Viscous Fingeringand

Aharonov-Bohm Effectin

Quantum Hall Regime

In collaboration with

• Oded Agam, Eldad Bettelheim, Anton Zabrodin

and acknowledgement to:

• M. Mineev-Weinstein, L. Kadanoff, L. Levitov,I. Krichever, B. Shraiman

Viscous (or Saffmann-Taylor) fingering

an unstable front between two immensible phase isdriven by a gradient of a harmonic field:

Laplacian growth;

diffusion driven patterns.

vn = −∇n P on the interface

∆P = 0 in oil,

P = σ × curvature on the interface

σ -surface tension=0

Hele-Shaw cell = 2D geometry.

1

Electrons in the Quantum Hall Regime

Spin polarized electrons in a plane in the lowest levelof a strong quantizing magnetic field B(x, y) > 0:

H =1

2m

((−ih ∇ − A)2 − hB

)

Quantizing magnetic field:

ωc = B2m

→ ∞, m → 0, B = fixed.

Magnetic length

l =√

2hB

2

Aharonov-Bohm fluxes

A nonuniform magnetic field outside of the droplet

W (z, z) =∫ z

0

�Ad�l =

=|z|22�2

−∑

a

qa log |z − ζa| =

=|z|22�2

− Re∑

k

tkzk,

Harmonic moments of the nonuniform part of magneticfield

tk = −1

k

∫outside

(x + iy)−kB(x, y)dxdy

Vacuum amplitude

τN(t1, t2, . . .) =∑

a

eiqanaNa

Na - number of loops winding na times around a point ζa .

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Quantized Hele-Shaw problem =Electronic Droplet in Q H regime

• A semiclassical droplet on a plane in a quantizingmagnetic field B, is governed by

the same equations as viscous fingering,

scaled to a nanometer scale 10−9.

• The surface tension = the first quantumcorrection= magnetic length

σ = 23π

4√

2πl.

• The semiclassical limit:

N → ∞, l ≡√

2hB

→ 0, Area = �2N = fixed.

5

Growth:

• the number of electrons: N → N + 1;

• quantized area: t → t + πl2 = discretized time.

6

Numbers

Effective capillary number in Hele-Shaw cell

Length =2π

q

b2

12ησ ∼ 500-800 nanometers

- q - the flow rate;

- b - the thickness of the cell;

- η - the viscosity and

- σ - the surface tension.

To be compared with the capillary effects of fromquantum corrections — It is the magnetic length

� ∼ 50 nm, at B = 2T

.

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Conservation of harmonic momentsat zero surface tension

At zero surface tension σ = 0 harmonic moments areconserved:

tk = − 1

πk

∫oil

(x + iy)−kdxdy = do not change in time

⇓

F ind the form of the domain whose area increases intime while all moments remain fixed

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Laplacian growth (zero surface tension)= an evolution of conformal maps

vn = −∇n P on the interface

∇2P = 0 in oil, P = 0 in water

• w(z) - a conformal map of the domain to theunit disk

• pressure p = − log |w(z, t)|

• velocity is the harmonic measure

vn(z, t) = |w′(z, t)|, z ∈ interface

• Evolution of conformal map under a deformation ofthe domain.

9

Integrability: Toda lattice hierarchy

Dynamics of the interface with no surface tension isintegrable:

• a map: z(w) = r(t)w +∑

k uk(t)w−k;

• evolution: ∂tkz(w) = {z(w), Hk(w)};

• flows Hk(w) =(zk(w)

)+;

• zero-curvature: ∂tkHp − ∂tpHk + {Hk, Hp} = 0;

• string equation {z(w, t), z(w, t)} = 1.

The Poisson bracket {f, g} = w(∂wf∂tg − ∂wg∂tf)is defined w.r.t a pair: log w and the area πt:

{logw, t} = 1.

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Toda hierarchy

• Dispersionless limit of the Toda hierarchy(the first equation)

r(t) = e∂tϕ

∂t1∂t1ϕn = ∂te∂tϕ

• Toda hierarchy (first equation)

rn = eϕn−ϕn+1

∂t1∂t1ϕn = eϕn−ϕn+1 − eϕn+1−ϕn

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Finite-time singularities

As a result of the scale invariance some fingers develop

cusp-like singularities occurred at a finite time.

A small capillary effect curbs the singularities andintroduces a scale:

pressure on the interface = p(z)∣∣C = −σ × curvature

σ = surface tension

Stochastic, noise

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Quantization ⇔ resolving singularities

A natural mechanism which introduces a scale butcaptures both physical and mathematical aspects ofthe problem is just the quantization.

• Zero surface tension Saffman-Taylor problemarises as a semiclassical limit.

• Naturally the surface tension appears as the firstquantum correction.

• A quantized problem respects the mathematicalstructures of the ” classical” problem. It isintegrable in a similar manner as the ”classicalproblem”:

Matrix elements of the quantum problem evolveaccording to the Toda integrable hierarchy .

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• The form of Laplacian growth ready for thequantization

{z(w, t), z(w, t)} = 1, |w| = 1.

The Poisson bracket {, } is defined with respect toa pair log w and the area πt:

{logw, t} = 1.

• Quantization implies:

– coordinates z, z ⇒ operators Z, Z†

– Poisson brackets ⇒ the commutator:

[Z, Z†] = h

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Electrons in a quantizing magnetic field

The algebra of coordinates

[Z, Z†] = �2 = 2hB

is the algebra of

electrons confined on the Landau level by astrong magnetic field.

H =1

2m

(−ih ∇ − A

)2

• Quantizing magnetic field:

ωc = B2m

→ ∞, m → 0, B = fixed.

• Magnetic length

l =√

2hB

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Semiclassical orbits

• A uniform magnetic field=Orbits are circular

Ψ(z1, ..., zN) =N∏

n<m

(zn − zm)e−∑

n|zn|24�2

At large N all zn are uniformly distributed inside adisk of the radius �

√2N .

Area=number of electrons × (magnetic length)2

• Nonuniform magnetic field:

e−|zn|2

4�2 → e∫ z0

�A�dl

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If magnetic field is nonuniform only outside of thedroplet

∫ z

0

Ad l =|z|22�2

−∑

k

tkzk

tk = −1

k

∫outside

(x + iy)−kB(x, y)dxdy

Orbits are deformed, but their moments remain tk.The orbits are differed by the quantized Area.

Ψ(z1, ..., zN) =N∏

n<m

(zn − zm)∏n

e−|zn|2

4�2+Re

∑k tkzk

n

17

A probabilty to add a particle to a droplet at apoint z is an overlap < N |N + 1 >= ΨN(z)

- String-orthogonal polynomials

ΨN(z) = e−1hV (z)

∫|∆(ξ)|2

N∏1

(z − ξn) · e−1hV (ξn)

∆(z) =∏

n<m

(zn − zm) - VanderMonde determinant;

- Potential

V (ξn) = 2Re

∫ z

0

Ad l =|z|2�2

−∑

k

(tkzk + tkzk)

- Non-Hermitian Random Matrix model

PN(z) =∫

Det(z − M)e−1htrV (M)[DM ]N×N

M = U−1diag(z1, . . . , zN)U.

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Toda hierarchy

Pn(z) = rnzn + . . .

rn = eϕn−ϕn+1

∂t1∂t1ϕn = eϕn−ϕn+1 − eϕn+1−ϕn

Dispersionless limit:

rn → r(t), at h → 0, t = nh

r(t) = e∂tϕ

∂t1∂t1ϕn = ∂te∂tϕ

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Semiclassical wave function

• The wave function=Baker-Akhiezer function of theToda hierarchy.

• Semiclassical analysis of the wave function =Witham method.

A saddle point:

∑m

2

zn − zm

= zn −∑

k

ktkzk−1

All eigenvalues are uniformly distributed inside a dropletwith moments tk and area Nh.

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ψN(z) =√w′(z)e−1

hS(z)e− ihA(z)

S(z) - an action of a particle moving along theorbit in a phase space with a given tk and area Nh;

A(z) - an area covered by a particle.

|ψN(z)|2 � |w′(z)|δC(z)

Orbits grow with the Area according to theLaplacian Growth Law.

velocity= probability rate of growth =

vn = |ψN(z)|2 = |w′(z)|

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Conclusions

QHE:

• Electronic droplet in Quantum Hall regime developsfingering instability.

This is quantum interference effect. It is importantnear a plateau transition.

• Inevitable singularities near a plateau transition;

• Singularities - regime where regular semiclassicalexpansion fails; Failure of Edge states;

• Is fingering observable in semiconductornanostructures? Special geometries?

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Viscous Fingering:

• Quantization provides a mechanism of stabilizingsingularities on a microscale.

• Capillary effects - nonlinear aspects of Navie-Stocksequation = Semiclassical correction.

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Other aspects (not covered)

- Relation to 2D quantum gravity and topologicalfield theories

- Integrability,

- Evolution of Conformal maps and Riemann mappingtheorem,

- Plateau transition in QH effect.

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