N ON - LINEAR T HEORY OF FQH E DGE : F RACTIONALLY C HARGED S OLITONS - PowerPoint PPT Presentation

N ON - LINEAR T HEORY OF FQH E DGE : F RACTIONALLY C HARGED S OLITONS , E MERGENT T OPOLOGY IN N ON - LINEAR W AVES , Q UANTUM H YDRODYNAMICS OF FQH LIQUID P . Wiegmann (Discussions with friends: Abanov, Bettelheim, Cappelli) Phys. Rev. Lett. 108, 206810 (2012) Florence May 25, 2012 1 / 1

Messages - Waves on the Edge of FQH are essentially non-linear; - Emergence of quantization in non-linear dynamics; - FQH - hydrodynamics "Hall-viscosity" in the bulk propels to the boundary (a universal corrections to Chern-Simon "theory") - Relation between FQHE and CFT - revised Only Laughlin’s states (for now). 2 / 1

FQHE -L AUGHLIN ’ S STATE ( S ) Particles on a plane in a quantized magnetic field (with a strong Coulomb Interaction) Ψ 0 ( z 1 ,..., z N ) = � ∆( z 1 ,..., z N ) � β e − � i | z i | 2 / 4 ℓ 2 B � - ∆ = i � = j ( z i − z j ) - VanDerMonde determinant - ℓ B -magnetic length; - ν = 1 /β - is a filling fraction; - β = 1 - IQHE; β = 3- FQHE. Important features: - Wave-function is holomorphic; - Degree of zero at z i → z j is larger than 1; 3 / 1

S PECTRUM IN THE BULK : - The ground state is β = ν − 1 - degenerate; - All excitations are gapped: ∆ 1 / 3 ∼ 10 − 30 K , kT ≪ ∆ 1 / 3 ≪ ħ h ω c - Coherent States: deformation of Laughlin’s state by a holomorphic function Ψ 0 ( z 1 ,..., z N ) = � ∆( z 1 ,..., z N ) � β e − 1 � i | z i | 2 / 2 ℓ 2 B 2 � i V ( z i ) , Ψ V ( z 1 ,..., z N ) = Ψ 0 e - Singularities of V are vortices (or "quasi-holes") σ = − 1 4 π ∆ V = Real 4 / 1

E DGE STATES : FQHE IN A CONFINING POTENTIAL - Potential well lifts a degeneracy: � H 0 → H = H 0 + U ( | r i | ) i - Low energy states emerge. They are localized on an edge → Edge States; - Smooth potential: Curvature of the potential is small compared to the gap but a slope is larger than electric field ℓ 2 B ∇ 2 ℓ 2 B ∇ y U ≫ e 2 y U ≪ ∆ ν , F IGURE : Boundary waves: the boundary layer is highlighted y y(x) x 5 / 1

L INEAR E DGE S TATES THEORY (W EN , 1991) - Density is a chiral field: � e ikx ρ k , ρ k = ρ † ρ ( x ) = − k , k [ ρ k , ρ l ] = ν k δ k + l , ( ∂ t − c 0 ∇ x ) ρ = 0, h − 1 ℓ 2 - c 0 = ħ B |∇ y U | is a slope of the potential well (non-universal); - factor ν proliferates to the exponent in edge tunneling - Common believe ( I disagree with): c = 1 - CFT of free bosons with a compactification radius ν = β − 1 . 6 / 1

Propagation of a wave-packet The linear theory does not answer a question: how does a smooth non-equilibrium state (a wave packet) propagate? ρ − c 0 ∇ ρ = 0, ˙ wave equation, ρ ( x , t ) = ρ ( x − c 0 t , t = 0 ) The shape does not change !? A new scale must be included in the theory; New scale: ∆ 1 / 3 ≪ ħ h ω c - energy of a hole or a vortex (non-universal scale); Most phenomena do not depend on the scale and are universal; 7 / 1

N ON - LINEAR THEORY OF E DGE S TATES - Linearized version (fails in 10 ps ): ( ∂ t − c 0 ∇ x ) ρ = 0 - Universal description of non-linear chiral boson at FQHE edge � � 2 ρ 2 − 1 − ν 1 ( ∂ t − c 0 ∇ x ) ρ − κ ∇ 4 π ∇ ρ H = 0 [ ρ ( x ) , ρ ( x ′ )] = ν ∇ δ ( x − x ′ ) , f ( x ′ ) � f H ( x ) = 1 x − x ′ dx ′ π P . V . - New scale: κ ∼ ∆ ν ℓ 2 B / ħ h - energy of a quasi-hole less cyclotron energy (non-universal scale) but the form of equation is universal; - The universal coefficient (important!) 8 / 1

O VERSHOOT : DIPOLE MOMENT OF FQHE DROPLET β   � e − 1 � i | z i | 2 / 4 ℓ 2 Ψ 0 = ( z i − z j ) B , 2   i > j � R ( r − R ) ρ ( r ) dr = 1 − 2 ν d = 8 π 0 Dipole moment of a spherical droplet � � � i | z i | 2 / 2 ℓ 2 | z − z i | 2 β | ∆ | 2 β e − d 2 z i 〈 ρ 〉 = lim B N →∞ i 9 / 1

No overshoot at β = 1 ν ≤ 1. 10 / 1

ρ" 1.2 1.0 ρ I " 0.8 0.6 0.3 ρ"ρ I # 0.2 0.4 η(r)" 0.1 0.2 0.0 2 4 6 8 10 - 0.1 0.0 - 0.2 6 7 8 9 10 ρ ( y ) ≈ ρ I ( y ) + ηδ ′ ( y ) � y ( ρ − ρ I ) dy = 1 − ν η = 4 π 11 / 1

Benjamin-Ono Equation: Properties � � 2 ρ 2 − 1 − ν 1 ( ∂ t − c 0 ∇ ) ρ − κ ∇ 4 π ∇ ρ H = 0 - Classical Benjamin-Ono equation describes surface waves of interface of stratified fluids; - Integrable (despite being non-local); 12 / 1

B ENJAMIN -O NO E QUATION : F RACTIONALLY QUANTIZED SOLITONS � 2 ρ 2 − 1 − ν � 1 ( ∂ t − c 0 ∇ ) ρ − κ ∇ 4 π ∇ ρ H = 0 - Two branches of solitons: - subsonic : holes propagating to the left; � Charge ν = 1 /β : ρ h dx = integer × ν - ultrasonic : particles propagating to the right: � ρ p dx = integer Charge 1: ρ = q A q = 1, − ν , V q = q κ A ( x − V q t ) 2 + A 2 π - Classical Benjamin-Ono equation has only one branch - particles Benjamin-Ono is the only integrable equation with a quantized charge of solitons 13 / 1

Two branches of excitations: Separation between holes (moving right) and particles (moving left) ii 14 / 1

Q UANTIZATION THROUGH EVOLUTION 15 / 1

Input: Quantum Hydrodynamics - Laughlin’s coherent states: (i) Analyticity, (ii) Degree of zeros β = 3 � � � i | z i | 4 / 2 ℓ 2 ( z i − z j ) β e − B + i V ( z i ) Ψ V = i < j - Galilean Invariance H = m h ħ 2 v † ρ v , ∆ 1 / 3 ∼ m ℓ 2 B - Velocity v = v x − i v y v | Ψ 0 〉 = 0, β i hm ν v i = ∂ z i − e β = 1 � 2 cA ( z i ) − , ν . 2 ħ z i − z j j � = i 16 / 1

Incompressible Chiral Quantum Fluid - Laughlin’s coherent states: � � i | z i | 2 / 2 ℓ 2 � ( z i − z j ) β e − B + i V ( z i ) Ψ V = i < j - Velocity β β = 1 i hm ν v i = ∂ z i − e � 2 cA ( z i ) − , ν . 2 ħ z i − z j j � = i - Velocity matrix elements m v | Ψ V 〉 = − 2 i ∂ z V | Ψ V 〉 - Incompressibility ∇ · v = 0, v = ∇ × Ψ - Edge states dynamics - irrotational flow (no vortices) ∇ × v = 0, ∆Ψ = 0 17 / 1

Subtleties and main steps 18 / 1

Chiral Constraint ( Property of Laughlin’s states) Relation between velocity and density ν ( ∇ × v ) = ρ − ρ I + 1 − ν 4 π ∆ log ρ (Wiegmann, Zabrodin, 2006) 19 / 1

Potential flow and the Boundary Waves � ρ ν − ρ I d 2 z ′ 2 πν + 1 − ν ∆ v = 0, im v = 4 πν ∂ log ρ z − z ′ Boundary value of velocities m v y = 1 − ν m ( v x − c 0 ) = − ν − 1 ¯ 4 πν y H ρ y ( x ) , xx Kinematic Boundary Condition ˙ y + v x ∇ x y + v y = 0 leads to Quantum Benjamin Ono Equation. 20 / 1

Q UANTUM H YDRODYNAMICS IN THE BULK - Laughlin’s state reformulated as a hydrodynamics if the bulk: [ v ( r ) , ρ ( r ′ )] = − i ∇ δ ( r − r ′ ) ; canonical hydro-variables, ρ + ∇ ( ρ v ) = 0, ˙ continuity equations ∇ · v = 0; incompressibility, [ v ( r ) × v ( r ′ )] = 4 πν − 1 i δ ( r − r ′ ) , Heisenberg algebra ν ( ∇ × v ) = ρ − ρ I + 1 − ν 4 π ∆ log ρ Chiral constraint 21 / 1

S UBTLETIES - Short-distance anomaly or OPE 〈 ρ v 〉 = 〈 ρ 〉〈 v 〉 − 1 4 ν ∇ ∗ 〈 ρ 〉 - Dipole moment and singularity on the boundary � R ( r − R ) ρ ( r ) dr = 1 − 2 ν d = 8 π 0 22 / 1

Subtleties: Stress energy tensor - Input: Gallilean invariance E = m ρ v 2 2 - Outcome: Stress energy tensor - "Hall-viscositiy" h T xx = m ρ v x v x + ħ � � 4 ν ρ ∇ x v y + ∇ y v x 23 / 1

B ENJAMIN -O NO E QUATION AS A D EFORMED B OUNDARY CFT - Boundary CFT exterior of the droplet; - Boundary stress energy tensor component 2 ( ∇ n ϕ ) 2 + ν − 1 T nn = 1 4 ν ∇ n ∇ s ϕ , −∇ ϕ = ρ - Benjamin-Ono Equation is a deformation of CFT: ρ = ∇ T nn ˙ - Deformation of a boundary is generated by the normal components of the stress-energy tensor. 24 / 1

CFT AND FQHE - FQHE Edge hydrodynamics is a deformation of Boundary CFT with c = 1 − 6 � � ν − 1 / � ν � 2 < 1 - CFT lives outside of a droplet; Contrary to a common believe that FQHE c = 1 bulk CFT. 25 / 1

S UMMARY - Boundary waves in FQHE are essentially nonlinear; - Two branches of solitons with charges 1 and - ν ; - Deformation of the boundary are generated by a stress energy tensor of CFT situated outside of the dropletwith c = 1 − 6 ν − 1 ( ν − 1 ) 2 < 1 - An origin of shifting the central charge is a dipole moment located on the boundary of the droplet 26 / 1