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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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