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

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

Bulk-edge correspondence Edge void Bulk In a nutshell: Termination of bulk of a topological insulator implies edge states

Bulk-edge correspondence 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:

Bulk-edge correspondence 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, resp. to the Edge.

Bulk-edge correspondence 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, resp. to the Edge. ◮ Bulk-edge duality: Can it be shown that the two indices agree?

Bulk-edge correspondence. Done? 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, resp. 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.; Essin & Gurarie

Bulk-edge correspondence. Today 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, resp. to the Edge. Done differently. ◮ Bulk-edge duality: Can it be shown that the two indices agree? Done differently.

Introduction Rueda de casino Hamiltonians Indices Further results

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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

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

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. 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) ◮ translation invariant in the vertical direction

Bulk Hamiltonian Hamiltonian on the lattice Z × Z (plane) ◮ translation invariant in the vertical direction ◮ period may be assumed to be 1:

Bulk Hamiltonian 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.

Bulk Hamiltonian 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, . . . )

Bulk Hamiltonian 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

Bulk Hamiltonian 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 ∈ S 1 := R / 2 π Z

Bulk Hamiltonian 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 ∈ S 1 := R / 2 π Z End up with wave-functions ψ = ( ψ n ) n ∈ Z ∈ ℓ 2 ( Z ; C N ) and Bulk Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V n ( k ) ψ n � � H ( k ) ψ with V n ( k ) = V n ( k ) ∗ ∈ M N ( C ) (potential) A ( k ) ∈ GL ( N ) (hopping)

Bulk Hamiltonian 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 ∈ S 1 := R / 2 π Z End up with wave-functions ψ = ( ψ n ) n ∈ Z ∈ ℓ 2 ( Z ; C N ) and Bulk Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V n ( k ) ψ n � � H ( k ) ψ with V n ( k ) = V n ( k ) ∗ ∈ M N ( C ) (potential) A ( k ) ∈ GL ( N ) (hopping): Schr¨ odinger 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 , . . . } ◮ translation invariant as before (hence Bloch reduction) Wave-functions ψ ∈ ℓ 2 ( N ; C N ) and Edge Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V ♯ H ♯ ( k ) ψ � � n ( k ) ψ n

Edge Hamiltonian Hamiltonian on the lattice N × Z (half-plane) with N = { 1 , 2 , . . . } ◮ translation invariant as before (hence Bloch reduction) Wave-functions ψ ∈ ℓ 2 ( N ; C N ) and Edge Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V ♯ H ♯ ( k ) ψ � � n ( k ) ψ n which ◮ agrees with Bulk Hamiltonian outside of collar near edge (width n 0 ) V ♯ n ( k ) = V n ( k ) , ( n > n 0 )

Edge Hamiltonian Hamiltonian on the lattice N × Z (half-plane) with N = { 1 , 2 , . . . } ◮ translation invariant as before (hence Bloch reduction) Wave-functions ψ ∈ ℓ 2 ( N ; C N ) and Edge Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V ♯ H ♯ ( k ) ψ � � n ( k ) ψ n which ◮ agrees with Bulk Hamiltonian outside of collar near edge (width n 0 ) V ♯ n ( k ) = V n ( k ) , ( n > n 0 ) ◮ has Dirichlet boundary conditions: for n = 1 set ψ 0 = 0

Edge Hamiltonian Hamiltonian on the lattice N × Z (half-plane) with N = { 1 , 2 , . . . } ◮ translation invariant as before (hence Bloch reduction) Wave-functions ψ ∈ ℓ 2 ( N ; C N ) and Edge Hamiltonian n = A ( k ) ψ n − 1 + A ( k ) ∗ ψ n + 1 + V ♯ H ♯ ( k ) ψ � � n ( k ) ψ n which ◮ agrees with Bulk Hamiltonian outside of collar near edge (width n 0 ) V ♯ n ( k ) = V n ( k ) , ( n > n 0 ) ◮ has Dirichlet boundary conditions: for n = 1 set ψ 0 = 0 Note: σ ess ( H ♯ ( k )) ⊂ σ ess ( H ( k )) , but typically σ disc ( H ♯ ( k )) �⊂ σ disc ( H ( k ))

Graphene as an example Hamiltonian is nearest neighbor hopping on honeycomb lattice a ) b ) B A n 1 n 2 n 2 n 1 � a 1 + � a 2 � � � a 2 � a 2 a 1 a 1 (a) zigzag, resp. (b) armchair boundaries Dimers ( N = 2).

Graphene as an example Hamiltonian is nearest neighbor hopping on honeycomb lattice a ) b ) B A n 1 n 2 n 2 n 1 � a 1 + � a 2 � � � a 2 � a 2 a 1 a 1 (a) zigzag, resp. (b) armchair boundaries Dimers ( N = 2). For (b): � 0 � ψ A � 1 � � 0 1 � ∈ C N = 2 , n ψ n = A ( k ) = − t , V n ( k ) = − t ψ B e i k 0 1 0 n 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 ∈ S 1 : µ / ∈ σ ( H ( k ))

General assumptions ◮ Gap assumption: Fermi energy µ lies in a gap for all k ∈ S 1 : µ / ∈ σ ( H ( k )) ◮ Fermionic time-reversal symmetry: Θ : C N → C N ◮ Θ is anti-unitary and Θ 2 = − 1; ◮ Θ induces map on ℓ 2 ( Z ; C N ) , pointwise in n ∈ Z ; ◮ For all k ∈ S 1 , H ( − k ) = Θ H ( k )Θ − 1 Likewise for H ♯ ( k )

Elementary consequences of H ( − k ) = Θ H ( k )Θ − 1 ◮ σ ( H ( k )) = σ ( H ( − k )) . Same for H ♯ ( k ) .

Elementary consequences of H ( − k ) = Θ H ( k )Θ − 1 ◮ 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 ) = Θ H ( 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 = Θ H Θ − 1 ( H = H ( k ) or H ♯ ( k )) Hence any eigenvalue is even degenerate (Kramers).

Elementary consequences of H ( − k ) = Θ H ( 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 = Θ H Θ − 1 ( H = H ( k ) or H ♯ ( k )) Hence any eigenvalue is even degenerate (Kramers). Indeed ⇒ H (Θ ψ ) = E (Θ ψ ) H ψ = E ψ = and Θ ψ = λψ , ( λ ∈ C ) is impossible: − ψ = Θ 2 ψ = ¯ λ Θ ψ = ¯ λλψ ( ⇒⇐ )

Elementary consequences of H ( − k ) = Θ H ( 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 = Θ H Θ − 1 ( H = H ( k ) or H ♯ ( k )) Hence any eigenvalue is even degenerate (Kramers). k ∈ S 1 π 0 E ∈ R µ − 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 µ ��������� ��������� ��������� ��������� 0 k π Bands, Fermi line, edge states Definition: Edge Index I ♯ = parity of number of eigenvalue crossings

The edge index The spectrum of H ♯ ( k ) ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� symmetric on − π ≤ k ≤ 0 µ ��������� ��������� ��������� ��������� 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 Schr¨ odinger equation ( H ( k ) − z ) ψ = 0 (as a 2nd order difference equation) has 2 N solutions ψ = ( ψ n ) n ∈ Z , ψ n ∈ C N .

Towards the bulk index Let z ∈ C . The Schr¨ odinger equation ( H ( k ) − z ) ψ = 0 (as a 2nd order difference equation) has 2 N solutions ψ = ( ψ n ) n ∈ Z , ψ n ∈ C N . ∈ σ ( H ( k )) . Then Let z / E z , k = { ψ | ψ solution, ψ n → 0 , ( n → + ∞ ) } has ◮ dim E z , k = N .

Towards the bulk index Let z ∈ C . The Schr¨ odinger equation ( H ( k ) − z ) ψ = 0 (as a 2nd order difference equation) has 2 N solutions ψ = ( ψ n ) n ∈ Z , ψ n ∈ C N . ∈ σ ( H ( k )) . Then Let z / E z , k = { ψ | ψ solution, ψ n → 0 , ( n → + ∞ ) } has ◮ dim E z , k = N . ◮ E ¯ z , − k = Θ E z , k

The bulk index k ∈ S 1 Im z π 0 µ E = Re z − π Loop γ and torus T = γ × S 1 Vector bundle E with base T ∋ ( z , k ) , fibers E z , k , and Θ 2 = − 1.

The bulk index k ∈ S 1 Im z π 0 µ E = Re z − π Loop γ and torus T = γ × S 1 Vector bundle E with base T ∋ ( z , k ) , fibers E z , k , and Θ 2 = − 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 ∈ S 1 Im z π 0 µ E = Re z − π Loop γ and torus T = γ × S 1 Vector bundle E with base T ∋ ( z , k ) , fibers E z , k , and Θ 2 = − 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 ∈ S 1 Im z π 0 µ E = Re z − π Loop γ and torus T = γ × S 1 Vector bundle E with base T ∋ ( z , k ) , fibers E z , k , and Θ 2 = − 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 ∈ S 1 Im z π 0 µ E = Re z − π Loop γ and torus T = γ × S 1 Vector bundle E with base T ∋ ( z , k ) , fibers E z , k , and Θ 2 = − 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 ◮ torus ϕ = ( ϕ 1 , ϕ 2 ) ∈ T = ( R / 2 π Z ) 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 ◮ torus ϕ = ( ϕ 1 , ϕ 2 ) ∈ T = ( R / 2 π Z ) 2 with cut (figure) ϕ 2 ϕ ϕ 2 = π ϕ 2 = 0 ϕ 1 cut − ϕ