Vignettes

J. Avron

Dept. of Physics, TechnionIsrael

August 28, 2016

Happy 70-th birthday, Barry

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1 Comparison of projections (ASS-1994)

• P , Q orthogonal projections.

• Tr (P −Q) = Tr (P −Q)3

Theorem 1. Suppose (P −Q)2n+1 trace class, then

Tr(P −Q)2n+1 = Tr(P −Q)2n+3 = . . .

= dim ker(P −Q− 1)− dim ker(P −Q+ 1)

∈ Z

-1 10−λ λ

Spect (P −Q)

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1.1 Anti-commutative Pythagoras

• P + P⊥ = 1, Q+Q⊥ = 1

• (P − P⊥︸ ︷︷ ︸C−S

)2 = 1, (Q⊥ −Q︸ ︷︷ ︸C+S

)2 = 1

• C = P −Q, S = P⊥ −Q

• C2 + S2 = 1︸ ︷︷ ︸Pythagoras

, CS + SC = 0︸ ︷︷ ︸anti−commutative

-1 10−λ λ

Spect C

Proof.

C |ψ〉 = λ |ψ〉 =⇒ SC |ψ〉 = λ(S |ψ〉) = −C(S |ψ〉)︸ ︷︷ ︸anti−commutative

Fail ifS |ψ〉 = 0 =⇒ C |ψ〉 = ± |ψ〉︸ ︷︷ ︸

Pythagoras

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1.2 Index of quasi-diagonal unitaries (Kitaev)

• Quasi-diagonal:

U =

∗ ∗ ∗ 0 0 00 ∗ ∗ ∗ 0 00 0 ∗ ∗ ∗ 00 0 0 ∗ ∗ ∗

=

(U++ U+−U−+ U−−

)Theorem 2. Suppose U quasi-diagonal, then

Tr(|U+−|2)− Tr(|U−+|2) ∈ Z

Example 1.1. Index (Right shift) = Index

0 1 0 00 0 1 0

0 0 0 10 0 0 0

= 1

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Proof.Tr(|U+−|2)− Tr(|U−+|2) = Tr(P − UPU ∗)3

• C2 = P − PQ−QP +Q = PQ⊥ +QP⊥

• C3 = C2P − C2Q = PQ⊥P −QP⊥Q

• P =

(1 00 0

)PUP⊥ =

(0 U+−0 0

)• Tr(|U+−|2) = Tr(P UP⊥U

∗︸ ︷︷ ︸Q⊥

P )

• Tr(UPU ∗︸ ︷︷ ︸Q

P⊥ UPU∗︸ ︷︷ ︸

Q

) = Tr(PU ∗P⊥UP )

= Tr(P⊥UPU∗P⊥) = Tr(|U−+|2)

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1.3 Application: Semi-Thouless pump

• Semi-infinite, period 3 chain

•J12J23 J31

• The Hamiltonian H(J) =

0 J12 0 0 . . .J12 0 J23 0 . . .0 J23 0 J31 . . .. . . . . . . . . . . . . . .

• Essential spectrum=3 Band

P

• Since H is gapped: A full band is a nominal insulator

• Number of electrons in full band: dimP =∞

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1.4 Pumping

• J controls

• U pump cycle

• P − UPU ∗: charge transported to infinity

• Pumping cycle for a disconnected chain

J12

J31

J23

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1.5 Disconnected chain

• Matrix :

0 1 01 0 00 0 0

Eigensystem:

• Matrix :

0 0 00 0 10 1 0

Eigensystem

• Adiabatic+ superposition=Quantum

Eigenvalue

1 2 3

+

−

+

−1 2 3J12

J31

J23

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1.6 Spectral flow

• Charge transport

• Spectral flow: Monitor the gap

• Bulk-Edge duality

P⊥

P

pumping cycle

energy

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2 Fredholm index and zero modes

• F an m× n matrix:

IndexF = dim kerF ∗F − dim kerFF ∗ ∈ Z

•

IndexF = dim kerF ∗F︸︷︷︸n×n

− dim kerFF ∗︸︷︷︸m×m

= Tr(1n×n − F ∗F )− Tr(1m×m − FF ∗)= n−m

0

Spec F ∗F

0

Spec FF ∗

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2.1 Stable Zero modes

• Bi-partite graph: |A|, |B|

• H =

(0 FF ∗ 0

), A ↔︸︷︷︸

hop

B

• H

(1|A|×|A| 0

0 −1|B|×|B|

)= −

(1|A|×|A| 0

0 −1|B|×|B|

)H

•

#zero modes = dim kerF ∗F + dim kerFF ∗

≥∣∣ dim kerF ∗F − dim kerFF ∗

∣∣=∣∣|A| − |B|∣∣

0

Spec H

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3 Broken translation invariance in homogeneous fields

3.1 Magnetic translations

• AHS 78

• H(vj), vj = −i∂j − bjkxk︸ ︷︷ ︸gauge invariant

, B = b12 − b21

• Conserved generator of translations: tj = −i∂j − bkjxk

• [tj, vk] = 0, [v1, v2] = −iB, [t1, t2] = iB

• Ta = e−it·a, (Taψ)(x) = ei(xjbjka

k)︸ ︷︷ ︸re-Gauge

ψ(x− a)

• Weyl algebra: TaTa′ = eiΦ︸︷︷︸magnetic flux

Ta′Ta

a

a′

Φ

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3.2 Charge particle on a torus

• Torus: R2/Z2

• Standard Periodic boundary conditions

ψ(x− a) 6= eikaψ(x)︸ ︷︷ ︸[pj ,vk]6=0

inconsistent with H(v)

• “Gauge periodic boundary” conditions,

Ta |ψ〉 = eika |ψ〉 ,

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3.3 Dirac quantization

• Periodic Boundary conditions:TaTa′ |ψ〉 = ei(ka+ka′)︸ ︷︷ ︸

global phase

|ψ〉 = Ta′Ta |ψ〉

• Weyl: TaTa′ = eiΦ Ta′Ta

• Φ︸︷︷︸flux

∈ 2πZ

ψ(0, 0)

eiΦ/2ψ(0, 0)

e−iΦ/2ψ(0, 0)

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3.4 Eigenstates are not translation invariant

• Phase of 〈x|ψ〉 winds B times around the torus.

• The density |ψ(x)|2 has n zeros (can’t be uniform!)

• Who broke translation symmetry?

ψ(0, 0)

eiΦ/2ψ(0, 0)

eiΦψ(0, 0)

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3.5 Aharonov-Bohm fluxes

• A determines B and also

• Aharonov-Bohm fluxes: φj =∫γjA

• φj are gauge invariant.

• φj are are not translation invariant.

φ1

γ1 φ2

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4 Generic crossings and Chern numbers

Theorem 3 (Wigner von Neuman). In the linear space ofHermitian matrices, 2-fold degeneracies have co-dimension 3.

• H(x) = H(0) +∑

j (∂jH) (0)xj +O(x2)

• Generic 2-level crossing:

H(x) = g0(x)1 +∑j

gjk xjσk︸ ︷︷ ︸2gjk=Tr(∂jH)σk

+O(x2)

• Pauli: σ1 =

(0 11 0

), σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)• Conic: det g 6= 0.

• Stable

x2

E

x1

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4.1 Barry’s formula for Chern numbers of generic crossings

• Berry-Barry 1983

• H(x) =∑3

j,k=1 xjgjkσ

k

• H(x) = H(x)‖H(x)‖

• Projection on ground state:

P (x) = 1−H(x)2

• Chern(P |S2) =

{sgn det g S2 encloses origin

0 otherwise

x2

E

x1

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4.2 Wigner von Neuman for closed 3-manifolds

Theorem 4. Generic crossing in closed 3-D manifolds come inpairs

k

E

k

E

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4.3 Stability of pairs

• Projection on lowest band P (k)

• kj the points of conic singularities

• gj the matrix associated with conic

Theorem 5. For generic crossings of the lowest band∑j

sgn det gj = Chern(P |S2) = 0 = 0

• Nielsen-Nynomia

kx

kz

ky

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4.4 Insulators, metals and Weyl semi-metal

Efk

Insulator

Efk

Metal

Efk

Weyl semi-metal

Empty Ball Fermi arc

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4.5 Fermi arcs

• Chern(P |Cylinder) = ±1

• Zero modes on surface=Fermi arc

kx

kz

ky

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5 Acknowledgment

M. Aizenman, J. Bellissard, M. Berry, A. Elgart, M. Fraas,J. Frohlich, G.M. Graf, A. Grossman, I. Herbst, O. Kenneth,

Y. Last, L. Sadun, L. Schulman, R. Seiler, B. Simon, E. Wigner

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