A bird’s-eye view on Z 2 topology Domenico Monaco ETH Z¨ urich September 5th, 2018
Motivation Topological insulators in class AII Kitaev’s periodic table Symmetry Dimension AZ T C S 1 2 3 4 5 6 7 8 A 0 0 0 0 Z 0 Z 0 Z 0 Z AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 D 0 1 0 0 0 0 0 Z 2 Z Z Z 2 DIII -1 1 1 0 0 0 0 Z 2 Z 2 Z Z AII -1 0 0 0 Z 2 Z 2 Z 0 0 0 Z CII -1 -1 1 Z 0 Z 2 Z 2 Z 0 0 0 C 0 -1 0 0 0 0 0 Z Z 2 Z 2 Z CI 1 -1 1 0 0 0 0 Z Z 2 Z 2 Z D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 1 / 21 Z 2
Motivation Topological insulators in class AII Kitaev’s periodic table Symmetry Dimension AZ T C S 1 2 3 4 5 6 7 8 A 0 0 0 0 Z 0 Z 0 Z 0 Z AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 D 0 1 0 0 0 0 0 Z 2 Z Z Z 2 DIII -1 1 1 0 0 0 0 Z 2 Z 2 Z Z AII -1 0 0 0 Z 2 Z 2 Z 2 Z 2 Z 0 0 0 Z CII -1 -1 1 Z 0 Z 2 Z 2 Z 0 0 0 C 0 -1 0 0 0 0 0 Z Z 2 Z 2 Z CI 1 -1 1 0 0 0 0 Z Z 2 Z 2 Z D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 1 / 21 Z 2
Motivation Topological insulators in class AII 2D AII: quantum spin Hall insulator z y x Spin Hall effect L m v F v jx m F 0 Z 2 classification [Fu–Kane–Mele 2005–07] normal insulator (trivial phase) vs topological insulator (QSH phase) � � FKM := 1 F − 1 A mod 2 ∈ Z 2 2 π 2 π EBZ ∂ EBZ D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 2 / 21 Z 2
Outline of the presentation 1 TRS topological insulators 2 FKM as a topological obstruction The FMP index The GP index 3 FKM and WZW amplitudes The GT+ index WZW amplitude and square root 4 More on the Z 2 invariant
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Time-reversal symmetric topological insulators d -dimensional TRS topological insulator (class AII) A map P : R d → B ( C M ) (possibly M = ∞ ) such that ◮ P ( k ) = P ( k ) ∗ = P ( k ) 2 is a rank- m orthogonal projection, m = 2 n ◮ k �→ P ( k ) is smooth (at least C 1 ) ◮ k �→ P ( k ) is Z d -periodic: P ( k + λ ) = P ( k ) for λ ∈ Z d � k ∈ BZ ≃ T d ◮ odd/fermionic time-reversal symmetry (TRS): M = 2 N and ∃ antiunitary Θ: C M → C M , Θ 2 = − 1 , such that Θ P ( k ) Θ − 1 = P ( − k ) � k ∈ EBZ ≃ “half a T d ” Example P ( k ) = family of Fermi projections of a gapped, periodic, TRS Hamiltonian, in the Bloch–Floquet representation: P ( k ) = 1 ( −∞ , E F ] ( H ( k )) D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 3 / 21 Z 2
TRS topological insulators Bloch frames Bloch frame A collection Φ( k ) = { φ 1 ( k ) , . . . , φ m ( k ) } ⊂ C M , k ∈ R d , of orthonormal vectors such that m � P ( k ) = | φ a ( k ) � � φ a ( k ) | a =1 Φ is called ◮ smooth if each k �→ φ a ( k ) is smooth ◮ periodic if each k �→ φ a ( k ) is Z d -periodic � 0 � 1 n ◮ TRS if Φ( − k ) = [ΘΦ( k )] ε , with ε := − 1 n 0 D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 4 / 21 Z 2
TRS topological insulators Bloch frames Bloch frame A collection Φ( k ) = { φ 1 ( k ) , . . . , φ m ( k ) } ⊂ C M , k ∈ R d , of orthonormal vectors such that m � P ( k ) = | φ a ( k ) � � φ a ( k ) | a =1 Φ is called ◮ smooth if each k �→ φ a ( k ) is smooth ◮ periodic if each k �→ φ a ( k ) is Z d -periodic � 0 � 1 n ◮ TRS if Φ( − k ) = [ΘΦ( k )] ε , with ε := − 1 n 0 D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 4 / 21 Z 2
TRS topological insulators Bloch frames Bloch frame A collection Φ( k ) = { φ 1 ( k ) , . . . , φ m ( k ) } ⊂ C M , k ∈ R d , of orthonormal vectors such that m � P ( k ) = | φ a ( k ) � � φ a ( k ) | a =1 Φ is called ◮ smooth if each k �→ φ a ( k ) is smooth ◮ periodic if each k �→ φ a ( k ) is Z d -periodic � 0 � 1 n ◮ TRS if Φ( − k ) = [ΘΦ( k )] ε , with ε := − 1 n 0 D. Monaco (Roma Tre) A bird’s-eye view on Z 2 Z 2 topology 5/09/2018 4 / 21 Z 2
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