Motivations The model Solution Conductance and numerics Conclusion Edge states at spin quantum Hall transitions Roberto Bondesan (LPTENS/IPhT Saclay) With: I. Gruzberg (Chicago), J. Jacobsen (Paris), H. Obuse (Karlsruhe), H. Saleur (Saclay) Workshop at GGI, Florence: New quantum states of matter in and out of equilibrium 24/04/2012 Based on BJS: arXiv:1101.4361, BGJOS: arXiv:1109.4866 Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Outline 1 Motivations Anderson transitions: IQHE, SQHE Edge states 2 The model Quantum network models Quantum-classical localization 3 Solution Superspins and σ -models Universality and critical exponents 4 Conductance and numerics Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Anderson transitions • Disordered non-interacting electrons • Symmetry classification [Dyson ’62, Altland,Zirnbauer ’97] • 10 classes (WD, Chiral, BdG) • E. g. : IQHE in class A, e it H ∈ U( N ) • Escaping localization in 2d [Evers,Mirlin ’08] • MIT Localized Extended • Topological phases Localized Localized Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Integer Quantum Hall Effect • 2DEG at high B , low T 1 Disorder h • Plateaus ρ xy = e 2 ν [Von Klitzing Nobel ’85] 2 Chiral edge states ν = 2 Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Integer Quantum Hall Effect • 2DEG at high B , low T 1 Disorder h • Plateaus ρ xy = e 2 ν [Von Klitzing Nobel ’85] 2 Chiral edge states ν = 2 Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Spin quantum Hall effect (SQHE) • d + id disordered superconductors [Senthil,Marston,Fisher ’99] • e it H ∈ Sp(2 N ), class C • Topological superconductor in 2d: σ spin ∈ 2 Z • � = QSH ( ∈ A II , T 2 = − 1, [Kane,Mele ’05] ) • Some aspects exactly solvable! Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Low-energy field theories • Non linear σ -model [Wegner ’79, Efetov ’83] : 1 � d 2 z ∂ † µ Z † S = α ∂ µ Z α 2 g 2 σ • IQHE criticality: topological term [Pruisken ’84] S top = i θ N [ Z ] • At θ = π (mod 2 π ), g σ = O (1), LogCFT c = 0 (?) σ H ∼ θ σ 0 ∼ 1 / g 2 σ • Bulk exponents, ξ , etc. indep. of θ (if θ = π (mod 2 π )) Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Edge states and the θ -angle • Bulk-boundary σ L σ R #(edge states) = σ R − σ L • Edge states as θ increased • 1D QED: quark-antiquark screen F ∝ θ [Coleman ’75, Affleck ’85] • Boundary properties dep. on exact value θ [Xiong,Read,Stone ’97] Aim of the talk Edge states for SQHE Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Edge states and the θ -angle • Bulk-boundary σ L σ R #(edge states) = σ R − σ L • Edge states as θ increased • 1D QED: quark-antiquark screen F ∝ θ [Coleman ’75, Affleck ’85] • Boundary properties dep. on exact value θ [Xiong,Read,Stone ’97] Aim of the talk Edge states for SQHE Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Network models: quantum percolation • Chalker-Coddington model [Chalker,Coddington ’88] : + + − − − + + � √ 1 − t 2 � t • , √ : S = 1 − t 2 − t A B ⇒ U e , e ′ = S e ′ , e U e , U e ∈ U (1) Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Edge states in CC model • Extreme limits 1 t c = √ t = 0: QH state ν = 1 t = 1: Insulator 2 • Higher plateaus: chiral extra edge channels [BGJOS ’12] : Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Edge states in CC model • Extreme limits 1 t c = √ t = 0: QH state ν = 1 t = 1: Insulator 2 • Higher plateaus: chiral extra edge channels [BGJOS ’12] : Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Network model SQHE with edge channels • U e ∈ SU (2) � � � 1 − t 2 • , , , t x : S x = 1 ⊗ x � 1 − t 2 − t x x A B L R • L = m , R = n : m n 2 L Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Network model SQHE with edge channels • U e ∈ SU (2) � � � 1 − t 2 • , , , t x : S x = 1 ⊗ x � 1 − t 2 − t x x A B L R • L = m , R = − n : m n 2 L Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Network model SQHE with edge channels • U e ∈ SU (2) � � � 1 − t 2 • , , , t x : S x = 1 ⊗ x � 1 − t 2 − t x x A B L R • L = − m , R = n : m n 2 L Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Network model SQHE with edge channels • U e ∈ SU (2) � � � 1 − t 2 • , , , t x : S x = 1 ⊗ x � 1 − t 2 − t x x A B L R • L = − m , R = − n : m n 2 L Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Disorder average I [Gruzberg,Read,Ludwig ’99,Beamond,Cardy,Chalker ’02,Mirlin,Evers,Mildenberger ’03,Cardy ’04] • G ( e , e ′ , z ) = � e | (1 − z U ) − 1 | e ′ � = � γ ( e , e ′ ) · · · zU e j s j · · · • SUSY path integral • x σ ( e ) , η σ ( e ) , σ = ↑ , ↓ • Lattice action: W [ x , η ] = zx ∗ ( e ′ ) U e ′ , e x ( e ) + z η ∗ ( e ′ ) U e ′ , e η ( e ) • �•� = � D µ ( x , η ) • exp( W [ x , η ]) ⇒ G ( e , e ′ , z ) = � x ( e ) x ∗ ( e ′ ) � Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Disorder average II d U = projects S ( C 2 ⊗ C 1 | 1 ) onto SU(2)-inv. : � 1 � � 1 Truncation: Ψ = 1 , 2 ( x ↑ η ↓ − x ↓ η ↑ ) , η ↑ η ↓ √ 2 Dilute-dense mapping: e ′ e 2 1 : e ′ e 1 2 i ( e ′ � � � j Ψ ∗ � i Ψ α ′ i ) δ α 1 ,α ′ 1 δ α 2 ,α ′ 2 S 11 S 22 − δ α 1 ,α ′ 2 δ α 2 ,α ′ 1 S 12 S 21 α j ( e j ) • Example: Ψ = 1: (1 − t 2 ) + t 2 = 1 • IQH: U(1)-inv. is infinite-dim. space, but [Ikhlef,Fendley,Cardy ’11] Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Disorder average II d U = projects S ( C 2 ⊗ C 1 | 1 ) onto SU(2)-inv. : � 1 � � 1 Truncation: Ψ = 1 , 2 ( x ↑ η ↓ − x ↓ η ↑ ) , η ↑ η ↓ √ 2 Dilute-dense mapping: e ′ e 2 1 : e ′ e 1 2 i ( e ′ � � � j Ψ ∗ � i Ψ α ′ i ) δ α 1 ,α ′ 1 δ α 2 ,α ′ 2 S 11 S 22 − δ α 1 ,α ′ 2 δ α 2 ,α ′ 1 S 12 S 21 α j ( e j ) • Example: Ψ = 1: (1 − t 2 ) + t 2 = 1 • IQH: U(1)-inv. is infinite-dim. space, but [Ikhlef,Fendley,Cardy ’11] Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion New geometrical model ⇒ Classical loops (fug. = 1): • Decomposition: = (1 − t 2 + t 2 A ) A A = (1 − t 2 + t 2 B ) B B = (1 − t 2 + t 2 L ) L L = (1 − t 2 + t 2 R ) R R Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion New geometrical model ⇒ Classical loops (fug. = 1): • Decomposition: = (1 − t 2 + t 2 A ) A A = (1 − t 2 + t 2 B ) B B = (1 − t 2 + t 2 L ) L L = (1 − t 2 + t 2 R ) R R Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Conductance • Landauer: PCC g = Tr tt † , t = � e out | (1 − U ) − 1 | e in � • No replicas: g = 2 � η ↓ ( e out ) η ↑ ( e out ) η ∗ ↑ ( e in ) η ∗ ↓ ( e in ) � = 2 P ( e in , e out ) e ′ ∈ C out P ( e , e ′ ) g = 2 � � ⇒ e ∈ C in ⇒ Loops ≡ transport What next: Solve loop model, then go back to SQH. Roberto Bondesan Edge states at SQH transitions
Motivations The model Solution Conductance and numerics Conclusion Conductance • Landauer: PCC g = Tr tt † , t = � e out | (1 − U ) − 1 | e in � • No replicas: g = 2 � η ↓ ( e out ) η ↑ ( e out ) η ∗ ↑ ( e in ) η ∗ ↓ ( e in ) � = 2 P ( e in , e out ) e ′ ∈ C out P ( e , e ′ ) g = 2 � � ⇒ e ∈ C in ⇒ Loops ≡ transport What next: Solve loop model, then go back to SQH. Roberto Bondesan Edge states at SQH transitions
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