Bulk-edge duality for topological insulators Gian Michele Graf ETH - - PowerPoint PPT Presentation
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
Gian Michele Graf ETH Zurich Symposium on Statistical Mechanics: Many-Body Quantum Systems University of Warwick 17-21 March 2014
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
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Time-reversal invariant fermionic system
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Time-reversal invariant fermionic system
spin down Edge void Bulk spin up Edge void
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Time-reversal invariant fermionic system
spin down Edge void Bulk spin up Edge void
◮ 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.
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Time-reversal invariant fermionic system
spin down Edge void Bulk spin up Edge void
◮ 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 = sphere (differ by genus).
◮ Insulator in the Bulk: Excitation gap
For independent electrons: band gap at Fermi energy
◮ Time-reversal invariant fermionic system
spin down Edge void Bulk spin up Edge void
◮ 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 = sphere (differ by genus). Contributors to the field: Kane, Mele, Zhang, Moore; Fr¨
Hasan
Material: InAs/GaSb (quantum well); AlSb (barrier)
Courtesy: S. M¨ uller, K. Ensslin
Courtesy: S. M¨ uller, K. Ensslin
Courtesy: S. M¨ uller, K. Ensslin
Courtesy: S. M¨ uller, K. Ensslin
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ−
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ−
spin currents
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant.
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant. Indeed: Θψ, Vψ =
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant. Indeed: by Θ anti-unitary, Θψ, Vψ = ΘVψ, Θ2ψ =
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant. Indeed: by Θ anti-unitary, Θ2 = −1, [Θ, V] = 0, Θψ, Vψ = ΘVψ, Θ2ψ = − VΘψ, ψ =
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant. Indeed: by Θ anti-unitary, Θ2 = −1, [Θ, V] = 0, V = V ∗ Θψ, Vψ = ΘVψ, Θ2ψ = − VΘψ, ψ = − Θψ, Vψ
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant.
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant.
Edge void Bulk
|ψ Θ|ψ µ− µ+ Θ time-reversal µ+ > µ− disorder V
spin currents
disorder Θψ|V|ψ = 0 if V is time-reversal invariant.
Deformation as interpolation in physical space:
interpolating material
◮ Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
Deformation as interpolation in physical space:
void interpolating material
◮ Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
◮ Ordinary insulator void: Edge states
Deformation as interpolation in physical space:
void interpolating material
◮ Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
◮ Ordinary insulator void: Edge states ◮ Bulk-edge correspondence: Termination of bulk of a
topological insulator implies edge states.
Deformation as interpolation in physical space:
void interpolating material
◮ Gap must close somewhere in between. Hence: Interface
states at Fermi energy.
◮ Ordinary insulator void: Edge states ◮ Bulk-edge correspondence: Termination of bulk of a
topological insulator implies edge states. (But not conversely!)
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
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”
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
◮ Goal: State the (intrinsic) topological property
distinguishing different classes of insulators. More precisely:
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
◮ Goal: State the (intrinsic) topological property
distinguishing different classes of insulators. More precisely:
◮ Express that property as an Index relating to the Bulk,
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
◮ Goal: State the (intrinsic) topological property
distinguishing different classes of insulators. More precisely:
◮ Express that property as an Index relating to the Bulk,
◮ Bulk-edge duality: Can it be shown that the two indices
agree?
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
◮ Goal: State the (intrinsic) topological property
distinguishing different classes of insulators. More precisely:
◮ Express that property as an Index relating to the Bulk,
◮ Bulk-edge duality: Can it be shown that the two indices
agree? Schulz-Baldes et al.; Essin & Gurarie
Edge void Bulk
In a nutshell: Termination of bulk of a topological insulator implies edge states
◮ Goal: State the (intrinsic) topological property
distinguishing different classes of insulators. More precisely:
◮ Express that property as an Index relating to the Bulk,
◮ Bulk-edge duality: Can it be shown that the two indices
agree? Done differently.
Introduction Rueda de casino Hamiltonians Indices Further results
Dancers
◮ start in pairs, anywhere ◮ end in pairs, anywhere (possibly elseways & elsewhere) ◮ are free in between ◮ must never step on center of the floor
Dancers
◮ start in pairs, anywhere ◮ end in pairs, anywhere (possibly elseways & elsewhere) ◮ are free in between ◮ must never step on center of the floor ◮ are unlabeled points
Dancers
◮ start in pairs, anywhere ◮ end in pairs, anywhere (possibly elseways & elsewhere) ◮ are free in between ◮ must never step on center of the floor ◮ are unlabeled points
There are dances which can not be deformed into one another. What is the index that makes the difference?
A snapshot of the dance
A snapshot of the dance Dance D as a whole
D
A snapshot of the dance Dance D as a whole
D
A snapshot of the dance Dance D as a whole
D
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
Hamiltonian on the lattice Z × Z (plane)
Z Z −2 −1 1 n = −3 2 3 4
Hamiltonian on the lattice Z × Z (plane)
Z Z −2 −1 1 n = −3 2 3 4
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1:
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f.
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . )
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ), totalling N
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ), totalling N
◮ Bloch reduction by quasi-momentum k ∈ S1 := R/2πZ
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ), totalling N
◮ Bloch reduction by quasi-momentum k ∈ S1 := R/2πZ
End up with wave-functions ψ = (ψn)n∈Z ∈ ℓ2(Z; CN) and Bulk Hamiltonian
with Vn(k) = Vn(k)∗ ∈ MN(C) (potential) A(k) ∈ GL(N) (hopping)
Hamiltonian on the lattice Z × Z (plane)
◮ translation invariant in the vertical direction ◮ period may be assumed to be 1: sites within a period as
labels of internal d.o.f., along with others (spin, . . . ), totalling N
◮ Bloch reduction by quasi-momentum k ∈ S1 := R/2πZ
End up with wave-functions ψ = (ψn)n∈Z ∈ ℓ2(Z; CN) and Bulk Hamiltonian
with Vn(k) = Vn(k)∗ ∈ MN(C) (potential) A(k) ∈ GL(N) (hopping): Schr¨
difference equation
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
N Z 1 n = 2 3 4
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
◮ translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ ℓ2(N; CN) and Edge Hamiltonian
n(k)ψn
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
◮ translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ ℓ2(N; CN) and Edge Hamiltonian
n(k)ψn
which
◮ agrees with Bulk Hamiltonian outside of collar near edge
(width n0) V ♯
n(k) = Vn(k) ,
(n > n0)
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
◮ translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ ℓ2(N; CN) and Edge Hamiltonian
n(k)ψn
which
◮ agrees with Bulk Hamiltonian outside of collar near edge
(width n0) V ♯
n(k) = Vn(k) ,
(n > n0)
◮ has Dirichlet boundary conditions: for n = 1 set ψ0 = 0
Hamiltonian on the lattice N × Z (half-plane) with N = {1, 2, . . .}
◮ translation invariant as before (hence Bloch reduction)
Wave-functions ψ ∈ ℓ2(N; CN) and Edge Hamiltonian
n(k)ψn
which
◮ agrees with Bulk Hamiltonian outside of collar near edge
(width n0) V ♯
n(k) = Vn(k) ,
(n > n0)
◮ has Dirichlet boundary conditions: for n = 1 set ψ0 = 0
Note: σess(H♯(k)) ⊂ σess(H(k)), but typically σdisc(H♯(k)) ⊂ σdisc(H(k))
Hamiltonian is nearest neighbor hopping on honeycomb lattice n2 n1 n1 A B
a1
a2 a) b) n2 (a) zigzag, resp. (b) armchair boundaries Dimers (N = 2).
Hamiltonian is nearest neighbor hopping on honeycomb lattice n2 n1 n1 A B
a1
a2 a) b) n2 (a) zigzag, resp. (b) armchair boundaries Dimers (N = 2). For (b): ψn = ψA
n
ψB
n
A(k) = −t 1 eik
Vn(k) = −t 1 1
∈ GL(N) Also: Extensions with spin, spin orbit coupling leading to topological insulators (Kane & Mele)
◮ Gap assumption: Fermi energy µ lies in a gap for all
k ∈ S1: µ / ∈ σ(H(k))
◮ Gap assumption: Fermi energy µ lies in a gap for all
k ∈ S1: µ / ∈ σ(H(k))
◮ Fermionic time-reversal symmetry: Θ : CN → CN
◮ Θ is anti-unitary and Θ2 = −1; ◮ Θ induces map on ℓ2(Z; CN), pointwise in n ∈ Z; ◮ For all k ∈ S1,
H(−k) = ΘH(k)Θ−1 Likewise for H♯(k)
◮ σ(H(k)) = σ(H(−k)). Same for H♯(k).
◮ σ(H(k)) = σ(H(−k)). Same for H♯(k). ◮ Time-reversal invariant points, k = −k, at k = 0, π.
◮ 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).
◮ 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 = Θ2ψ = ¯ λΘψ = ¯ λλψ (⇒⇐)
◮ 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).
E ∈ R π −π k ∈ S1 µ
Bands, Fermi line (one half fat), edge states
Introduction Rueda de casino Hamiltonians Indices Further results
The spectrum of H♯(k)
k symmetric on −π ≤ k ≤ 0 π
Bands, Fermi line, edge states Definition: Edge Index I♯ = parity of number of eigenvalue crossings
The spectrum of H♯(k)
k symmetric on −π ≤ k ≤ 0 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.
Let z ∈ C. The Schr¨
(H(k) − z)ψ = 0 (as a 2nd order difference equation) has 2N solutions ψ = (ψn)n∈Z, ψn ∈ CN.
Let z ∈ C. The Schr¨
(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
◮ dim Ez,k = N.
Let z ∈ C. The Schr¨
(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
◮ dim Ez,k = N. ◮ E¯ z,−k = ΘEz,k
π −π k ∈ S1 µ E = Re z Im z
Loop γ and torus T = γ × S1 Vector bundle E with base T ∋ (z, k), fibers Ez,k, and Θ2 = −1.
π −π k ∈ S1 µ E = Re z Im z
Loop γ and torus T = γ × S1 Vector bundle E with base T ∋ (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)
π −π k ∈ S1 µ E = Re z Im z
Loop γ and torus T = γ × S1 Vector bundle E with base T ∋ (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)
π −π k ∈ S1 µ E = Re z Im z
Loop γ and torus T = γ × S1 Vector bundle E with base T ∋ (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?
π −π k ∈ S1 µ E = Re z Im z
Loop γ and torus T = γ × S1 Vector bundle E with base T ∋ (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 . . .
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
◮ torus ϕ = (ϕ1, ϕ2) ∈ T = (R/2πZ)2
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
◮ torus ϕ = (ϕ1, ϕ2) ∈ T = (R/2πZ)2 with cut (figure) cut ϕ2 = 0 ϕ2 = π ϕ1 ϕ2 ϕ −ϕ
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
◮ torus ϕ = (ϕ1, ϕ2) ∈ T = (R/2πZ)2 with cut (figure) cut ϕ2 = 0 ϕ2 = π ϕ1 ϕ2 ϕ −ϕ ◮ a (compatible) section of the frame bundle of E
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
◮ torus ϕ = (ϕ1, ϕ2) ∈ T = (R/2πZ)2 with cut (figure) cut ϕ2 = 0 ϕ2 = π ϕ1 ϕ2 ϕ2 −ϕ2 ◮ a (compatible) section of the frame bundle of E ◮ the transition matrices T(ϕ2) ∈ GL(N) across the cut
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 Sketch of proof: Consider
◮ torus ϕ = (ϕ1, ϕ2) ∈ T = (R/2πZ)2 with cut (figure) cut ϕ2 = 0 ϕ2 = π ϕ1 ϕ2 ϕ2 −ϕ2 ◮ a (compatible) section of the frame bundle of E ◮ the transition matrices T(ϕ2) ∈ GL(N) across the cut
Θ0T(ϕ2) = T −1(−ϕ2)Θ0 , (ϕ2 ∈ S1) with Θ0 : CN → CN antilinear, Θ2
0 = −1
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
ϕ2 = 0 ϕ2 = π ϕ2 −ϕ2
◮ Θ0T(ϕ2) = T −1(−ϕ2)Θ0 ◮ Only half the cut (0 ≤ ϕ2 ≤ π) matters for T(ϕ2) ◮ At time-reversal invariant points, ϕ2 = 0, π,
Θ0T = T −1Θ0
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
ϕ2 = 0 ϕ2 = π ϕ2 −ϕ2
◮ Θ0T(ϕ2) = T −1(−ϕ2)Θ0 ◮ Only half the cut (0 ≤ ϕ2 ≤ π) matters for T(ϕ2) ◮ At time-reversal invariant points, ϕ2 = 0, π,
Θ0T = T −1Θ0 Eigenvalues of T come in pairs λ, ¯ λ−1: Θ0(T − λ) = T −1(1 − ¯ λT)Θ0
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
ϕ2 = 0 ϕ2 = π ϕ2 −ϕ2
◮ Θ0T(ϕ2) = T −1(−ϕ2)Θ0 ◮ Only half the cut (0 ≤ ϕ2 ≤ π) matters for T(ϕ2) ◮ At time-reversal invariant points, ϕ2 = 0, π,
Θ0T = T −1Θ0 Eigenvalues of T come in pairs λ, ¯ λ−1: Θ0(T − λ) = T −1(1 − ¯ λT)Θ0 Phases λ/|λ| pair up (Kramers degeneracy)
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
ϕ2 = 0 ϕ2 = π ϕ2 −ϕ2
◮ Θ0T(ϕ2) = T −1(−ϕ2)Θ0 ◮ Only half the cut (0 ≤ ϕ2 ≤ π) matters for T(ϕ2) ◮ At time-reversal invariant points, ϕ2 = 0, π,
Θ0T = T −1Θ0 Eigenvalues of T come in pairs λ, ¯ λ−1: Θ0(T − λ) = T −1(1 − ¯ λT)Θ0 Phases λ/|λ| pair up (Kramers degeneracy)
◮ For 0 ≤ ϕ2 ≤ π, phases λ/|λ| form a rueda, D
Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1
ϕ2 = 0 ϕ2 = π ϕ2 −ϕ2
◮ Θ0T(ϕ2) = T −1(−ϕ2)Θ0 ◮ 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.
Theorem Bulk and edge indices agree: I = I♯
Theorem Bulk and edge indices agree: I = I♯ I = +1: ordinary insulator I = −1: topological insulator
◮ For this slide only: N = 1.
◮ For this slide only: N = 1. Schr¨
difference) equation on the half-line (H♯ − z)ψ♯ = 0 (no b.c. at n = 0)
◮ For this slide only: N = 1. Schr¨
difference) equation on the half-line (H♯ − z)ψ♯ = 0 (no b.c. at n = 0) with solution ψ♯
n ∈ C, (n = 0, 1, 2) decaying at n → ∞
1 −1 n ψ# 2 3 4 5 ψ#
1
◮ For this slide only: N = 1. Schr¨
difference) equation on the half-line (H♯ − z)ψ♯ = 0 (no b.c. at n = 0) with solution ψ♯
n ∈ C, (n = 0, 1, 2) decaying at n → ∞
1 −1 n ψ# 2 3 4 5 ψ#
1
◮ Solution is unique up to multiples
◮ For this slide only: N = 1. Schr¨
difference) equation on the half-line (H♯ − z)ψ♯ = 0 (no b.c. at n = 0) with solution ψ♯
n ∈ C, (n = 0, 1, 2) decaying at n → ∞
1 −1 n ψ# 2 3 4 5 ψ#
1
◮ Solution is unique up to multiples ◮ ψ♯ 0 = 1 picks a unique solution,
◮ For this slide only: N = 1. Schr¨
difference) equation on the half-line (H♯ − z)ψ♯ = 0 (no b.c. at n = 0) with solution ψ♯
n ∈ C, (n = 0, 1, 2) decaying at n → ∞
1 −1 n ψ# 2 3 4 5 ψ#
1
◮ Solution is unique up to multiples ◮ ψ♯ 0 = 1 picks a unique solution, except if n = 0 is a node
E = Re z Im z
◮ ◮ ◮
E = Re z Im z k
◮ ◮ ◮
E = Re z k
◮ ◮ ◮
k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ ◮
Im z k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ ◮
Im z k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ Choose solution ψ♯(z, k) such that ψ♯
0(z, k) = 1
◮
Im z k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ Choose solution ψ♯(z, k) such that ψ♯
0(z, k) = 1
◮ Continue ψ♯(z, k) inside upper/lower part of the disk: ψ±(z, k)
Im z k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ Choose solution ψ♯(z, k) such that ψ♯
0(z, k) = 1
◮ Continue ψ♯(z, k) inside upper/lower part of the disk: ψ±(z, k)
ψ+(µ, k) = t(k)ψ−(µ, k) (t(k) = 0)
Im z k µ E(k)
◮ E = E(k): Dirichlet eigenvalue, solution has node ψ♯
0(E, k) = 0
◮ Choose solution ψ♯(z, k) such that ψ♯
0(z, k) = 1
◮ Continue ψ♯(z, k) inside upper/lower part of the disk: ψ±(z, k)
ψ+(µ, k) = t(k)ψ−(µ, k) (t(k) = 0)
A B k 1 A, B t(k) − + k0
Winding number is sgn E′(k0) = ±1
π −π k ∈ S1 µ E = Re z Im z
Fermi line (one half fat) edge states torus
π −π k ∈ S1 µ E = Re z Im z
Fermi line (one half fat) edge states torus
◮ ψ, ψ♯ solutions (bulk, edge) at z, k decaying at n → +∞ ◮ Bijective map ψ → ψ♯, so that ψn = ψ♯
n (n > n0)
◮ ∃ψ = 0 | ψ♯
n=0 = 0 ⇔ z ∈ σ(H♯(k)) ⇔ ψ → ψ♯ n=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 S1
k = π k S1
Edge rueda: edge eigenvalues
1 k = 0 k k = π S1
Bulk rueda: eigenvalues of T(k)
k = π k S1
Edge rueda: edge eigenvalues
1 k = 0 k k = π S1
Bulk rueda: eigenvalues of T(k) Ruedas share intersection points.
k = π k S1
Edge rueda: edge eigenvalues
1 k = 0 k k = π S1
Bulk rueda: eigenvalues of T(k) Ruedas share intersection points. Hence indices are equal
Introduction Rueda de casino Hamiltonians Indices Further results
◮ Alternate formulation of bulk index ◮ Direct link to edge picture ◮ Application to graphene
So far, only periodicity along edge assumed (quasi-momentum k).
So far, only periodicity along edge assumed (quasi-momentum k). Now: doubly periodic case (quasi-momenta k, κ): Brillouin zone serves as torus
κ k
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
single bands can not be isolated; but pairs can.
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
single bands can not be isolated; but pairs can. If so: Bloch solutions for pair (2j − 1, 2j) form Bloch bundle Ej over Brillouin zone
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
single bands can not be isolated; but pairs can. If so: Bloch solutions for pair (2j − 1, 2j) form Bloch bundle Ej over Brillouin zone Theorem I =
I(Ej) with product over filled pairs.
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
single bands can not be isolated; but pairs can. If so: Bloch solutions for pair (2j − 1, 2j) form Bloch bundle Ej over Brillouin zone Theorem I =
I(Ej) with product over filled pairs. Note: Bulk solution are decaying to n → +∞, Bloch solutions are bounded
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:
−π π ε2j−1(κ) ε2j (κ) κ ε
single bands can not be isolated; but pairs can. If so: Bloch solutions for pair (2j − 1, 2j) form Bloch bundle Ej over Brillouin zone Theorem I =
I(Ej) with product over filled pairs. Note: Bulk solution are decaying to n → +∞, Bloch solutions are bounded Proof using Bloch variety (Kohn)
A direct link between indices of Bloch bundles and the edge index.
A direct link between indices of Bloch bundles and the edge
k π −π
Definition: Edge Index N ♯ = signed number of eigenvalue crossings
A direct link between indices of Bloch bundles and the edge
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 between indices of Bloch bundles and the edge
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 ♯ =
ch(Ej) with sum over filled bands.
A direct link between indices of Bloch bundles and the edge
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 ♯ =
ch(Ej) with sum over filled bands. (cf. Hatsugai) Here via scattering and Levinson’s theorem.
κ k
Brillouin zone ∋ (κ, k)
κ− κ k κ+ k
Minima κ−(k) and maxima κ+(k) of energy band εj(κ, k) in κ at fixed k
κ+ −π κ π εj(κ)
κ− κ k κ+
Minima κ−(k) and maxima κ+(k) of energy band εj(κ, k) in κ at fixed k
κ k κ+
Maxima κ+(k)
k κ+
Maxima κ+(k) with semi-bound states (to be explained)
k κ+
κ π εj(κ)
At fixed k: Energy band εj(κ, k) and the line bundle Ej of Bloch states
κ π
Line indicates choice of a section |κ
band). No global section in κ ∈ R/2πZ is possible, as a rule.
κ+ −π κ π
States |κ above the solid line are left movers (ε′
j(κ) < 0)
κ+ −π κ π
They are incoming asymptotic (bulk) states for scattering at edge (from inside)
κ+ −π κ π
Scattering determines section |out
|out
κ+ −π κ π
|out Scattering matrix |out = S+|κ as relative phase between two sections of the same fiber (near κ+)
κ+ −π κ π
|out Scattering matrix |out = S+|κ as relative phase between two sections of the same fiber (near κ+) Likewise S− near κ−.
κ+ −π 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.
κ+ −π κ π
As κ → κ+, whence |in = |κ → |κ+ |out = S+|κ → |κ+ (up to phase) their limiting span is that of |κ+, d|κ dκ
(bounded, resp. unbounded in space). The span contains the limiting scattering state |ψ ∝ |in + |out. If (exceptionally) |ψ ∝ |κ+ then |ψ is a semi-bound state.
κ+ −π κ π
k κ+
As a function of k, semi-bound states occur exceptionally.
Recall from two-body potential scattering: The scattering phase at threshold equals the number of bound states E σ(p2 + V) arg S
Recall from two-body potential scattering: The scattering phase at threshold equals the number of bound states E σ(p2 + V) arg S
N changes with the potential V when bound state reaches threshold (semi-bound state ≡ incipient bound state)
Spectrum of edge Hamiltonian
k ε(k) = εj(κ+(k), k)
Spectrum of edge Hamiltonian
k ε(k) = εj(κ+(k), k) k1 k2
lim
δ→0 arg S+(ε(k) − δ)
k1
= ±2π
k π −π
j0 + 1 j0
N ♯ = N(S(j0)
+ )
−
)
j0
N(S(j)
+ ) − N(S(j) − )
=
j0
ch(Ej) (N(S(1)
− ) = 0)
Hamiltonian: Nearest neighbor hopping with flux Φ per plaquette.
Spectrum in black
E Φ (mod 2π)
Spectrum in black
E Φ (mod 2π)
What is the Hall conductance (Chern number) in any white point?
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:
◮ r number of bands below Fermi energy ◮ s, t integers
s is so determined only modulo q.
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:
◮ r number of bands below Fermi energy ◮ s, t integers
s is so determined only modulo q. For square lattice, s ∈ (−q/2, q/2).
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:
◮ r number of bands below Fermi energy ◮ s, t integers
s is so determined only modulo q. For square lattice, s ∈ (−q/2, q/2). Not for other lattices.
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:
◮ r number of bands below Fermi energy ◮ 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)
What is the Hall conductance (Chern number) in any white point?
What is the Hall conductance (Chern number) in any white point?
Bulk = Edge I = I♯
Bulk = Edge I = I♯
◮ The bulk and the indices of a topological insulator (of
reduced symmetry) are indices of suitable ruedas
◮ In case of full translational symmetry, bulk index can be
defined and linked to edge in other ways
◮ Application (Quantum Hall): graphene ◮ Three dimensions ... ◮ Open questions: No periodicity (disordered case)?