Bulk-edge duality for topological insulators Gian Michele Graf ETH - - PowerPoint PPT Presentation

Bulk-edge duality for topological insulators

Gian Michele Graf ETH Zurich Statistical Mechanics Seminar University of Warwick 8 November, 2012

joint work with Marcello Porta

Topological insulators: first impressions

◮ 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¨

• hlich

Bulk-edge correspondence

Deformation as interpolation in physical space:

• topological insulator
• rdinary insulator

interpolating material

◮ Gap must close somewhere in between. Hence: Interface

states at Fermi energy.

Bulk-edge correspondence

Deformation as interpolation in physical space:

• topological insulator

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!)

Bulk-edge correspondence

Termination of bulk of a topological insulator implies edge states But:

◮ What is the (intrinsic) topological property distinguishing

different classes of insulators? More precisely:

◮ Can that property be expressed as an Index relating to the

Bulk, or to the Edge?

◮ Bulk-edge duality: Can it be shown that the two indices

agree?

Edge void Bulk Edge void

Bulk-edge correspondence

Termination of bulk of a topological insulator implies edge states But:

◮ What is the (intrinsic) topological property distinguishing

different classes of insulators? More precisely:

◮ Can that property be expressed as an Index relating to the

Bulk, or to the Edge?

◮ Bulk-edge duality: Can it be shown that the two indices

agree?

Edge void Bulk

Bulk-edge correspondence. Done?

Termination of bulk of a topological insulator implies edge states But:

◮ What is the (intrinsic) topological property distinguishing

different classes of insulators? More precisely:

◮ Can that property be expressed as an Index relating to the

Bulk, or to the Edge? Yes, e.g. Kane and Mele.

◮ Bulk-edge duality: Can it be shown that the two indices

agree? Schulz-Baldes et al.. . . .

Edge void Bulk

Bulk-edge correspondence. Today

Termination of bulk of a topological insulator implies edge states But:

◮ What is the (intrinsic) topological property distinguishing

different classes of insulators? More precisely:

◮ Can that property be expressed as an Index relating to the

Bulk, or to the Edge? Done differently.

◮ Bulk-edge duality: Can it be shown that the two indices

agree? Done differently.

Edge void Bulk

Rules of the dance

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. Which is the index that makes the difference?

Not the winding number

For this slide only: Dancers exempted from pairing up at ends. Dance: D = (D(t))a≤t≤b with D(t) a collection of points on the circle.

a b t

Winding w(D) := (2π)−1

dancers

turning angles If extreme positions are (collectively) the same, then N(D) := w(D) defines the (integer) winding number N(D)

The index of a Rueda

Dance D

D

Let dance ˜ D bring back the united pairs to initial positions.

The index of a Rueda

Dance D

D ˜ D

Let dance ˜ D bring back the united pairs to initial positions. Concatenation D#˜ D has winding number N(D#˜ D) = w(D) + w(˜ D) ˜ D is not unique, but w(˜ D1) − w(˜ D2) ∈ 2Z Hence I(D) := (−1)N(D#˜

D)

is a well-defined index for the Rueda

The index of a Rueda

Dance D

D

I(D) = parity of number of crossings of fiducial line

Bulk Hamiltonian

Hamiltonian on the lattice Z × Z (plane)

◮ translation invariant in the second direction (soon to

become the direction of the edge)

◮ 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

• 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): Schr¨

• dinger eq. is the 2nd order

difference equation

Edge Hamiltonian

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

• H♯(k)ψ
• n = A(k)ψn−1 + A(k)∗ψn+1 + V ♯

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))

Graphene as an example

Hamiltonian is nearest neighbor hopping on honeycomb lattice n2 n1 n1 A B

• a2

a1

• a1
• a2
• a1 +

a2 a) b) n2 (a) zigzag, resp. (b) armchair boundaries Dimers (N = 2). For (b): ψn = ψA

n

ψB

n

• ∈ CN=2 ,

A(k) = −t 1 eik

• ,

Vn(k) = −t 1 1

• For (a): too, but A(k) /

∈ GL(N) Also: Extensions with spin, spin orbit coupling leading to topological insulators (Kane & Mele)

General assumptions

◮ 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; ◮ For all k ∈ S1,

H(−k) = ΘH(k)Θ−1 where Θ also denotes the map induced on ℓ2(Z; CN). Likewise for H♯(k)

Elementary consequences of H(−k) = ΘH(k)Θ−1

◮ 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

The edge index

The spectrum of H♯(k)

• µ

k symmetric on −π ≤ k ≤ 0 π

Bands, Fermi line, edge states Definition: Edge Index I♯ = parity of number of eigenvalue crossings At fixed k, map gap to S1 \ {1} and bands to 1 ∈ S1: Edge Index is index of a rueda.

Towards the bulk index

Let z ∈ C. The Schr¨

• dinger 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

◮ dim Ez,k = N. ◮ E¯ z,−k = ΘEz,k

The bulk index

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 involution Θ. Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1 (besides of N = dim E)

Idea: Cut torus to cylinder. Consider transition matrix across cut. On half the cut its eigenvalues form a rueda (endpoint condition from Kramers).

The bulk index

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 involution Θ. 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)

Main result

Theorem Bulk and edge indices agree: I = I♯ I = +1: ordinary insulator I = −1: topological insulator

Time reversal invariant bundles (E, T, Θ)

ϕ1 ϕ2 (0, π) (0, 0) (π, −π) T ϕ −ϕ (π, 0) ◮ T ∋ ϕ = (ϕ1, ϕ2) ◮ Time-reversal invariant points, ϕ = −ϕ at

ϕ = (0, 0), (π, 0), (0, π), (π, π)

◮ Θ : Eϕ → E−ϕ, Θ antilinear with Θ2 = −1 ◮ Frame bundle F(E) has fibers F(E)ϕ ∋ v = (v1, . . . vN)

consisting of bases v of Eϕ.

Classification of time reversal invariant bundles

Consider the cut torus:

cut ϕ1 ϕ2 (0, π)

Lemma On the cut torus the frame bundle admits a section ϕ → v(ϕ) ∈ F(E)ϕ which is time-reversal invariant: v(−ϕ) = (Θv(ϕ))ε with ε the block diagonal matrix with blocks 0 −1

1 0

• Idea: At a time reversal invariant point, that means (N = 2)

v2 = Θv1 v1 = −Θv2

Classification of time reversal invariant bundles

Consider the cut torus:

− +

cut ϕ1 ϕ2

Lemma On the cut torus the frame bundle admits a section ϕ → v(ϕ) ∈ F(E)ϕ which is time-reversal invariant: v(−ϕ) = (Θv(ϕ))ε with ε the block diagonal matrix with blocks 0 −1

1 0

• Transition matrix T(ϕ2) ∈ GL(N)

v+(ϕ2) = v−(ϕ2)T(ϕ2) , (ϕ2 ∈ S1)

Classification of time reversal invariant bundles

Consider the cut torus:

− +

cut ϕ1 ϕ2

Lemma On the cut torus the frame bundle admits a section ϕ → v(ϕ) ∈ F(E)ϕ which is time-reversal invariant: v(−ϕ) = (Θv(ϕ))ε with ε the block diagonal matrix with blocks 0 −1

1 0

• Transition matrix T(ϕ2) ∈ GL(N)

v+(ϕ2) = v−(ϕ2)T(ϕ2) , (ϕ2 ∈ S1) There exists a relation between T(ϕ2) and T(−ϕ2) Θ0T(ϕ2) = T −1(−ϕ2)Θ0 with Θ0 = εC, (C complex conjugation on CN)

Classification of time reversal invariant bundles

Family of transition matrices Θ0T(ϕ2) = T −1(−ϕ2)Θ0 , (ϕ2 ∈ S1) with Θ0 = εC, hence Θ2

0 = −1. ◮ Only 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 (with equal multiplicities): Θ0(T − λ) = T −1(1 − ¯ λT)Θ0

◮ Phases λ/|λ| form a rueda for 0 ≤ ϕ2 ≤ π

Definition: The Index of the bundle E is that of that rueda D I(E) := I(D) The statement of theorem is now complete.

Proof of Theorem (sketch)

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))

◮ 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 λ(k) making one turn of S1

Proof of Theorem: Dual ruedas

Edge rueda

1

Bulk rueda Ruedas share intersection points Hence indices are equal.

Further results

So far, only periodicity along edge assumed (quasi-momentum k). Now: doubly periodic case (quasi-momenta k, κ) 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 =

• j

I(Ej) with product over filled pairs. Note: Bulk solution are decaying to n → +∞, Bloch solutions are bounded Proof by Kohn’s theorem

Final remarks

Further (tentative) results:

◮ A direct link between indices of Bloch bundles and the

edge index via Levinson’s theorem.

◮ 3d topological insulators

Open questions:

◮ No periodicity (disordered case)?

Summary

Bulk = Edge I = I♯