting

Confined zero modes

Super Critical fluxons; Φa > 1 Aharonov-Bohm abelian phases

Φ1 e2πiΦ2 Φ2

Localized zero modes

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 13 / 34

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SLIDE 14

Braiding fluxes

Deconfined modes

Anyons

Holonomy–Abelian & non-abelian curvature & topological

curvature

Topological if: D = N − 1 Identical fluxes 1 − 1

N < Φ < 1

Burau rep of braid group : 1 − ν ν 1

  • ,

ν = e−2πiΦ

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 14 / 34

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SLIDE 15

Braiding fluxes Zero modes

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 15 / 34

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SLIDE 16

Braiding fluxes Zero modes

Aharonov and Casher

Fluxons

Log-Superposition: ¯ ∂z log ψ = i ¯ A = ⇒ (A1 + A2, ψ1ψ2)

ζ2 =position (Φ1, ζ1) Φ3 =flux (Φ4, ζ4)

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 16 / 34

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SLIDE 17

Braiding fluxes Zero modes

Weak individuals, Φa < 1, strong community, ΦT > 1

Point fluxes

(Φ2, ζ2) (Φ1, ζ1) (Φ5, ζ5) (Φ4, ζ4)

Zero modes; 0 < Φa < 1 ψ(z; ζ) = P(z)

  • polynom
  • a

(z − ζa)−Φa, deg(P) < ΦT − 1

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 17 / 34

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SLIDE 18

Braiding fluxes Adiabatically Moving fluxes

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 18 / 34

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SLIDE 19

Braiding fluxes Adiabatically Moving fluxes

Bad defrosting

Dead frozen

Defrosted Hamiltonian ζ → ζ(t)

  • control

H(Aζ) → H

  • Aζ(t)
  • Wrong sources

A = J (Φ, ζ) E E Current fluxon

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 19 / 34

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Braiding fluxes Adiabatically Moving fluxes

Gauge fields of moving flux

Defrosting and Gauge freedom

Motion generates weak electric fields E = −v × B

  • localized on fluxon

E fluxon v Defrosted potentials A = A(z − ζ(t)), A0 = −v · A(z − ζ(t))

  • Inertial frame

Closed path in control ζa = ⇒ closed path in (A0, A)

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 20 / 34

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SLIDE 21

Braiding fluxes Adiabatically Moving fluxes

Topology in Gappless Adiabatic evolution

What is the time scale?

  • Gapless. Distance between fluxon defines time scale:

time scale = m h (distance)2

  • dim analysis

, distance = |ζa − ζb|

  • length scale

Control Energy Imζ Reζ

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 21 / 34

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SLIDE 22

Braiding fluxes Adiabatically Moving fluxes

Parallel transport

Connection

Zero modes: PD

  • projection

: Span{zjψ0|j = 0, . . . , D − 1}

  • zero modes

, z|ψ0 =

  • a

(z − ζa(t))−Φa

  • ζa=ζb=···⇒|ψ0=∞

Evolution within PD ψ(z, t) = P(z, t)

polynom

ψ0, P(z, t) =

D

  • pj(t)zj,

Connection PDDtψ = 0

  • No motion

, Dt = ∂t − iA0

  • covariant derivative

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 22 / 34

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SLIDE 23

Braiding fluxes Metric and connection

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 23 / 34

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SLIDE 24

Braiding fluxes Metric and connection

The connection

Metric

Geometric–independent of time schedule: iPDd|ψ = PD   

  • a

dxa

  • flux displace

· Aa|ψ

i∂a|ψ

   A (non-orthogonal) basis zj|ψ0, j = 0, . . . , D − 1 Hilbert space metric (g)jk (ζ, ¯ ζ)

control

= ψ0|¯ zjzk|ψ0 Diverges when fluxons collide: (g)jk(ζa = ζb = . . . ) = ∞

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 24 / 34

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SLIDE 25

Braiding fluxes Metric and connection

Beauty parlor

Connection P(z, t) = D

0 pj(t)zj =

⇒ p(t) = (p0, . . . , pD−1) 0 = (d + A) p, A = g−1(∂ζg)

  • semi pure gauge

Semi-pure A = g−1(∂g)

  • semi pure gauge

= g−1dg

pure gauge

, d = ∂ + ¯ ∂

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 25 / 34

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SLIDE 26

Braiding fluxes Metric and connection

Factorization

holomorphic × anti-holomorphic

Heuristics (g)jk(ζ, ¯ ζ) = ψ0(ζ)|

anti−holomorphic

¯ zjzk |ψ0(ζ)

holomorphic

, Factorization of metric g(ζ, ¯ ζ, Φ)

  • D×D

= Ψ∗(ζ; Φ)

  • D×(N−1)

G(Φ)

(N−1)×(N−1)

Ψ(ζ; Φ)

(N−1)×D

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 26 / 34

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SLIDE 27

Braiding fluxes Metric and connection

Branch structure Ψ

Fluxons and cuts

The matrix Ψ Ψak(ζ) = ζa

ξN

dz zkψ0(z; ζ), a ∈ 1, . . . , N − 1, k ∈ 0, . . . , D − 1

t t t

ζ1 Σ1 ∞1 ζ2 Σ2 ∞2 ζ3 Σ3 ∞3 ∞0 = ∞3

✲ ✻

x y

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 27 / 34

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SLIDE 28

Braiding fluxes Magic

Outline

1

Motivation Non abelian anyons Aharonov Casher

2

Braiding fluxes Zero modes Adiabatically Moving fluxes Metric and connection Magic

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 28 / 34

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SLIDE 29

Braiding fluxes Magic

The magic when D = N − 1

Conservation laws

0 =

  • d + g−1(∂g)
  • connection
  • p =

⇒ 0 =

  • gd + (∂g)
  • p

=

  • Ψ∗GΨ d + ∂(Ψ∗GΨ)
  • p
  • factorization

= Ψ∗G

  • Ψ d + (∂Ψ)
  • p
  • holomorphy

= Ψ∗G

  • Ψ d +

(dΨ)

holomorphy

  • p =

Ψ∗G

  • D×(N−1)

d(Ψ p) D = N − 1 Ψ∗G = a square matrix Invertible (since g > 0) d(Ψ p) = 0 = ⇒ p(ζ) a function on control space Curvature localized at branch points ζa

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 29 / 34

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SLIDE 30

Braiding fluxes Magic

Monodromy

Branched surface

t t t

ζ1 Σ1 ∞1 ζ2 Σ2 ∞2 ζ3 Σ3 ∞3 ∞0 = ∞3

✲ ✻

x y

d(Ψ p) = 0 Ψ a function on branched control space Monodromy of p induced from Ψ Holonomy is topological

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 30 / 34

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SLIDE 31

Braiding fluxes Magic

Monodromy

Ψaj(ζ) = ζa

ζN dξ ξj N b=1 (ξ − ζb)−Φb

  • branched

What happens to Ψb as fluxon a goes around it:

a a b b na

The monodromy matrix, non-abelian M(νa, νb) = 1 − νa + νaνb νa(1 − νb) 1 − νa νa

  • ,

det M = νaνb Eigenvalues(M) = {1, νaνb}

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 31 / 34

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SLIDE 32

Summary

Summary

Pauli Anyons Point-like fluxes are non-abelian anyons When Φ = N − 1 braiding of fluxes is topological Outlook

Spin connection Conic Anyons (Kenneth)

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 32 / 34

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SLIDE 33

Appendix Further Reading

Further Reading I

  • J. Preskill, Lecture Notes
  • O. Kenneth and J. Avron, ArXiv & Ann. Phys. 2014.
  • Y. Aharonov and R. Casher, Phys. Rev A, 1979

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 33 / 34

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SLIDE 34

Appendix Further Reading

Integrals: dz ∧ d¯ z

D = N − 1: p a function on branched control space. p has the monodromy of Ψ Stokes Cuts and more

Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 34 / 34