Bulk-edge duality for topological insulators

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Topological Phases of Quantum Matter Erwin Schrödinger Institute, Vienna September 8-12, 2014

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Topological Phases of Quantum Matter Erwin Schrödinger Institute, Vienna September 8-12, 2014 joint work with Marcello Porta thanks to Yosi Avron

Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

Topological insulators: first impressions Insulator in the Bulk: Excitation gap For independent electrons: band gap at Fermi energy

Topological insulators: first impressions Insulator in the Bulk: Excitation gap For independent electrons: band gap at Fermi energy Time-reversal invariant fermionic system

Topological insulators: first impressions Insulator in the Bulk: Excitation gap For independent electrons: band gap at Fermi energy Time-reversal invariant fermionic system void Edge Bulk Edge void spin up spin down 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.

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

Bulk-edge correspondence void Edge 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. Done? void Edge 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 void Edge 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 Quantum Hall effect

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

Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

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,... ), totalling N Bloch reduction by quasi-momentum k S 1 := R/2πZ End up with wave-functions ψ = (ψ n ) n Z l 2 (Z; C N ) and Bulk Hamiltonian ( H(k)ψ ) n = A(k)ψ n 1 + A(k) ψ n+1 + V n (k)ψ n with V n (k) = V n (k) M N (C) (potential) A(k) GL(N) (hopping)

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 ( H (k)ψ ) n = A(k)ψ n 1 + A(k) ψ n+1 + V 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 )

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) = ΘH(k)Θ 1 Likewise for H (k)

Elementary consequences of H( k) = ΘH(k)Θ 1 σ(h(k)) = σ(h( k)). Same 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.

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). Indeed Hψ = Eψ = H(Θψ) = E(Θψ) and Θψ = λψ, (λ C) is impossible: ψ = Θ 2 ψ = λθψ = λλψ ( )

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). k S 1 π 0 µ E R π Bands, Fermi line (one half fat), edge states

Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

The edge index The spectrum of H (k) symmetric on π k 0 00000000 11111111 000000000 111111111 000000000 111111111 000 111 000 111 µ 000000000 111111111 000000000 111111111 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 00000000 11111111 000000000 111111111 000000000 111111111 000 111 000 111 µ Definition: Edge Index 000000000 111111111 000000000 111111111 0 k π Bands, Fermi line, edge states 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 = (ψ n ) n Z, ψ n C 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 = (ψ n ) n Z, ψ n C N. Let z / σ(h(k)). Then has E z,k = {ψ ψ solution, ψ n 0, (n + )} dim E z,k = N.

The bulk index Im z π k S 1 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 Im z π k S 1 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)

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/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 torus ϕ = (ϕ 1, ϕ 2 ) T = (R/2πZ) 2 with cut (figure) ϕ 2 ϕ 2 ϕ 1 ϕ 2 = π ϕ 2 = 0 cut ϕ 2 a (compatible) section of the frame bundle of E the transition matrices T (ϕ 2 ) 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 ϕ 2 ϕ 2 = π ϕ 2 = 0 ϕ 2 Θ 0 T (ϕ 2 ) = T 1 ( ϕ 2 )Θ 0 Only half the cut (0 ϕ 2 π) matters for T (ϕ 2 ) At time-reversal invariant points, ϕ 2 = 0, π, Θ 0 T = T 1 Θ 0 Eigenvalues of T come in pairs λ, λ 1 : Θ 0 (T λ) = T 1 (1 λt )Θ 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 ϕ 2 ϕ 2 = π ϕ 2 = 0 ϕ 2 Θ 0 T (ϕ 2 ) = T 1 ( ϕ 2 )Θ 0 For 0 ϕ 2 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

Idea of proof of I = I Im z k S 1 0 π µ E = Re z Fermi line (one half fat) edge states torus π

Idea of proof of I = I Im z k S 1 0 π µ E = Re z Fermi line (one half fat) edge states torus π Edge states define a rueda

Idea of proof of I = I Im z k S 1 0 π µ E = Re z Fermi line (one half fat) edge states torus π Edge states define a rueda Bulk torus cut along Fermi line;

Idea of proof of I = I Im z k S 1 0 π µ E = Re z Fermi line (one half fat) edge states torus π Edge states define a rueda Bulk torus cut along Fermi line; frame bundle admits a section, which is aware of the edge ;

Proof of Theorem: Dual ruedas S 1 000000 111111 000000 111111 1 000000 111111 0000000 1111111 k = 0 k = 0 k = π k Edge rueda: edge eigenvalues; Bulk rueda: eigenvalues of T (k) k k = π

Proof of Theorem: Dual ruedas S 1 000000 111111 000000 111111 1 000000 111111 0000000 1111111 k = 0 k = 0 k = π k Edge rueda: edge eigenvalues; Bulk rueda: eigenvalues of T (k) The two ruedas share intersection points. k k = π

Proof of Theorem: Dual ruedas S 1 000000 111111 000000 111111 1 000000 111111 0000000 1111111 k = 0 k = 0 k = π k Edge rueda: edge eigenvalues; Bulk rueda: eigenvalues of T (k) k k = π The two ruedas share intersection points. Hence indices are equal

Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

A simpler result in a simpler setting: Quantum Hall effect in doubly periodic lattices

A simpler result in a simpler setting: Quantum Hall effect in doubly periodic lattices 00000000 11111111 000000000 111111111 000000000 111111111 0000 1111 000 111 µ π 000000000 111111111 000000000 111111111 k π Definition: Edge Index N = signed number of eigenvalue crossings

A simpler result in a simpler setting: Quantum Hall effect in doubly periodic lattices Definition: Edge Index N = signed number of eigenvalue crossings Bulk: Brillouin zone (torus) of quasi-momenta (κ, k). Energy bands j = 0, 1,...; eigenstates κ, k j form a line bundle E j with torus as base space (Bloch bundle). Chern number ch(e j ) is computed by cutting torus to cylinder taking global section of E j on cylinder defines transition matrix T (phase factor) along circle ch(e j ) = winding number of T.

A simpler result in a simpler setting: Quantum Hall effect in doubly periodic lattices Definition: Edge Index N = signed number of eigenvalue crossings Bulk: ch(e j ) is the Chern number of the Bloch bundle E j of the j-th band. Bulk index is sum over filled bands.

Duality via scattering k Brillouin zone (κ, k) Energy band ε j (κ, k) κ

Duality via scattering k Minima κ (k) and maxima κ + (k) of energy band ε j (κ, k) in κ at fixed k ε j (κ) k κ π κ κ + κ π 0000000 1111111

Duality via scattering k Minima κ (k) and maxima κ + (k) of energy band ε j (κ, k) in κ at fixed k ε j (κ) k κ κ + κ π κ κ + κ π 0000000 1111111

Duality via scattering k Minima κ (k) and maxima κ + (k) of energy band ε j (κ, k) in κ at fixed k κ κ + κ

Duality via scattering k Maxima κ + (k) κ + κ

Duality via scattering k 01 01 00 11 00 11 Maxima κ + (k) with semi-bound states (to be explained) 01 κ + κ

Duality via scattering k 01 01 00 11 00 11 01 κ + κ

Duality via scattering ε j (κ) At fixed k: Energy band ε j (κ, k) and the line bundle E j of Bloch states π κ π 0000000 1111111

Duality via scattering π κ π 0000000 1111111 Line indicates choice of a section κ of Bloch states (from the given band). No global section in κ R/2πZ is possible, as a rule.

Duality via scattering States κ above the solid line are left movers (ε j (κ) < 0) π κ κ + κ π 0000000 1111111

Duality via scattering They are incoming asymptotic (bulk) states for scattering at edge (from inside) π κ κ + κ π 0000000 1111111 01 01 01 01 01 01 01 01 01 01 κ in

Duality via scattering 0 01 1 11 00 01 π κ κ + κ π 0000000 1111111 Scattering determines section out of right movers above line 01 01 01 01 01 01 01 01 01 01 out κ in

Duality via scattering 0 01 1 11 00 01 π κ κ + κ π 0000000 1111111 01 01 01 01 01 01 01 01 01 01 out κ in Scattering matrix out = S + κ as relative phase between two sections of the same fiber (near κ + )

Duality via scattering 0 01 1 11 00 01 π κ κ + κ π 0000000 1111111 Chern number computed by sewing ch(e j ) = N (S + ) N (S ) with N (S ± ) the winding of S ± = S ± (k) as k = π... π.

Duality via scattering 00 11 0 01 1 00 11 01 π κ κ + κ π 0000000 1111111 As κ κ +, whence in = κ κ + out = S + κ κ + (up to phase) their limiting span is that of κ +, d κ dκ κ+ (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.

Duality via scattering k 00 11 0 01 1 11 00 01 01 01 00 11 00 11 π κ κ + κ π 0000000 1111111 01 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(p 2 + V ) 0 E arg S E=0+ = 2πN

Levinson s theorem Recall from two-body potential scattering: The scattering phase at threshold equals the number of bound states σ(p 2 + 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 000 111 000 111 0000 1111 000 111 k k ε(k) = ε j (κ + (k), k)

Levinson s theorem (relative version) Spectrum of edge Hamiltonian 000 111 000 111 0000 1111 000 111 k k 1 k 2 ε(k) = ε j (κ + (k), k) k lim arg S +(ε(k) δ) k 2 = ±2π δ 0 k 1

Proof j 0 + 1 000000000 111111111 000000000 111111111 000000000 111111111 0000 1111 000 111 µ 00 11 000000000 111111111 0000000000 1111111111 j 0 π k π N = N (S (j 0) + ) ( (j = N (S 0 +1) ) ) = = j 0 j=0 j 0 j=0 N (S (j) + ) N (S(j) ) ch(e j ) (N (S (1) ) = 0)

Summary 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