P. Wiegmann- Quantum Shock Waves and Emergence of Quantized Edge States

8/3/2019 P. Wiegmann- Quantum Shock Waves and Emergence of Quantized Edge States

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Quantum Shock Waves andEmergence of Quantized Edge States

Together with Alexander Abanov and Eldad Bettelheim

1

P. Wiegmann


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FQHE - topological quantum liquid: a gapped state on a closed manifold with a monodromy.

The ground state is degenerate on a multiply-connected closed maniol


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Probing is possible only through a boundary

Point like, or extended

Extended boundary - an Edge

Topological liquid is not gapped any longer -

Edge states propagate as a sound - subject to decoherence


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Edge state carries the information of the topology:

Edge tunneling, noise measurements.


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Topological aspects:

fractional charge and braidingcan be measured in edge states out-of- equlibrium

Quantum Shock Wave


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A smooth bump in density propagating:

all gradients


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(i) transport on the edge is essentially non-linear,


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Disipativeless shock waves


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Each Soliton carries a FRACTIONAL CHARGE

Soliton Train

time

space


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How to describe quantum dynamics of Edge State?

Step 1: Conjecture:

Edge state (Abelian state) is effectivelydescribed by Calogero model


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V (x ) = (x ) 1

x 2,

1sinh 2 x

,

Interpolates between Lattinger liquid and Calogeromodel - quantum wires, edge states of FQHE

Model Hamiltonian: Calogero model


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Shock wave: Riemann-Hopf Equation

+ ( x )2 = 0

Classical Riemann equations are ill-dened (no single valued solutions)


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Stabilization of shock waves by:

(i) by quantum corrections;

(ii) by interaction


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V (x ) = (x ) 1

x 2,

1sinh 2 x

,

=12

( x )2 + 2x

=1

L cotx x

L (x )dx

Interaction -- stabilizes shock wave through dispersion

(x + iL ) = (x )


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=12

( x )2 + 2x

=1

L cotx x

L (x )dx

L 0 : =12

( x )2 + 3x

ILW-equation

KdV-equation

L : = 12

( x )2 + 2x (x )x x dx

Benjamin-Ono equation


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(i) Lattinger Liquid - short range interaction:

+ ( x )2

+ 3x

= 0 KdV-equation

(ii)Fractional Hall Edge State - long range interaction (Calogero model)

+ ( x )2 + 2x H = 0 Benjamin-Ono equation

H (x ) = (z )x z dz2 i


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(i) KdV-equation: Non-linear waves in shallow water

+ ( x )2

+ 3x

= 0

(ii) Benjamin-Ono equation: Non-linear waves in deep water

+ ( x )2 + 2x H = 0

H (x ) = (z )x z dz2 i


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On the relation between Calogero model and CFT

=

xT

T = ( x )2 + 0 2x

0 =

1 /

( 1)(x i x j )2


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Period of oscillations is

(interaction) () 1 >> k 1F

Quantum Non-linear Equations can be treated semiclassically


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Conclusions:

Transport is 1D-electronic systems is essentially non-linear,

Any smooth semiclassical excitation evolving in timeis unstable, and eventually sharp,

will be fragmented to individual soliton-likeparticles,

Fermi-surface is an essentially semiclassical , it is destroyed inevolution.


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M i Gl


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Morning Glory


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Chain of rolling cloudsMorning glory

South Australia Believed to be Benjamin-Ono eq


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P. Wiegmann- Quantum Shock Waves and Emergence of Quantized Edge States

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