Bulk-edge duality for topological insulators

Transcription

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

2 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

3 Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

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

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

6 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

7 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.

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

9 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:

10 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.

11 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 duality: Can it be shown that the two indices agree?

12 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

13 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.

14 Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

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30 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

31 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

32 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?

33 The index of a Rueda A snapshot of the dance

34 The index of a Rueda A snapshot of the dance Dance D as a whole D

35 The index of a Rueda A snapshot of the dance Dance D as a whole D

36 The index of a Rueda A snapshot of the dance Dance D as a whole D

37 The index of a Rueda A snapshot of the dance Dance D as a whole D

38 The index of a Rueda A snapshot of the dance Dance D as a whole D

39 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

40 Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

41 Bulk Hamiltonian Hamiltonian on the lattice Z Z (plane) Z Z n =

42 Bulk Hamiltonian Hamiltonian on the lattice Z Z (plane) Z Z n =

43 Bulk Hamiltonian Hamiltonian on the lattice Z Z (plane) translation invariant in the vertical direction

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

45 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.

46 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,... )

47 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

48 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

49 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)

50 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) : Schrödinger eq. is the 2nd order difference equation

51 Edge Hamiltonian Hamiltonian on the lattice N Z (half-plane) with N = {1, 2,...} Z N n =

52 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

53 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 )

54 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 ) any (self-adjoint, local) boundary conditions

55 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 ) any (self-adjoint, local) boundary conditions Note: σ ess (H (k)) σ ess (H(k)), but typically σ disc (H (k)) σ disc (H(k))

56 General assumptions Gap assumption: Fermi energy µ lies in a gap for all k S 1 : µ / σ(h(k))

57 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)

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

59 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.

60 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).

61 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 ( )

62 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

63 Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

64 The edge index The spectrum of H (k) symmetric on π k µ k π Bands, Fermi line, edge states Definition: Edge Index I = parity of number of eigenvalue crossings

65 The edge index The spectrum of H (k) symmetric on π k µ Definition: Edge Index 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.

66 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.

67 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.

68 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. E z, k = ΘE z,k

69 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.

70 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)

71 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) Definition: Bulk Index I = I(E)

72 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) Definition: Bulk Index I = I(E) What s behind the theorem? How is I(E) defined?

73 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) Definition: Bulk Index I = I(E) What s behind the theorem? How is I(E) defined? Aside... a rueda...

74 Time-reversal invariant bundles on the torus Theorem In general, vector bundles (E, T, Θ) can be classified by an index I(E) = ±1

75 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

76 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

77 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 ϕ 1 ϕ 2 = π ϕ 2 = 0 cut ϕ

78 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 ϕ 1 ϕ 2 = π ϕ 2 = 0 cut ϕ a (compatible) section of the frame bundle of E

79 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

80 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 Θ 0 T (ϕ 2 ) = T 1 ( ϕ 2 )Θ 0, (ϕ 2 S 1 ) with Θ 0 : C N C N antilinear, Θ 2 0 = 1

81 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

82 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

83 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 Phases λ/ λ pair up (Kramers degeneracy)

84 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 Phases λ/ λ pair up (Kramers degeneracy) For 0 ϕ 2 π, phases λ/ λ form a rueda, D

85 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.

86 Main result Theorem Bulk and edge indices agree: I = I

87 Main result Theorem Bulk and edge indices agree: I = I I = +1: ordinary insulator I = 1: topological insulator

88 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

89 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

90 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;

91 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 ;

92 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 ; transition matrix defines rueda.

93 Proof of Theorem: Dual ruedas S k = 0 k = 0 k = π k Edge rueda: edge eigenvalues; Bulk rueda: eigenvalues of T (k) k k = π

94 Proof of Theorem: Dual ruedas S 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

95 Proof of Theorem: Dual ruedas S 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

96 Introduction Rueda de casino Hamiltonians Indices Quantum Hall effect

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

98 A simpler result in a simpler setting: Quantum Hall effect in doubly periodic lattices µ π k π Definition: Edge Index N = signed number of eigenvalue crossings

99 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).

100 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).

101 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 )

102 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.

103 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.

104 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: N = j ch(e j )

105 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: N = j ch(e j ) (cf. Hatsugai) Here via scattering and Levinson s theorem.

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

107 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

108 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

109 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

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

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

112 Duality via scattering k κ + κ

113 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

114 Duality via scattering π κ π 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.

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

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

117 Duality via scattering π κ κ + κ π Scattering determines section out of right movers above line out κ in

118 Duality via scattering π κ κ + κ π out κ in Scattering matrix out = S + κ as relative phase between two sections of the same fiber (near κ + )

119 Duality via scattering π κ κ + κ π out κ in Scattering matrix out = S + κ as relative phase between two sections of the same fiber (near κ + ) Likewise S near κ.

120 Duality via scattering π κ κ + κ π 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.

121 Duality via scattering π κ κ + 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 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.

122 Duality via scattering k π κ κ + κ π κ + κ As a function of k, semi-bound states occur exceptionally.

123 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

124 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)

125 Levinson s theorem (relative version) Spectrum of edge Hamiltonian k k ε(k) = ε j (κ + (k), k)

126 Levinson s theorem (relative version) Spectrum of edge Hamiltonian 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

127 Proof j µ 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)

128 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