Platonic QHE Chern numbers J. Avron Department of Physics, - - PowerPoint PPT Presentation

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Platonic QHE Chern numbers J. Avron Department of Physics, - - PowerPoint PPT Presentation

Dec 06, 2022 210 likes •546 views

Platonic QHE Chern numbers J. Avron Department of Physics, Technion ESI, 2014 Hofstadter butterfly: Chern numbers phase diagram Avron (Technion) Platonic QHE ESI 2014 1 / 32 Outline Prelims 1 Physics Math Platonic QHE 2 Virtual work

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SLIDE 1

Platonic QHE

Chern numbers

  • J. Avron

Department of Physics, Technion

ESI, 2014

Hofstadter butterfly: Chern numbers phase diagram

Avron (Technion) Platonic QHE ESI 2014 1 / 32

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SLIDE 2

Outline

1

Prelims Physics Math

2

Platonic QHE

3

Virtual work & Hall conductance

4

Chern=Kubo

Avron (Technion) Platonic QHE ESI 2014 2 / 32

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SLIDE 3

Prelims Physics

Outline

1

Prelims Physics Math

2

Platonic QHE

3

Virtual work & Hall conductance

4

Chern=Kubo

Avron (Technion) Platonic QHE ESI 2014 3 / 32

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SLIDE 4

Prelims Physics

Aharonov-Bohm flux tubes

Quantum flux

  • A · dx =
  • φ

winds origin

  • therwise

Φ0 = 2π

  • e
  • fundamental

= 2π Flux through a micro-organism φ 100µ Amoeba

Avron (Technion) Platonic QHE ESI 2014 4 / 32

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SLIDE 5

Prelims Physics

AB Periodicity

Flux tube modifies boundary condition: |ψ+ = eiφ |ψ− Aθ = φδ(x ∈ cut) b.c. 2π periodic in φ: |ψ+ = eiφ |ψ− AB periodicity H(φ + 2π) = UH(φ)U∗

Avron (Technion) Platonic QHE ESI 2014 5 / 32

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SLIDE 6

Prelims Math

Outline

1

Prelims Physics Math

2

Platonic QHE

3

Virtual work & Hall conductance

4

Chern=Kubo

Avron (Technion) Platonic QHE ESI 2014 6 / 32

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SLIDE 7

Prelims Math

Projections: P

Orthogonal projections P2 = P

projection

, P = P∗

  • rthogonal

P =

d

  • 1
  • ψj

ψj

  • ,

ψi|ψj = δij P⊥ = ✶ − P

  • complementary

PP⊥ = P⊥P = 0

Avron (Technion) Platonic QHE ESI 2014 7 / 32

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SLIDE 8

Prelims Math

Family of projections

Paradigm

Berry: Spin in magnetic field B P(B) = ✶ + H(ˆ B) 2 , ˆ B = B |B| H(B) = B · σ = 1 2

  • B3

B1 − iB2 B1 + iB2 −B3

  • P(B) sick at B = 0 ⇔ H(0) degenerates

B

Avron (Technion) Platonic QHE ESI 2014 8 / 32

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SLIDE 9

Prelims Math

Family of projections

Parameter=control space

φ ∈ parameter space=control space P(φ) : (parameter space) → smooth projections Parameter space Hilbert space φ

moving frame

P(φ) P⊥(φ)

Avron (Technion) Platonic QHE ESI 2014 9 / 32

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SLIDE 10

Prelims Math

dP

Motion of projections

P2 = P

matrices

P dP + dP P = dP Corrolary dP P = (✶ − P)dP = P⊥dP P dP P = P P⊥

=0

dP = 0 Kato P dP P = 0

Avron (Technion) Platonic QHE ESI 2014 10 / 32

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SLIDE 11

Prelims Math

Kato evolution

Unitary evolution in evolving subspaces

Kato’s unitary evolution P = U P0 U∗

  • Notion of parallel transport

, U = U(φ), P = P(φ) Who generates U? i dU = A

  • generator

U φ φ′

Avron (Technion) Platonic QHE ESI 2014 11 / 32

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SLIDE 12

Prelims Math

Kato’s evolution

Commutator equation

Generator satisfies commutator equation dP = i[A, P] Proof: P0 = U∗ P U = ⇒ 0 = (dU∗) P U + U∗ dP U + U∗ P dU 0 = U(dU∗)

−(dU)U∗

P + dP + P (dU)U∗

−iA

A: Not unique! Ambiguity: commutant (P)

Avron (Technion) Platonic QHE ESI 2014 12 / 32

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SLIDE 13

Prelims Math

Kato’s evolution

Generator

Commutator equation for A: dP = i[A, P] A Generator A = i(dU)U∗

  • Definition

= −i[dP, P]

  • Generator

Verify: i[A, P] =

  • [dP, P], P
  • = (dP)P − 2 P(dP)P
  • =0

+P dP = dP

Avron (Technion) Platonic QHE ESI 2014 13 / 32

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SLIDE 14

Prelims Math

Parallel transport

Connection

Parallel transport: No motion in P |ψ = P |ψ

  • vector∈P

, 0 = P d |ψ

  • no−motion

d |ψ = d

  • P |ψ
  • = (dP) |ψ + Pd |ψ

=0

= (dP) P |ψ = [dP, P]

iA

|ψ parameter space Covariant derivative: D =

  • d − iA
  • ,

D |ψ = 0 ⇔ Pd |ψ = 0

Avron (Technion) Platonic QHE ESI 2014 14 / 32

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SLIDE 15

Prelims Math

Parallel transport

Berry’s phase

1-D projection: P = |ψ ψ| Parallel transport: 0 = P d |ψ = |ψ ψ|dψ Parallel transport = ⇒ No local Berry’s phase 0 = ψ|dψ − 1

2d (ψ|ψ) =1

= ψ|dψ − dψ|ψ 2 = i Im ψ|dψ

  • Berry′s phase

|ψ0 |ψ1 D |ψ = 0 |ψ0 eiβ |ψ0

Avron (Technion) Platonic QHE ESI 2014 15 / 32

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SLIDE 16

Prelims Math

Curvature

Failure of parallel transport

Parallel transport is path dependent: |ψ0 |ψ1 = |ψ1

Curvature

Curvature=Failure of parallel transport Ωjk = i [Dj, Dk]

  • definition

=

  • ∂jAk − ∂kAj
  • − i[Aj, Ak]
  • Non−abelian magnetic fields

Avron (Technion) Platonic QHE ESI 2014 16 / 32

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SLIDE 17

Prelims Math

Curvature for projections

P(dP)(dP)P

Curvature=iP(dp)(dP)P Ωjk = i [Dj, Dk]

  • definition

= i [∂jP, ∂kP] Proof:

  • Dj, Dk
  • P = [Pdj, Pdk]P

= P(∂jP)(∂kP) − P(∂kP)(∂jP) = P[∂jP, ∂kP]

Avron (Technion) Platonic QHE ESI 2014 17 / 32

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SLIDE 18

Prelims Math

Curvature

1-D projection

1-D: P = |ψ ψ| ΩjkP = i [∂jP, ∂kP]P =

Berry′s curvature

  • i
  • ∂jψ|∂kψ − ∂kψ|∂jψ
  • P

Example: Spin 1/2 H = Φ · σ, P = ✶ + ˆ H 2 ΩjkP = εjkℓ ΦℓdΦjdΦk 4|Φ|3

  • 1/2 spherical angle

P

Avron (Technion) Platonic QHE ESI 2014 18 / 32

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SLIDE 19

Prelims Math

Gauss Bonnet

Geometry meets topology

Gauss-Bonnet: Gaussian curvature & genus 1 2π

  • Ω
  • Curvature

dS = 2(1 − genus)

Avron (Technion) Platonic QHE ESI 2014 19 / 32

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SLIDE 20

Prelims Math

Chern numbers

Proof for torus (TKNN)

P(φ1, φ2) periodic |ψ(φ1, φ2) periodic up to phase: |ψ(0, 0) = e−iα |ψ(2π, 0) = e−iγ |ψ(0, 2π) = e−iβ |ψ(2π, 2π) Angle counted mod 2π eiα |ψ |ψ eiβ |ψ eiγ |ψ φ1 φ2 (α − 0)mod 2π + (β − α)mod 2π + (γ − β)mod 2π + (0 − γ)mod 2π = 0 Chern numbers i

  • T

dψ|dψ = i

  • ∂T

ψ|dψ ∈ 2πZ

Avron (Technion) Platonic QHE ESI 2014 20 / 32

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SLIDE 21

Prelims Math

Chern numbers

Projections

Chern numbers Chern(P, M) = i 2π

  • M

Tr P[∂jP, ∂kP] dΦjdΦk ∈ Z, M: 2-D compact manifold (no bdry= ∂M = 0) Chern(P, M) invariant under smooth deformations of P P singular at eigenvalue crossing–dim P jumps

Avron (Technion) Platonic QHE ESI 2014 21 / 32

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SLIDE 22

Prelims Math

Chern numbers

Facts

Chern(0, M) = Chern(✶, M) = 0 Chern(P1 ⊕ P2, M) = Chern(P1, M) + Chern(P2, M)

Avron (Technion) Platonic QHE ESI 2014 22 / 32

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SLIDE 23

Prelims Math

Chern numbers

From sphere to ball

H = B · σ, B = (Bx, By, Bz)

  • 3−D space

Linear map of parameter space: B = g Φ, Φ = (Φx, Φy, Φz) det g = 0 Chern H(Φ) =

3

  • j,k=1

gjkΦkσj, P(Φ) = ✶ + ˆ H 2 Chern(P) = sgn det g Φ B

Avron (Technion) Platonic QHE ESI 2014 23 / 32

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SLIDE 24

Prelims Math

Chern numbers

What is counted?

Contracting into the solid torus + − + − Simon Chern(P, T) =

  • sgn det g(Φd)
  • degeneracies

Avron (Technion) Platonic QHE ESI 2014 24 / 32

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SLIDE 25

Prelims Math

QHE

Driving and response

φ2 Hall current loop emf loop φ1 Hall bar Platonic Driving: emf = ˙ φ1 Response: Hall current = ∂H

∂φ2

H(φ1, φ2): Periodic, nondegenerate, hermitian matrix

Avron (Technion) Platonic QHE ESI 2014 25 / 32

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SLIDE 26

Prelims Math

Variations on a theme

Bloch momenta & controls

φ2 φ1

Periodic (k1, k2) conserved Bloch momenta Brillouin Zone ∞ noninteracting (gapped) fermions Multiply connected (φ1, φ2) controls Fluxes Aharnonov-Bohm period Interacting (finite)

Avron (Technion) Platonic QHE ESI 2014 26 / 32

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SLIDE 27

Prelims Math

Example: 3 × 3 matrix function

Hofstadter Butterfly with flux 1/3

H(φ) = eiφ1 T

  • translation

+eiφ2 S

  • shift

+h.c. T =   1 1 1  

  • lattice translation

, S =   1 ω ¯ ω  

  • S=FTF ∗

ω = e2πi/3 1/3

Hofstadter Model B = 1/3 Avron (Technion) Platonic QHE ESI 2014 27 / 32

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SLIDE 28

Virtual work & Hall conductance

Virtual work

Q-observable

H : (parameter space φ) → Hamiltonian Virtual work δH = dH(φ) dφ

  • bservable

δφ Loop current: Virtual work of Aharonov-Bohm flux I = dH dφ φ

Avron (Technion) Platonic QHE ESI 2014 28 / 32

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SLIDE 29

Virtual work & Hall conductance

Charge transport

Time dependent Feynman-Hellman

Virtual work=Rate of Berry’s phase ψ| ∂φH |ψ

  • Virtual work

= ∂t

  • iψ|∂φψ
  • Berry′s phase

Schrödinger i∂t |ψ = H(φ) |ψ Pf: ψ| ∂φH |ψ = ∂φ ψ| H |ψ

time−independent=0

  • − ∂φψ| H |ψ

i|∂tψ

− ψ| H

−i∂tψ|

|∂φψ = ∂t

  • iψ|∂φψ
  • Avron (Technion)

Platonic QHE ESI 2014 29 / 32

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SLIDE 30

Virtual work & Hall conductance

Hall conductance

Definition

Def: Flux averaged Hall conductance: 2πσ = 1 2π π

π

dφ2 ∞

∞

dt ψt| ∂2H |ψt dt

  • flux average charge transport

time φ1 = π φ1 = −π Flux φ1 Control averaging⇐ ⇒ Filled band. Averaging: Gets rid of persistent currents

Avron (Technion) Platonic QHE ESI 2014 30 / 32

φ2

Current loop emf

φ1

Hall bar

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SLIDE 31

Chern=Kubo

Chern=Kubo

Geometry of transport

In adiabatic limit: flux average transport=Chern 2πσ − →

adiabatic

i 2π π

π

dφ1dφ2 Tr P[∂1P, ∂2P]

  • Chern

Loop currents: ψ| ∂φH |ψ = ∂t

  • iψ|∂φψ
  • Adiabatic limit: |ψ → |ψA ,

H → A Adiabatic loop currents: ∂t

  • iψA|∂φψA
  • = ψA| ∂φA |ψA

i ψA| ∂φ[ ˙ P, P] |ψA = iTr P∂φ[∂tP, P] = iTr P[∂φP, ∂tP]

Avron (Technion) Platonic QHE ESI 2014 31 / 32

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SLIDE 32

Chern=Kubo

Critique

Why Platonic?

φ = control :Too general φ = Bloch momenta: Too special Gap condition: Too strong–localization Where is 2-D? Where is thermodynamic limit Why average? Sample appear to be less important than connecting circuit What about fractions?

Avron (Technion) Platonic QHE ESI 2014 32 / 32