Edge universality in interacting 2 d topological insulators Marcello Porta Joint with: G. Antinucci (UZH) and V. Mastropietro (Milan)
Summary • Introduction: edge transport in noninteracting quantum Hall systems and time-reversal invariant systems. Bulk-edge duality. • Many-body quantum systems. Results: Edge transport coefficients for quantum Hall and TRI systems. Interacting bulk-edge correspondence, Haldane relations. • Sketch of the proof: Renormalization group and Ward identities. • Conclusions. Marcello Porta Edge universality October 3, 2018 1 / 22
Introduction Introduction: noninteracting systems Marcello Porta Edge universality October 3, 2018 1 / 22
Introduction Integer quantum Hall effect • Bulk topological order in condensed matter systems is deeply related to the emergence of gapless edge modes. • Example. Integer quantum Hall effect [von Klitzing et al. ’80] 2 d insulators exposed to strong magnetic field and in-plane electric field. Marcello Porta Edge universality October 3, 2018 2 / 22
Introduction Integer quantum Hall effect • Bulk topological order in condensed matter systems is deeply related to the emergence of gapless edge modes. • Example. Integer quantum Hall effect [von Klitzing et al. ’80] 2 d insulators exposed to strong magnetic field and in-plane electric field. Linear response: J = σE + o ( E ) with σ = conductivity matrix: � � n 0 σ = 2 π , n ∈ Z . − n 0 2 π Marcello Porta Edge universality October 3, 2018 2 / 22
Introduction Integer quantum Hall effect: theory • Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ 2 ( Z 2 ; C M ). Suppose that σ ( H ) is gapped, µ = Fermi level ∈ gap( H ). µ R Marcello Porta Edge universality October 3, 2018 3 / 22
Introduction Integer quantum Hall effect: theory • Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ 2 ( Z 2 ; C M ). Suppose that σ ( H ) is gapped, µ = Fermi level ∈ gap( H ). � ⊕ T 2 dk ˆ • For simplicity, H ( x ; y ) ≡ H ( x − y ). Bloch decomp.: H = H ( k ) Let ˆ P µ ( k ) = χ ( ˆ H ( k ) ≤ µ ) = Fermi projector. Thouless et al. ’82: � dk P µ ( k )] ∈ 1 (2 π ) 2 Tr C M ˆ P µ ( k )[ ∂ k 1 ˆ P µ ( k ) , ∂ k 2 ˆ 2 π Z σ 12 = i T 2 σ 12 = Chern number of Bloch bundle: E B = { ( k, u ) ∈ T 2 × C M | u ∈ Ran ˆ P µ ( k ) } Marcello Porta Edge universality October 3, 2018 3 / 22
Introduction Integer quantum Hall effect: theory • Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ 2 ( Z 2 ; C M ). Suppose that σ ( H ) is gapped, µ = Fermi level ∈ gap( H ). � ⊕ T 2 dk ˆ • For simplicity, H ( x ; y ) ≡ H ( x − y ). Bloch decomp.: H = H ( k ) Let ˆ P µ ( k ) = χ ( ˆ H ( k ) ≤ µ ) = Fermi projector. Thouless et al. ’82: � dk P µ ( k )] ∈ 1 (2 π ) 2 Tr C M ˆ P µ ( k )[ ∂ k 1 ˆ P µ ( k ) , ∂ k 2 ˆ 2 π Z σ 12 = i T 2 σ 12 = Chern number of Bloch bundle: E B = { ( k, u ) ∈ T 2 × C M | u ∈ Ran ˆ P µ ( k ) } • IQHE for general (disordered) systems: Bellissard et al. ’94. σ 12 = Noncommutative Chern number. Avron-Seiler-Simon ’94. σ 12 = index of a pair of projections. Aizenman-Graf ’98. Strong disorder ⇒ Hall plateaux. Marcello Porta Edge universality October 3, 2018 3 / 22
Introduction Edge states in quantum Hall systems • Halperin ’82. Hall phases must come with robust edge currents. • Intuition. For a weak, slowly varying vector potential A , Z ( A ) � e iσ 12 Ω A ∧ dA +irr. = (gap assumption) Z (0) � e iσ 12 Ω ( A + dα ) ∧ d ( A + dα )+irr. = (gauge inv.) Z ( A ) � Z (0) e iσ 12 ∂ Ω dα ∧ A +irr. = (Stokes) σ 12 � = 0 ⇒ The gap assumption cannot be true! Marcello Porta Edge universality October 3, 2018 4 / 22
Introduction Edge states in quantum Hall systems: more precise • Let H be a lattice Schr¨ odinger operator on the cylinder: (0 , L ) ( L, L ) (0 , 0) ( L, 0) Figure: Dotted lines: Dirichlet boundary conditions. Identify vertical sides. Marcello Porta Edge universality October 3, 2018 5 / 22
Introduction Edge states in quantum Hall systems: more precise • Let H be a lattice Schr¨ odinger operator on the cylinder: • Let H p the counterpart of H with periodic b.c.. Hyp.: H p is gapped. Marcello Porta Edge universality October 3, 2018 5 / 22
Introduction Edge states in quantum Hall systems: more precise • Let H be a lattice Schr¨ odinger operator on the cylinder: • Let H p the counterpart of H with periodic b.c.. Hyp.: H p is gapped. σ ( H ) might differ from σ ( H p ) by the presence of edge states. � ⊕ T 1 dk 1 ˆ H ( k 1 ), ˆ H ( k 1 ) = 1 d Hamiltonian. Spectrum of ˆ Figure: H = H ( k 1 ). • Red curve: eigenvalue branch ε ( k 1 ), with eigenstates (edge modes) with ξ x 2 ( k 1 ) ∼ e − cx 2 . ϕ x ( k 1 ) = e ik 1 x 1 ξ x 2 ( k 1 ) , Marcello Porta Edge universality October 3, 2018 5 / 22
Introduction The bulk-edge correspondence • Bulk-edge duality: relation between σ 12 of H p and the edge states of H . ω e � σ 12 = 2 π e with ω e = ± 1 (chirality of the edge state.) 1 ( b ) : σ 12 = − 1 Figure: ( a ) : σ 12 = 2 π , 2 π , ( c ) : σ 12 = 0. Marcello Porta Edge universality October 3, 2018 6 / 22
Introduction The bulk-edge correspondence • Bulk-edge duality: relation between σ 12 of H p and the edge states of H . ω e � σ 12 = 2 π e with ω e = ± 1 (chirality of the edge state.) 1 ( b ) : σ 12 = − 1 Figure: ( a ) : σ 12 = 2 π , 2 π , ( c ) : σ 12 = 0. • Rigorous results for noninteracting systems: Hatsugai, ’93: Translation invariant systems. Schulz-Baldes et al. ’00: Disordered systems (with bulk gap). Graf et al. ’02: Anderson localization regime. Marcello Porta Edge universality October 3, 2018 6 / 22
Introduction Time-reversal invariant systems • Quantum Hall systems are an example of topological insulators. Necessary condition for σ 12 � = 0: breaking of TRS (magnetic field). • Unbroken TRS: charge transport is trivial but spin transport is possible. Edge Edge void Bulk void spin up spin down • Spin Hall effect: Murakami-Nagaosa-Zhang ’03, ... (Fr¨ ohlich et al. ’93.) Model: Kane-Mele ’05 . Discovery: Bernevig-Hughes-Zhang ’06 (theory), K¨ onig et al. ’07 . Marcello Porta Edge universality October 3, 2018 7 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ Figure: σ ( ˆ H ( k 1 )) for the Kane-Mele model. σ ( ˆ H ( k 1 )) = σ ( ˆ H ( − k 1 )). • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . Marcello Porta Edge universality October 3, 2018 8 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ π k • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . Marcello Porta Edge universality October 3, 2018 8 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ π k • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . Marcello Porta Edge universality October 3, 2018 8 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ π k • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . Marcello Porta Edge universality October 3, 2018 8 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ π k • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . Marcello Porta Edge universality October 3, 2018 8 / 22
Introduction Edge spin transport � ⊕ T 1 dk 1 ˆ • Gapped TRI model on a cylinder, Hamiltonian H = H ( k 1 ). H ( k 1 ) = Θ − 1 ˆ H ( − k 1 )Θ, with Θ 2 = − 1, Θ antiunitary. TRS: ˆ π k • Eigenvalues at k 1 = − k 1 are even degenerate (Kramers degeneracy). ⇒ (edge) Z 2 classification of H : parity of # of pairs of edge modes at µ . • Bulk Z 2 classif. is also possible (no direct connection with transport). • Graf-P. ’13: bulk-edge duality for TRI systems. Marcello Porta Edge universality October 3, 2018 8 / 22
Many-body quantum systems Many-body quantum systems Marcello Porta Edge universality October 3, 2018 8 / 22
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