1. Assa Auerbach, Ilya Finkler, Bert Halperin and Amir Yacoby
Self Organization in Non Equilibrium: The Quantum Hall Zero Conductance State
Assa Auerbach, Ilya Finkler, Bert Halperin and Amir Yacoby
Phys. Rev. Lett. 94, (2005)
Organizing Principle: Lyapunov Functional
Zero Conductance Domains Phase.
Introducing Long Range Disorder .
Experimental effects of disorder potentials.

2. ZRS Dark magneto resistance: Shubnikov-de-Haas oscillations at large B
Hall resistance (classical)
Dark magneto resistance:
Shubnikov-de-Haas oscillations
at large B
Microwave photo resistance.
oscillations with period:
ZRS

3. Microscopic Theory: negative conductivity
Ando+Uemura.
No radiation, weak magnetic field: (SCBA)
Durst et. al. (Yale).
Displacement Photoconductivity (DP)
---> negative conductivity
Dimitriev, Mirlin, Polyakov, Aleiner, Vavilov
Distribution Function (DF) photoconductivity.

4. Magneto Transport theory
j(E) is determined by microscopics at lengthscales < lc
If:
Hall current drops out!
Since
there exists a scalar functional

5. The Lyapunov functional
If exists, then
1. Extrema of G are Kirchoff Steady States.
2. G is a spontaneously decreasing function of time!
(like a ‘Free Energy’).
Electrostatics:

6. The Global Lyapunov Minimum: (Ground Steady-State)

7. The Zero Conductance state (no long range disorder)
Negative conductivity is unstable toward creation of finite field domains
corbino
Free motion of domain wall under external bias: Zero Conductance

8. Domains in two dimensions: Clean system
unstable regime
Lyapunov function
domain walls bisect field directions
photoinduced electric potential
What are the effects of a long range potential disorder?

9. Coupling a slowly varying internal field
electrochemical field
photocurrent
Microscopics: the different role of electric and electrochemical fields
in the current equation:
More Generally: a weak internal field modifies g:
The induced field would align with the disorder, but it is frustrated by the stability constraint

10. White noise disorder (Imry-Ma argument)
In D=2 white noise disorder is irrelevant to the Zero Conductance State. Mesoscopic current fluctuations

11. One dimensional disorder
Resistance is finite:
2. Disorder induces a photovoltaic effect:

12. stability constraints on domain walls with disorder
1. The field magnitude at domain walls is marginally stable .
2. E is a gradient: domain walls locally constrained by
domain wall
3. Currents at domain walls are
4. Domains enclose zero ``2D charge’’ of the disorder potential

13. The separable `Egg Carton’ potential
Charge density

14. correlated disorder f(x)+g(y)

15. 2D disorder (non separable)
radiation
Field E cannot fully align
with disorder field Ed:
Frustration

16. Charge and dissipative currents
1. Domain walls are pinned. 2. Frustration drives circulating dissipative currents. 3. I(V) is discontinuous, walls jump between steady states ==> glassiness.

17. conclusions
For constant Hall conductivity, there is a Lyapunov functional which organizes the stability of steady states
The Zero Conductance State survives weak 2D white noise disorder.
The disorder fields produces a photo voltage and circulating dissipative currents.
1D and correlated or strong 2D disorder pins domain walls and produce finite resistivity.
Soon: A fuller microscopic theory for j(E): the magnitude of induced fields,
detailed phase diagram and frequency dependent magnetoresistance.