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 Edge Edge void Bulk 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. Analogy: torus � = sphere (differ by genus). Contributors to the field: Kane, Mele, Zhang, Moore; Fr¨ ohlich
Bulk-edge correspondence Deformation as interpolation in physical space: �� �� ��� ��� �� �� �� �� ��� ��� �� �� ��� ��� ��� ��� �� �� ��� ��� �� �� �� �� ��� ��� �� �� ��� ��� ��� ��� �� �� ��� ��� �� �� �� �� ��� ��� �� �� topological insulator interpolating material ordinary insulator ◮ Gap must close somewhere in between. Hence: Interface states at Fermi energy.
Bulk-edge correspondence Deformation as interpolation in physical space: �� �� ��� ��� �� �� �� �� ��� ��� �� �� ��� ��� ��� ��� �� �� ��� ��� �� �� �� �� ��� ��� �� �� ��� ��� ��� ��� �� �� ��� ��� �� �� �� �� ��� ��� �� �� topological insulator interpolating material void ◮ 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 Edge void Bulk 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. Winding turning w ( D ) := ( 2 π ) − 1 � angles a t b dancers 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 (˜ D 1 ) − w (˜ D 2 ) ∈ 2 Z 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 ∈ 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 , . . . } ◮ 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). 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 )) ◮ Fermionic time-reversal symmetry: Θ : C N → C N ◮ Θ is anti-unitary and Θ 2 = − 1; ◮ For all k ∈ S 1 , H ( − k ) = Θ H ( k )Θ − 1 where Θ also denotes the map induced on ℓ 2 ( Z ; C N ) . Likewise 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 . 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
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 At fixed k , map gap to S 1 \ { 1 } and bands to 1 ∈ S 1 : Edge Index is index of a rueda.
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 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 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 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
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