Platonic QHE Index J. Avron Department of Physics, Technion ESI, - PowerPoint PPT Presentation

Platonic QHE Index J. Avron Department of Physics, Technion ESI, 2014 Hofstadter butterfly: Chern numbers phase diagram Avron (Technion) Platonic QHE ESI 2014 1 / 25

Outline Laughlin pump 1 Fredholm and Relative index 2 Avron (Technion) Platonic QHE ESI 2014 2 / 25

Laughlin pump Laughlin pump φ emf: − ˙ φ ˙ Radial current: σ φ ���� Hall current Charge transported by fluxon: B emf � ˙ σ φ dt = 2 π σ ∈ Z Losing charge in Hilbert space Avron (Technion) Platonic QHE ESI 2014 3 / 25

Laughlin pump Landau in the plane y ¯ z = x + iy , z = x − iy 2 ¯ 2 ∂ = ∂ x − i ∂ y , ∂ = ∂ x + i ∂ y z ∂ + Bz D = ¯ x 4 � � 2 H = 1 p − 1 = 2 D ∗ D + B [ D , D ∗ ] = B 2 B × x 2 , 2 2 � �� � � �� � ladder Landau B Avron (Technion) Platonic QHE ESI 2014 4 / 25

Laughlin pump Holomorphic tricks Lowest Landau level � � ∂ + Bz ¯ 0 = D | ψ � = | ψ � 4 | ψ 0 � = e − B | z | 2 / 4 , D | ψ 0 � = 0 | ψ m � = z m | ψ 0 � , D | ψ m � = 0 , m ∈ Z + m + 1 m Avron (Technion) Platonic QHE ESI 2014 5 / 25

Laughlin pump Spectral Flow Flux as bc φ m + 1 m | ψ � + = e i φ | ψ � − Spectral flow: m → m + φ 2 π Avron (Technion) Platonic QHE ESI 2014 6 / 25

Laughlin pump Spectral flow Hilbert hotel Energy m Flux insertion pushes unit charge (per Landau level) to infinity Avron (Technion) Platonic QHE ESI 2014 7 / 25

Laughlin pump Fredholm Index IndexT = dim ker T − dim ker T ∗ = dim ker T ∗ T − dim ker TT ∗ T ∗ T = ✶ = ⇒ dim ker T ∗ T = 0 TT ∗ = ✶ − | 0 � � 0 | T ∗ ⇒ dim ker TT ∗ = 1 = T Index T = 0 − 1 = − 1 Avron (Technion) Platonic QHE ESI 2014 8 / 25

Laughlin pump Fredholm Index Careless commutators Spec ( TT ∗ ) \ 0 = Spec ( T ∗ T ) \ 0 � 1 − | t j | 2 � � Tr ( ✶ − T ∗ T ) = j dim ker T ∗ T − dim ker TT ∗ = Tr ( ✶ − T ∗ T ) − Tr ( ✶ − TT ∗ ) If ✶ − T ∗ T has a trace Index T = Tr [ T ∗ , T ] Avron (Technion) Platonic QHE ESI 2014 9 / 25

Fredholm and Relative index Relative index Non-commutative Pythagoras C = P − Q , S = P − Q ⊥ � �� � � �� � cos sin Non-commutative Pythagoras C 2 + S 2 = ✶ CS + SC = 0 , � �� � � �� � non − commutative Pythagoras CS = ( P − Q )( P − Q ⊥ ) = P − PQ ⊥ − QP = PQ − QP C 2 = ( P − Q ) 2 = P − QP − PQ + Q = Q ⊥ P + P ⊥ Q C ↔ S ⇔ Q ↔ Q ⊥ Avron (Technion) Platonic QHE ESI 2014 10 / 25

Fredholm and Relative index ∞ − ∞ ∈ Z Comparing infinite projections Relative index: Assume Tr | P − Q | 2 m < ∞ .Then ∀ n ≥ m 2 n + 1 = dim ker ( P − Q − 1 ) − dim ker ( P − Q + 1 ) Tr ( P − Q ) � �� � C Proof: C | ψ � = λ | ψ � ⇒ SC | ψ � = λ S | ψ � � � − C S | ψ � = − λ S | ψ � -1 − λ 0 λ 1 What if S | ψ � = 0 0 = � S ψ | S ψ � = � ψ | ✶ − C 2 | ψ � = 1 − λ 2 � Tr ( P − Q ) 2 n + 1 = λ 2 n + 1 = deg ( 1 ) − deg ( − 1 ) j Avron (Technion) Platonic QHE ESI 2014 11 / 25

Fredholm and Relative index Fredholm=Relative index ( P − Q ) 2 P = ( P − Q ) Q ⊥ P = PQ ⊥ P Theorem: = Tr ( P − Q ) 3 , Q = U ∗ PU Index PUP � �� � U : Range P → Range P T ∗ T = PQP T = PUP , dim ker T ∗ T − dim ker TT ∗ = Tr ( P − UQPQU ∗ ) − PQP ) − Tr ( P ���� ���� ✶ ✶ = Tr PQ ⊥ P − Tr QP ⊥ Q = Tr ( P − Q ) 3 Avron (Technion) Platonic QHE ESI 2014 12 / 25

Fredholm and Relative index Quantized flows Kitaev | U − 1 , 0 | 2 e i α − 1 e i α 0 e i α 1 e i α 2 | U 0 , − 1 | 2     e i α − 1 0 0 0 0 1 0 0 e i α 0 0 0 0 0 0 1 0         e i α 1 0 0 0 0 0 0 1     e i α 2 0 0 0 0 0 0 0 � �� � � �� � Ind = 1 Ind = 0 Quantized flow � � | U jk | 2 − | U jk | 2 ∈ Z j < 0 , k ≥ 0 j ≥ 0 , k < 0 Avron (Technion) Platonic QHE ESI 2014 13 / 25

Fredholm and Relative index Driving and Response Hall effect H ( φ 1 , φ 2 ) = U ( φ 1 , φ 2 ) HU ∗ ( φ 1 , φ 2 ) U ( φ 1 , φ 2 ) = e i ( φ 1 Λ 1 + φ 2 Λ 2 ) V y − → I x Finite voltage, not field Λ j x Avron (Technion) Platonic QHE ESI 2014 14 / 25

Fredholm and Relative index φ 2 control: Voltage U ( φ 2 ) = e i φ 2 Λ 2 ( y ) Λ 2 Driving voltage: x � ∞ � � E · dy = ˙ φ 2 Λ 2 ( ∞ ) − Λ 2 ( −∞ ) ˙ φ 2 ∞ � �� � = 1 = ˙ φ 2 t Voltage Linear response=Adiabatic driving Robust Precise shape of Λ( x ) and φ ( t ) irrelevant Avron (Technion) Platonic QHE ESI 2014 15 / 25

Fredholm and Relative index Currents Conserved currents Unitary family ˙ Λ 1 H ( φ 1 ) = U ( φ 1 ) HU ∗ ( φ 1 ) , U ( φ 1 ) = e i φ 1 Λ 1 Current: Conservation law of charge Λ 1 rigid box − ˙ ∂ 1 H = i [Λ 1 , H ( φ 1 )] = Λ 1 ���� Λ 1 soft box rate of charge in box Avron (Technion) Platonic QHE ESI 2014 16 / 25

Fredholm and Relative index Adiabatic curvature Adiabatic curvature for unitary families Ω 12 = i Tr P (Λ 1 P ⊥ Λ 2 − Λ 2 P ⊥ Λ 1 ) P Proof: � � P [ ∂ 1 P , ∂ 2 P ] P = − P [Λ 1 , P ] , [Λ 2 , P ] P � � = − P [Λ 1 , P ⊥ ] , [Λ 2 , P ⊥ ] P = P (Λ 1 P ⊥ Λ 2 − Λ 2 P ⊥ Λ 1 ) P Avron (Technion) Platonic QHE ESI 2014 17 / 25

Fredholm and Relative index Adiabatic curvature=2 π Kubo P short range ∧ translation invariant, L → infty 2 π Kubo � �� � i Ω 12 = lim L 2 Tr L × L P ( X 1 P ⊥ X 2 − X 2 P ⊥ X 1 ) P L →∞ � L / 2 3 = i � � � � � d 2 x j � P x j � x j + 1 )( x 2 × x 3 ) , x 4 = x 1 L 2 − L / 2 j = 1 Λ 2 0 1 / 2 L x − L / 2 L / 2 0 − 1 / 2 L Avron (Technion) Platonic QHE ESI 2014 18 / 25

Fredholm and Relative index 2 π Kubo=Index Index ( PUP ) P=Fermi-Projection U = z Index ( PUP ) = 2 π Kubo | z | � �� � AB fluxon Landau : | ψ m � ∼ z m e −| z | 2   0 0 0 0 c 1 0 0 0   emf PUP =   B 0 c 2 0 0   0 0 c 3 0 radial current Avron (Technion) Platonic QHE ESI 2014 19 / 25

Fredholm and Relative index Relative index in 2-D U = z Q = UPU ∗ , P − Q = [ P , U ] U ∗ , | z | � � 1 − U ( z 1 ) � z 1 | ( P − Q ) | z 2 � = � z 1 | P | z 2 � U ( z 2 ) 3 � � � � U ( z j ) � Tr ( P − Q ) 3 = � � � � P dz j z j � z j + 1 1 − , z 4 = z 1 U ( z j + 1 ) j = 1 Avron (Technion) Platonic QHE ESI 2014 20 / 25

Fredholm and Relative index P Translation invariant (Ergodic) Relative index= Kubo � Tr ( P − Q ) 3 = d 2 y d 2 z � 0 | P | y � � y | P | z � � z | P | 0 � y × z � �� � 2 π Kubo Elements of proof: Both sides cubic in P Translation invariance: 3 3 � � � � � � � � � � � P � P z j � z j + 1 = z j + a � z j + 1 + a j = 1 j = 1 2 (of 6) integration over an explicit function: 3 � � � U ( z j + a ) � d 2 a 1 − U ( z j + 1 + a ) j = 1 Avron (Technion) Platonic QHE ESI 2014 21 / 25

Fredholm and Relative index Relative index= Kubo The world most complicated formula for area of triangles 3 � � � � U ( z j − a ) �� � � d 2 a d 2 a 1 − = i sin γ j = 2 π i Area ( z 1 , z 2 , z 3 ) U ( z j + 1 − a ) � �� � j = 1 � �� � magic computation Connes: z 2 a Convergence, γ 3 Dimension analysis γ 1 γ 2 Translation invariance Colin de Verdier: � 2 π a in triangle � z 3 γ j = z 1 0 a outside Avron (Technion) Platonic QHE ESI 2014 22 / 25

Fredholm and Relative index Platonic Omissions Localization FQHE Chern Simons K-theory Bulk Edge duality Hofstadter butterfly Diophantine equations Hall viscosity Spin transport Open Systems Avron (Technion) Platonic QHE ESI 2014 23 / 25

Fredholm and Relative index References Thouless, Niu, Kohmoto Avron, Seiler, Simon, Fraas, Graf Frohlich, Wen, Stone Bellissard, Schultz-Baldes, Prodan Aizenman, Graf, Wartzel Hastings and Michalakis Read, Taylor, Haldane Avron (Technion) Platonic QHE ESI 2014 24 / 25

Fredholm and Relative index Acknowledgments R. Seiler, B. Simon, G.M. Graf, M. Fraas, L. Sadun, O. Kenneth, J. Bellissard Avron (Technion) Platonic QHE ESI 2014 25 / 25