Bulk-edge duality for topological insulators
Gian Michele Graf
ETH Zurich
Symposium on Statistical Mechanics:
Many-Body Quantum Systems
University of Warwick
17-21 March 2014
Bulk-edge duality for topological insulators
Gian Michele Graf
ETH Zurich
Symposium on Statistical Mechanics:
Many-Body Quantum Systems
University of Warwick
17-21 March 2014
joint work with Marcello Porta
thanks to Yosi Avron
Introduction
Rueda de casino
Hamiltonians
Indices
Further results
Topological insulators: first impressions
I
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
Topological insulators: first impressions
I
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
I
Time-reversal invariant fermionic system
Topological insulators: first impressions
I
Time-reversal invariant fermionic system
void
spin up
spin down
Bulk
Edge
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
Edge
I
void
Topological insulators: first impressions
I
Time-reversal invariant fermionic system
void
Bulk
Edge
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
Edge
I
void
spin up
spin down
I
Topology: In the space of Hamiltonians, a topological
insulator can not be deformed in an ordinary one, while
keeping the gap open and time-reversal invariance.
Topological insulators: first impressions
I
Time-reversal invariant fermionic system
void
Bulk
Edge
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
Edge
I
void
spin up
spin down
I
Topology: In the space of Hamiltonians, a topological
insulator can not be deformed in an ordinary one, while
keeping the gap open and time-reversal invariance.
Analogy: torus 6= sphere (differ by genus).
Topological insulators: first impressions
I
Time-reversal invariant fermionic system
void
Bulk
Edge
Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
Edge
I
void
spin up
spin down
I
Topology: In the space of Hamiltonians, a topological
insulator can not be deformed in an ordinary one, while
keeping the gap open and time-reversal invariance.
Analogy: torus 6= sphere (differ by genus).
Contributors to the field: Kane, Mele, Zhang, Moore; Fröhlich;
Hasan
Pictures
Material: InAs/GaSb (quantum well); AlSb (barrier)
Courtesy: S. Müller, K. Ensslin
Pictures
Courtesy: S. Müller, K. Ensslin
Pictures
Courtesy: S. Müller, K. Ensslin
Pictures
Courtesy: S. Müller, K. Ensslin
A technological application
void
Edge
µ−
|ψi
µ+ > µ−
Bulk
Θ|ψi
µ+
Θ time-reversal
A technological application
void
Edge
µ−
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
Indeed:
hΘψ, V ψi =
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
Indeed: by Θ anti-unitary,
hΘψ, V ψi = hΘV ψ, Θ2 ψi =
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
Indeed: by Θ anti-unitary, Θ2 = −1, [Θ, V ] = 0,
hΘψ, V ψi = hΘV ψ, Θ2 ψi = − hV Θψ, ψi =
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
Indeed: by Θ anti-unitary, Θ2 = −1, [Θ, V ] = 0, V = V ∗
hΘψ, V ψi = hΘV ψ, Θ2 ψi = − hV Θψ, ψi = − hΘψ, V ψi
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
• Backscattering to other edge suppressed by separation
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
• Backscattering to other edge suppressed by separation
• Autobahn principle (Zhang);
A technological application
µ−
void
Edge
disorder V
|ψi
µ+ > µ−
Bulk
Θ|ψi
Θ time-reversal
µ+
• On the two edges (net): parallel charge currents, anti-parallel
spin currents
• Stable against backscattering, |ψi → Θ|ψi, induced by
disorder
hΘψ|V |ψi = 0
if V is time-reversal invariant.
• Backscattering to other edge suppressed by separation
• Autobahn principle (Zhang); U.S. patent 20120138887
Bulk-edge correspondence
Deformation as interpolation in physical space:
topological insulator
I
000
111
000
111
11
00
00
11
00
11
00
11
000
111
000
111
000
111
000
111
00
11
0011
11
00
00
11
000
111
000
111
000
111
000
111
00 11
11
0011
00 11
00
interpolating material
ordinary insulator
Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
Bulk-edge correspondence
Deformation as interpolation in physical space:
topological insulator
000
111
000
111
11
00
00
11
00
11
00
11
000
111
000
111
000
111
000
111
00
11
0011
11
00
00
11
000
111
000
111
000
111
000
111
00 11
11
0011
00 11
00
interpolating material
void
I
Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
I
Ordinary insulator
void: Edge states
Bulk-edge correspondence
Deformation as interpolation in physical space:
topological insulator
000
111
000
111
11
00
00
11
00
11
00
11
000
111
000
111
000
111
000
111
00
11
0011
11
00
00
11
000
111
000
111
000
111
000
111
00 11
11
0011
00 11
00
interpolating material
void
I
Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
I
Ordinary insulator
I
Bulk-edge correspondence: Termination of bulk of a
topological insulator implies edge states.
void: Edge states
Bulk-edge correspondence
Deformation as interpolation in physical space:
topological insulator
000
111
000
111
11
00
00
11
00
11
00
11
000
111
000
111
000
111
000
111
00
11
0011
11
00
00
11
000
111
000
111
000
111
000
111
00 11
11
0011
00 11
00
interpolating material
void
I
Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
I
Ordinary insulator
I
Bulk-edge correspondence: Termination of bulk of a
topological insulator implies edge states. (But not
conversely!)
void: Edge states
void
Edge
Bulk-edge correspondence
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
From a talk by Z. Hasan
Experimental Challenge:
How to experimentally “measure” topological invariants
that do not give rise to quantization of charge or spin transport
cannot be done via transport in Z2 topological insulators
(transport is still interesting and becoming possible)
Experimentally IMAGE (see) boundary/edge/surface states
Experimentally Probe BULK--BOUNDARY CORRESPONDENCE
Experimental prove “topological order”
Spectroscopy is capable of probing
BULK--BOUNDARY correspondence,
Determine the topological nature of boundary/surface states &
experimentally prove “topological order”
From a paper by Z. Hasan
void
Edge
Bulk-edge correspondence
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
void
Edge
Bulk-edge correspondence
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
I
Goal: State the (intrinsic) topological property
distinguishing different classes of insulators.
More precisely:
void
Edge
Bulk-edge correspondence
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
I
Goal: State the (intrinsic) topological property
distinguishing different classes of insulators.
More precisely:
I
Express that property as an Index relating to the Bulk,
resp. to the Edge.
void
Edge
Bulk-edge correspondence
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
I
Goal: State the (intrinsic) topological property
distinguishing different classes of insulators.
More precisely:
I
Express that property as an Index relating to the Bulk,
resp. to the Edge.
I
Bulk-edge duality: Can it be shown that the two indices
agree?
void
Edge
Bulk-edge correspondence. Done?
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
I
Goal: State the (intrinsic) topological property
distinguishing different classes of insulators.
More precisely:
I
Express that property as an Index relating to the Bulk,
resp. to the Edge. Yes, e.g. Kane and Mele.
I
Bulk-edge duality: Can it be shown that the two indices
agree? Schulz-Baldes et al.; Essin & Gurarie
void
Edge
Bulk-edge correspondence. Today
Bulk
In a nutshell: Termination of bulk of a topological insulator
implies edge states
I
Goal: State the (intrinsic) topological property
distinguishing different classes of insulators.
More precisely:
I
Express that property as an Index relating to the Bulk,
resp. to the Edge. Done differently.
I
Bulk-edge duality: Can it be shown that the two indices
agree? Done differently.
Introduction
Rueda de casino
Hamiltonians
Indices
Further results
Rueda de casino. Time 00 1500
Rueda de casino. Time 00 2500
Rueda de casino. Time 00 3500
Rueda de casino. Time 00 4400
Rueda de casino. Time 00 44.2500
Rueda de casino. Time 00 44.5000
Rueda de casino. Time 00 44.7500
Rueda de casino. Time 00 4500
Rueda de casino. Time 00 45.2500
Rueda de casino. Time 00 45.5000
Rueda de casino. Time 00 4600
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Rueda de casino. Time 00 5500
Rueda de casino. Time 10 1600
Rueda de casino. Time 30 2300
Rules of the dance
Dancers
I
start in pairs, anywhere
I
end in pairs, anywhere (possibly elseways & elsewhere)
I
are free in between
I
must never step on center of the floor
Rules of the dance
Dancers
I
start in pairs, anywhere
I
end in pairs, anywhere (possibly elseways & elsewhere)
I
are free in between
I
must never step on center of the floor
I
are unlabeled points
Rules of the dance
Dancers
I
start in pairs, anywhere
I
end in pairs, anywhere (possibly elseways & elsewhere)
I
are free in between
I
must never step on center of the floor
I
are unlabeled points
There are dances which can not be deformed into one another.
What is the index that makes the difference?
The index of a Rueda
A snapshot of the dance
The index of a Rueda
A snapshot of the dance
Dance D as a whole
D
The index of a Rueda
A snapshot of the dance
Dance D as a whole
D
The index of a Rueda
A snapshot of the dance
Dance D as a whole
D
The index of a Rueda
A snapshot of the dance
Dance D as a whole
D
I(D) = parity of number of crossings of fiducial line
Introduction
Rueda de casino
Hamiltonians
Indices
Further results
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
Z
Z
n = −3 −2 −1
0
1
2
3
4
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
Z
Z
n = −3 −2 −1
0
1
2
3
4
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1:
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f.
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . )
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ),
totalling N
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ),
totalling N
I
Bloch reduction by quasi-momentum k ∈ S 1 := R/2πZ
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ),
totalling N
I
Bloch reduction by quasi-momentum k ∈ S 1 := R/2πZ
End up with wave-functions ψ = (ψn )n∈Z ∈ `2 (Z; CN ) and Bulk
Hamiltonian
H(k)ψ n = A(k )ψn−1 + A(k)∗ ψn+1 + Vn (k)ψn
with
Vn (k) = Vn (k)∗ ∈ MN (C) (potential)
A(k) ∈ GL(N) (hopping)
Bulk Hamiltonian
Hamiltonian on the lattice Z × Z (plane)
I
translation invariant in the vertical direction
I
period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ),
totalling N
I
Bloch reduction by quasi-momentum k ∈ S 1 := R/2πZ
End up with wave-functions ψ = (ψn )n∈Z ∈ `2 (Z; CN ) and Bulk
Hamiltonian
H(k)ψ n = A(k )ψn−1 + A(k)∗ ψn+1 + Vn (k)ψn
with
Vn (k) = Vn (k)∗ ∈ MN (C) (potential)
A(k) ∈ GL(N) (hopping): Schrödinger eq. is the 2nd order
difference equation
Edge Hamiltonian
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
Z
N
n=
0
1
2
3
4
Edge Hamiltonian
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
I
translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ `2 (N; CN ) and Edge Hamiltonian
H ] (k)ψ
n
= A(k )ψn−1 + A(k)∗ ψn+1 + Vn] (k)ψn
Edge Hamiltonian
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
I
translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ `2 (N; CN ) and Edge Hamiltonian
H ] (k)ψ
n
= A(k )ψn−1 + A(k)∗ ψn+1 + Vn] (k)ψn
which
I
agrees with Bulk Hamiltonian outside of collar near edge
(width n0 )
Vn] (k) = Vn (k) ,
(n > n0 )
Edge Hamiltonian
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
I
translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ `2 (N; CN ) and Edge Hamiltonian
H ] (k)ψ
n
= A(k )ψn−1 + A(k)∗ ψn+1 + Vn] (k)ψn
which
I
agrees with Bulk Hamiltonian outside of collar near edge
(width n0 )
Vn] (k) = Vn (k) ,
(n > n0 )
I
has Dirichlet boundary conditions: for n = 1 set ψ0 = 0
Edge Hamiltonian
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
I
translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ `2 (N; CN ) and Edge Hamiltonian
H ] (k)ψ
n
= A(k )ψn−1 + A(k)∗ ψn+1 + Vn] (k)ψn
which
I
agrees with Bulk Hamiltonian outside of collar near edge
(width n0 )
Vn] (k) = Vn (k) ,
(n > n0 )
I
has Dirichlet boundary conditions: for n = 1 set ψ0 = 0
Note: σess (H ] (k)) ⊂ σess (H(k)), but typically
σdisc (H ] (k )) 6⊂ σdisc (H(k))
Graphene as an example
Hamiltonian is nearest neighbor hopping on honeycomb lattice
a)
n2
BA
n1
b)
n2
n1
~a1 + ~a2
~a2
~a1
~a2 ~a1
(a) zigzag, resp. (b) armchair boundaries
Dimers (N = 2).
Graphene as an example
Hamiltonian is nearest neighbor hopping on honeycomb lattice
a)
n2
BA
n1
b)
n2
n1
~a1 + ~a2
~a2
~a1
~a2 ~a1
(a) zigzag, resp. (b) armchair boundaries
Dimers (N = 2). For (b):
A
0
ψn
N=2
ψn =
∈C
, A(k ) = −t ik
B
e
ψn
1
,
0
0 1
Vn (k ) = −t
1 0
For (a): too, but A(k ) ∈
/ GL(N)
Also: Extensions with spin, spin orbit coupling leading to
topological insulators (Kane & Mele)
General assumptions
I
Gap assumption: Fermi energy µ lies in a gap for all
k ∈ S1:
µ∈
/ σ(H(k))
General assumptions
I
Gap assumption: Fermi energy µ lies in a gap for all
k ∈ S1:
µ∈
/ σ(H(k))
I
Fermionic time-reversal symmetry: Θ : CN → CN
I
I
I
Θ is anti-unitary and Θ2 = −1;
Θ induces map on `2 (Z; CN ), pointwise in n ∈ Z;
For all k ∈ S 1 ,
H(−k ) = ΘH(k )Θ−1
Likewise for H ] (k)
Elementary consequences of H(−k ) = ΘH(k)Θ−1
I
σ(H(k)) = σ(H(−k )). Same for H ] (k).
Elementary consequences of H(−k ) = ΘH(k)Θ−1
I
I
σ(H(k)) = σ(H(−k )). Same for H ] (k).
Time-reversal invariant points, k = −k , at k = 0, π.
Elementary consequences of H(−k ) = ΘH(k)Θ−1
I
I
σ(H(k)) = σ(H(−k )). Same for H ] (k).
Time-reversal invariant points, k = −k , at k = 0, π. There
H = ΘHΘ−1
(H = H(k) or H ] (k))
Hence any eigenvalue is even degenerate (Kramers).
Elementary consequences of H(−k ) = ΘH(k)Θ−1
I
I
σ(H(k)) = σ(H(−k )). Same for H ] (k).
Time-reversal invariant points, k = −k , at k = 0, π. There
H = ΘHΘ−1
(H = H(k) or H ] (k))
Hence any eigenvalue is even degenerate (Kramers).
Indeed
Hψ = Eψ =⇒ H(Θψ) = E(Θψ)
and Θψ = λψ, (λ ∈ C) is impossible:
−ψ = Θ2 ψ = λ̄Θψ = λ̄λψ
(⇒⇐)
Elementary consequences of H(−k ) = ΘH(k)Θ−1
I
I
σ(H(k)) = σ(H(−k )). Same for H ] (k).
Time-reversal invariant points, k = −k , at k = 0, π. There
H = ΘHΘ−1
(H = H(k) or H ] (k))
Hence any eigenvalue is even degenerate (Kramers).
k ∈ S1
π
0
µ
E ∈R
−π
Bands, Fermi line (one half fat), edge states
Introduction
Rueda de casino
Hamiltonians
Indices
Further results
The edge index
The spectrum of H ] (k)
symmetric on −π ≤ k ≤ 0
1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
µ
111111111
000000000
000000000
111111111
0
k
Bands, Fermi line, edge states
Definition: Edge Index
I ] = parity of number of eigenvalue crossings
π
The edge index
The spectrum of H ] (k)
symmetric on −π ≤ k ≤ 0
1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
µ
111111111
000000000
000000000
111111111
0
k
π
Bands, Fermi line, edge states
Definition: Edge Index
I ] = parity of number of eigenvalue crossings
Collapse upper/lower band to a line and fold to a cylinder: Get
rueda and its index.
Towards the bulk index
Let z ∈ C. The Schrödinger equation
(H(k ) − z)ψ = 0
(as a 2nd order difference equation) has 2N solutions
ψ = (ψn )n∈Z , ψn ∈ CN .
Towards the bulk index
Let z ∈ C. The Schrödinger equation
(H(k ) − z)ψ = 0
(as a 2nd order difference equation) has 2N solutions
ψ = (ψn )n∈Z , ψn ∈ CN .
Let z ∈
/ σ(H(k)). Then
Ez,k = {ψ | ψ solution, ψn → 0, (n → +∞)}
has
I
dim Ez,k = N.
Towards the bulk index
Let z ∈ C. The Schrödinger equation
(H(k ) − z)ψ = 0
(as a 2nd order difference equation) has 2N solutions
ψ = (ψn )n∈Z , ψn ∈ CN .
Let z ∈
/ σ(H(k)). Then
Ez,k = {ψ | ψ solution, ψn → 0, (n → +∞)}
has
I
dim Ez,k = N.
I
Ez̄,−k = ΘEz,k
The bulk index
k ∈ S1
Im z
π
0
µ
E = Re z
−π
Loop γ and torus T = γ × S 1
Vector bundle E with base T 3 (z, k ), fibers Ez,k , and Θ2 = −1.
The bulk index
k ∈ S1
Im z
π
0
µ
E = Re z
−π
Loop γ and torus T = γ × S 1
Vector bundle E with base T 3 (z, k ), fibers Ez,k , and Θ2 = −1.
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1 (besides of N = dim E)
The bulk index
k ∈ S1
Im z
π
0
µ
E = Re z
−π
Loop γ and torus T = γ × S 1
Vector bundle E with base T 3 (z, k ), fibers Ez,k , and Θ2 = −1.
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1 (besides of N = dim E)
Definition: Bulk Index
I = I(E)
The bulk index
k ∈ S1
Im z
π
0
µ
E = Re z
−π
Loop γ and torus T = γ × S 1
Vector bundle E with base T 3 (z, k ), fibers Ez,k , and Θ2 = −1.
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1 (besides of N = dim E)
Definition: Bulk Index
I = I(E)
What’s behind the theorem? How is I(E) defined?
The bulk index
k ∈ S1
Im z
π
0
µ
E = Re z
−π
Loop γ and torus T = γ × S 1
Vector bundle E with base T 3 (z, k ), fibers Ez,k , and Θ2 = −1.
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1 (besides of N = dim E)
Definition: Bulk Index
I = I(E)
What’s behind the theorem? How is I(E) defined? Aside . . . a rueda . . .
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
I torus ϕ = (ϕ1 , ϕ2 ) ∈ T = (R/2πZ)2
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
I torus ϕ = (ϕ1 , ϕ2 ) ∈ T = (R/2πZ)2 with cut (figure)
ϕ2
ϕ
cut
ϕ1
−ϕ
ϕ2 = π
ϕ2 = 0
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
I torus ϕ = (ϕ1 , ϕ2 ) ∈ T = (R/2πZ)2 with cut (figure)
ϕ2
ϕ
cut
ϕ1
I
ϕ2 = π
−ϕ
a (compatible) section of the frame bundle of E
ϕ2 = 0
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
I torus ϕ = (ϕ1 , ϕ2 ) ∈ T = (R/2πZ)2 with cut (figure)
ϕ2
ϕ2
ϕ2 = π
ϕ2 = 0
cut
ϕ1
−ϕ2
I
I
a (compatible) section of the frame bundle of E
the transition matrices T (ϕ2 ) ∈ GL(N) across the cut
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
Sketch of proof: Consider
I torus ϕ = (ϕ1 , ϕ2 ) ∈ T = (R/2πZ)2 with cut (figure)
ϕ2
ϕ2
ϕ2 = 0
ϕ2 = π
cut
ϕ1
−ϕ2
I
I
a (compatible) section of the frame bundle of E
the transition matrices T (ϕ2 ) ∈ GL(N) across the cut
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0 ,
with Θ0 : CN → CN antilinear, Θ20 = −1
(ϕ2 ∈ S 1 )
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
ϕ2
ϕ2 = π
ϕ2 = 0
−ϕ2
I
I
I
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0
Only half the cut (0 ≤ ϕ2 ≤ π) matters for T (ϕ2 )
At time-reversal invariant points, ϕ2 = 0, π,
Θ0 T = T −1 Θ0
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
ϕ2
ϕ2 = π
ϕ2 = 0
−ϕ2
I
I
I
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0
Only half the cut (0 ≤ ϕ2 ≤ π) matters for T (ϕ2 )
At time-reversal invariant points, ϕ2 = 0, π,
Θ0 T = T −1 Θ0
Eigenvalues of T come in pairs λ, λ̄−1 :
Θ0 (T − λ) = T −1 (1 − λ̄T )Θ0
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
ϕ2
ϕ2 = π
ϕ2 = 0
−ϕ2
I
I
I
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0
Only half the cut (0 ≤ ϕ2 ≤ π) matters for T (ϕ2 )
At time-reversal invariant points, ϕ2 = 0, π,
Θ0 T = T −1 Θ0
Eigenvalues of T come in pairs λ, λ̄−1 :
Θ0 (T − λ) = T −1 (1 − λ̄T )Θ0
Phases λ/|λ| pair up (Kramers degeneracy)
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
ϕ2
ϕ2 = π
ϕ2 = 0
−ϕ2
I
I
I
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0
Only half the cut (0 ≤ ϕ2 ≤ π) matters for T (ϕ2 )
At time-reversal invariant points, ϕ2 = 0, π,
Θ0 T = T −1 Θ0
Eigenvalues of T come in pairs λ, λ̄−1 :
Θ0 (T − λ) = T −1 (1 − λ̄T )Θ0
I
Phases λ/|λ| pair up (Kramers degeneracy)
For 0 ≤ ϕ2 ≤ π, phases λ/|λ| form a rueda, D
Time-reversal invariant bundles on the torus
Theorem In general, vector bundles (E, T, Θ) can be classified
by an index I(E) = ±1
ϕ2
ϕ2 = π
ϕ2 = 0
−ϕ2
I
Θ0 T (ϕ2 ) = T −1 (−ϕ2 )Θ0
I
For 0 ≤ ϕ2 ≤ π, phases λ/|λ| form a rueda, D
Definition (Index): I(E) := I(D)
Remark: I(E) agrees (in value) with the Pfaffian index of Kane and Mele.
. . . aside ends here.
Main result
Theorem Bulk and edge indices agree:
I = I]
Main result
Theorem Bulk and edge indices agree:
I = I]
I = +1: ordinary insulator
I = −1: topological insulator
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1.
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1. Schrödinger (2nd order
difference) equation on the half-line
(H ] − z)ψ ] = 0
(no b.c. at n = 0)
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1. Schrödinger (2nd order
difference) equation on the half-line
(H ] − z)ψ ] = 0
(no b.c. at n = 0)
with solution ψn] ∈ C, (n = 0, 1, 2) decaying at n → ∞
ψ0#
ψ1#
n
−1
0
1
2
3
4
5
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1. Schrödinger (2nd order
difference) equation on the half-line
(H ] − z)ψ ] = 0
(no b.c. at n = 0)
with solution ψn] ∈ C, (n = 0, 1, 2) decaying at n → ∞
ψ0#
ψ1#
n
−1
I
0
1
2
3
Solution is unique up to multiples
4
5
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1. Schrödinger (2nd order
difference) equation on the half-line
(H ] − z)ψ ] = 0
(no b.c. at n = 0)
with solution ψn] ∈ C, (n = 0, 1, 2) decaying at n → ∞
ψ0#
ψ1#
n
−1
0
1
2
3
I
Solution is unique up to multiples
I
ψ0] = 1 picks a unique solution,
4
5
Proof of Theorem (preliminary remark)
I
For this slide only: N = 1. Schrödinger (2nd order
difference) equation on the half-line
(H ] − z)ψ ] = 0
(no b.c. at n = 0)
with solution ψn] ∈ C, (n = 0, 1, 2) decaying at n → ∞
ψ0#
ψ1#
n
−1
0
1
2
3
4
5
I
Solution is unique up to multiples
I
ψ0] = 1 picks a unique solution, except if n = 0 is a node
A winding number
Im z
E = Re z
I
I
I
A winding number
Im z
k
E = Re z
I
I
I
A winding number
k
E = Re z
I
I
I
A winding number
k
1
0
0
1
E(k)
µ
I
I
I
E = Re z
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
A winding number
Im z
k
1
0
0
1
E(k)
µ
I
I
I
E = Re z
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
A winding number
Im z
k
1
0
0
1
E(k)
µ
E = Re z
I
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
I
Choose solution ψ ] (z, k ) such that ψ0] (z, k ) = 1
I
A winding number
Im z
k
1
0
0
1
E(k)
µ
E = Re z
I
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
I
Choose solution ψ ] (z, k ) such that ψ0] (z, k ) = 1
I
Continue ψ ] (z, k) inside upper/lower part of the disk: ψ ± (z, k)
A winding number
Im z
k
1
0
0
1
E(k)
µ
E = Re z
I
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
I
Choose solution ψ ] (z, k ) such that ψ0] (z, k ) = 1
I
Continue ψ ] (z, k) inside upper/lower part of the disk: ψ ± (z, k)
ψ + (µ, k ) = t(k )ψ − (µ, k )
(t(k) 6= 0)
A winding number
Im z
k
1
0
0
1
E(k)
µ
E = Re z
I
E = E(k ): Dirichlet eigenvalue, solution has node ψ0] (E, k) = 0
I
Choose solution ψ ] (z, k ) such that ψ0] (z, k ) = 1
I
Continue ψ ] (z, k) inside upper/lower part of the disk: ψ ± (z, k)
ψ + (µ, k ) = t(k )ψ − (µ, k )
+
A − B k
k0
(t(k) 6= 0)
t(k)
Winding number is sgn E 0 (k0 ) = ±1
0
A, B
1
Proof of Theorem (sketch)
k ∈ S1
Im z
π
0
−π
µ
E = Re z
Fermi line (one half fat)
edge states
torus
Proof of Theorem (sketch)
k ∈ S1
Im z
π
0
µ
E = Re z
Fermi line (one half fat)
edge states
torus
−π
I
I
I
I
I
I
I
ψ, ψ ] solutions (bulk, edge) at z, k decaying at n → +∞
Bijective map ψ 7→ ψ ] , so that ψn = ψn] (n > n0 )
]
]
∃ψ 6= 0 | ψn=0
= 0 ⇔ z ∈ σ(H ] (k )) ⇔ ψ 7→ ψn=0
not 1-to-1.
There is a section of the frame bundle F (E), global on T, except
at edge eigenvalue crossings
Cut the torus along the Fermi line; let T (k) be the transition
matrix
There T (k) = IN , except near eigenvalue crossings
As k traverses one of them, T (k ) has eigenvalues 1 (multiplicity
N − 1) and t(k) making one turn of S 1
Proof of Theorem: Dual ruedas
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
S1
k =0
k
k =π
Edge rueda: edge eigenvalues
S1
1
k =0
k
k =π
Bulk rueda: eigenvalues of T (k )
Proof of Theorem: Dual ruedas
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
S1
S1
k =0
k
1
k =π
Edge rueda: edge eigenvalues
k =0
k
k =π
Bulk rueda: eigenvalues of T (k )
Ruedas share intersection points.
Proof of Theorem: Dual ruedas
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
S1
k =0
k
k =π
Edge rueda: edge eigenvalues
S1
1
k =0
k
k =π
Bulk rueda: eigenvalues of T (k )
Ruedas share intersection points. Hence indices are equal Introduction
Rueda de casino
Hamiltonians
Indices
Further results
Further results
I
Alternate formulation of bulk index
I
Direct link to edge picture
I
Application to graphene
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ).
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k
κ
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
ε
ε2j (κ)
ε2j−1 (κ)
κ
−π
0
π
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
single bands can not be
ε
isolated; but pairs can.
ε2j (κ)
ε2j−1 (κ)
κ
−π
0
π
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
single bands can not be
ε
isolated; but pairs can.
ε2j (κ)
If so: Bloch solutions for pair
(2j − 1, 2j) form Bloch
ε2j−1 (κ)
bundle Ej over Brillouin zone
κ
−π
0
π
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
single bands can not be
ε
isolated; but pairs can.
ε2j (κ)
If so: Bloch solutions for pair
(2j − 1, 2j) form Bloch
ε2j−1 (κ)
bundle Ej over Brillouin zone
κ
−π
0
π
Theorem
I=
Y
j
with product over filled pairs.
I(Ej )
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
single bands can not be
ε
isolated; but pairs can.
ε2j (κ)
If so: Bloch solutions for pair
(2j − 1, 2j) form Bloch
ε2j−1 (κ)
bundle Ej over Brillouin zone
κ
−π
0
π
Theorem
I=
Y
I(Ej )
j
with product over filled pairs.
Note: Bulk solution are decaying to n → +∞, Bloch solutions
are bounded
Alternate formulation of bulk index
So far, only periodicity along edge assumed (quasi-momentum
k ). Now: doubly periodic case (quasi-momenta k , κ): Brillouin
zone serves as torus
k = 0, π:
single bands can not be
ε
isolated; but pairs can.
ε2j (κ)
If so: Bloch solutions for pair
(2j − 1, 2j) form Bloch
ε2j−1 (κ)
bundle Ej over Brillouin zone
κ
−π
0
π
Theorem
I=
Y
I(Ej )
j
with product over filled pairs.
Note: Bulk solution are decaying to n → +∞, Bloch solutions
are bounded
Proof using Bloch variety (Kohn)
A direct link to the edge
A direct link between indices of Bloch bundles and the edge
index.
A direct link to the edge
A direct link between indices of Bloch bundles and the edge
index. Simpler setting: Quantum Hall effect.
1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
µ
111111111
000000000
000000000
111111111
−π
k
π
Definition: Edge Index
N ] = signed number of eigenvalue crossings
A direct link to the edge
A direct link between indices of Bloch bundles and the edge
index. Simpler setting: Quantum Hall effect.
Definition: Edge Index
N ] = signed number of eigenvalue crossings
Bulk: ch(Ej ) is the Chern number of the Bloch bundle Ej of the
j-th band.
A direct link to the edge
A direct link between indices of Bloch bundles and the edge
index. Simpler setting: Quantum Hall effect.
Definition: Edge Index
N ] = signed number of eigenvalue crossings
Bulk: ch(Ej ) is the Chern number of the Bloch bundle Ej of the
j-th band.
Duality:
N] =
X
j
with sum over filled bands.
ch(Ej )
A direct link to the edge
A direct link between indices of Bloch bundles and the edge
index. Simpler setting: Quantum Hall effect.
Definition: Edge Index
N ] = signed number of eigenvalue crossings
Bulk: ch(Ej ) is the Chern number of the Bloch bundle Ej of the
j-th band.
Duality:
N] =
X
ch(Ej )
j
with sum over filled bands.
(cf. Hatsugai) Here via scattering and Levinson’s theorem.
Duality via scattering
k
Brillouin zone 3 (κ, k )
κ
Duality via scattering
Minima κ− (k) and maxima κ+ (k) of
energy band εj (κ, k) in κ at fixed k
k
εj (κ)
k
κ−
κ+
κ
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
−π0000000
κ−
κ+ π
Duality via scattering
k
Minima κ− (k) and maxima κ+ (k) of
energy band εj (κ, k) in κ at fixed k
κ−
κ+
κ
Duality via scattering
k
Maxima κ+ (k )
κ+
κ
Duality via scattering
k
0
1
0
1
11
00
00
11
Maxima κ+ (k ) with semi-bound
states (to be explained)
1
0
κ+
κ
Duality via scattering
k
0
1
0
1
11
00
00
11
1
0
κ+
κ
Duality via scattering
εj (κ)
At fixed k: Energy band εj (κ, k) and
the line bundle Ej of Bloch states
κ
1111111
−π0000000
π
Duality via scattering
κ
1111111
−π0000000
π
Line indicates choice of a section |κi
of Bloch states (from the given
band). No global section in
κ ∈ R/2πZ is possible, as a rule.
Duality via scattering
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
−π0000000
κ−
κ+ π
States |κi above the solid line are
left movers (ε0j (κ) < 0)
Duality via scattering
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
−π0000000
κ−
κ+ π
They are incoming asymptotic (bulk)
states for scattering at edge (from
inside)
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
|κi ≡ |ini
Duality via scattering
1
0
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
0000000
−π κ−
κ+ π
Scattering determines section |outi
of right movers above line
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
|outi
|κi ≡ |ini
Duality via scattering
1
0
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
0000000
−π κ−
κ+ π
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
|outi
|κi ≡ |ini
Scattering matrix
|outi = S+ |κi
as relative phase between two
sections of the same fiber (near κ+ )
Duality via scattering
1
0
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
0000000
−π κ−
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
|outi
|κi ≡ |ini
Scattering matrix
|outi = S+ |κi
κ+ π
as relative phase between two
sections of the same fiber (near κ+ )
Likewise S− near κ− .
Duality via scattering
1
0
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
0000000
−π κ−
κ+ π
Chern number computed by sewing
ch(Ej ) = N (S+ ) − N (S− )
with N (S± ) the winding of
S± = S± (k) as k = 0 . . . π.
Duality via scattering
11
00
0
1
00
11
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
−π0000000
κ−
κ+ π
As κ → κ+ , whence
|ini = |κi → |κ+ i
|outi = S+ |κi → |κ+ i (up to phase)
their limiting span is that of
|κ+ i,
d|κi dκ κ+
(bounded, resp. unbounded in space). The span contains the
limiting scattering state |ψi ∝ |ini + |outi.
If (exceptionally) |ψi ∝ |κ+ i then |ψi is a semi-bound state.
Duality via scattering
k
11
00
0
1
00
11
11
00
0
1
11
00
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 κ
1
0
1
1111111
0000000
−π κ−
κ+ π
1
0
0
1
00
11
00
11
1
0
κ+
κ
As a function of k , semi-bound states occur exceptionally.
Levinson’s theorem
Recall from two-body potential scattering: The scattering phase
at threshold equals the number of bound states
σ(p2 + V )
0
arg S E=0+ = 2πN
E
Levinson’s theorem
Recall from two-body potential scattering: The scattering phase
at threshold equals the number of bound states
σ(p2 + V )
0
E
arg S E=0+ = 2πN
N changes with the potential V when bound state reaches
threshold (semi-bound state ≡ incipient bound state)
Levinson’s theorem (relative version)
Spectrum of edge Hamiltonian
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
k∗
k
ε(k) = εj (κ+ (k), k)
Levinson’s theorem (relative version)
Spectrum of edge Hamiltonian
1111111
0000000
0000000
1111111
0000000
1111111
0000000
1111111
k1 k∗ k2
ε(k) = εj (κ+ (k), k)
k
k2
lim arg S+ (ε(k) − δ) = ±2π
δ→0
k1
Proof
111111111
000000000
000000000
111111111
000000000
111111111
000000000
j0 + 111111111
1
µ
1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
j0
−π
k
(j )
(j +1)
N ] = N (S+0 )
=
j0
X
= N (S−0
(j)
j0
X
j=0
(1)
(N (S− ) = 0)
(j)
N (S+ ) − N (S− )
j=0
=
π
ch(Ej )
)
An application: Quantum Hall in graphene
An application: Quantum Hall in graphene
Hamiltonian: Nearest neighbor hopping with flux Φ per
plaquette.
An application: Quantum Hall in graphene
Spectrum in black
Φ (mod 2π)
E
An application: Quantum Hall in graphene
Spectrum in black
Φ (mod 2π)
E
What is the Hall conductance (Chern number) in any white
point?
An application: Quantum Hall in graphene
What is the Hall conductance (Chern number) s in any white
point?
Bulk approach (Thouless): If Φ = p/q, (p, q coprime) then
r = sp + tq
where:
I
r number of bands below Fermi energy
I
s, t integers
s is so determined only modulo q.
An application: Quantum Hall in graphene
What is the Hall conductance (Chern number) s in any white
point?
Bulk approach (Thouless): If Φ = p/q, (p, q coprime) then
r = sp + tq
where:
I
r number of bands below Fermi energy
I
s, t integers
s is so determined only modulo q.
For square lattice, s ∈ (−q/2, q/2).
An application: Quantum Hall in graphene
What is the Hall conductance (Chern number) s in any white
point?
Bulk approach (Thouless): If Φ = p/q, (p, q coprime) then
r = sp + tq
where:
I
r number of bands below Fermi energy
I
s, t integers
s is so determined only modulo q.
For square lattice, s ∈ (−q/2, q/2). Not for other lattices.
An application: Quantum Hall in graphene
What is the Hall conductance (Chern number) s in any white
point?
Bulk approach (Thouless): If Φ = p/q, (p, q coprime) then
r = sp + tq
where:
I
r number of bands below Fermi energy
I
s, t integers
s is so determined only modulo q.
For square lattice, s ∈ (−q/2, q/2). Not for other lattices.
→ Edge approach (with Agazzi, Eckmann)
The colors of graphene
What is the Hall conductance (Chern number) in any white
point?
The colors of graphene
What is the Hall conductance (Chern number) in any white
point?
Summary
Bulk = Edge
I = I]
Summary
Bulk = Edge
I = I]
I
The bulk and the indices of a topological insulator (of
reduced symmetry) are indices of suitable ruedas
I
In case of full translational symmetry, bulk index can be
defined and linked to edge in other ways
I
Application (Quantum Hall): graphene
I
Three dimensions ...
I
Open questions: No periodicity (disordered case)?