Braiding fluxes in Pauli Hamiltonians Anyons for anyone J. Avron O. Kenneth Department of Physics, Technion Montreal, 2014 Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 1 / 34
Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 2 / 34
Motivation Non abelian anyons Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 3 / 34
Motivation Non abelian anyons Gates Unitary: | ψ � �→ U| ψ � U ⇒ dim ( H ) = 2 n n − qubits = Universal single qubit gates: � 1 � 1 � � 1 0 1 √ = , = H e i π/ 4 e i π/ 4 1 − 1 0 2 Universal two qubits: • Z Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 4 / 34
Motivation Non abelian anyons Anyons and quantum computing Desiderata Fault tolerance Gap H = Protected subspace Topological quantum computing—non-abelain anyons Lindner & Stern, Science Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 5 / 34
Motivation Non abelian anyons Non abelian anyons Theory and experiment Theory Localized modes of interacting) fermions or spins Theoretical realization Anyons in FQHE √ Majoranas: electron / 2 Experiment Fractional charges in FQHE, Evidence for Majorana Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 6 / 34
Motivation Aharonov Casher Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 7 / 34
Motivation Aharonov Casher Aharonov Casher Topological Zero modes Geometric setting: Φ T Pauli equation: spin 1 / 2, g = 2 � Φ T = 1 � 2 ≥ 0 , � ( − i ∇ − A ) · σ B dx ∧ dy 2 π Zero modes: Zero modes Continuum Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 8 / 34
Motivation Aharonov Casher Aharnonov Casher holomorphy Decoupling in 2-D: � � 0 ∂ z − iA z ( − i ∇ − A ) · σ = − 2 i ∂ z − i ¯ ¯ A z 0 Zero modes: � � ( ψ, 0 ) t = 0 , = ⇒ ( ¯ ∂ z − i ¯ ( − i ∇ − A ) · σ A ) ψ = 0 � �� � 1 − st order pde Holomorphy: z ) ∈ Ker ( ¯ ∂ z − i ¯ ψ ( z , ¯ ψ ( z , ¯ A ) ∋ P ( z ) z ) ���� holomorphic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 9 / 34
Motivation Aharonov Casher Aharonov and Casher Index Poissons’ equation–source B log ψ 0 = i ∂ z ¯ ∂ z ¯ ∂ z A ���� ���� ∆ B Polynomial decay: ∆ − 1 = 1 z →∞ | z | − Φ T , ψ 0 = exp (∆ − 1 B ) − → 2 π log z Aharonov-Casher Index theorem: Number of zero modes D = ⌈ Φ T ⌉ − 1 Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 10 / 34
Motivation Aharonov Casher Confined and free zero modes Φ a > 1 vs Φ a < 1 Two types of Charge-Flux composite Φ a > 1 Φ b = Φ b ′ = 3 / 4 Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 11 / 34
Braiding fluxes Braiding fluxes Gates from braiding fluxes curvature Φ a Φ b What gates can you make by braiding fluxons? Catch 22: Holonomy without curvature! Φ a ∈ R ; Think of 1 / 2 < Φ a < 1 No gap protection Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 12 / 34
Braiding fluxes Adiabatic evolution AB-Anyons Adiabatic evolution for moving fluxes Gapless Gauge issues Defrosting Confined zero modes Super Critical fluxons; Φ a > 1 Aharonov-Bohm abelian phases e 2 π i Φ 2 Φ 1 Φ 2 Localized zero modes Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 13 / 34
Braiding fluxes Deconfined modes Anyons Holonomy–Abelian & non-abelian curvature & topological curvature Topological if: D = N − 1 Identical fluxes 1 − 1 N < Φ < 1 � 1 − ν � ν ν = e − 2 π i Φ Burau rep of braid group : , 1 0 Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 14 / 34
Braiding fluxes Zero modes Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 15 / 34
Braiding fluxes Zero modes Aharonov and Casher Fluxons Log-Superposition: ∂ z log ψ = i ¯ ¯ A = ⇒ ( A 1 + A 2 , ψ 1 ψ 2 ) (Φ 1 , ζ 1 ) ζ 2 = position Φ 3 = flux (Φ 4 , ζ 4 ) Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 16 / 34
Braiding fluxes Zero modes Weak individuals, Φ a < 1, strong community, Φ T > 1 Point fluxes (Φ 1 , ζ 1 ) (Φ 2 , ζ 2 ) (Φ 5 , ζ 5 ) (Φ 4 , ζ 4 ) Zero modes; 0 < Φ a < 1 � ( z − ζ a ) − Φ a , ψ ( z ; ζ ) = P ( z ) deg ( P ) < Φ T − 1 ���� a polynom Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 17 / 34
Braiding fluxes Adiabatically Moving fluxes Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 18 / 34
Braiding fluxes Adiabatically Moving fluxes Bad defrosting Dead frozen Defrosted Hamiltonian � � ζ �→ ζ ( t ) H ( A ζ ) �→ H A ζ ( t ) � �� � control Wrong sources � A = J E fluxon (Φ , ζ ) E Current Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 19 / 34
Braiding fluxes Adiabatically Moving fluxes Gauge fields of moving flux Defrosting and Gauge freedom E Motion generates weak electric fields E = − v × B v � �� � fluxon localized on fluxon Defrosted potentials A = A ( z − ζ ( t )) , A 0 = − v · A ( z − ζ ( t )) � �� � Inertial frame Closed path in control ζ a = ⇒ closed path in ( A 0 , A ) Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 20 / 34
Braiding fluxes Adiabatically Moving fluxes Topology in Gappless Adiabatic evolution What is the time scale? Gapless. Distance between fluxon defines time scale: time scale = m h ( distance ) 2 , distance = | ζ a − ζ b | � �� � � �� � length scale dim analysis Re ζ Energy Im ζ Control Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 21 / 34
Braiding fluxes Adiabatically Moving fluxes Parallel transport Connection Zero modes: � : Span { z j ψ 0 | j = 0 , . . . , D − 1 } ( z − ζ a ( t )) − Φ a P D , � z | ψ 0 � = ���� � �� � a projection zero modes � �� � ζ a = ζ b = ···⇒| ψ 0 � = ∞ Evolution within P D D � p j ( t ) z j , ψ ( z , t ) = P ( z , t ) ψ 0 , P ( z , t ) = � �� � 0 polynom Connection P D D t ψ = 0 , D t = ∂ t − iA 0 � �� � � �� � No motion covariant derivative Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 22 / 34
Braiding fluxes Metric and connection Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 23 / 34
Braiding fluxes Metric and connection The connection Metric Geometric–independent of time schedule: � i P D d | ψ � = P D d x a · A a | ψ � ���� � �� � a flux displace i ∂ a | ψ � A (non-orthogonal) basis z j | ψ � 0 , j = 0 , . . . , D − 1 Hilbert space metric ( g ) jk ( ζ, ¯ z j z k | ψ 0 � = � ψ 0 | ¯ ζ ) � �� � control Diverges when fluxons collide: ( g ) jk ( ζ a = ζ b = . . . ) = ∞ Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 24 / 34
Braiding fluxes Metric and connection Beauty parlor Connection P ( z , t ) = � D 0 p j ( t ) z j = ⇒ p ( t ) = ( p 0 , . . . , p D − 1 ) g − 1 ( ∂ ζ g ) 0 = ( d + A ) p , A = � �� � semi pure gauge Semi-pure d = ∂ + ¯ g − 1 ( ∂ g ) � = g − 1 d g A = , ∂ � �� � � �� � pure gauge semi pure gauge Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 25 / 34
Braiding fluxes Metric and connection Factorization holomorphic × anti-holomorphic Heuristics z j z k | ψ 0 ( ζ ) � ( g ) jk ( ζ, ¯ ¯ ζ ) = � ψ 0 ( ζ ) | , � �� � � �� � anti − holomorphic holomorphic Factorization of metric g ( ζ, ¯ = Ψ ∗ ( ζ ; Φ) ζ, Φ) G (Φ) Ψ( ζ ; Φ) � �� � � �� � � �� � � �� � D × D D × ( N − 1 ) ( N − 1 ) × ( N − 1 ) ( N − 1 ) × D Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 26 / 34
Braiding fluxes Metric and connection Branch structure Ψ Fluxons and cuts The matrix Ψ � ζ a dz z k ψ 0 ( z ; ζ ) , Ψ ak ( ζ ) = a ∈ 1 , . . . , N − 1 , k ∈ 0 , . . . , D − 1 ξ N ∞ 3 ✻ y Σ 3 t ζ 3 ∞ 2 Σ 2 t ζ 2 ∞ 1 Σ 1 t ζ 1 ∞ 0 = ∞ 3 ✲ x Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 27 / 34
Braiding fluxes Magic Outline Motivation 1 Non abelian anyons Aharonov Casher Braiding fluxes 2 Zero modes Adiabatically Moving fluxes Metric and connection Magic Avron, Kenneth (Technion) Braiding fluxes Montreal 2014 28 / 34
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