Platonic QHE Chern numbers J. Avron Department of Physics, - PowerPoint PPT Presentation

Platonic QHE Chern numbers J. Avron Department of Physics, Technion ESI, 2014 Hofstadter butterfly: Chern numbers phase diagram Avron (Technion) Platonic QHE ESI 2014 1 / 32

Outline Prelims 1 Physics Math Platonic QHE 2 Virtual work & Hall conductance 3 Chern=Kubo 4 Avron (Technion) Platonic QHE ESI 2014 2 / 32

Prelims Physics Outline Prelims 1 Physics Math Platonic QHE 2 Virtual work & Hall conductance 3 Chern=Kubo 4 Avron (Technion) Platonic QHE ESI 2014 3 / 32

Prelims Physics Aharonov-Bohm flux tubes Quantum flux φ � � winds origin φ A · dx = 0 otherwise � Φ 0 = 2 π = 2 π e ���� fundamental Amoeba Flux through a micro-organism 100 µ Avron (Technion) Platonic QHE ESI 2014 4 / 32

Prelims Physics AB Periodicity Flux tube modifies boundary condition: A θ = φδ ( x ∈ cut ) | ψ � + = e i φ | ψ � − b.c. 2 π periodic in φ : | ψ � + = e i φ | ψ � − AB periodicity H ( φ + 2 π ) = UH ( φ ) U ∗ Avron (Technion) Platonic QHE ESI 2014 5 / 32

Prelims Math Outline Prelims 1 Physics Math Platonic QHE 2 Virtual work & Hall conductance 3 Chern=Kubo 4 Avron (Technion) Platonic QHE ESI 2014 6 / 32

Prelims Math Projections: P Orthogonal projections P 2 = P P = P ∗ , � �� � � �� � projection orthogonal d � � � � � � , P = � ψ j ψ j � ψ i | ψ j � = δ ij 1 P ⊥ = ✶ − P � �� � complementary PP ⊥ = P ⊥ P = 0 Avron (Technion) Platonic QHE ESI 2014 7 / 32

Prelims Math Family of projections Paradigm Berry: Spin in magnetic field B B P ( B ) = ✶ + H (ˆ B ) B = B ˆ , | B | 2 � � H ( B ) = B · σ = 1 B 3 B 1 − iB 2 B 1 + iB 2 − B 3 2 P ( B ) sick at B = 0 ⇔ H ( 0 ) degenerates Avron (Technion) Platonic QHE ESI 2014 8 / 32

Prelims Math Family of projections Parameter=control space φ ∈ parameter space=control space P ( φ ) : ( parameter space ) �→ smooth projections Parameter space Hilbert space P ( φ ) φ moving frame P ⊥ ( φ ) Avron (Technion) Platonic QHE ESI 2014 9 / 32

Prelims Math dP Motion of projections P 2 = P � �� � matrices P dP + dP P = dP Corrolary dP P = ( ✶ − P ) dP = P ⊥ dP P dP P = P P ⊥ dP = 0 � �� � = 0 Kato P dP P = 0 Avron (Technion) Platonic QHE ESI 2014 10 / 32

Prelims Math Kato evolution Unitary evolution in evolving subspaces Kato’s unitary evolution P = U P 0 U ∗ , U = U ( φ ) , P = P ( φ ) � �� � Notion of parallel transport φ Who generates U ? i dU = A U ���� generator φ ′ Avron (Technion) Platonic QHE ESI 2014 11 / 32

Prelims Math Kato’s evolution Commutator equation Generator satisfies commutator equation dP = i [ A , P ] Proof: P 0 = U ∗ P U = ⇒ 0 = ( dU ∗ ) P U + U ∗ dP U + U ∗ P dU 0 = U ( dU ∗ ) P + dP + P ( dU ) U ∗ � �� � � �� � − ( dU ) U ∗ − i A A : Not unique! Ambiguity: commutant ( P ) Avron (Technion) Platonic QHE ESI 2014 12 / 32

Prelims Math Kato’s evolution Generator Commutator equation for A : dP = i [ A , P ] A Generator A = i ( dU ) U ∗ = − i [ dP , P ] � �� � � �� � Definition Generator Verify: � � i [ A , P ] = [ dP , P ] , P = ( dP ) P − 2 P ( dP ) P + P dP = dP � �� � = 0 Avron (Technion) Platonic QHE ESI 2014 13 / 32

Prelims Math Parallel transport Connection Parallel transport: No motion in P | ψ � = P | ψ � 0 = P d | ψ � , � �� � � �� � vector ∈ P no − motion � � d | ψ � = d P | ψ � = ( dP ) | ψ � + Pd | ψ � � �� � = 0 parameter space = ( dP ) P | ψ � = [ dP , P ] | ψ � � �� � i A Covariant derivative: � � D = d − i A , D | ψ � = 0 ⇔ Pd | ψ � = 0 Avron (Technion) Platonic QHE ESI 2014 14 / 32

Prelims Math Parallel transport Berry’s phase | ψ 1 � 1-D projection: P = | ψ � � ψ | Parallel transport: 0 = P d | ψ � = | ψ � � ψ | d ψ � D | ψ � = 0 Parallel transport = ⇒ No local Berry’s phase | ψ 0 � 0 = � ψ | d ψ � − 1 2 d ( � ψ | ψ � ) e i β | ψ 0 � | ψ 0 � � �� � = 1 = � ψ | d ψ � − � d ψ | ψ � 2 = i Im � ψ | d ψ � � �� � Berry ′ s phase Avron (Technion) Platonic QHE ESI 2014 15 / 32

Prelims Math Curvature Failure of parallel transport Parallel transport is path dependent: | ψ 1 � � = | ψ 1 � Curvature | ψ 0 � Curvature=Failure of parallel transport � � Ω jk = i [ D j , D k ] = ∂ j A k − ∂ k A j − i [ A j , A k ] � �� � � �� � definition Non − abelian magnetic fields Avron (Technion) Platonic QHE ESI 2014 16 / 32

Prelims Math Curvature for projections P ( dP )( dP ) P Curvature= iP ( dp )( dP ) P Ω jk = i [ D j , D k ] = i [ ∂ j P , ∂ k P ] � �� � definition Proof: � � D j , D k P = [ Pd j , Pd k ] P = P ( ∂ j P )( ∂ k P ) − P ( ∂ k P )( ∂ j P ) = P [ ∂ j P , ∂ k P ] Avron (Technion) Platonic QHE ESI 2014 17 / 32

Prelims Math Curvature 1-D projection 1-D: P = | ψ � � ψ | Berry ′ s curvature � �� � � � Ω jk P = i [ ∂ j P , ∂ k P ] P = i � ∂ j ψ | ∂ k ψ � − � ∂ k ψ | ∂ j ψ � P Example: Spin 1/2 P = ✶ + ˆ H H = Φ · σ, 2 Φ ℓ d Φ j d Φ k Ω jk P = ε jk ℓ P 4 | Φ | 3 � �� � 1 / 2 spherical angle Avron (Technion) Platonic QHE ESI 2014 18 / 32

Prelims Math Gauss Bonnet Geometry meets topology Gauss-Bonnet: Gaussian curvature & genus � 1 Ω dS = 2 ( 1 − genus ) 2 π ���� Curvature Avron (Technion) Platonic QHE ESI 2014 19 / 32

Prelims Math Chern numbers Proof for torus (TKNN) P ( φ 1 , φ 2 ) periodic e i γ | ψ � e i β | ψ � | ψ ( φ 1 , φ 2 ) � periodic up to phase: | ψ ( 0 , 0 ) � = e − i α | ψ ( 2 π, 0 ) � φ 2 = e − i γ | ψ ( 0 , 2 π ) � = e − i β | ψ ( 2 π, 2 π ) � e i α | ψ � | ψ � φ 1 Angle counted mod 2 π ( α − 0 ) mod 2 π + ( β − α ) mod 2 π + ( γ − β ) mod 2 π + ( 0 − γ ) mod 2 π = 0 Chern numbers � � � d ψ | d ψ � = i � ψ | d ψ � ∈ 2 π Z i T ∂ T Avron (Technion) Platonic QHE ESI 2014 20 / 32

Prelims Math Chern numbers Projections Chern numbers � i Chern ( P , M ) = Tr P [ ∂ j P , ∂ k P ] d Φ j d Φ k ∈ Z , 2 π M M : 2-D compact manifold (no bdry= ∂ M = 0) Chern ( P , M ) invariant under smooth deformations of P P singular at eigenvalue crossing–dim P jumps Avron (Technion) Platonic QHE ESI 2014 21 / 32

Prelims Math Chern numbers Facts Chern ( 0 , M ) = Chern ( ✶ , M ) = 0 Chern ( P 1 ⊕ P 2 , M ) = Chern ( P 1 , M ) + Chern ( P 2 , M ) Avron (Technion) Platonic QHE ESI 2014 22 / 32

Prelims Math Chern numbers From sphere to ball H = B · σ, B = ( B x , B y , B z ) Φ � �� � 3 − D space Linear map of parameter space: B = g Φ , Φ = (Φ x , Φ y , Φ z ) det g � = 0 Chern B 3 P (Φ) = ✶ + ˆ H � g jk Φ k σ j , H (Φ) = 2 j , k = 1 Chern ( P ) = sgn det g Avron (Technion) Platonic QHE ESI 2014 23 / 32

Prelims Math Chern numbers What is counted? Contracting into the solid torus − + − + Simon � Chern ( P , T ) = sgn det g (Φ d ) � �� � degeneracies Avron (Technion) Platonic QHE ESI 2014 24 / 32

Prelims Math QHE Driving and response φ 1 emf loop Hall bar φ 2 Hall current loop Platonic Driving: emf = ˙ φ 1 Response: Hall current = ∂ H ∂φ 2 H ( φ 1 , φ 2 ) : Periodic, nondegenerate, hermitian matrix Avron (Technion) Platonic QHE ESI 2014 25 / 32

Prelims Math Variations on a theme Bloch momenta & controls φ 1 φ 2 Periodic Multiply connected ( k 1 , k 2 ) conserved ( φ 1 , φ 2 ) controls Bloch momenta Fluxes Brillouin Zone Aharnonov-Bohm period ∞ noninteracting (gapped) Interacting (finite) fermions Avron (Technion) Platonic QHE ESI 2014 26 / 32

Prelims Math Example: 3 × 3 matrix function Hofstadter Butterfly with flux 1 / 3 H ( φ ) = e i φ 1 + e i φ 2 T S + h . c . ���� ���� translation shift     0 1 0 1 0 0 1/3 T = 0 0 1 S = 0 0 , ω     1 0 0 0 0 ω ¯ � �� � � �� � lattice translation S = FTF ∗ Hofstadter Model B = 1 / 3 ω = e 2 π i / 3 Avron (Technion) Platonic QHE ESI 2014 27 / 32

Virtual work & Hall conductance Virtual work Q-observable H : ( parameter space φ ) �→ Hamiltonian Virtual work δ H = dH ( φ ) δφ d φ � �� � φ observable Loop current: Virtual work of Aharonov-Bohm flux I = dH d φ Avron (Technion) Platonic QHE ESI 2014 28 / 32

Virtual work & Hall conductance Charge transport Time dependent Feynman-Hellman Virtual work=Rate of Berry’s phase � � � ψ | ∂ φ H | ψ � = ∂ t i � ψ | ∂ φ ψ � � �� � � �� � Virtual work Berry ′ s phase Schrödinger i ∂ t | ψ � = H ( φ ) | ψ � Pf: time − independent = 0 � �� � � ψ | ∂ φ H | ψ � = ∂ φ � ψ | H | ψ � − � ∂ φ ψ | H | ψ � − � ψ | H | ∂ φ ψ � � �� � � �� � i | ∂ t ψ � − i � ∂ t ψ | � � = ∂ t i � ψ | ∂ φ ψ � Avron (Technion) Platonic QHE ESI 2014 29 / 32