Universal edge transport in interacting Hall systems Marcello Porta - PowerPoint PPT Presentation

Universal edge transport in interacting Hall systems Marcello Porta University of Z¨ urich, Institute for Mathematics Joint work with G. Antinucci (UZH) and V. Mastropietro (Milan)

Hall effect • Hall effect (Edwin Hall 1879): J 1 E J 2 B • Linear response (weak E ): J 1 = σ 11 E , J 2 = σ 21 E . σ 11 = longitudinal conductivity, σ 21 = − σ 12 = Hall conductivity. Marcello Porta Edge transport July 6, 2017 1 / 17

Integer quantum Hall effect • von Klitzing ’80. Experiment on GaAs-heterostructures (insulators). σ 21 3 measurement 2 1 classical prediction ρ σ 11 ( ρ = density of charge carriers.) IQHE: σ 21 = e 2 h · n , n ∈ Z . • Theory for noninteracting systems: Laughlin ’81, Thouless et al. ’82 ... Rigorous results: Avron-Seiler-Simon ’83, Bellissard et al. ’94, Aizenman-Graf ’98 ... Marcello Porta Edge transport July 6, 2017 2 / 17

Bulk-edge correspondence • Halperin ’82. Hall phases come with robust edge currents. Figure: Magnetic field points out of the screen. • Edge currents are necessary to preserve gauge invariance. Essential feature of the gauge theory of states of matter [Fr¨ ohlich ’91] Marcello Porta Edge transport July 6, 2017 3 / 17

Bulk-edge correspondence: rigorous results • Halperin ’82. Hall phases come with robust edge currents. • Hatsugai ’93; Schulz-Baldes et al ’00, Graf et al. ’02: bulk-edge duality. σ 12 = e 2 � ω e ω e = (chirality of the edge state) ∈ {− 1 , +1 } h e Figure: ( a ) : σ 12 = e 2 ( b ) : σ 12 = − e 2 h , h , ( c ) : σ 12 = 0. • Graf-P. ’13: extension to quantum spin Hall systems. Marcello Porta Edge transport July 6, 2017 4 / 17

Bulk-edge correspondence: rigorous results • Halperin ’82. Hall phases come with robust edge currents. • Hatsugai ’93; Schulz-Baldes et al ’00, Graf et al. ’02: bulk-edge duality. σ 12 = e 2 � ω e ω e = (chirality of the edge state) ∈ {− 1 , +1 } h e Figure: ( a ) : σ 12 = e 2 ( b ) : σ 12 = − e 2 h , h , ( c ) : σ 12 = 0. • Graf-P. ’13: extension to quantum spin Hall systems. • Many-body interactions? Marcello Porta Edge transport July 6, 2017 4 / 17

Interacting systems Interacting systems Marcello Porta Edge transport July 6, 2017 4 / 17

Interacting systems Lattice fermions • Interacting electron gas on Λ L = [0 , L ] 2 ⊂ Z 2 . Fock space Hamiltonian: � � � � a + y ) a − H = x,ρ H ρρ ′ ( � x, � x,ρ ′ + λ n � x,ρ v ρρ ′ ( � x, � y ) n � y,ρ ′ − µ N � � ρ,ρ ′ ρ,ρ ′ � x,� y � x,� y H, v finite-ranged, ρ ∈ { 1 , . . . , M } = internal degree of freedom. • H is equipped with cylindric boundary conditions: (0 , L ) ( L, L ) (0 , 0) ( L, 0) Figure: Dotted lines: Dirichlet boundary conditions. • Translation invariance in x 1 direction: H ρρ ′ ( � y ) ≡ H ρρ ′ ( x 1 − y 1 ; x 2 , y 2 ). x, � Marcello Porta Edge transport July 6, 2017 5 / 17

Interacting systems Lattice fermions • Interacting electron gas on Λ L = [0 , L ] 2 ⊂ Z 2 . Fock space Hamiltonian: � � � � a + y ) a − y,ρ ′ − µ N H = x,ρ H ρρ ′ ( � x, � x,ρ ′ + λ n � x,ρ v ρρ ′ ( � x, � y ) n � � � � x,� y ρ,ρ ′ � x,� y ρ,ρ ′ H, v finite-ranged, ρ ∈ { 1 , . . . , M } = internal degree of freedom. • Assumption. For periodic b.c., σ ( H (per) ) is gapped. Instead, edge states might appear in σ ( H ). • ε ( k 1 ) = eigenvalue branch of ˆ H ( k 1 ). The corresponding edge state is: with ξ x 2 ( k 1 ) ∼ e − cx 2 . x ( k 1 ) = e ik 1 x 1 ξ x 2 ( k 1 ) , ϕ � ˆ Marcello Porta Edge transport July 6, 2017 5 / 17

Interacting systems Edge transport coefficients • Let µ ∈ σ ( H (per) ). • Edge transport. Perturb at distance ≤ a from x 2 = 0. Linear response? ( L, L ) (0 , L ) (0 , a ′ ) (0 , a ) (0 , 0) ( L, 0) • Interesting physical observables: charge density and current density, � � � a + x,ρ a − � e i [ ia + x ) a − n � x = x,ρ , j � x = � e i ,ρ H ρρ ′ ( � x + � e i , � x,ρ ′ +h.c.] . � � � x + � � ρ i =1 , 2 ρ,ρ ′ Their support will be x 2 ≤ a ′ , with L ≫ a ′ ≫ a ≫ 1. Marcello Porta Edge transport July 6, 2017 6 / 17

Interacting systems Edge transport coefficients • Let µ ∈ σ ( H (per) ). • Edge transport. Perturb at distance ≤ a from x 2 = 0. Linear response? • Edge charge susceptibility: � 0 dt e tη � κ a,a ′ ( η, p 1 ) n ≤ a ′ n ≤ a � [ˆ � − p 1 ] � � � ∞ = i p 1 ( t ) , ˆ −∞ p 1 = � n ≤ a � ∞ = lim β,L →∞ L − 1 Tr · e − β ( H− µ N ) / Z β,L . � � ˆ x 2 ≤ a ˆ n p 1 ,x 2 , � · � � Marcello Porta Edge transport July 6, 2017 6 / 17

Interacting systems Edge transport coefficients • Let µ ∈ σ ( H (per) ). • Edge transport. Perturb at distance ≤ a from x 2 = 0. Linear response? • Edge charge susceptibility: � 0 dt e tη � κ a,a ′ ( η, p 1 ) n ≤ a ′ n ≤ a � � [ˆ − p 1 ] � � ∞ � = i p 1 ( t ) , ˆ −∞ p 1 = � n ≤ a � ∞ = lim β,L →∞ L − 1 Tr · e − β ( H− µ N ) / Z β,L . � � ˆ x 2 ≤ a ˆ n p 1 ,x 2 , � � · � • Charge conductance and Drude weight: � 0 dt e tη � j ≤ a ′ G a,a ′ ( η, p 1 ) n ≤ a p 1 ( t ) , ˆ = i � � [ˆ 1 , − p 1 ] � � � ∞ −∞ � � 0 � ∞ + ∆ a � dt e tη � D a,a ′ ( η, p 1 ) j ≤ a j ≤ a ′ � [ˆ 1 ,p 1 ( t ) , ˆ � � = − i 1 , − p 1 ] � −∞ with ∆ a = � � [ X ≤ a j ≤ a 1 , ˆ � � 1 , 0 ] � � ∞ . Spin transport: n → n ↑ − n ↓ , j → j ↑ − j ↓ . Marcello Porta Edge transport July 6, 2017 6 / 17

Interacting systems Bulk transport coefficients • The response to bulk perturbations is expected to be edge-independent. • Kubo formula: bulk conductivity matrix. � � 0 � i dt e ηt � � (per) � (per) σ ij = lim � [ j i ( t ) , j j ] � � � + � � � [ X i , j j ] � � ∞ ∞ η η → 0 + −∞ j = � � x � � (per) = lim β,L →∞ L − 2 Tr · e − β ( H (per) − µ N ) / Z (per) � � � · � � j � x , ∞ � β,L • Bachmann-de Roeck-Fraas ’17, Monaco-Teufel ’17: derivation of Kubo formula for gapped many-body lattice models. • Hastings-Michalakis ’14, Giuliani-Mastropietro-P. ’15: quantization of σ 12 for λ � = 0. [HM]: gapped H . [GMP]: λ ≪ gap( H ). • Giuliani-Jauslin-Mastropietro-P. ’16: Hall transitions in the Haldane-Hubbard model: λ ≫ gap( H ). Marcello Porta Edge transport July 6, 2017 7 / 17

Interacting systems Noninteracting edge transport • Let λ = 0. Define p = ( η, p 1 ). Edge transport coefficients: ω e p 1 � κ a,a ′ ( p ) + R a,a ′ = ( p ) (susceptivity) κ 2 π − iη + v e p 1 e ω e − iη � + R a,a ′ G a,a ′ ( p ) = ( p ) (conductance) G 2 π − iη + v e p 1 e | v e | − iη � D a,a ′ ( p ) + R a,a ′ = D ( p ) (Drude weight) − iη + v e p 1 2 π e v e = velocity of edge state, ω e = sgn( v e ), lim a,a ′ →∞ lim p → 0 R a,a ′ ( p ) = 0. ♯ • Bulk-edge correspondence: ω e � p 1 → 0 G a,a ′ ( η, p 1 ) G = a,a ′ →∞ lim lim η → 0 + lim = 2 π e = σ 12 Marcello Porta Edge transport July 6, 2017 8 / 17

Interacting systems Interacting edge transport: Bosonization • Edge transport coefficients ∼ correlations of ∂ x 0 φ e , ∂ x 1 φ e , with φ e = bosonic free field with covariance [Wen, ’90]: − p � = δ ee ′ ω e 1 � ˆ p ˆ φ e ′ − φ e + p 1 ( − iη + v e p 1 ) . 2 π • Bosonization. Mapping of interacting, relativistic fermions in free bosons with interaction-dependent parameters: “ n e → ∂ x 1 φ e ”. • The mapping in free bosons breaks down for nonrelativistic models. Nonlinearities produce quartic interactions among bosons. • Exact computation of the transport coefficients for λ � = 0? • From now on: one edge state per edge (spin degenerate). Marcello Porta Edge transport July 6, 2017 9 / 17

Interacting systems Reference model: Chiral Luttinger liquid • Chiral Luttinger liquid. Massless 1 + 1-dim. Grassmann field: � dk S (ref) (2 π ) 2 ψ + k,σ ( − ik 0 + vk 1 ) ψ − ( ψ ) = Z k,σ . 0 R 2 Noninteracting density-density correlation function: − ip 0 − vp 1 1 � (ref) = − � � � ˆ n p ; ˆ n − p � � . 2 π | v | Z 2 − ip 0 + vp 1 • Expect. Lattice edge states effectively described by interacting χLL : � � dp dk S (ref) ( ψ ) = Z (2 π ) 2 ψ + k,σ ( − ik 0 + vk 1 ) ψ − k,σ + gZ 2 (2 π ) 2 ˆ w ( p )ˆ n p ˆ n − p R 2 R 2 for suitable bare parameters Z , v , g (and suitable regularization scheme). Marcello Porta Edge transport July 6, 2017 10 / 17